math-pedagogy

[twin shell]

yes, in an ideal setting, teaching math would allow students to develop into informed voters
on the other hand, you could argue this already belongs to the "daily life" category

but sanity-checking 0 * 420 = 0 is important as mentioned

[tawny slate]

i think it should be in the daily life category

yes

[tardy ember]

but anyway yeah i think my answer to a lot of this is "well that sounds like it would be good if schools were doing that, which they aren't"

[tawny slate]

yeah, i think i was just trying to say that if schools aren't doing that, being clear about what the goals of education should be helps us help ourselves where the system fails us

that's all

ironically, the goal isn't that clear

[surreal lily]

I’m a TA for a Calc 1 class. A lot of students with weak algebra skills come to my office hours for homework help, but it difficult to help when they are missing multiple key skills. What are some ways to better support students struggling with calc because of algebra?

[heavy trail]

Make them slow down and focus on what they're doing and why when they do algebra. If you're trying to play a song on the piano and keep making mistakes, you slow the tempo until you can get the mistakes under control.

Everything else depends on what, specifically, they're confused about and why they can't self-correct.

They might be annoyed because it will feel like you're "not answering the question". Some professors might object, saying your job is to help with the content and not remediate. So YMMV.

Slow is smooth, smooth is fast.

[halcyon glade]

I would specifically point out to them the roadblock they're experiencing and get them to focus on the skills that they're struggling with. If that means going back two courses in content, then so be it, I meet people at their level.

[heavy trail]

I'd say it depends on the situation. I think that could be the right call if it's a triage situation, e.g., there's a long line of other students waiting to get help.

If they're the only student around, I see no reason not to do what you can. Whatever time and energy someone "saves" by avoiding attempts at remediation now will probably be paid when the time comes to grade their confused output — with interest!

[heavy trail]

I think there's a separate question of what can or should be done in the full classroom context, if anything, but I thought Anne was talking about a situation like a TA during office hours or other open-ended "help time".

[surreal lily]

Both questions get discussed a lot in my department. I don’t have much say in curriculum, but I do hold a lot of office hours that don’t have long lines where it is totally feasible to help with prerequisite knowledge

[halcyon glade]

I would also point to additional resources (e.g. KhanAcademy) where they can learn from

[tardy jasper]

I wonder if instead of teaching subtraction and division we only taught adding negative numbers and multiplying by reciprocals.

For me a lot of the problems I had learning algebra were caused by subtraction and division. It may be initially harder to teach students but it may make it easier for people to understand later on. I wonder if there have been any studies on this

[lethal leaf]

I have lost my tablet. Any good last minute teaching solutions for sharing writing while tutoring online?

worst case I do an awful mix of live tex + drawing with mouse

[storm hawk]

rip, an alternative would be to use your phone as a webcam pointed at paper as you write. Shouldn't be too hard to macgyver something to hold it in place. But probably drawing with mouse is the best option.

[austere inlet]

I've seen profs live TeX on classes and it's doable if you type at a reasonable speed

drawing with mouse though sounds like a pain

[austere inlet]

I would go for this if you have decent writing

maybe with a marker or a thick pen

[halcyon glade]

I hold office hours this way. Hidden benefit is that it forces you to go slow so that the students have time to absorb.

[spice matrix]

If you must live tex. I really recommend live Typst instead because it’s a lot faster to type a lot of stuff. It has good shorthands.

E.g. f : A -> B will render as LaTeX would render f : A \to B
a / b will render the same as \frac{a}{b}

[lethal leaf]

I don't wanna learn a new thing as I tutor 💀

[halcyon glade]

Yeah I probably LaTeX faster than I handwrite atp, unless for some sort of complicated diagram/picture

[austere delta]

Guess writing -> isn't much faster than \to, but having : be replaced by \colon and := by \coloneqq would be nice

[high quarry]

texmacs is probably better than live tex

[tawdry venture]

it is a huge pain. like at least get a graphics tablet lol

[tall bolt]

If you use something like VSCode you can already TeX most things faster than even in typst with all the shortcuts. \frac{}{} is just @ enter etc etc and you’ve got the benefit of it actually just being LaTeX

[heavy trail]

How do I control whether the fraction is displayed inline or as a block offset on its own line in Typst?

[north wraith]

what are obvious advantages of latex over typst tho?

recompilation time of 10 sec vs .5 ms?

[north wraith]

$1/2$ and $ 1/2 $?

spaces matter

[north wraith]

u get much more than that

in typst u would write:
{ x in A | P(x) }

compared to latex:
\{ x \in A \, | \, P(x) \}

[heavy trail]

😉

To the extent Typst is about "being easier" than LaTeX, it has its fair share arbitrary syntactical decisions like that. I didn't find it saved me any real time when typing.

The speed of compilation is nice, though.

[north wraith]

I mean, u also have to account for how long u have used typst

if u have been using latex for 15 years but typst only for 6 months, u will still be typesetting in latex faster

[north wraith]

meant to reply here

[heavy trail]

I agree Typst has a shallower learning curve, if that's the argument.

[north wraith]

that's not what I meant but I agree with this statement either way

another reasonable argument I'd say is that typst is modern lol

it's a very active project, and since it doesn't have to maintain so much backward compatibility as latex, it can actually implement things that a lot of people didn't like in latex

for how many decades has latex existed?

[heavy trail]

eh

[austere delta]

It's 40 this year 🎉

(40 years not decades)

[heavy trail]

Nothing says "exciting" like wondering whether my documents will compile in 5 years.

[north wraith]

u pin the compiler version

[heavy trail]

See? It's easy, you just grombulate the recalcitron.

My point is all these arguments are mostly symmetric.

The LaTeX person says "You don't have to type \{ x \in A \, | \, P(x) \}, you can make a macro." Or "You can configure your editor to help."

[north wraith]

then u gotta make a macro for everything xD

I mean, if u don't want, don't use it

I'm just advocating for an amazing tool, that I (and a big community) really enjoy

[heavy trail]

BTW, you "should" be typing:

\{ x \in A \mid P(x) \}

Anyhow, yes, use whatever tools you prefer. Some people like apples, some people like oranges.

[north wraith]

by the way:

typst | latex
< | \le
"a" | \text{a}
(x) | \left( x \right)

[north wraith]

fair

[austere delta]

$x < y, x \le y$

[burnt vessel]

jagr2808

[heavy trail]

heh

[austere delta]

Wait, so how do you write < ?

[north wraith]

weeeeelllllll xD
my bad

it should have been '<=' for typst

[austere delta]

I see

[north wraith]

and just '!=' for not equal sign

[austere delta]

You're telling me I've been using \leq my whole life, when I could have been writing \le

[halcyon glade]

\set{x \in A; P(x)} 🙂

[heavy trail]

Typst's main selling point is the compilation speed and that's real.

The syntax differences are all superficial, IMO, and they appeal to folks who find LaTeX's syntax "surprising", i.e., people coming to any typesetting system for the first time.

But LaTeX's syntax is more uniform.

[north wraith]

syntax differences get significantly more important when we get to scripting and templating

I'll post some code samples when I get home

hard to type code on the go with coffee in another hand

[heavy trail]

I know enough Typst to know what you mean. I helped a friend typeset significant portions of Rudin using it.

[north wraith]

altho to be completely fair, ppl familiar with C-style (more like Rust-style) syntax will benefit from it the most

[heavy trail]

I've lived through enough "new systems" to know what the churn of a nascent ecosystem means. It's exciting and fun and you pay the piper a few years down the road.

See, e.g., trying to run any Node project you haven't touched in 2-3 years.

[north wraith]

also true

for that case we have another new and exciting system called Nix

[heavy trail]

Mmmhmm

And round and round it goes

[north wraith]

software is like that, yeah

[heavy trail]

The computer was suddenly revealed as palimpsest. The machine that is everywhere hailed as the very incarnation of the new had revealed itself to be not so new after all, but a series of skins, layer on layer, winding around the messy, evolving idea of the computing machine.

...

And down under all those piles of stuff, the secret was written: We build our computers the way we build our cities — over time, without a plan, on top of ruins.

— Ellen Ullman (1998)
https://www.salon.com/1998/05/12/feature_321/

[north wraith]

a reply to the highlighted quote:

I think someone estimated that to change a "single place" in a "reasonably sized" open source project, it takes ~6 months

so yeah, that's why it's the way it is

[heavy trail]

uhh

[north wraith]

another way of saying that, is that in coding, complexity of verifying a piece of code does what it is intended to, doesn't grow linearly with respect to added lines of code

u have to go back and check almost everything before, if u ever indent to use / replace some old code with new one

[heavy trail]

👍

[spice matrix]

To do inline you just escape. $x \/ y$ will render as x / y

[spice matrix]

I’m aware that snippets exist. These can also make typst faster though. snippets are not a LaTeX exclusive feature lol

[tall bolt]

I’m not claiming they are, I’m just doubtful that writing typst is any faster or easier, what it is is less standard

The compile times are nice though, latex with a big bibliography can be a bit of a bitch lol

[spice matrix]

I’ve written a lot of both. With a good editor setup. Typst requires a lot less friction. And I’m just as fast on Typst without snippets as I am on LaTeX with snippets.

The only reason I still use LaTeX is to communicate to others/collaborate.

[spice matrix]

oh this might also be it depending on what cufflink meant

[spice matrix]

I just learned too lol

[tall bolt]

I’m also not massively convinced by this, like I legitimately think that if you know maths in English and someone tells you the syntax for latex maths commands is , you can write about 90% of what you need by literally just guessing the obvious thing

At the end of the day it doesn’t really matter, use whatever you like, but yeah just not personally convinced of the arguments for typst

[steady jetty]

...until you decide to draw something more complicated with TikZ 😛

[sacred wren]

anyone use mathematica or nauh

[storm hawk]

Try #computing-software

[dry comet]

Someone teach me maths

[austere inlet]

this isn't the channel to ask this

[dry comet]

Then what is lol

[austere inlet]

you already asked in the social channels

[dry comet]

And was directed here

U weird lol

[midnight scarab]

Idk who told you this, but if you just look at the channel description you'll see this is the wrong channel

[austere delta]

!help

[coral copper]

To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.

[queen anchor]

Hi! I'm about to become a math helper at my university, and I'm eager to learn as much as I can to be a good helper. I know it probably won't be very effective to just ask for tips here, but at least it should be worth a try. That aside, what else can I do? Are there any youtube channels that share content for educators? Bibliography I can study from? It is not expected from "second hand helpers" to actually know about pedagogy so I have zero background, and I'm not planning to become a teacher as in high-school teacher, but my uni offers courses on general didactics, psychology and so, for those that do want to be teachers. It feels like too much of an effort for my job, but it also seems kind of worth if I want to stand out as a good helper.

[rapid tusk]

are there other helpers at your school that you can talk to?

[queen anchor]

Yes, there are a lot. Most don't care that much about those topics, but there are a few that do mention some things they do to keep a class going forward

[deep kindle]

What would your regular responsibilities look like?

[deep kindle]

If you just want random tips I find it benefits students when you're as patient as possible, don't put time pressure on their problem solving, pay attention to their thought process because while a given problem will have one or a few correct answers there are countless solution paths one could take to get there. If you have access to any brief notes on given subjects it may help, I have a quickstudy notes sheet for stats for when students come in for stats tutoring and I haven't touched the content in months

I find that understanding common misconceptions helps me better approach student misinterpretations of given concepts

This book in particular is really concise with misconceptions that occur in secondary level mathematics

[queen anchor]

The students can come with me during the break to ask anything. There might be a few times where I get to explain exercises in the blackboard. And I'll have to correct exams at the end of the semester.

[deep kindle]

In that case familiarity with whatever math may be involved in the course is paramount, on top of previous tips

Not much more embarrasing than being the one who's supposed to know stuff and getting stumped, I've been there lol

[queen anchor]

Yes, the courses I'll be in are the ones I've already passed. Definitely going to take a look at things before, but it won't be that much of a struggle

[deep kindle]

If you'd like, look into various presentation styles or different ways to represent or model a given problem. Sometimes students struggle to understand a concept not because of the complexity of the concept itself but as a result of incompatibility with the instructor's method of presenting it

[queen anchor]

Oh, that's one I've been in

[deep kindle]

As far as technical integration goes, chatgpt is a no no which I imagine you know. Having desmos, geogebra, or wolfram|alpha handy always helps depending on problem context

[queen anchor]

One time my professor used induction in a very hand wavy way, and I had just passed the subject that teaches it

[deep kindle]

I had my fair share of "the proof is by witchcraft" professors so I know how that feels

[queen anchor]

Geogebra seems like a good choice for analysis courses, but always having it premade

[deep kindle]

I was robbed of the opportunity to take analysis myself so I don't know anything specific of those courses other than they are very abstract and students will struggle with that

A good approach to problem solving with abstract problems is to first consider simpler cases, which may help students who you find struggle

Though I don't know how readily that applies to analysis

[queen anchor]

It's a good thing to know anyway

[deep kindle]

I don't know if you've seen 3blue1brown on youtube, but he has a video on problem solving strategies that has some really good ideas. It's long but worth it

[queen anchor]

Oh yeah I've seen it

[storm hawk]

I recently took an intensive course for teaching assistants/demonstrators/tutors/helpers. I'll give a brief outline of the modules:

For modules 1 and 2: There is a difference between instructing and facilitating. While facilitating may take more time and energy, students will benefit more from this. Facilitating examples are like giving tools and asking relevant questions to students to make them think for themselves an approach to a problem.

For module 3: The most important thing is that your student understands your feedback. Be also conscious not to overspend time that could be going to your research.

Making concise feedback is good. Also instead of saying "you made a mistake here" saying "there is a mistake here" is better.

For module 4: Different tools one can use during class like Kahoot, Introductions, etc which you can research on.

Finally for module 5: Be aware that some students might have circumstances that are not apparent. For example, one can think that a student that is always late and doesn't participate in class might not care about the class. But what if the student has 2 jobs, has anxiety, etc?

There's a lot of things I didn't mention which we covered in the class, so I think it's a worthwhile time investment if you can to participate in such a class.

Okay this was a good review for me, so I guess I'll mention some of the biggest tips we got to refresh my memory:

-Give more time to students to think on a problem if the same students are always answering (maybe none)

-Grade consistently

-Know where to redirect students in case they need help (mental/academic/etc)

[storm hawk]

-Adult learning is different. They want to have motivation to learn something, and why and how learning something will help them.

[queen anchor]

Thanks a lot, that's all super helpful

[rain hawk]

How many fundamental theorems of calculus should there be

[deep kindle]

Infinitely many

I'm not understanding the question, is there more than one?

[austere delta]

I've heard people argue that it should be the mean value theorem instead

[modern trench]

The fundamental theorem of calculus is Los' theorem for the hyperreals

[tawny slate]

i like this

[quasi musk]

What an interesting perspective. Doesn't FTC sort of subsume Mean Value Theorem?

[austere delta]

How so?

Like both imply eachother, but I guess that's true for any pair of true theorems.

[quasi musk]

https://math.stackexchange.com/questions/91561/can-the-fundamental-theorem-of-calculus-be-proved-without-an-appeal-to-mean-valu

From this MSE

But overall, my intuition is FTC gives you a very precise answer

[austere delta]

So this is essentially the argument for why MVT is the true fundamental theorem yeah

[quasi musk]

Whereas MVT is a more coarse idea

[quasi musk]

Not quite ~ it lays a lot of groundwork, but doesn't have the same level of precision as FTC does

That being said, a lot of the ideas in MVT and the proof is an amazing example of great ideas

I guess it depends on what you think is more "fundamental"

Do you think a rougher idea, with less precision is more fundamental than a more refined approach?

[austere delta]

Well, I think a fundamental theorem should be a simple theorem with wide implications and applications.

So for example I think the first isomorphism theorem should be the fundamental theorem of algebra.

I don't recall so much what you prove and how in calculus, so I don't really have strong opinions there.

[quasi musk]

I think FTC being FTC is correct because it has both wide applications, and is a very precise relation between derivatives and integrals

[midnight scarab]

With the Riemann integral doesn't the FTC require slightly stronger assumptions than the MVT?

[quasi musk]

Although I love the proof of MVT, because you get to take a difference of two functions then analyze the graphs

[midnight scarab]

Though the FTC is often more useful

Just like the integral form of the Taylor remainder

Also Taylor series by repeared IbP is much neater than the adaptation of the MVT proof by applying Rolle to a well-chosen function

[quasi musk]

I just taught the theorem & rough proof of MVT to my high schoolers last saturday (After school program)

[midnight scarab]

(Which requires knowing the answer)

[quasi musk]

I didn't get into the weeds with rolle's theorem, and go on and on, but the major steps

[midnight scarab]

Nice!

[versed vapor]

Since i couldn't find a psychological aspect of math type channel I'm going to post this here. Is it normal to feel like your math abilities are gone when you're back from a significant break? I'm returning to school after 4 years, and I can't seem to do any math problem any more, it's almost like all my powers have vanished, can people even recover from a 4 year break/interruptance? I'm starting to think not ...

[deep kindle]

Math in general is very abstract and is easy to lose if you don't practice it for a while

I've you've interacted with given concepts before, you'll pick them back up quicker when you see them again

[austere inlet]

it's to be expected, just as if you took a 4 year break from any other hobby profession or human activity, get slowly back into it and maybe review the basics (e.g. undergrad analysis and algebra) if you feel you need that

for the record questions like this are fine in #math-discussion or #advanced-lounge (which is a channel about academic life broadly speaking) this channel is more like a place for people to discuss and share teaching experiences

[versed vapor]

thank you for pointing me in the right direction, regarding channel info 👍

[rain hawk]

yeah sometimes people list it as two

[wispy slate]

X-Pen

[halcyon glade]

Probably not the right place to ask this question, maybe ask in one if the discussion channels like #discussion or #serious-discussion

Oh I see

[wispy slate]

https://www.storexppen.fr/

I use it, because my favorite (french) youtubeur use it

Enchanté

[wispy slate]

Tu connais Scientia Egregia ? Il utilise xpen

Hmmmm

Okay

[wary thunder]

I want to ask about Lang’s Undergraduate Algebra and Dummit and Foote.

They seem pedagogically different in order. For example, Lang’s 2nd chapter is on Mappings. Is there a reason that this interlude is included, but excluded in D&F? Lang also goes into Rings much, much earlier.

Two undergraduate algebra books, two very different organizations and timings. What are the pedagogical differences/intents and the difference in learning outcomes?

If this isn't an acceptable place for this question, let me know. I figured it's fine since it is about pedagogy.

[versed geyser]

just came up with an insanely powerful way to teach linear coordinate systems intuitively

forgive my handwriting

you can hand-wave higher dimension coordinate systems as "the city plans of n+1 dimensional beings"

[versed geyser]

this also transitions nicely to curvilinear coordinates, in the rare case youre teaching both of those in the same class

[dapper flume]

Yeah it's a good one. Happens to be the approach in this book

https://understandinglinearalgebra.org/sec-bases.html

[brittle karma]

hi guys! i'm a tutor at my university's quantitative resource center and i was thinking about talking to the director about hosting an ai kliteracy workshop. i want to teach students about ways to use chatgpt and other ai sites to help with math and coding in a resposible way. like, i want to help avoid cheating/academic dishonesty as well as showing them how to use ai and know when it's wrong. especially because i find that chatgpt is wrong a lot of the time when it comes to proof-based math like analysis and graph theory. just wondering if anyone has any ideas about things i should include? i would of course reach out to professors for guidance about how they would prefer their students to utilize ai. but i want to make sure i have a comprehensive proposal before lol

[tawny slate]

i hope my input isnt stuff that is too obviously going to be covered because i dont know what is or isnt standard for this topic, and also im shooting from the hip with just my limited experience with this so also take it with a grain of salt i guess

i think an important approach is to understand what exactly AI even is at a high level, and to talk about the common pitfalls of AI in the same way we talk about common logical fallacies in humans

talk about how AI is basically just a hyperparameterized function that iterates on data to try and curve fit, that its basically statistics and not going conscious entity

then show things like overfitting, or how AI can reinforce biases rather than avoid them, inner alignment problems, etc

then maybe point out that a LLM like chatgpt is not meant to be a knowledge source but an attempt to model human speech, so it is specifically trained to sound smart rather than be factual, and that it essentially cannot verify facts because it cannot interface with the real world beyond just the data it is being fed

[rocky berry]

(I’m going to be a killjoy but I just want to mention that ai queries have a pretty large carbon imprint, like 10 chatgpt queries a day amounts to 0.15 tons of carbon emissions a year)

[tawny slate]

it probably helps to give explicit examples of how AI can be abused, to give students a direct feel for its impacts and dangers, like using AI to parse resumes, an AI to tell if someone is hetero/homo from photos alone, dead internet theory and model collapse, etc

as a bonus, if you're being more comprehensive and thorough, might also help to explain that AI cant explain its thought process or how it comes to an answer, like how neural networks just have weighted nodes but no obvious qualitative descriptors, and so oversight and accountability is also a big problem

[halcyon glade]

It would be good to include positive examples of how using AI might be beneficial to understanding too, so that they have some sort of model behavior

Is this about AI literacy in general or specifically the use of AI in academics?

You might want to narrow the scope

[versed sable]

hi, I actually am preparing a mathematics project with my high school students and I've been inspired by this article https://www.edutopia.org/article/using-chatgpt-support-student-led-inquiry if you wanna use it.

[brittle karma]

yes! i’m definitely going to touch on this! i go to a pretty “crunchy” school where a lot of people are very eco-conscious. but i also understand that realistically, people are going to use chatgpt and other ai models regardless. i thought maybe by showing them that ai was wrong a lot and giving them other resources (like this discord server), i could maybe redirect questions that they would’ve asked ai

[brittle karma]

i want to talk specifically about ai use for quantitative disciplines like stats, math, cs, etc. so i probably won’t talk about essay writing and stuff like that just because it’s not really related to what the resource center i tutor at talks about

[brittle karma]

thank you!

[turbid zenith]

What kind of ton?

[turbid zenith]

Here's my current AI policy from my syllabus in my liberal arts mathematics class:

Recent advances in artificial intelligence have provided a number of tools that can be used (or misused) for many purposes. However, most of the writing we do in this class requires personal reflection — no matter how sophisticated a computer is, it can’t read your mind to recount your experiences with learning mathematics or to elaborate on your own convictions on important issues. Learning to use AI is an emerging skill, so if you do plan to use AI to aid your writing, you need to let me know about your plan ahead of time, so we can explore how the technology can be used as a tool for good while keeping you, the human writer, as the central voice of what you create.

[brittle karma]

thank you!

[turbid zenith]

And here's the policy from my calculus class — some of the same but some different

Recent advances in artificial intelligence have provided a number of tools that can be used (or misused) for many purposes. You can use generative AI to explain concepts and brainstorm ideas, but you need to be careful of the accuracy of the information, and you must document this use clearly and describe how it supported your learning. Submitting AI-generated solutions without attribution is a violation of academic integrity. If you want to use AI for a specific purpose, please let me know in advance so we can discuss appropriate use.

I've had students use it on my assignments anyway, and with a many of the kinds of questions I ask nowadays it doesn't do very well 😛

So like ... I just give them the bad grade

For example I had a problem where they analyzed a casino game I made up (sort of a combination of roulette and sic bo), and the AI consistently "didn't see" the black 0 spaces so a bunch of students' answers had probabilities with a denominator of 18 🤔

[brittle karma]

i think ai can be good to get feedback for questions you’ve already done if they’re practice sets that the professor won’t correct or grade themselves. and even then, chatgpt is wrong half the time so it needs some probing with “well i got this answer because of xyz”

[brittle karma]

yikes..

[turbid zenith]

The associated problems btw (though with a bit more scaffolding than I had last year)

[turbid zenith]

I would say AI is only good for math if you have the ability to sniff out when it's leading you astray.

[brittle karma]

exactly! i also think being able to point out when ai is wrong builds confidence. not only that but being able to explain your thinking and how you got your answer is a very good skill to have

[turbid zenith]

Yes. But the flip side is that Dunning-Kruger ends up leading the students who don't know it astray.

[brittle karma]

yeah unfortunately... that's why i really want to teach ai literacy in my school

[heavy trail]

This is true, but the same could be said for a solution manual. Solution manuals can be used to good effect for self-practice, after all. Why don't folks feel compelled to teach "solution manual literacy"?

Maybe it's not about this-or-that "literacy", but about the ability to effectively seek out, receive, and integrate different sources of feedback.

For example, most math classes have a wealth of slightly-to-very-wrong attempts: other students' attempts. I suspect those mistakes are more ecologically representative, too. That is, being able to recognize + correct another students' mistakes is probably more transferable to my future attempts than doing the same with an LLM's mistakes.

[brittle karma]

that’s fair and i get where you’re coming from. i guess what i mean specifically is that if someone is going to use ai, i would rather it be to check an answer they’ve already come to themselves rather than asking ai to do it for them and then try to understand the question from the ai explanation. i do agree that looking at other students’ mistakes is much more valuable than looking at ai’s mistakes. the thing is that classes are structured really differently depending on the professor. i had one professor who would release all of the answers to the homeworks the day before it was due so people would have time to self-correct. but then i have other classes where we would get practice sets and no answers (since it was for our own practice and no feedback was to be given by the prof). but i do agree with everything you’re saying about being able to seek out and use feedback that is given. i just find that sometimes, feedback isn’t given

also, there’s no need to teach solution manual literacy since it’s coming from a human-checked source and it’s extremely rare that the solution manual would be wrong. whereas, ai is wrong a lot of the time with math especially

[heavy trail]

if someone is going to use ai, i would rather it be to check an answer they’ve already come to themselves rather than asking ai to do it for them and then try to understand the question from the ai explanation.

Sure, but you'd say the same about a solutions manual, I assume.

i just find that sometimes, feedback isn’t given

Sorry, yeah, I'm using "feedback" in a very expansive way. I don't just mean the professor or TA or whoever giving structured feedback to a particular student. I mean any sort of comparison one can reflect on, e.g., a computer gives you "feedback" when you try to run an invalid program by displaying an error, the wall gives you "feedback" when you try to walk through it and it stops you in your tracks, etc.

[heavy trail]

Yes, but there are productive and an unproductive ways to use a solution manual.

Copying the answer thoughtlessly is unproductive, at least as far as learning is concerned. Maybe not as far as getting a good grade is concerned. Hah.

Comparing my own earnest attempt to the official solution and reflective on the differences is some level of productive.

[brittle karma]

i guess i’m mainly concerned with students learning incorrectly from ai rather than learning inefficiently. i 100% agree with you that simply getting an answer from anywhere—textbook, solution manual, ai—is not helpful if you don’t put in the effort to understand it first. i guess i was just mainly thinking of the instances when ai gives wrong answers and explanations. but yeah, i 100% agree with you on how to effectively learn

i also guess when i use the term “ai literacy” i don’t mean being able to effectively use ai. i mean it in the same way it’s used when talking about media literacy—being able to tell when media is accurate vs. when it’s saying something bogus. i think we may be misunderstanding what the other is meaning haha

[heavy trail]

I suppose my assumption is that we care about students "recognizing mistakes" in large part because we want them to recognize (and hopefully correct) mistakes in their own judgement, process, etc.

[brittle karma]

yeah, i definitely agree with you there! but i guess when it comes to using ai, i’m thinking a little smaller. i’m assuming that if a student is using ai to get an answer quickly or to explain a topic, they probably don’t understand the question or topic enough to discern how right their own logic is. hence why i want to direct them to other sources such as their classmates, their professor’s office hours, and even this discord server

but i’m not disagreeing with you at all because i 100% agree that ultimately, students should be able to figure out whether or not they’re making mistakes and how to avoid them on their own. but sometimes they may not necessarily have the tools to do so which is why they go to other sources (typically ai nowadays)

[heavy trail]

In my mind, it's simpler still. They need to find ways to hold up their own judgment up against other sources of judgment and (attempt to) reconcile the differences.

Either they have their own judgment or they don't.

If they don't then the question is how to (begin to) develop it. If they do then the question is how to find other sources of judgment and how to compare + reconcile.

For example, in my experience, if a student is "stuck" in an introductory course then ~50% of the time the right habit is for them to look back at theorems and definitions from the current chapter. How do they know the AI is even using the same definition as their textbook, for example? Do they even realize that the same mathematical term can be defined differently between texts?

IME the answer is for novice math students is no, they think all mathematical terms are absolutely immutable and enshrined in some perfect Platonic museum, never to be disturbed.

The only "solution" to that is for them to tell the LLM what the definition is, which involves them doing what they should've done sans LLM, anyhow.

Typing that out now, though, I see it might be a way to "trick" students into doing the right thing.

Like, maybe they're more willing to tell the AI because they know it "doesn't know". Whereas if they ask a more-expert person they might get frustrated if they're asked "What's your definition of XYZ?" because they assume the expert "should" know.

(That happens on this Discord all the time, for example.)

e.g., "AI Pro Tip: Tell the AI what definitions you're using, you'll get better answers."

But that's not really an "AI" pro tip, that's something you should think to do no matter who you're communicating with. Maybe not up front, but at least at when you first suspect you might not be on the same page.

[brittle karma]

i feel like we’re getting at the same thing here lol. i do not think ai should be used if a student is stuck to the point where the solution is to look over the course material. which is why i think that ai should only be used if the student has already finished the problem and is looking for reinforcement (if they are right, ai will tell them and then they can either see “oh yeah, i was wrong” or they can debate with the ai until either they realize they are in fact wrong or the ai is wrong.) but i don’t think ai should ever be used to teach a student a subject or to teach a student how to solve a certain problem. on the topic of using the same definitions, a lot of people i know will upload their textbooks to whatever llm they’re using lol

this is why i think it’s best to steer students away from ai but also show them that ai can be helpful in certain circumstances

but i still think that the best resource (if you can’t figure it out yourself) is your professor or another expert in the field rather than ai

[heavy trail]

Well, AI pro tip: don't upload the whole textbook, you'll (probably) blow out the context window and have reduced confidence it "knows" the relevant definition.

It'll know some shadow of the relevant definition, but then you're just back to square one.

[brittle karma]

good tip lmao (i’ve never uploaded my textbook personally)

[tawny slate]

so I think the generative use case here is somewhat limited, due to a meta reason

you are trying to learn about X, so you ask the AI, but the AI could be wrong about X, and you need to know X to tell where it is wrong, but you dont know X

this logical argument would imply that AI is effectively useless here because it could be misleading, compared to a textbook whose content is verified (excluding small errors missed by the editors)

at least math is better in the sense that all the results can be checked formally, so a student who is independent in math may still be able to find use in it, but by then they would probably find it way faster and more reliable to simply use the texts anyways because they can read them

really feel like the math use cases are very limited

[sharp beacon]

i've used AI to generate ideas for solving math problems before

successfully

but it can and will also spit back something wrong with the exact same level of apparent confidence instead

[round drift]

i'm starting to become increasingly skeptical about grading in graduate level courses

where i do my phd is a strong r1 and i used to attend for my master's is another r1

most graders aren't qualified to even grade the courses they sign up to grade for at my old institution

imo

and the new one is the graders seem weak and inflexible, at least being a co-worker of them

my honest evaluation from the way my boss seems to demand us grade undergrad/master's level courses is to segregate rather than educate, and in non-core smaller phd level courses it seems like the professor's do tend to say "A for effort"

like do it this way, or you get -2 points for a 5 point problem even for a correct solution

[dapper flume]

The only useful thing I've been able to use AI for is to recall the names of things that have a definition, or to formulate definitions clearly.

A funny thing I've done recently is to ask it to invent new math ideas that incorporate consciousness, and then scrutinize it for the obvious nonsense it spits out lol

[gray shard]

Do you have an after exam meeting where the students can attend and discuss the grading?

[gray shard]

U can use some ais to find scripts which contain information you are looking for.

[round drift]

only in some courses, and normally they discuss with professors

[potent dragon]

I’m tutoring a kid in algebra 2, and I can tell he’s definitely challenged, but I can’t tell if he’s just naturally slow or he has a disability—it really changes how i go about it

[austere delta]

How should one differentiate those two situations?

Or what would you do differently I mean

[potent dragon]

if i’m helping someone with a disability like autism, i use sketches and diagrams to get to the point, if they’re just naturally slow, i use verbal examples and things that they can picture

[lethal leaf]

Does it matter what they have, or just you need some more tools to help them learn?

[austere delta]

Feel like non-autistic people also get benefits from sketches, but I don't know anything about it so

[potent dragon]

I’m sorry, i’m not sure what you’re asking.

[potent dragon]

I’ve been doing this for nearly a year and I’ve found that they do, but imagery is more effective

[lethal leaf]

Like why spend the energy trying to determine what exactly they have (whatever "naturally slow" means) and why not instead just try whatever tools you do have (diagrams, verbal examples, etc)?

What is the difference between sketches / diagrams and imagery?

[potent dragon]

I do this because it helps me to figure out how I pace it and how much I explain it. I do use whatever tools I have, but I haven’t been able to achieve consistent results with one or the other.

[lethal leaf]

Ok so it seems like the answer to my question is that you are looking for more tools.

Hm, I mean do they seem to be just missing background?

[potent dragon]

sketches relating directly to the problem vs word-problems and real-word examples

[lethal leaf]

Or is it something more?

[potent dragon]

They aren’t, where I live Algebra 1 and Geo are requirements for alg 2

i gtg i’ll be back in 2 hours

thanks for the help 🙏

[lethal leaf]

There's a difference in having taken Algebra 1 and Geometry and actually understanding Algebra 1 and Geometry. Far too often people get moved on from one class to the other with vast holes in knowledge which only makes future classes harder.

[halcyon glade]

I feel like this really just depends person-to-person

How long have you known this kid for?

I don't think it's helpful to try to fit them into a category like this

[raw sedge]

Has anyone read / used the Gelman Stats bag of tricks book? I have skimmed it but not really used it. It looks really helpful though

[wooden condor]

Hey

[unborn hill]

i mean

it doesn't get much more competitive than the program I teach in, but I feel like a lot of college students are just here to get an A

I want to structure some assignments as more exploratory , but I feel like they will just be cheated anyway

[unreal island]

what course r u teaching

[unborn hill]

systems programming

[unreal island]

cooked

[unborn hill]

it's not math but im making it really mathy

rn

[austere delta]

The trick is to not make the grade depend on the assignment

[unreal island]

dont these courses usually have a course project? maybe that area can be exploratory

[empty gull]

Tbh I think the only real way to avoid this is to give very obscure problems that will be hard to find
Because I can tell you there will always be cheating and that's just unavoidable
So your options are A) ignore it and just focus on the students that DO care or B) try to make life hell for everybody in the class

[unborn hill]

right, the projects don't cover all intuition though

and conceptuals

[unborn hill]

right but then

what's the point of doing it lol?

[austere delta]

To learn

[unborn hill]

But then not every student will care about it

Like this is essentially is the issue

[unreal island]

their problem is that the students are there to get an A

[austere delta]

Doesn't sound like your problem

[unborn hill]

Right

[unreal island]

so why would the students do non graded assignments

[empty gull]

I feel like you're focusing too much on trying to please everybody

[unborn hill]

I want them to experience my experience

More than anything

Not in its entirety but the important things

[empty gull]

You can't please everybody there will always be slackers
And as jagr said that's a thing you have to come to terms with
If you try to fight it you're just gonna be some grumpy professor that everyone will hate

[unborn hill]

I'm a TA , not a prof

[empty gull]

same thing

[unborn hill]

i teach the labs

but right

[unreal island]

my prof assigns homework that gives intuition and practice. its not counted in final grade but u must get >60% in average of all homeworks in order to pass the course. Essentially forcing us to put some effort in the homework

[unborn hill]

Hmmm

Do students not half-ass it then? @unreal island

[unborn hill]

I think an exceptional teacher converts slackers into non-slackers

That's something I dream about sometimes

[empty gull]

this is very much not true
Movies portray it that way

but that's like

a massive lie

[unborn hill]

what makes you think it's a lie?

I think that's the seminal case of inspiration

[austere delta]

Some do. But adults are allowed to be responsible for their own actions. University is not kindergarten

[unborn hill]

That is true actually

Good point

[south ice]

It would be exceptional. Unrealistically so except in a small amount of cases.

[unborn hill]

😔

I mean yea obviously right

But ugh

Maybe because it's a public school ?

[empty gull]

Exceptional teachers don't motivate unmotivated students to do more
Exceptional teachers push those who have any level of motivation above 0 to do more than they think they could before

[unreal island]

the problem is that a few people who r actually passionate about math solve the questions and then share it to everyone else

i dont think u can solve that problem fundamentally

its almost impossible to track that behaviour

[austere delta]

Like by all means, motivate your students. But if they don't care, forcing them to do exercises isn't exactly gonna spark joy for the subject

[unreal island]

in the end the more u force them, the cleverer ways they find to bypass that

[unborn hill]

I think the first claim is a subset of the second

I don't think 0 motivation is possible lol

that's crazy

[unborn hill]

yea definitely, it shouldn't be a matter of force

[south ice]

Some people sign up for classes and literally never attend at all

[empty gull]

Believe me some people really couldn't give any less of a shit in some classes

[south ice]

Maybe it takes nonzero motivation

To sign up

[unreal island]

i lowkey see like 80 people on the exam day whereas on a lecture day, its like 20 people

i make new friends on the exam day

[south ice]

But it's so close it'e negligible lol

[unborn hill]

The more you force an assignment down a student's throat, the more likely they will be to get through it "fast" (e.g. entering random answers until they get it right or straight up cheating)
But if the assignment is optional for example, some gems might overlook it because they have other mandatory stuff to do 24/7

[unborn hill]

LOL

maybe I should have sit down and come up with baseline pedagogical axioms or something 😹

the main issue is there is too much external variety with students

[halcyon glade]

If you present the content in an interesting and informative manner, then you've done your half; they need to do their half

[unborn hill]

it is half and half i suppose

[halcyon glade]

I teach a recitation where few students show up, but the students who show up have all found it to be incredibly useful

That's good enough for me

[unborn hill]

did you make it extra credit/

?

The thing is when I took this course, I also did my fair share of slacking, but as the course progressed I realized how vital these concepts were for my professional success in general

But there is no algorithm for inspiration right, it's just something that happens, right synapses fire in your brain at the right time then boom

[halcyon glade]

No

People just came because it was helpful

It was at 8:30 AM too

I think in uni, students should be expected to be responsible enough to take charge of their own learning. You provide a scaffolding and help as needed, but you can't climb the ladder for them.

[halcyon glade]

I didn't actually mind at all

But it seemed to deter students from coming

[timid shard]

8:15 classes are better than 20:15 classes

[assuming this isn't a night program and there's a full day of studying before]

[timid shard]

yup

[timid shard]

it's interesting that trig is in the standard curriculum in most high schools around the world

like sure trig is important but I think there's more fundamental stuff that gets skipped in school so I'm not sure that's the reason

maybe it's the intersection of algebra and geometry that makes it a good topic to teach?

cherry on top of those two subjects?

[austere delta]

Yeah, I think this is one reason.

I think also, sin, cos and exp are important examples of smooth functions when going into calculus. And they are quite fundamental for studying differential equations, which is an important subject for the sciences.

[timid shard]

they're quite fundamental in most maths and science

but so are induction and logic and they're not typically taught

afaik

calculus isn't standard in America either

although I don't know the relationship between taking trig and taking calculus

maybe you have to do both

[austere delta]

Calculus is not standard??

[timid shard]

in high school? I don't think so

[austere delta]

Damn

I don't see induction or logic being particularly useful for the sciences.

Induction is fairly useless unless you want to do math. And I guess it depends what you mean by logic, people have to be able to reason, but I don't know that learning what the contrapositive is would help much with that

[timid shard]

true

however this is maths class

although maybe that's the answer, they want it to be as broad as possible

[austere delta]

Well it's called math class, but that doesn't mean "preparations for studying math at University"

[timid shard]

but we were taught trig and not stats

very little probability

not sure trig is more important

[austere delta]

Stats is something that should be taught, that I can agree with

[timid shard]

unless there's a good pedagogical reason, which is what I was curious about

[abstract grove]

calculus is a standard last math class for US high schoolers. For the ones qualified to take it

[timid shard]

how many "qualify"?

[abstract grove]

it is an advanced class, so not that many

but it’s still standard

I guess the most common last high school math class is probably precalc/trig

[timid shard]

I'll be honest I'm not sure I consider that standard

nevertheless, the same question applies, why is trig there

[abstract grove]

fair enough, bit of a semantic point

I don’t really know the story behind how the standard US math curriculum came to be

trig definitely has a lot of applied use in the sciences and engineering and so on

but I agree with you logic should be in there a lot more, and earlier on

[halcyon glade]

I don't really agree, I think if anything there should be more statistics, unless by logic you mean teaching logical fallacies like begging the question and the Wason selection task

Statistics is pretty unique in that it teaches inductive reasoning

I think trigonometry is incredibly applicable across math and physics, and it makes sense to require it before calculus

I also don't think there's any reason why people should have to take calculus, unless they want to pursue math or the sciences in uni, whereas skills learned in statistics are broadly applicable

[timid shard]

I don't disagree, my question was why is trig preferred over say statistics

true!

honestly though calculus without trig isn't that bad

you can still do a lot of interesting stuff

yeah? we learned it at the same time

doesn't feel like one relies on the other

just ignore trig functions

identities, geometric problems

equations

[halcyon glade]

Yeah it's totally possible, I think it might hurt you later on if you go on to study physics/engineering though, I saw this happen a bit with my Finnish friend attending an American uni; my Finnish friend had not encountered trigonometric functions until calculus and was surprised by some of the topics that were standard in American high schools

[timid shard]

I'm coming off here as really anti trig, I honestly don't care, it just seemed like and odd choice somehow

[halcyon glade]

(For most students, who don't go on to study math-intensive fields, I don't think this really matters)

[timid shard]

we never got that far in school but maybe that's just my country

Israel

[timid shard]

physics seems like the best answer to my question

I think that's it

I mean maybe but I'm not sure how that's related

[halcyon glade]

I think trigonometry is also a subject which had much more historical relevance (think navigation before GPSes)

Same with Euclid-style geometry

[timid shard]

which apparently is going out of fashion

[halcyon glade]

It's pretty lodged into place in the US system (unfortunately, IMO)

[timid shard]

id be happy to hear why you think that!

[halcyon glade]

The "two-column" proof style that's taught in US schools is a terrible misrepresentation of how mathematicians actually communicate and think about problems and teaches students that mathematics is just about proving obvious things using obscure acronyms and terminology

Most of the time it's just angle-chasing with a triangle, and there's no real creative element

It's also just not useful, either from the perspective of applications or for learning more mathematics

[timid shard]

I hadn't thought of that

I always thought it was cool there was a more proof based subject

but it is kinda different

[halcyon glade]

It's also just weird to me that geometry would be the only math subject that involves reasoning, surely algebra and calculus also involve reasoning? But it's very rare that students ever write arguments for those classes, even though there's a wealth of simple and interesting problems

It seems to me there's an artificial separation between calculation and reasoning about things, when really almost any type of calculation involves some sort of reasoning and any reasoning involves calculation of some sort

[timid shard]

our national maths final exam has been gradually moving from just "differentiate, integrate, find minima/maxima etc" to "show that" style questions in the calculus part and people (teachers) have been complaining the test is much harder for students

personally I enjoyed it much more

[halcyon glade]

I think that's healthy, it's good to put high expectations on students (as long as they're reasonable)

I've always thought a mentor-mentee pair is really good when the mentee realizes they can achieve something they hadn't thought to be possible before. A lot of students have learned helplessness, and it can be difficult to get out of that mindset.

[timid shard]

doesn't iPad have it built in?

idk

you can probably crop the video when editing it

there's free

no, don't have experience

you don't need anything heavy

you're not going to be doing FX work

CGI

crazy blockbuster film editing

[austere delta]

At least when I use goodnotes with Zoom, the toolbar is not displayed (I see the toolbar, but zoom doesn't).

So it might work similar with recording.

Can also use zoom or similar software for recording

Then you can also use audio from your computer or a microphone or whatever

[rapid tusk]

holy shit so much this 😭 i think it'd be much more enjoyable for students to learn geo if the things they were proving weren't so insultingly trivial

we don't need five lines to prove smth as obvious as vertical angles

[pure light]

i think the idea is that an introduction to proofs is best done if the things you're proving seem intuitively true

not that it's executed very well

[wary ether]

I'm doing some marking and I have seen the same mistake come up a stunning number of times

People keep calculating the magnitude of a vector by summing its components

Where do you reckon this comes from?

[long pelican]

My guess: Not doing their homework :)

[midnight scarab]

It's cause (1,-1) is obviously the zero vector

[wary ether]

They didn't sqrt either

And idk, this has happened several times

[midnight scarab]

I think this is similar to why people add numerators and denominators of fractions

Just do the simplest possible thing without any regard for the meaning

[heavy trail]

My guess is two-fold:

1. They are following a folk conception of "norm" that is more like the L1 norm
2. If they ever use concrete vectors in their reasoning or as a sanity check, they are particularly nice vectors, e.g., integer coordinates, all positive/non-negative, etc.

To figure out students' folk conceptions I put them in a position where they have to answer quickly, without real time to reflect. I want their most reflexive, unreflective judgements.

Like, on the count of 3 I'm going to reveal a vector on the whiteboard. You have 5 seconds to write down its norm on a piece of paper and then hold it up in the air. Make the first vector (1,0) and folks will say 1. Make the second vector (0,10) and folks will say 10.

Make the third vector (1,1) and see how many folks say 2.

[wary ether]

Hmm that's a great theory

I might try that sometime

[heavy trail]

I wouldn't comment on the 2, but I'd take stock.

Then I might ask about (-2, 2). Does anyone answer 0?

If someone has a halfway-reasonable conception of the L2 norm they might still write 0, but as soon as they do they should go "Wait...that can't be right."

And if someone says 0 without thinking twice then their conception of "norm" is way out of whack.

If they're just working abstractly, it's easier to say |(x,y,z)| = x + y + z without thinking twice. There's nothing immediate which even affords them the opportunity to realize they were mistaken.

[heavy trail]

I really enjoy Wiio's Laws: https://jkorpela.fi/wiio.html

Taken literally, the laws are tongue in cheek. The first four:

1. Communication usually fails, except by accident.
   a. If communication can fail, it will
   b. If communication cannot fail, it still most usually fails
   c. If communication seems to succeed in the intended way, there's a misunderstanding
   d. If you are content with your message, communication certainly fails
2. If a message can be interpreted in several ways, it will be interpreted in a manner that maximizes the damage
3. There is always someone who knows better than you what you meant with your message
4. The more we communicate, the worse communication succeeds

The page ends with a "pedagogic corollary":

The Pedagogic Corollary: Give the student a chance to realize he misunderstood it all

[vocal phoenix]

Those are incredibly accurate, I've experienced this both in education and in business

[tawny slate]

yeah i think thats a good way to think about it

humans do tend to subconsciously prefer simplicity

[heavy trail]

I don't think it's about simplicity per se, it's just about whatever first comes to mind given a student's prior understanding, experience, and current context.

For example, I teach colleges students to code — students who are absolute, never-seen-code novices — and they'll read code like this early on:

x = 10
print(x)

They see = and the entire complex of thoughts, habits, training, etc. around high school algebra get activated. It doesn't even occur to them there's something they could reflect on.

I sometimes drive it home by talking about things like:

[heavy trail]

I think about (4) every time someone suggests adding more content or details to their syllabus as a way to head off common student mistakes, misconceptions, etc.

I'll just explain it in even more detail and it'll be clear!

The checklist will continue to grow until the mistakes stop.

(Current size of checklist: 832 items. Don't miss any!)

[midnight scarab]

And then comes if x = 10

[heavy trail]

Hehe.

Rather than exploding the level of detail by elaborating on the difference, I try to emphasize from the start that these "facts" are choices humans made and they could've been otherwise. The computer doesn't care.

I'll show them some Pascal, ML, Smalltalk, Scheme, FORTRAN, etc. so they get an early sense of how arbitrary the choices are.

Why is double equals so common? Because C did it that way and C was so popular that doing anything too different in future languages was an impediment to adoption.

“We build our computer systems like we build our cities: over time, without a plan, on top of ruins.

— Ellen Ullman

[tawny slate]

aaaaand wilos law at work

[heavy trail]

Those Finns, always lurking

[dark phoenix]

hello

[pallid quail]

hi there

[barren shore]

guys

someone with generalized anxiety disorder keep dming me asking for math help, for their functional maths level 2

but i have no idea how to teach someone like that

they struggle a lot with basic fractions

im not a teacher to begin with

but i cant ignore them

what do i do

😭

i dont even have patience

is there any teacher here who would really love to work with this dude and teach him some maths?

[timid shard]

if they're not your friend and/or paying you, you have no obligation to help

they can ask in this server

or many other online sources

like flatly say "I don't have the time/skills to help you, but these sources might help" if you don't want to just leave them on their own

and link to help servers yt channels, khan academy etc

[barren shore]

that's what i did..... he still keeps dming 😭

[thin frigate]

That is called harassment, straight up

I would block him

Like, he doesnt listen, clearly

So you dont have many options left

[halcyon glade]

Agreed, block, and report to ModMail if they're from this server

[heavy trail]

You have peers who attended lecture. Talk to them.

[timid shard]

not having a strong basis of algebra and arithmetic really fucks students over

it's hard to teach and progress in integration if we need to go over power rules most of the lesson

[halcyon glade]

I think at that point you just have to tutor them in algebra instead

[timid shard]

yup, hence the getting fucked over, it's hard to close the gap

[halcyon glade]

Yeah :|

At least if you're aware of it and it's one-on-one, you can adjust your lesson plans

If you have a group of really mixed students, that really sucks

[timid shard]

yeah, can't imagine that's fun

it's also sad to see people struggle over their advanced material and not understand why they can't do it when the problem isn't even in that material

[potent dragon]

If there is a student who incessantly asks questions irrelevant to the topic (do i know how to do a back flip, can i be a penguin, etc etc), and wont get back on topic no matter what i say, is it better to answer them and take up class time or ignore them?

[halcyon glade]

Are there other students there or just them?

What's the context here, are they being forced to be there?

[potent dragon]

I’m teaching taekwondo, and while i am trying to teach the class of about 15, i am being tugged, interrupted, yelled at, etc etc and it is disrupting the learning of the other students, all of whom are paying to be there.

The issue is: He’s paying too, so i can’t just deny his learning. I believe my boss has brought it up with his father several times, but it doesn’t seem to be having an effect.

[halcyon glade]

It sounds like your boss is aware of the issue, and it's pretty severe. Have they considered removing the child from the class?

It definitely seems unfair to the other students to entertain the whims of this one student

[heavy trail]

This is up to your boss, the training/school/dojang culture he wants to embody, and his feelings about customers.

Make sure he knows you feel like it's beyond your capacity to handle effectively and the issue has become acute. Do you have permission to tell the child to stop? Can you tell the student to sit out?

It's a martial arts studio after all. The student could be asked to sit out the remainder of the session if he isn't disciplined enough to engage non-disruptively.

The "worst" that happens is your boss loses the customer, which your boss might be fine with. Or maybe your boss gets buy-in from the parent to take more forceful measures (like being asked to sit out, etc) as part of the discipline/training.

You have a lot of wiggle room here, much more than the typical, academic classroom environment. Convince your boss that it's a real issue for you and for the other students and come up w/ a plan together. Hopefully they're down with it and don't just shrug and say, "Well, I talked with his father, not much else I can do."

It's called a dojang in Taekwondo, right? I did Judo, so only know Japanese martial arts terms. Apologies if I got that wrong.

[potent dragon]

I’ll keep this in mind today. I appreciate the help from both of you.

[potent dragon]

It is! Not a lot of people know that, always nice to see someone with knowledge on taekwondo

[heavy trail]

Speaking as a business owner, there is such a thing as "firing your customers".

If you have one customer that takes up 90% of your support time and generates 0.1% of your revenue then it can make good business sense to stop servicing them (aka "fire" them).

I realize there are other cultural and social factors at play. I'm just pointing that out because your boss might be more ok with simply removing the student from class than you think.

(Assuming your boss is the person ultimately responsible for those sorts of decisions)

[acoustic basin]

Good people here

[deep kindle]

Changing the sign of a in a(bx-h)^n + k is always taught to be a reflection about the x-axis but I fail to see how it is unless k is 0. If k is non-zero, the reflection is about the line y=k, not the x-axis. Am I crazy?

[halcyon glade]

You could also interpret it as a reflection about the x-axis then a translation by k.

[deep kindle]

Yet if you start with a translated function and then change the sign of a, it's not a reflection about the x-axis

Is there an order of transformations?

What benefits does that interpretation have over mine other than being a little less abstract?

[halcyon glade]

There is a natural order in this case just because of how algebraic expressions are nested

The +k happens after the a*

[abstract grove]

yeah, if you purely want to reflect y = a(x-h)^2 + k over the x axis, first factor out a: y = a[(x-h)^2 + k/a]. Then the reflected version is y = -a[(x-h)^2 + k/a]

[night vessel]

Hey, teachers i am student of standard 9 in india. I would like one of you (if comfortable and free) to teach me some advance maths for International Maths Olympiad. I have put forth my request and even if nobody's willing, i feel honoured to ask you this.
Thanks

[turbid zenith]

F's in chat please for grading absolutely terrible calculus tests

[midnight scarab]

Oof

Wishing you strength

[turbid zenith]

Like "how do I even give meaningful feedback" level

Did I miss something and make this too difficult somehow?

[midnight scarab]

Seems like a good exam (assuming they know what marginal revenue etc means)

(I would have wasted 30s figuring out the meme though 😂)

[midnight scarab]

Perhaps not the easiest (as in it requires some thought, which is good)

Did most students manage to do 1a&b, 2a-c/d and 3?

[turbid zenith]

Well they should… the class is for economics/business/accounting majors

So I use marginal stuff all the time in my examples

[turbid zenith]

Nope

Lots of people thinking the derivative of 6π^2 is 12π, or that 9/x = x^-9

And lots of derivatives of fg being f’g’

[tall bolt]

This looks pretty on par with the calculus I did in highschool so no I wouldn’t say it looks to hard

I feel like the lack of understanding of the product rule and power rule is a clear indication that people just didn’t learn the material, it can’t get much more basic than that

[lethal leaf]

it seems long?

idk how long they had to do it

it seems like the type of exam where if you know everything coming in, you have the time to write it all out

[halcyon glade]

If the mistakes are widespread enough, I would probably just go over common mistakes in class instead of giving detailed feedback to each person

[lethal leaf]

but if you have to think about most problems you're screwed

but again idk how long they had

[turbid zenith]

They had 1 hour

#1 should have been pretty quick — just do the derivative rules

#2 I can see giving people problems if they forgot the formulas ... but they were allowed to bring a notesheet

So I have people doing stuff like revenue = price * cost, because I gave them price and I gave them cost, so obviously they must need to either add, subtract, multiply, or divide them!

It's the same thing that has people learn that the way to get through math class in elementary school is grab whatever two numbers you see in the problem and slap one of The Four Operations between them

[lethal leaf]

yea but like reading and writing everything takes time, not even counting needing to think about the problems if you don't remember it exactly

right but they've been doing elementary school math for alot longer than calculus

it might be worth prodding your class and asking if they felt that time was an issue

[turbid zenith]

I'm saying it's the same kind of thinking

[lethal leaf]

same kind != same comfort

[turbid zenith]

Yeah, I will ... but I already distilled it a bunch to make it shorter

[turbid zenith]

I mean it's the same kind of wrong thinking

[turbid zenith]

Thinking about it, can you imagine a student NOT saying yes to this if asked?

[lethal leaf]

yes in that maybe the issue was that they didn't know the material

in which case giving them 10 hours wouldn't have helped

[turbid zenith]

That's a possibility, but maybe I'm just cynical and feel like many students would jump at the chance to say "yes it's your fault"

[lethal leaf]

fair point

[turbid zenith]

Considering how I've had students tell me "we never learned that in [class before this one]"

[lethal leaf]

maybe I'm being too hopeful in that they can self-reflect honestly like this

[turbid zenith]

And I'm thinking "you were literally my student in [class before this one]"

Self-reflection is that skill we wish students had

I'm starting to grade #3, and I've already had one student put 65/1 for (a)

dM/dt? Just put M/t, that d doesn't mean anything anyway

. . . ugh, thinking about it, is that theoretically a plausible but really inexact answer?

Since you traveled 65 miles in 1 hour?

It's an average rate of change at the VERY least...

Bruh, I've graded three tests, and nobody has gotten 3a. Literally miles per hour.

I think so far 65/1 is the most common answer I've gotten

[quasi musk]

I guess you're going for a slope of the secant line to approximate the slope of the tangent line

So the most relevant data points would be t = 0.75, 1, and 1.25 with d = 55, 65, and 78; respectively. So the guesstimate would be something like $$\frac{78-55}{1.25-0.75} = \frac{23}{0.5} = 46 \frac{mi}{hr}?$$

[burnt vessel]

MoonBears-C-

[quasi musk]

Is that your intended solution?

[turbid zenith]

Sure that would be one possible solution

You could also replace either one of them with directly at 65

Any of those show you're using the idea of a slope between two close-by points, and if you've got the right unit you're gold

[quasi musk]

So technically, 65/1 miles/hour is a correct solution because it is the slope of a secant line

I think maybe if the question wrote something like "Write down a good approximation" or something like that to avoid the trivial solution that students didn't think about

I think the exam is reasonable for 1 hour. If a student is ready there are 3 questions, and each question takes sub 10 minutes

at a maximum, the exam takes a good student 30 minutes. Multiply by two and you get one hour

[turbid zenith]

I decided to give credit if they put (65 - 0) / (1 - 0)

(Nobody did)

[quasi musk]

I agree 65/1 isn't an acceptable answer

[feral swan]

I think it's too long for 1 hour

It feels like 1.5 for a decent time, 2 to really be generous

[timid shard]

I agree

[feral swan]

I remember teaching geography calc and this would kill them. The derivatives don't become rote

[timid shard]

people probably got stressed

[feral swan]

But not speaking from a lot of experience, mostly from seeing my fellow students

Like my probability prof said: " if it takes me more than 15 minutes to solve a 3 hour test the test is too hard"

[tall bolt]

I agree an hour is a little tight, but it seems the issues were much more fundamental than a time pressure issue

[heavy trail]

1 hour is too short. There are 12 questions, which means 5 minutes per question including the time it takes to understand what the question is asking.

Take a question like (1) which is just calculating the derivatives. Let's say someone answers (a)-(c) perfectly. They give accurate mathematical reasoning. They show exactly what rules they use and justify their use. Maybe they go above and beyond and specify where the functions are differentiable.

But they leave (d) and (e) blank. Do they get 6XP since they did 6 out of 10?

Does that student understand the material better or worse than someone who got 8XP but their answers had minor errors, they misapplied some procedures, had mostly-but-not-quite-correct reasoning, etc.?

Would they have extended their "perfect" work from (a)-(c) to (d) and (e) given 10-15 more minutes? Are they being evaluated on how fast they can produce an answer?

[heavy trail]

For example, how many students fell into the misdirection/trap set in 1a by misreading $\pi$ as $x$ when asked to differentiate $$y = 5x^3 + 6\pi^2 + 7x - x + \frac{9}{x}$$
\
The structure of the very first question screams: "Read very carefully, or else. You won't have time to come back and fix a mistake if you realize too late that you misread something. But also don't dilly dally because there's 11 more where this came from."

[burnt vessel]

Cufflink

[heavy trail]

That's a genuine question, BTW. You should have ways of checking the construct validity of your assessment, however crude.

Say 30% of the students made that mistake and answered as if you wrote x insyead of \pi in 1(a).

What is that information supposed to tell you about a students' understanding of derivatives? What proportion of those 30% are confused about how to differentiate a constant versus misreading the question because it is all-but-identical to other polynomials they've seen?

If a student correctly differentiated constants elsewhere on the exam I'd probably assume they just misread the first question.

If the goal is to get them to "look closely" or really isolate whether they understand the difference, I'd ask a different question, e.g.,

Do these two functions have the same derivative? Calculate the derivative for both and say whether they're the same or different:

f(x) = 5x^3 + 6π^2 + 7x - x + 9/x
g(x) = 5x^3 + 6x^2 + 7x - x + 9/x

[turbid zenith]

I evaluate their scores holistically. If they gave amazing perfect mathematically accurate reasoning on, say, 1a-c, then they would likely get 15 XP for Product and Quotient Rule at the very least. I'm not looking to take off points, I'm looking for evidence of what they do know.

[turbid zenith]

I get this, except they've already seen exactly that trap in class at least twice. We talked about it specifically.

We've specifically discussed the importance of reading carefully, of taking a few seconds to think before you just launch into a procedure, and so on.

It shouldn't take 5 minutes to take the derivative of a polynomial-like expression. If that's taking you 5 minutes, you haven't practiced enough. Period.

I'm not sure how much more stripped-down I could have made this test to fit in the 1-hour time block I've been given. I would have loved to have it be a take-home problem set, but, yeah, ChatGPT is RAMPANT now.

Should I just have, like, killed #2 or something?

[heavy trail]

I see. Is this the first exam you've given them? Or the first one you've graded like this?

[turbid zenith]

Yep. And they have the chance to completely replace the XP from the exam using the final.

It's the first time I've given an in-person exam in forever, I used to do all problem sets and such.

But now it's near-impossible to ensure that the work is theirs anymore

[heavy trail]

Well, I have thoughts, but I'm not sure if you're looking for feedback. I shouldn't have presumed.

[turbid zenith]

I don't really know what I'm looking for. :/ Part of it is venting, part of it is of course wanting to fix things

But this already comes from having to change what used to be a very generous mastery-based lots-of-tries-until-you've-gotten-it grading system to what seem to be the realities of the available technology and people's incentives

So I don't know how much more I can cut things down without starting to just not teach certain material, and even THAT is already gutted a good bit because this is a business calculus class, for economics/business/acccounting majors

So yeah I guess if you want to share your thoughts that's fine ... I don't know if I have much room to make drastic changes at this point because of the syllabus etc

Yeah. 1 hour classes feel short for this class. I guess I could split across two days, but that means I'd have to get rid of another day of class.

4 people earned 3 badges (out of 4, which combines these scores with the autograded homework on Edfinity).
1 person earned 2 badges.
4 people earned 1 badge.
10 people earned 0 badges.

Nope.

One person will likely earn the 4th badge once they finish some missing work

10 XP possible for Edfinity (autograded HW)
15 XP possible from exams

25 XP total

You need 20 XP to earn a badge

It sure is interesting to see students earning 10/10 on the homework but 7/15 on the exam

Unfortunately it's very difficult (for me at least) to come up with questions that ChatGPT can't do in this class

(Plus I only have so much time to do anything for this, because I'm teaching 3 other preps and finishing my dissertation)

You need 20 out of 25 XP to get a Badge.

10 XP can come from Edfinity, 15 XP can come from the exam.

Each Badge has its own Edfinity assignment, and on the exam there are subscores on how many XP you earned toward each Badge

So for this one it was:

• D-1: Derivative Concepts
• D-2: Derivative Computations
• D-3: Chain Rule
• D-4: Product & Quotient Rules

Yup, it's a WeBWorK wrapper that's easier to set up

The questions on there are harder than the ones on this exam, and expose them to pretty much all the things that were on the exam

Example question and solution.

I guess I have the power since I'm the only one teaching it, but wouldn't that have made it even worse? 😂 Since the in-person exams were the part people flailed on.

[heavy trail]

I think there's a lot of low-hanging fruit on this exam itself, just from an instructional/experimental design perspective. I don't think you have to test for fewer things.

I can't say whether what I have in mind would improve outcomes, but (I believe) they'd improve controls for other factors so you'd at least be sure a student doing poorly indicated something about their understanding and not something about the exam.

I think one should focus on what happened here and now and get a handle on it first, before flipping a bunch of switches and turning a bunch of knobs based on vibes.

[turbid zenith]

That'e essentially what they already have :/

They get 3 tries per question on any one attempt but they get unlimited attempts.

Not that I've ever heard.

[turbid zenith]

I'm interested.

[midnight scarab]

But the students who messed up wrote down 12 pi, not 12 x. Which means that surely they noticed it was a pi not an x

What's stopping them from wanting to get it over and done with using gpt?

[turbid zenith]

This is a good point

If a student wrote 12x I think I would be more okay with that

[subtle field]

im not a teacher/ta/prof, but i figure its fine to ask some questions here:
ive agreed to give like an intro-group theory talk to a community im in (specifically a community around rubiks cubes), so group actions are particularly important. does this seem like a reasonable progression of ideas to go over? and are there any tips i should know about teaching group theory or just math in general? i dont have much math teaching experience. if this is the wrong place to ask lmk (and if someone answers please ping)

[feral swan]

No you can't and the new model is reportedly even better

I was told it can solve real analysis 1 exams

[turbid zenith]

Yeah. Like every time a new model comes out and someone goes "tee hee it still can't count the R's in 'strawberry', it thinks 9.11 is bigger than 9.9, tee hee"

But that leaves out all the things it CAN do, and they're often things that can way too easily undermine learning

I can much more easily write stuff for my liberal arts class that ChatGPT isn't as good at, but for something standard like calculus, that's much harder

[subtle field]

hmm, i was planning on doing it then so i could cover orbit/stabilizer thm which is important but ig i can just cover that after quotient groups? makes sense

[quasi musk]

It also makes standardized testing/in person tests that much more valuable when evaluating students ability, knowledge, and experience

[turbid zenith]

Yup

[quasi musk]

I hadn't had that realization until you just said it. I know for a long time educators have pushed against these in person exams

But now I can't see a valid argument against them

[turbid zenith]

I mean I see plenty of arguments against them

But unfortunately AI has made it so there's not really another way in many cases :/

[quasi musk]

Exactly, but the validity/soundness of their arguments have been significantly decreasing due to AI. There are definitely aspects of learning & being a student that aren't captured on a test. But there's no good way to properly individualize education when we're assigned 40+ students/class

[turbid zenith]

That depends on what arguments they were making.

For something basic like algebra or calculus, its arguments are often fine.

I ran it through a bunch of my problems, and while they weren't ALL right, a whole ton of them were.

However, I will probably cancel my Chegg subscription pretty soon

Because nowadays nobody is using it so I don't have to watch for my questions to pop up on there 🙂

[halcyon glade]

wait how long is this talk??

[subtle field]

as long as it takes

how long do you think itll take?

[halcyon glade]

With 3 hours a week, around a month or two

[subtle field]

its online so people can come and go as they please

i wasnt planning to be super thorough

its group theory for a specific application so i dont need to cover things super deeply

[turbid zenith]

Stuff like this, though, ChatGPT tends to do very poorly on XD

[elfin sleet]

Your list is basically the whole content of my school's introductory group theory course, which has a typical 3 hours per week, 3 months of lectures structure. Granted, that course moves slowly and you won't be as thorough, but it gives a point of reference

[subtle field]

okay, interesting

thanks

i might split this into multiple talks then

[feral swan]

@turbid zenith wanted to say that if I got you as my prof I'd feel blessed

Its so hard to teach early year classes because of their difficulties and mentalities and so many just give up. But one teacher that doesn't do it can change the whole degree, and indeed, life, for a student

I really respect the effort and care you can see through the tests and pages and general discussion you put up here sometimes.

[flat token]

Me and my profer talked about an axiomatic approach to teach math from the ground up, obviously doent work for the average student but was interested on everyone's ideas on that.

[rapid tusk]

its one of those things that sounds better on paper than it actually works in practice

[tardy ember]

honestly i don't think it sounds that good even on paper

it's a good way to teach maths to a computer, but the first thing a human needs to know isn't the details of the logical foundation, it's the entire concepts of logic, abstraction, etc.

[rapid tusk]

the time you waste going over pedantic formalisms could be better spent

proofs are important especially when a result seems to go against our intuition

but if you're going from first principles on everything

your students will have completely tuned out by week 2

[tardy ember]

also given the whole "humans aren't as good as abstraction as computers" thing, it's often not really reasonable to attempt to learn about something without seeing any examples of it

so if you want to know how to prove things, it isn't necessarily completely useless to have an actual definition of what a proof is, but most of what you want is a lot of examples of proofs

[rapid tusk]

there's a reason we don't start the grade school math curriculum with real analysis

[tardy ember]

...also i think a lot of students, if you give them a completely formal definition of proof, will either not understand the definition, or at best, use it correctly but be left with no idea how to actually solve anything except when they successfully stumble onto the right symbolic manipulation

[pure light]

we did try something resembling that with new math. famously didn't go over very well though

[rapid tusk]

every time i see someone overromanticizing learning math fRom fiRST prinCIpLeS

[tardy ember]

an important part of humans performing mathematics is the extent to which, on a day-to-day basis, you should not be thinking about exactly how the foundations are defined, you can re-use a lot of your pre-existing intuitions of how reasoning works and just refine them in the areas where they don't quite work

[rapid tusk]

i die a little inside

[tardy ember]

also related to this, the fact that proofs are in practice mostly written in natural language

[flat token]

Hence why i mean it wouldnt work for most

but i feel like this is the approach i take when i try to teach myself anything

obvioulsy an axiomatic teaching method does not need to mean going from logical foundations

but more so, teaching the idea that if we define this then this follows blah blah

i feel like thats the main issue most people sturgled with when going from high school to uni

is just that concept it s self

Like for example i was tutoring someone in basic calc, like limits and stuff and it really seemed to help to write stuff out with strict definitions instead of just doing stuff otherwise

definitions are usually stated but their importance are not really conveyed in early math teaching

i think that lack of importance/emphasis is really what can make math confusing for a lot of people

i taught my gf, who did not take a math class for 2 years the basics of analysis using this method and she understood it quite well so, i try to use it every time i teach anyone anything

Even for proofs at my level i absolutley hate it when some assumptions are pulled out of no where and their justification is not explained, my dream proof would have a line by line explanation where everything follows from

I have a professor who does this everytime and i absolutley love it, it is so easy to understand the most complex subjects

[rapid tusk]

ah you're talking about the more meta stuff like

what motivates us to take this step?

etc

[flat token]

not really

[rapid tusk]

if not exactly

it seems related

[flat token]

just emphazizing prior definitions

[rapid tusk]

oh many a good prof will make sure to do that

whether the students are paying any attention is another story

[flat token]

yea but many dont and ESPECIALLY at lower level maths like high school

im not really talking about proofs here just using it as an example of how this applies to my current situation

[rapid tusk]

oh at the high school level

yea i had quite a few teachers like that

[flat token]

Well i, and most of the people i know didnt

[rapid tusk]

as in

they failed to explain things fully

etc

[flat token]

yea i get it

math should be easy because its just a huge chain of implications, if A then B

you get lost in the sauce when that chain breaks

[rapid tusk]

i love reading [every late undergrad textbook ever]

where steps are skipped left and right

😃

thanks rudin

really helps my understanding !!

[flat token]

when steps are skipped i lose my crap

i have yet to find a perfect textbook

[rapid tusk]

or when they're poorly justified

if going into full detail on everything was easy someone would've done it by now 😭

[flat token]

very easy to write (Through def 2.x) next to your work

litterally a (Through algebra) would help sometimes

[rapid tusk]

my ultimate pet peeve

"by definition 31452346.12859.13883989"

it's much easier to remember stuff if you actually give it a name

give it character to remember it by

[flat token]

yea okay obviously

through x's theoreom, or through associativity wtv

naming can be left as an exercise for the reader

[vocal phoenix]

At least these days you can do hyperlinking in PDFs, so that you can click the "31452346.12859.13883989" to navigate to that place in the document.

But yeah, I also favour giving some kind of descriptive label to things that are referenced often

[timid shard]

thank you amsthm

[turbid zenith]

Woo! I get to teach complex numbers next week in my college algebra support class! 😄

[timid shard]

I really liked roots of unity when I studied that

thought they were neat

hope your student do too!

[turbid zenith]

We're not quite at that level

But I am gonna show them how to graph them

I think it's a travesty that students learn about i without talking about rotation

Some slides in the works before the students start trying some problems

[timid shard]

Gauss' plane isn't part of the curriculum?

[turbid zenith]

Nope! You just learn that i is this made-up solution to a problem that doesn't have one

And learn to do algebraic manipulations on it

[timid shard]

that seems so sad

[turbid zenith]

And even if you do learn about the complex plane, you certainly don't ever mention rotation

I guess the thinking is, talking about rotation involves sines and cosines! We're not teaching that until precalculus! And yeah, once trig is introduced, THEN students (might) get to see De Moivre's theorem etc

But like ... you don't need ANY of that for a 90 degree rotation

[timid shard]

that seems very backwards

[heavy trail]

I think that's a great approach. Something I also do is point out that that a lot of the numbers they believe they know "inside and out" are stranger/less intuitive than they think.

Articulating ideas like 0 and negative numbers and irrational numbers and then figuring out how to think of them as numbers were hard-won battles over (literally) thousands of years.

I give examples along basic dates and moments, which usually surprise students.

[heavy trail]

The actual history is pretty compelling, too. For example, Cardano, who is usually credited with being one of the first people to write it down, was skeptical of them but saw they helped him find the roots of cubics.

But he also didn't have the contemporary concept of a negative number!

“Cardano stated very clearly that negative numbers should be avoided: “Subtraction is made only of the smaller from the bigger. In fact, it is entirely impossible to subtract a bigger number from a smaller one.” And indeed, negative numbers are essentially absent from his treatment either in the enunciation of the problems or in their solutions.”

From A Brief History of Numbers by Leo Corry

Modern algebraic notation is being developed around the same time, too, but wasn't widely used. For example, the equality sign = only first appears a few years before Cardano, Bombelli, etc. start talking about (what we'd now call) imaginary numbers.

I don't necessarily go into depth with the history, but the goal is to emphasize to the students that concepts they take for granted like 0, negative numbers, square roots, etc. are hard-won concepts, developed and refined and argued about over the course of centuries and centuries.

There's a human element, here, and trying to change the way you think about numbers so that i can somehow count as one is part of a process that's as old as numbers.

Go back 1000 years and however you feel about i, folks would feel about -1.

[turbid zenith]

Yup! I love the story of i and of the cubic.

Cardano, Tartaglia, all of it.

[heavy trail]

nice

[tawny slate]

i absolutely agree that complex numbers should be taught before trig, i think complex numbers are the core to understanding sum and difference formulas

complex numbers are incredibly natural i think, lots of motivated applications when solving problems in real domains, nowhere near as niche as say quaternions

and on that note i think another thing you should include, in accordance to the cardano stuff

just telling students that accepting complex numbers helps solve problems involving reals (like cubic roots) may not necessarily be very compelling on its own in the abstract

perhaps you might create an example where problems involving positive numbers can only be solved by using negatives or integral solutions can only be found by using fractions/reals

[austere delta]

I find the fact that second order linear differential equations are much easier to solve with complex numbers a somewhat compelling reason.

It's also the context in which I was taught complex numbers myself. But it might have been beneficial to have learned about then earlier...

[tawny slate]

i do agree

but i think the reason why i wanted to include a more elementary example to motivate domain extensions was to avoid the feeling like this was so abstract and complicated and obtuse and niche

differential equations, while i agree are more powerful and have more direct real world application than polynomials, feel more intimidating to students who dont understand either well

like if someone told you that this highly exotic nonstandard number system is awesome because it makes the Vlodovsky-Sakamoto theorem applied to homeomorphic Jordan-Hilbert spaces on F2 and F3 norms super intuitive, the second question youd ask is "why do i care"

bringing this idea to a much more tangibly basic ELI5 level helps motivate why this kind of domain extension is so useful even for the most basic concepts

otherwise, id point out how complex numbers are even useful in generating functions used for counting

[austere delta]

Okay, but what are these 5yo situations where complex numbers would be useful?

Useful for deriving trig identities, but kinda requires accepting Eulers formula

[turbid zenith]

I tend to give alternating current as my example

It's simplified but I imagine time passing in quarter-cycle "ticks". Each time the current is multiplied by i.

• Starts off with current flowing in the positive direction.
• Multiply by i. Now it's stopped.
• Multiply by i. Now it's flowing in the negative direction.
• Multiply by i. Now it's stopped.
• Multiply by i. Now it's positive again.

What I tell students is essentially this.

Is -2 a number? I mean you can't have -2 apples right?

• Well, no, but that just means negative numbers aren't used for counting. They're better at representing things that change.

Is i a number? I mean you can't have i apples right?

• Well, no, but that just means imaginary numbers aren't used for counting. They're better at representing things that rotate or fluctuate.

[austere delta]

Having a think about what numbers are actually for, and why for example -2 should be a number makes sense.

The AC analogy hopefully distills some intuition, even though it's not really saying anything precise. Sounds like a good start, and then one can return with more motivation later.

[halcyon glade]

I think computer graphics is a nice motivation (being able to stretch and rotate stuff by multiplying by complex numbers)

Personally I remember learning about roots of unity, etc. when I learned about complex numbers (US), but it was sort of an optional assignment. We mainly just talked about solving quadratics.

Also yeah waves/anything periodic are a good motivation too

[halcyon glade]

I'm curious, do they know what AC is?

It's not a very intuitive concept either tbh, as opposed to like a Ferris wheel or some other circular motion

[turbid zenith]

I can explain it in five seconds

And they use it any time they plug something into the wall.

[tawny slate]

nono, i meant give an example where you need say the real numbers to solve problems involving only integers

to show that domain extensions dont have to involve high powered math to solve niche problems

[turbid zenith]

Most of the time my students have at least heard their plug called an "AC Adapter"

[wary ether]

I did a little exercise in my class where I had a few true/false questions. I had everyone put their hands up, then asked people to put their hands down if they had no idea, then put their hands down if they thought it was false.

I think the principle of this is good but... I noticed that people were writing and I interrupted their writing by having them raise their hands. And I was also worried about a horrifying situation where maybe only one person has their hand up (which fortunately did not happen, but anyway).

Any suggestions on improvements?

[halcyon glade]

I think if they're not raising up their hands for very long, it shouldn't impact their writing that much, you could control the pace of this by asking questions slower or giving students more time for thought before prompting

For the one-person-alone issue, I don't think it's necessarily a problem unless you make it a problem. You can even actively normalize it like, "Alright who disagrees? I want to hear from someone who disagrees." That way you can sort of encourage the process of taking risks or expressing academic disagreements. In general, if you're comfortable with it, they'll be more at ease too.

This is just my limited experience though

[austere delta]

Had a situation like this when I was subbing the other day.

Was asking like "who thinks this / who thinks that / who has no idea etc" with show of hands.

Only one person raising there hand for the "have no idea" question, realized I maybe shouldn't have asked that 😛 but it seemed to go by totally fine

[wary ether]

Yeah, this is why I did the all hands up thing. I think it's less likely to make someone feel singled out. I guess if people aren't too worried, it doesn't matter too much

[halcyon glade]

You could also try subtler signals like thumbs up/down

It's a bit more anonymous

[wary ether]

Hm that's a good idea I think. The point is that people engage even a little bit, so that ought to work

I like that a lot actually

[halcyon glade]

Yeah I think you just got to gauge the energy level of the classroom

[turbid zenith]

Something I learned from a colleague is to have students hold up fingers to their chest

So like they can vote with 1, 2, 3 fingers etc, or they can give their level of understanding

That way only the prof can see it

[swift cargo]

Hi everyone! I'm a math teacher from Mexico, and I'm curious about how addition and subtraction of integers are typically taught in your countries, especially to beginners around middle school age.

In Mexico, we often teach a rule of thumb: "different signs, it's a subtraction; same signs, it's an addition" to find the result (for example, -6 + 12, it's a subtraction; -6 - 23, it's an addition). While this can lead to the correct answer in some cases, I'm wondering if this specific rule, or a similar one, is also used in your education system, or if a different pedagogical approach is preferred.

I've seen that at least the majority of textbooks in English present the correct rules showed in the image. However, I'm keen to know how you personally do or teach these fundamental operations. What methods or explanations are emphasized?

My main interest is to understand the pedagogical approach. Is the focus primarily on conceptual understanding, or is memorizing rules more common? Thank you in advance for sharing your insights!

[normal bronze]

Those rules feel unfamiliar as a Canadian, but things could have changed since I learned...

The main conceptual tool from my recollection was the number line. Positive number means moving right, negative number mean moving left, and adding two numbers means finding the overall movement left or right.
I believe there was also a concept introduced that there's not a lot of difference between "subtraction" and "addition" on the integers, because 5 - 6 = 5 + (-6), and more unnaturally you can do 5 + 6 = 5 - (-6 ).
Otherwise, I remember a lot of just teaching quick rules and memorizing how to apply them.

I'll talk about how I personally do the computations, hopefully it reflects some pedagogical approach or helps a bit.
When I personally do integer addition, I essentially do what rule 1 and 2 say, but I don't think of it in terms of adding absolute values, I think exclusively in terms of the "likeness" of the signs.
When the two signs are like, I do rule 1, but it's just a 3 apples + 2 apples = 5 apples thing, I'm not taking the absolute value and then adding.
If anything, I'm happy when they are like signs because I don't have to consider absolute distance from zero at all.
So in Example 3a, I'd just omit the middle computation with absolute values.

When two signs are unlike, if the negative is attached to the number with smaller absolute value, I arrange it so it looks like usual natural number subtraction. In Example 3b I feel like I basically skip the "absolute value step" because of this.
If the negative is attached to the higher absolute value, I make a note that overall we're going in the negative (left) direction, and then do the "associated" subtraction:
-84 + 14 -> 84 is bigger so do the computation 84 -14 =70 -> but we were going the other direction so -70 is the answer. Someone might prefer to think of it as factoring out -1 for a quick moment like in the example, but in my head I'm just trying to do my subtraction in my preferred order.

I'm realizing now that I just never ever want to use the absolute value function for these calculations, I always skip that step and think about it any other way I can
That could indicate it wasn't a huge part of my education in this area...
Maybe because absolute value feels like a huge leap across the number line and I try instead just identify the overall direction we're going so I can "forget" there were two directions in the first place.

[abstract grove]

yes, I think about it / try to teach it the same way. Decide if you want to do an ordinary addition or subtraction of positive integers, and then attach the right sign based on the starting data

the absolute value thing is okay conceptually but not how I think about it

I mean it's equivalent to how I think about it

I do implicitly take absolute values

that's formally correct but it feels like a distraction somehow

maybe only because I'm so used to it

but I'm pretty sure I learned how to do it before I learned about absolute value. Hard to recall

[normal bronze]

im almost certain I learned absolute value much later.
im sure theres a way to introduce it helpfully, maybe involving the distance between integers, but here i agree it seems to divert attention from more important algebraic methods

[feral spade]

To me atleast, we were introduced number lines simultaneously with the negative numbers. So, this became very intuitive for everyone

[pastel sundial]

why do colleges force students who don't have the required background to take calculus

thinking about this since I'm TAing what's basically an easier/slower version of intro calc. And some of these students have such fundamental misunderstandings/confusions about algebra that it seems a bit strange to me that the university is forcing them through a calc class where they probably won't learn anything of substance

I'm very uninformed about the education system in general. But my understanding is that in K-12 there is a lot of standardization about what a student "should" learn, so schools kinda have to pass people through math classes even if they would be better suited in a lower level class just to meet standards. But no one is forcing colleges to force everyone to take calculus. And even if the colleges decide that everyone should take some amount of math, no one is preventing them from offering more remedial classes rather than just one term of college algebra.

It would be like if schools had a foreign language requirement, but the lowest level classes they had already assumed some basic understanding of the language. If you put me in a Spanish literature class that was taught primarily in Spanish, it doesn't matter how good the lecturer is, I'll learn almost nothing since I'm operating on scraps of things I remember in highschool and basically nothing else. And it feels like some of these students are like that for math.

[timid shard]

No idea where you are from, but presumably "knowing algebra" is a criterion for graduating high school, so from the universities pov the students should know it

although that's not necessarily good assuming that, as your experience shows

(I think it's also fair to say that colleges should teach at college level, so anyone not at that level should catch up on their own. I think it becomes a problem if it's the majority)

[austere delta]

I guess designing university around the assumption that people learned nothing in K12 is attacking the problem from the wrong end

[dapper flume]

As a US middle school teacher and college tutor, the disconnect is real. The common core is pretty rigorous honestly, but with the "push everyone through no matter what" mentality, none of the rigor is achieved. I am often teaching my 8th graders at a higher level of algebra than I am remediating with college students who have to take an algebra class and just don't know what they've supposedly learned in K-12.

Hyper-standardization, teaching to tests, and the hard focus on a track to calculus seem like obvious culprits, but they're really hurting people, despite the better pedagogy

[tall bolt]

At my university everyone in STEM takes a placement test as soon as they arrive, and if you do sufficiently poorly you’re made to take a fundamentals of maths course to build those foundations

Though I don’t know anyone who’s had to take it, and it runs alongside all your normal courses anyway, so I’ve got no idea how effective it is

[dapper flume]

A lot of US universities do that, but I don't think they offer any credit for it, despite being the price of a course

[tall bolt]

Yeah it’s not for credit at my uni either, you just have to take it

You don’t pay for courses here though

[pastel sundial]

This is fair, but also the assumption I'm talking about is significantly weaker than that: it's that some people have significant gaps in their K12 major education. And it's also not just an assumption but also a fact

[pastel sundial]

This is fair, but the counterpoint would be why are these universities accepting people who aren't ready to be taught at a college level

Not trying to be harsh. Just saying that if a university requires everyone to take classes in a certain area, then I think they should make sure that everyone has the prerequisites to take their most basic class in that area. By either (a) adding lower level classes, (b) not admitting people who don't have those prerequisites, or (c) cutting the requirement

(a) or (c) I think are the better options. (b) would probably mean in practice not admitting people who scored below a certain level on the SAT/ACT math section which I think is problematic for a bunch of reasons

[timid shard]

not a counterpoint, kind of goes together

[abstract grove]

K-12 school administrators really really really don't like failing students and making them retake classes

which goes some way to explaining why people get to university level with such big gaps

as to why the universities accept them when there is likely no shortage of more qualified applicants, it's a good question

[rapid tusk]

k12 education in the US has been in a race to the bottom for a while now

covid + the rise of ch*tgpt have only accelerated that trend

a prof at my uni said they've had to run a lot more calc 1 + precalc (!!!) sections ever since covid bc people were coming in with such huge content gaps

[blazing pawn]

What makes a hint valuable and effective in the context of problem-solving? And , how can one give better hints?

• context:

This is my first year tutoring mainly high-school students in algebra and I've seen that when I give hints to my students to make them "discover" the solution themselves, when they come across a similar problem they can handle it way better than if I had just given them a walk through the solution. Sometimes , nevertheless I think I give bad hints in the context of either revealing too much so the solution is obvious or too little so it makes the problem more confusing. Thanks to anyone who took the time to read/ answer.

[dapper flume]

Your approach/philosophy described pretty much already aligns with what I think makes a good hint. Often times, the hardest part about problem solving is finding a sensible approach. A lot of high school algebra students have a lot of practice with arithmetic but little practice with creative problem solving. A good hint spurs creative thinking. Some hints can be affirmative to the student's thinking, narrowing down their view of the problem (good for students who are on the right track but lack confidence). Other hints can expand on student's thinking, widening their view to see a perspective they may not have noticed (good for students who struggle because they are overly rigid in their thinking).

A good example I've seen before is for problems like evaluating log_1.5 (8.5). This student already knew how to do problems with logs that had arbitrary bases, but the fact that the entries were decimals threw them off. The hint here was literally just "can you write a problem in a similar format that you do understand?" and then prompting the student to follow their own blueprint.

It's obviously important to let students struggle their way through problem solving. The role of hints to me is to lighten that load just a little bit in order to encourage perseverance and satsifaction rather than resignation and quitting.

[halcyon glade]

Definitely a good question!! I second what was said above

[turbid zenith]

I just made a related rates problem for my students

Too topical? 😂

[heavy trail]

Folks only understand new things in terms of something they already understand. You have to figure out something they already understand that connects to what

Take time to figure out what they already understand. Sometimes that's previous technical or subject-matter knowledge. Sometimes that's an analogue to something outside of the subject, e.g., another subject or something in everyday life.

Every hint you give is an experiment whose null hypothesis is they're thinking about it in a way that allows the hint to land. If it doesn't land then they must be thinking about it differently than you imagined.

But each hint given should (ideally) be given with some idea of "If they're thinking about it like X then this hint will have Y effect."

So, I don't think it's about "revealing too much" or "revealing too little".

Sometimes you can give a student a worked solution and they see exactly what they were missing. They make exactly the generalizations you hope for, they reflect on why their approach wasn't working and have ideas for how to approach things differently in the future, etc.

If a student is in that situation then I think it'd be a mistake to do anything else but give them the appropriate worked solution.

The hard thing is figuring out whether they're in that situation or not, and the cost of giving a student not in that situation a worked solution.

[tawny slate]

does anyone have any math problems that unexpectedly make use of number bases, like some base other than 10?

obviously every problem that could be solved with a number base can also be solved without them, but i would like, ideally, the strongest examples of where using different bases gives the best clearest intuition of the solution

ideally using some base other than 2 or 10

the only examples i can think of is the x+y card flipping game (but this is maybe a little weak) and the josephus problem (which can easily be extended to bases other than 2)

[shadow flower]

cantor set stuff in base 3

[tawny slate]

yeah but what math problem does it solve

[shadow flower]

great question!

[torpid dew]

You can use it to show that the orbit of the function 3x for x < 0.5 and 3-3x for x >= 0.5 diverges for values outside of the Cantor set and not values in the Cantor set

[tawny slate]

orbit?

[austere delta]

f(f(f(....(f(x))...)

[abstract grove]

the orbit of a point in the plane under the action of SO(2) is literally an orbit around the origin (sorta - it's not a parametrized curve, but I expect there's a natural way to make it one from the action)

[abstract grove]

Is the converse of the Intermediate Value Theorem true?

Conway's base 13 function is a counterexample

[supple abyss]

Love this!! Related rates is still one of my favorite techniques!
I got that revenue from egg sales is changing at a rate of -645 dollars per week with the given market variables

[tawny slate]

hmmmm ok

a little bit esoteric for what im doing but i guess it technically works

wonder if i can eli5 it somehow

thanks anyways, if anyone has additional ideas let me know

[turbid zenith]

Should I use this to teach ∪ and ∩ tomorrow, please advise :V

[hollow musk]

I love it

[storm hawk]

in class groupwork, is it still best practice to group people into groups of 3-4?

[rapid tusk]

this is obviously not rigorous but i've always felt like that's around the sweet spot

[vagrant meadow]

I struggle to get them to work in groups at all, so no way am I gonna introduce a cap lol

[heavy trail]

If someone says that don't know who that is you have to say "I'm sad you can't smell what I'm cookin'."

[tawny slate]

who here has some elegant examples of analytic geometry?

i want relatively easy to follow simple examples of problems for which either the alg form or geo form is easy but the other one is monstrously difficult

some examples i already have:

• show that a linear system of two lines must have either 0, 1, or infinite solutions
• point A and point B are given, C lies on a horizontal line between A and B, we want to minimize the total distance of AB + BC (this geo form is easy, the alg form using distance formula is hilariously awful)
• prove that the medians of a triangle intersect at the same point

i dont mind examples in which both alg and geo forms are easy either, but ideally id like the one-sided examples more

for instance, proving that the diagonals of a parallelogram bisect each other I think is fairly simple both ways

[timid shard]

proving the tangent to a circle makes a 90 degree angle

hard geometrically

easy/easier algebraically

[tawny slate]

i like this example but i dont think its actually that difficult geometrically

in any case thanks

anyone else got more?

[abstract grove]

Let ABCD be an arbitrary quadrilateral. Construct squares on the sides AB, BC, CD, and DA. Show that the line segments joining the centers of opposite squares (i.e. center of square on AB to center of square on CD, and similarly for the other two) are congruent and perpendicular

A similar result holds for arbitrary triangles: if ABC is any triangle, construct equilateral triangles on AB, BC, and CA. Then the segments joining the centers of those equilateral triangles forms another equilateral triangle

[tawny slate]

oooooh ive heard the bottom example before but not the top one, but I haven't done both sides for either example

ima work through these two sometime soon

[tawny slate]

i forgot, how do you tell if 3 points make an equilateral triangle algebraically?

im trying not to invoke like matrices and other fancy stuff

[halcyon glade]

You mean besides just checking that the distances between them are equal?

[pure light]

one way would be to check if any two angles are 60 degrees with dot products (since the cosine would be 1/2)

there is also this theorem in the complex numbers: https://math.stackexchange.com/questions/953129/equilateral-triangle-from-three-complex-points

[tawny slate]

oh im such an idiot

right

[storm hawk]

Hmmm what is a good way to approach when one student is asking me a lot of questions during worksheet time. Like I want to go around helping other people but this student keeps asking for me to verify work and so on. Well I guess I could tell them to cross verify with other people in their group as well as ask them questions. Yeah I don't think it is tenable to answer one student's specific questions at a time in an advanced class of 30 ppl.

I'm thinking of remixing groups and having one member from the really good group each go into weaker groups but I wonder if that really good group was good because it was the three of them together, or if they were good because they were independently good.

[clever obsidian]

is your native language Hebrew by the way?

I'm just asking cause it's almost never called this in English
either the Argand plane or just complex plane

[timid shard]

weirdest way I have ever doxxed myself ngl

I did actually check this term was used in English

Google seemed to know what I meant so I went with it

[clever obsidian]

Google is a lot smarter than the average native English speaker, after all

[tawny slate]

||especially the native english speaker||

[snow shoal]

my 2 cents is that if they're constantly asking then tell them to verify with each other and that you'll only verify a few questions. Generally I've found that students understand if you say something along the lines of "I have 30 people to keep track of, so I'm limited in how much time I can spend with you specifically"

You can also annoy them by forcing them to explain their reasoning/steps to you/their classmates. This both helps them to learn how to communicate/defend their answer and will tire them out from asking you. They're much less likely to constantly ask if they know it's going to be a big commitment to explain their whole process.

That's one solution, but you might have to change strategies based on how they respond

[storm hawk]

Thanks! I will try explaining this to them.

[twin hound]

I second this. Oftentimes, even just trying to formulate the question out loud is sufficient to find the answer to it. Plus, the kid will learn to present its answers and thoughts, which is kind of the point of doing math.

[heavy trail]

What age?

[lethal leaf]

yea I've always phrased it as "it's not fair to the others if I spend all my time on one student" and usually they get it

for context this was when I was a TA for some undergraduate courses

I'm sure it would be different if these were younger students

[storm hawk]

Half masters half 4th years

[heavy trail]

Oh. Just be straight with them, IMO.

It's great you're asking questions, but you need to find ways to get traction on your own. The more advanced your work, the less likely you'll have someone to ask questions as the first course of action. I want you to try these 2-3 things first before asking and see how it goes: (give 2-3 ideas).

And whatever those 2-3 things are, every time they ask, make sure you ask them what they did WRT those 2-3 things. Those 2-3 things (if appropriate) will both make them less likely to have a question and make it easier to answer their questions when they do ask.

Like, at that level their conduct should be somewhere between semi-proefessional and professional.

I don't think you want to say "Stop asking questions", because that's not the issue per se. It's that they're going to have a hard time getting traction on anything with even a little novelty if they don't have other means.

[storm hawk]

Yeah it's somewhat hard because I could end up killing the whole question flow for other people that are more hesitant to ask questions who I wanna hear more from.

But in principle I want them to ask their neighbors first, so I think that's a good general rule if the question is related to the worksheet.

[heavy trail]

I make it explicit for my whole class that:

1. Asking me is just one type of feedback or one way to get unstuck/get traction
2. Students tend to either ask for help too little or too much, relative to other things they could be doing.
3. I might give feedback to ask more questions, sooner or try other things first and hold off
4. But, please, err on the side of asking questions because I can't give feedback like (3) unless you actually ask

It's also, like, I need coverage. If I'm focusing on a small number of students but it turns out half the class is stuck w/ similar stuff then everyone is missing an opportunity. Don't think I'm "annoyed" because you're asking as your first instinct relative to other things.

We're all here to develop more expert-like habits.

[turbid zenith]

Y'all

How am I supposed to come up with an engaging motivational hook for this

I'm teaching a support for college algebra class, and like so far I've been able to come up with at least some reason to care. Like a real world application, or a meme, or something.

But this symbol-pushing torture? Like ... come on

[rapid tusk]

yeah uh there’s not really much window dressing you can do with this

[turbid zenith]

At the moment the ONLY angle I can think of is point out how it's really the same as what you do when you multiply fractions

So calling attention to the fact that there's structural similarity that carries from one thing to another

[rapid tusk]

there’s the standard graphical approach (identifying asymptotes/discontinuities) but idk if that’s relevant here

[turbid zenith]

That's what pains me, that's not in our course

That's in regular College Algebra

The support course is just supposed to help with "skills"

There we go, now it's an appropriate reaction

[tawny slate]

yeah thats perfect

its just such a generalized thing, a core skill

i use it in so many weird obscure ways that ive forgotten all of the applications of it, i can only say to others "trust me, when you need this you need this"

[halcyon glade]

I think also encouraging people not to be intimidated by scary-looking symbols is important; it's an important skill to be able to take a complicated-looking problem, carry out simple steps, and gradually work through it.

[thin kayak]

Hey there i'm writing a linear algebra textbook and I'd like some tips on how to make my material more engaging and friendly to the reader. I'm also a tutor so it'd be nice to have stuff that can be applicable really anywhere

[tawny slate]

ill comment that from personal experience, the biggest difficulty of the linear algebra course was once you got deep into matrices and eigenvectors and such, you kinda forget that lost in all of that abstraction, you're just making statements about linear systems, which we learned about in middle school

one thing i liked about 3blue1brown's series (sorry) is that even though he didnt connect it all the way back to those linear systems of equations, he connected it to something almost as basic: a graphical visualization, which really helps to not only hone intuition but also bring it back down to earth

[thin kayak]

i love 3b1b's videos but i just feel like they don't translate super well to a textbook format, since he doesn't ever really get super precise about what certain things are. I know it's to be accessible to a larger audience, but it remains that it doesn't really map well to a textbook

[quasi musk]

It's designed to be appealing to a large audience, but I'd argue that it's really well designed for people that had been through a computational linear algebra & a proof based linear algebra class

There's no shortage of precise calculations out there in the written text, nor in lectures on youtube. There is a shortage of good graphics explaining the ideas of linear algebra

[thin kayak]

that's a fair point

[turbid zenith]

Gave my second test in Applied Calculus

They were worse than the first 😭

[tall bolt]

How long did they have? I could maybe see an hour being tight but yeah I feel like people should really be getting this stuff. It’s not like there’s any weird tricks or anything to spot they’re all very straightforward if you’ve been learning the material

[turbid zenith]

1 hour. And #3 was optional, only if you wanted to retry it.

Like I sanitized it as much as I could, giving them the simplest thing I could think of for optimization without it being trivial like a parabola, and the number of bizarre derivatives I got was nuts

Someone rewrote 14400/x as x^-14400

[tall bolt]

Ok well yeah if it was just problems 1 and 2 I feel like that’s more than fair

[tall bolt]

I see where the mistake came from, but yeah not ideal, seems like there’s very foundational issues in the class

[turbid zenith]

Also they had a full page of notes and Desmos and could ask me if they were stuck

[tall bolt]

Oh

[midnight scarab]

"and Desmos" is an important addition

[turbid zenith]

Yeah. They could have literally graphed it.

[midnight scarab]

Also I'm curious how did the asking you part go?

[turbid zenith]

And looked and backsolved from there what they should have gotten.

A few people asked me really minor things and I gave hints where I could but past that not a lot

[midnight scarab]

Hm

Tbf I'm not sure I would've made much of the opportunity cause it has a bit if a cheating feeling? Idk

[turbid zenith]

It was meant to be more of a "keep you from spinning your wheels" thing

The only thing I can figure out that maybe I should've done was given dq/dp in #2 and said "use implicit differentiation to show that dq=dp = ..."

[midnight scarab]

(I'm also the kind of student who'd walk up to the teacher and ask them are you sure this is correct if I had gotten something different and had triple checked and all)

[midnight scarab]

You mean give them the answer?

[turbid zenith]

Just to dq/dp, and make them show how you go from the original equation to that

Since they need to use dq/dp for the rest of the problems, though honestly I would also be fine with them using an incorrect answer correctly going forward

[midnight scarab]

I have a single criticism: rounding down 2/3 to 6.66 (and also the trained physicist is sweating profusely about significant )

Jokes aside I think it was a good test and the students not being able to do it seems to show they somehow haven't learned the material?

Could you perhaps try a short quiz with pure calculations (without a "story"). As in just calculate a derivative, study the variations of f(x)
Sometimes students are thrown off by the setting when really it's just mechanical

[turbid zenith]

I mean that was just for my own notes 😛 I would have taken anything reasonable, I'm not sure what would have been the most reasonable rounding a person would have done in context