math-pedagogy

[long pelican]

Making it more similar to English standardized tests

The main problems right now trace back to teachers wanting to optimize for short term results, therefore training in solving specific types of test-like problems

If you really want to know of an example curriculum I can come up with, maybe:

elementary school = number sense, geometric sense, and intro to thinking, reading, and writing in general terms with variables;

middle school: variables, functions, geometry, elementary number theory, elementary combinatorics, mathematical literacy (by which I mean reading and comprehending theorems and proofs with mathematical notation);

high school: more rigorous treatment of variables, functions, geometry, and combinatorics including trigonometry and mathematical proofs and optionally calculus. You could say superficially that it's the same as what we have but the devil lies in the details about what's emphasized. And this is probably not the best or most detailed possible curriculum possible to create

[pastel horizon]

Ahh that preaches to the choir in the UK, the exam boards have a lot of those think outside the box type questions to differentiate higher ability students

I'd say the main problem here is primary school not having many maths subject specialists so they have a big hole when it comes to secondary

This was a question that caused controversy among students who sat this GCSE but I think it's a great question

[long pelican]

I saw this question on social media!!!

[pastel horizon]

This is what separates your grades 8/9 from 6/7

[long pelican]

I have a feeling if I put this exact question on my next exam without preparing them for it whatsoever (they shouldn't need preparation), the success rate on it would be like 10%

What about you?

[pastel horizon]

That's how it should be

You only expect your top students to get this right

[long pelican]

Hmmm

[pastel horizon]

Probably 50% can do a simple probability tree

You need a few of these in to assess who really understands the concepts

[long pelican]

So you don't view it as a failure if 90% don't understand the concepts? 😮

[pastel horizon]

It's kind of nuanced that question, if it was a high ability (8/9) class I'd be disappointed if they missed this question. For someone lower it's just tough luck and you hope they picked up marks elsewhere.

It would be a failure if they didn't even have a surface level understanding of probability of course. This is testing not only their knowledge, but also their problem solving and reasoning skills. They need that extra layer.

Of course they could also just miss it under exam pressure. That would be a problem with exams rather than teaching though

[long pelican]

I tend to say that the Sally's sweets problem is one of the only problems that actually manage to test if someone understands probability. A surface level question is easily gamed by teachers teaching to the test

So if someone gets the surface level questions but not the deep ones, it indicates they are good at having worked examples in short term memory

[strange bronze]

how much time do they get for that?

it seems reasonable if they have a few minutes per question

[pastel horizon]

General rule is a minute per mark

I imagine something like the one I posted would be 4-6 marks

Probably 4 it's not complicated if you know what you're doing

I think this question came from Edexcel and from what I can tell, their papers are 100 marks over 2 hours so that rule leaves 20 minutes to check answers

[topaz scarab]

Okay, I read it a bit more. I guess yes? Except with much less formalism, and more pretty pictures. More examples.

I also have a hot take >.<
I want more open ended questions.

[long pelican]

Every theorem's proof is an open ended question :)

[long pelican]

Formalism is actually a tool to aid in precise communication, so if you feel like less formalism makes it more precise, go for it

[meager shore]

How could I show price increases are better explained by a labor shortage than an increasing labor cost?

(for a lesson)

...to high school math students

[strange bronze]

that seems more like an economics question than a math one

[meager shore]

......during a pandemic

!!!

Frogfuscius!!!

@strange bronze

[strange bronze]

yes.

[meager shore]

you know I made a pretty sick demo of a SMRPG spinoff a couple years ago?

[strange bronze]

neat

[meager shore]

this server doesn't have an econ or game theory section yet, huh

[topaz scarab]

I don't disagree.

[topaz scarab]

I feel like less precision is a good trade off. (thus, less formalism)

[tepid smelt]

I agee with a lot of this. I think there should be more customization though at the HS level. Also your curriculum lacks a lot of stats/programming which should be emphasized more due to the use of it in industry.

The curriculum should also encourage more depth where now we have to cover so much its hard to go slow. The focus on state testing is a huge problem through your right about that.

We still will have the issues with systemic poverty that impacts primarily my students of color who have awful home lives and can't do practice at home and come into my class years behind. We need stronger math teachers at the elementary level and more intervention for kids struggling at this level instead of passing them along compounding the problem by the time I get them

[astral laurel]

Define 'stats' and 'programming'

I'd argue the world needs less frequentist throw-me-in stats and less python programming. There needs to be more formalism on why the Normdist is ubiquitous - a focus on the prevalence of (bounded expectation) additive errors probably

It is difficult to say what language is 'best' but preferably a highly-performant, efficient, fast language (there's quite the alphabet on this), so definitely no Python unless JIT-bundled. An issue is that software development is not computer science, and I feel like code+CS synergy in teaching is really not done well. For example, a focus on FOSS, GNU scientific library could be more useful than telling students to implement their inefficient Dijkstra for the 31415921-th time.

[pastel horizon]

Programming can also refer to mathematical programming, important to disambiguate the two

Which btw, is a very applicable tool for many real world problems

Yeah you're correct though software development is more of an art form. Something that is more efficient might not necessarily be a better design if it's hard to maintain and extend functionality

[tepid smelt]

I was thinking more prepackaged tools like R/sql/excell etc things that are handy for statistics or numerical analysis and many companies use.

I mean even for engineering minded students it would be nice to have some practice with autocad/solidworks for some interesting project ideas using some of the math they learn.

I think like project euler has the right mindset here also where they have interesting math problems that can be done with coding and its up to your preference which you prefer. I think earlier exposure with problems like that would be great

[grim spindle]

How would you intuitively explain linear (in)dependence of a set to someone who just learned the term? Like explaining linear indepence of a point from a set is easy, but i'm not sure how to best present the general notion

[astral laurel]

Why not just use R^n

[grim spindle]

still, how would you explain it intuitively in R^n

consider that they learn this before learning basis and dimension too so no using those concepts either

[frosty flame]

The intuition to me is just that in a linearly independent set none of the points can be written as a linear combination of the others.

[wispy slate]

I guess looking at R,R^2 and R^3 is the best approach at first. Say you have a vector in R^2. You say that you are interested in finding how many vectors you can produce using this vector if you can only stretch it and add two vectors together to form another one. Quickly you can see that this way you will only form a line and not entire plane. Thats how you can introduce linear independence of another vector - using those operations you start with two vectors and you are asked if with those 2 vectors you can build entire R^2

[grim spindle]

sure but that's just the definition, it doesn't really help if you don't understand what a linear combination is very well

[grim spindle]

Hmm yea, I guess starting from the 2d case where linear independence boils down to scalar multiplication and then generalising is the best way

[frosty flame]

Yeah, it's just the definition, but to me that is the most intuitive understanding it and I would emphasize explaining that part. I think overemphasizing a geometric idea won't explain it better. The geometric idea will automatically come into play if you explain span of vectors geometrically.

[grim spindle]

this is a supplementary instruction thing so i'm thinking of how to explain these concepts given they've been introduced to them rigorously once

so i think maybe stressing the geometric angle a bit more might be good

[wispy slate]

For sure

[frosty flame]

I meant I wouldn't explain the definition or concept geometrically. But you can look into it through examples would be my suggestion.

Give a vector x in R^2, ask them to draw the span of the vector, give them vectors y and ask if the two point set {x,y} would be lin ind. Repeat for R^3.

But I feel like the geometry more comes into interpreting span/linear combinations than the independence specifically.

[real mauve]

in 2d and 3d, they simply are not parallel

if thy point every so slightly in a diff direction, theyre lin indep

if you have already explained vector addition, thats enough to make some drawings

and say one vector cant be fully explained by the other

i think for first timers, doing it geometrically might be better, especially since anyway you dont seem keen on using lineae combinations

[grim spindle]

It's not rhat I don't want to use them, I just want to offer a different perspective

[wispy slate]

does anyone know if its possible to paste images onto zoom whiteboard?

[light pelican]

Yeah this isn’t the channel for that. Try a discussion channel

Honestly this isn’t the best server for this but if you’re gonna ask in this server, those channels are probably best for that

[wispy slate]

lol many teachers/students use zoom whiteboard so I think it's a proper question for this channel

[kindred stag]

Yes, but this channel is more about theory and methods of pedagogy, whereas your question is more of a "how do I use this software"

Something asking about how to refine your teaching methods for online education would be better suited for this channel

[long pelican]

I have tried every time I use a zoom whiteboard (which is well over 10 now) and still haven't found a way. Proof by exhaustion that it's impossible without a zoom update!

[astral laurel]

Ask the screen sharer to show the image might be easier?

If not you can use Zoom whiteboard on an auto-updated thing, like Google Docs

Then you can both type and write, as some suggestions

[vagrant meadow]

so in a few hours im going to be tutoring a student who has a test coming up very soon (in less than 2 days) and they have a very poor understanding of the very basics of the test material. however theyre insisting on working through a previous test from the professor. so far doing that has been like pulling teeth because theres just so much foundational understanding that isn't there (this is differential equations and we had to spend about 3 minutes going over the derivatives of sin and cos...). so when i explain a concept from a later section that a problem is based on, they have no idea what im talking about because they dont understand the previous material.
so i sent them a message recommending that she go over the crucial sections she clearly doesnt understand before we meet or otherwise spend our time together going over that material, but she seems averse to the idea "because [going over the previous test] gives me a better understanding on how to approach the next test".
is giving them "homework" before a session a good approach? on one hand its their time and money, but on the other the things they dont want to do would be much more beneficial. i mean... im being paid to help them, right? not just to drag them through problems they don't understand, i would think?
what would you all recommend?

[tepid smelt]

To be honest your in a no win situation the student should have been getting help from the beginning especially if they are in a differential equations class and didn't understand basic derivatives.

One question would be how many hours will you have with the student? You might have to identify things she does know and perfect that for the exam and they will hope thats enough to pass. That might require not best practice of memorizing more instead of a real conceptual understanding. So focus on old problems sge likely will have the best shot at memorizing a certain technique and drill similar problems. See if they are willing to commit to consistent time going forward to hekp fill in gaps of knowledge to hopefully do better on the next exam or final

I deal with students like this a lot who are failing and come asking before finals what they can do. I tell them the truth in that there is likely not much you can do and to use this experience to not make the same choices going forward. Failure is a powerful learning tool that many need to experience before developing better habits

[topaz scarab]

on one hand its their time and money, but on the other the things they dont want to do would be much more beneficial.

I'm on the camp who goes "it is their money, so they call the shots". You have given them your best advice, so there's not much more you can do. Professionals with much bigger stakes have done the same I think (financial advisor, doctors, lawyers).

[quasi musk]

just draw a hard boundary and say "You need to work on this and bring it in next time for you to fully understand this. If you don't do this by next time then we will spend the session going over this, and you will have the exam to go over by yourself. If you want to maximize the amount of exam material we get through, then you have to put in some work. If not, we can sit here all day."

I am manifestly not in the camp of "They pay me so they're in charge". Tutors have to take lead and initiative, if they don't want to do that then you can't help them

[pastel horizon]

Ask them if they would go to a doctor and ignore their advice. Then say "ok, so why ignore advice from a professional teacher/tutor"

[quasi musk]

yeah

That's like 100% what it is. Your job as a tutor is to serve as a guide through tricky material

The student still has to walk down the path themselves

[vagrant meadow]

Okay well update I suppose.
This was her word-for-word response when I warned her that not understanding the basic material will make everything else take a long time.

Listen, I might not know these terms perfectly lol but I know how to do math. I just need help solving so I know what to do on the test and then next midterm and final. I’m not trying to fall off the wagon, I’m taking 15 units while managing work and also wedding planning. Right now these tutoring sessions are helping me out. Especially be stress free, hopefully I’m not too slow or annoying for you but the way we are working right now helps and please let me know if we can work in the near future, thank you.
So it looks like she's not interested in advice from a professional tutor.

And let's just say that she did not adequately demonstrate her touted ability to "do math". In fact, not only was her differential equations knowledge lacking, but her calculus, trig knowledge, and basic understanding of exponential properties (which is so fundamental to diff eq) was fearfully inadequate. It's a really bad sign when a DE student tells you the end behavior of e^(-t) is negative infinity.

She actually joined the zoom session with the solutions to the previous test out in front of her, and so while I was guiding her through the questions she was just reading off what the professor did. And when I'd ask her to explain the logic behind the correct things she was saying or to explain the steps she was skipping, she'd freeze and just start spouting off irrelevant terminology. I never called her out on it because she got so defensive before. We only got through two problems in one hour. How that was worth $55 and an hour of her time is absolutely beyond me.

But it's really good money and pretty easy, so I'm probably not going to push her too hard. I think before I would have considered myself in the camp of "fighting to do what's most helpful for the student", but after this interaction I find myself approaching the camp of "You can lead a horse (that's paying you good money) to water, but you can't make it drink". Maybe I've just sold out. :/

i really want to thank everyone who responded btw. all your responses are very helpful

[vagrant meadow]

In terms of hours with this student, the answer is not nearly enough. I totally agree with failure being a powerful learning tool, but she has said multiple times that that's not an option for her. Unfortunately, she's probably going to have to face that reality after this next test is graded. I guess it's a matter of whether or not she blames me for not being able to get her to understand the last two months of material in less than a week.

[long pelican]

Viewing math problems as being about knowing what to do before you look at them is a misconception in itself too!

[pastel horizon]

Sure, but this student has a weak foundation for trying to solve differential equations when they struggle to differentiate sin x

If you have a solid foundation then yeah you can definitely learn just as much from a problem you can't solve

[long pelican]

That misconception usually comes in a bundle of misconceptions though. So if you think math is about knowing what to do ahead of time, you might also minimize the importance of active thinking

Not doing active thinking while solving problems during or away from an exam means you don’t learn much at all from each practice problem. It compounds over time

[vagrant meadow]

The scariest part of this whole situation is that this isn't some engineering major who only needs this class as a requirement, this is a pure math major. how one can be a math major and care so little about understanding what you learn is just... crazy to me.

[vagrant meadow]

Agreed. Thinking that math is all about just doing an algorithm to solve for an answer doesn't do the subject justice.
That said, if one lacks a lot of foundational knowledge, active thinking doesn't help very much.

[astral laurel]

I'm somewhat mixed on this. On real problems such as world hunger you don't want inertia fatigue and 'thinking,' (but not doing anything) you literally need to algorithmise it. It's why embedded optimisation is a thing.

It's not exactly out of this world to get engineers who are told what they need to do when they recognise a pattern

[astral laurel]

But problems are, in a sense, just instances for which an oracle should be called.

[long pelican]

I’m a little lost on where embedded optimization comes in

And what do you mean by problems are instances for which an oracle should be called?

[long pelican]

Root cause of loss of foundational knowledge may also be attributed to lack of active thinking… if the derivative of sin x is just a leaf or even worse an isolated node in the graph of your knowledge, you are a lot more likely to lose it

[astral laurel]

Basically I feel like you are speaking about how they should think about how a math problem arises, how it relates to other things, etc. I don't deny there is value in that, but driving muscle memory to solving certain tasks is not worthless. I think education helps to create a basic, standard perspective on certain problems.

And for some people doing problems in an algorithmic way is not all bad. Perhaps it's optimal?

[long pelican]

Nah I’m talking a lot more localized. Derivative of sin x, remember how one proved a related limit with the squeeze theorem? Why does sin x repeat? Does that have to do with the definition?

What’s the definition of sin x even? Etc etc

Does this problem use the value of c at all?

[astral laurel]

Not knowing definition sounds like poor foundations, can't be fixed other than going back to them.

[vagrant meadow]

Sure, and practice is crucial. But that isn't what math is all about imo, especially if you're simply practicing without understanding.

[long pelican]

And most importantly what should you do if you don’t know what to do on a problem?

[astral laurel]

You literally have to draw a right-angled triangle and label the sin inside

[vagrant meadow]

But yeah conceptual knowledge alone can't save you on a test, usually, unless you've practiced doing problems.

[long pelican]

The test could be conceptual now, which the tests here are

[astral laurel]

Some would say this is a 'higher-order' problem. But I think the standardisation of tests make it so that people go 'Rule of Conservation of Detail' which typically works, and assume they need to dump in everything

So if a problem has parameters a, ..., f, they would likely seek some function f(a, ..., f) rather than a minimal function f(...) requiring only necessary parameters.

And this works >50% of the time? So on a 'random basis' it's the go-to meta-strategy

[vagrant meadow]

Sure but it's very very common to put simple computational problems on tests just to verify that the student can do the computational steps.

[long pelican]

Yep and as you said she’s having trouble because she’s remembering 99 false things (conceptual) which are detrimental to her ability to finish the steps correctly, right?

[vagrant meadow]

Indeed. I meant more in general

[long pelican]

Really surprised to learn she’s a pure math major

[astral laurel]

Honestly with this problem student I'd just go slower rather than faster. Unless the student is devoting >2hrs she will not develop the necessary print-answer-from-memory thing that people do to ace tests

So in that sense, on that assumption that she'd fail I'd prefer to work on the foundation

If all she does is really to do well on tests, she does not need a tutor, she only needs an answer oracle (which should be available but w/e) and practise like crazy to write the 'correct stuff,' which presumably she does not understand.

[quasi musk]

If that's the case then just go as fast as possible through the material

Get through as much as possible

Skip as many steps as you can manage

Then when they say "Why don't you show your steps"

Just say "Well I think we just need to get through the material, y'know for the test"

When she asks for more & more steps

Just keep re-iterating that "You asked not for a fundamental understanding, but what the solutions are so here they are"

Drive home the point that these are useless without understanding

[pastel horizon]

Even in an engineering major though, they need to understand the maths they're doing otherwise all their assumptions and approximations fall apart and they get garbage in = garbage out

[tepid smelt]

I am about to present inequalities to my freshman how do you go about explaining when and why the inequality reverses direction when solving them?

Context we just finished up basic solving of various linear equations along with graphing/word problems and now need to to the same with inequalities

[strange bronze]

i mean, it only reverses when multiplying/dividing by a negative

so you can explain it

• formally with a proof
• visually by imagining multiplication as "stretching" the number line and noticing that it "inverts" once you go negative
• algebraically by noting that:

a > b
-a < -b

is actually just subtraction:

a > b
0 > b - a
-b > -a

[tepid smelt]

Oh I like the stitching and inverting idea

[strange bronze]

heres what i mean by that https://www.desmos.com/calculator/djkcjnmwvy

[winged urchin]

Real number examples can be useful too. Like 1 < 2 but -1 > -2

[quasi musk]

This is the way I do it

I got 1 < 2

Which is bigger -1 or -2

I go*

So the moral is that negative signs flip inequalities. Works for just about everyone I've helped

[winged urchin]

Yeah. Getting students to agree on real solid number examples helps them agree on the often confusing variable expressions

You could also highlight the other common problem at least I encounter

And that's, multiplying by something that's negative that doesn't have the negative sign explicitly written

Like if I know x < 0 and I multiply 1 < 2 by x then we do get x > 2x

And again we can return to the case where x = -1 to bring them back into concrete number world if they get confused by that

[tepid smelt]

Thanks for all the replies it is really helpful. I was going to do the examples but like the idea of the little proof and the visual meaning behind multiplication

[long pelican]

I think you could also consider trying barebones mathematical logic ... no fancy analogies, just straight up realizing "if and only if" relationships, such as x < 3 if and only if -x > -3

I have an idea that students get confused when one says "You reverse inequalities when multiplying by negative numbers when solving them" without explanation, and that's understandable because what even governs that rule? How do you know to come up with it without someone telling you?

But on the contrary, it should be possible to understand "for all real numbers x and y, x < y if and only if -x > -y" from first principles, given some time

[astral laurel]

But they aren't. A huge number of people ace tests without understanding by literal pattern recognition.

[pastel horizon]

Plus, that's literally what his student asked for, help going through the past test

[pastel horizon]

Sure

[tawny slate]

I use this as a way to introduce inequalities. Here's how I structure my inequalities class (do with it what you will):

First introduce the idea that inequalities are logical statements in the same way that equations are, so they understand they are true or false.

Give problems similar to:
"if x>y, then 1/x < 1/y."
Is this statement always true? Can you find a counterexample?

The goal isn't to have students actually prove or disprove these statements necessarily. The goal is to catch students off-guard so they understand that they need to be skeptical and careful. In the above problem, students might go "it is always true, I can't find a counterexample" and then you show them an explicit counterexample.

The kids tend to get really excited about this because it feels like it should be a really simple problem they can tackle and grasp, that their intuition shouldn't fail them, and yet if they don't think carefully, that intuition may fail them repeatedly. Once they miss one or two of these, they really start to lower their confidence levels in their answers (not themselves) because they understand they need to be more critical.

By having them iterate through a bunch of "first principles" inequality statements in this way, gradually explaining each one through the logic and the intuition, they can begin to appreciate that the rules for inequalities are different from those of equations.

And then if these students are talented, I would then go into clever ways in which these inequalities can be utilized to solve seemingly impossible problems like "prove 2^81 > 3^49".

[long pelican]

Very nice! That's exactly what I had in mind!

[quasi musk]

Classic I did well on the test so it's fine

[topaz scarab]

You make the tests right? Then simply write better questions?

nvm

I didn't read the thread

haha

soz

[long pelican]

Ah, the art of writing good test questions :^)

Although if we're being honest, if a test consists solely of good questions, the average score on it would be less than 50%

(Unless your group of students is amazing)

[topaz scarab]

I don't think so.
Maybe yes at first. But because they had years of training with bad questions.
If they have years of training with good questions, with teachers that also have decades of experience of teaching real math, things might be different

but anyway, time for:

ANOTHER HOT TAKE IN MATH!

I think people should learn about reading spectogram before learnining about fourier transform?
What do you guys think?

[long pelican]

Yes this is correct

[topaz scarab]

you know how like people in grade school / middle school have reading comprehensions on bar charts, pie charts, etc2

I think, before you get to learn fourier transform, doing the exact same thing with spectogram is a great idea

[long pelican]

Spectrograms are really cool. Watching a spectrogram of music that's playing could be a very fun lecture activity

(+ deconstructing it after)

[topaz scarab]

I think so too!

So I think audio is the most intuitive start, but I think it would be also fun to move on to analyzing other non-audio timeseries

[long pelican]

What other math topics become similarly easier to learn after seeing the corresponding "demonstration"?

[topaz scarab]

EVERYTHING!

hahaha

except when the demo is even more arcane

[real mauve]

how would you explain what the spectrogram is tho? just curious

i think that's going too far

you could start with something simple like those tone generators, playing around with the tone frequency and listening

[long pelican]

Yep, then listening to a combination of two tones and noticing the spectrogram shows both frequencies separately

Cue wondering how you can find the frequencies given only the waveform

[topaz scarab]

I think the axis is clear enough?
X is time, Y is frequency. It shows which frequency is playing at which time.
If that's too difficult, we could start from this kind of thing (idk what it is called: it's called piano roll)

[real mauve]

idk, i've tried this a lot with undergrad engineering students

it didn't work so well

[astral laurel]

Decades of experience of teaching real math
So your argument is that current whole industry sucks, which I don't disagree with, but unfortunately it is not practical to have any 'revolution in education'

[real mauve]

i can't rule out the possibility that i did it wrong, but i went through several batches of them and many iterations of how to present the demo

i was left with the impression that, if a fourier series wasn't clear enough on its own, nothing else would remedy it lol

we get a lot of people here that come asking questions about the different transforms flavors after watching that 3b1b video and they understood nothing

[astral laurel]

I recently spoke to someone who went through the French system, and from what I gather, it is very 'real math' based. However, I can't say if it is optimal from a stress perspective or that the curriculum is matured to the point where there is clear segregation on what is expected of students

[topaz scarab]

Okay, this is exciting. Can you give examples on how it failed?

Like, did they not understand spectogram?
Or they always managed to mis-understood spectogram?
Or the understood sepctorgram, but that doesn't translate to fourier transform at all?

[real mauve]

for whatever reason, the idea that the same thing can be equivalently represented in different domains seemed to be a great struggle

right

they could understand the spectrogram and the time domain plot each on their own, but not their relationship

so then something like a time domain plot of a square wave and a frequency domain plot of a filter that would be applied to that square wave seemed to have no relationship to each other

[topaz scarab]

IDK about French, but good segregation is a great idea I think

[topaz scarab]

huh, got it, make sense

[long pelican]

Aw man they watched the 3b1b video and still understood nothing about Fourier transforms? Damn I had better expectations for 3b1b's usefulness for the newcomer

[real mauve]

i know this is a hot take in this server, but i have never found 3b1bs videos to be good for newcomers

they're only useful if you already know the topic and want a different take on it

just my POV though

[topaz scarab]

CONTROVERSIAL!!!
(I think I agree though, however, it is less about "newcomers" but rather, "people who prefer to think in a certain way")

[topaz scarab]

What if you do a drill on this type of question?
In high school, you can do a mini test.
Maybe it is a bit difficult in college.
But how about, like in week 2 or 3, make a MCQ with like 30 to 50 questions, all about matching a waveform and a spectogram?
maybe make it worth like 10% of the whole class

[long pelican]

I fear that'll just result in low performance and then what?

[real mauve]

later on that's pretty much what it boils down to anyway. using linearity and a handful of common waveforms, one just composes them as needed, because transforming is usually just a tool or small step in a larger procedure

but it could be useful early on, sure

technically this is already done though

in engineering, these so-called "fourier transform pair tables" are often used, which basically aim to *achieve the same result

and these are brought in fairly early. one rarely computes more than a handful of transforms beyond the first couple of lectures

[topaz scarab]

Well, my hope was that, if they know such test is coming, they would somehow, managed to learn it properly, and won't result in low performance. In fact, I would be very surprised if it results in low performance.

[long pelican]

If the very idea of a spectrogram fundamentally makes zero sense to them, and they're already making maximal use of whatever resources they have, it doesn't seem like they'll just magically succeed in large numbers when pressure is added

Like what Edd described for his class sounds like there's more to the issue than lack of motivation

[topaz scarab]

I think it is less about motivation in general, but on motivating a certain kind of understanding.
I feel like, without such test, they will spend their time doing algebra, instead of the understanding the concepts.
Which is what I think Edd's students are doing

[real mauve]

very problematic indeed because we had specially designed labs for this, where they would set up electrical/electronic circuits to actually make the filters and signals with electricity and see the waveforms with oscilloscopes, along with matlab tasks where they would use the built-in transform functions with the aim of plotting them and comparing, plus the stuff done on paper

there was actually very little math involved because one mostly deals with simple LTI systems when this is introduced, so the key idea is "use whatever you want to transform into frequency domain, and then all the operations are simply multiplication"

so the math was very light

[topaz scarab]

I see...

[real mauve]

i will admit this was also my first teaching experience a while back, so it is very possible i just absolutely sucked at it. but it was indeed approached from many angles, to little avail

[topaz scarab]

My 2 cents are: Maybe they are not familiar with the circuits and matlab in the 1st place, so they struggle with it more, and don't have the time to actually think about what's going on.

[topaz scarab]

I think my conclusion is: math is hard, teaching is hard, teaching math is even harder than both combined.

[real mauve]

this was usually 2nd or 3rd year of branches of electrical engineering branches (mechatronics, biomedical eng, telecom). at this point, all of the students were required to have learned at least one programming language. also the filters part comes in electrical circuits 2, so they are required to have taken the first one, where they spent like 20 hrs in the lab making circuits, and physics was a prerequisite, where they had also set up electrical circuits and worked with this sort of stuff

[civic tree]

im in a fourier analysis class rn and i have No clue what a spectogram is

catThink

[real mauve]

maybe ou've seen it as short time fourier transform, or, after showing it makes sense to consider it, the transform of the periodogram

usually pops up with the name "spectrogram" in stochastic settings rather than deterministic

[topaz scarab]

Edd said that they understand spectogram, they understand the time domain plots, but not the relationship between both.
Based on my experience, most student's won't spend a lot of time learning the relationships, because these kind of questions won't appear in exams.
So adding the "pressure" will motivate them to learn this, instead of the algebra.

[topaz scarab]

get out, this is a teacher's lounge

jk

[topaz scarab]

do you understand "piano roll"

[civic tree]

yea

oh this will just be terms in fourier series

of course

e^int for different n

[real mauve]

if you follow through with this, report on your degree of success :x i worry that introducing the piano keys as the y axis, it'll add in a third, completely unrelated domain

e^int, e^float

[topaz scarab]

ABSOLUTE PROOF THAT MY IDEA WORKS!

• drop mic *
• bow down *
  Thank you * wave * thank you for coming to my TED talk

[civic tree]

lol

[topaz scarab]

okay, well then, increase the resolution in the frequency space to get more frequency than piano keys, you will get this.
(note that this will also display the harmonics, so, it won't look exactly the same)

[real mauve]

introducing harmonics from the get-go

i would really go for something more intuitive first

like single tone vs square wave

where you can recognize just by listening it's the same "tone", but something else is going on

[topaz scarab]

(that's why I put it in brackets, to be less intimidating, but still accurate)

[real mauve]

btw if you're gonna have them compare waveforms to their spectra, PLEASE emphasize how important it is to know the spectrum of a square wave

a lot of problems later on can be solved by noticing one implicitly multiplied by a square window

[topaz scarab]

Oh, these days I am only teaching ML and Data science
I never taught these stuff btw
I used to teach middle and highschool

and I'm only a tutor, not a lecturer

[real mauve]

i deal with sig proc, which overlaps in like 99% of the topics with ML and data science

[topaz scarab]

yayy

[real mauve]

supervising masters and bachelors students in their research

[topaz scarab]

my research is on time series, which is kinda sig proc haha

please supervise me senpai

[real mauve]

[stark pine]

What do you think the best way is to help other people review?
Context: There's a first year "advanced calculus" course at my uni that is basically just analysis on R. Before the midterm, some other students and I ran a little study session for them where we gave them a challenging problem and then split them into groups so they could discuss it, and we went around and discussed the problems with their groups + gave them hints when they got stuck. The problem used almost all the concepts they needed to know.

They had fun and many of them said they got something out of it, but I saw a lot of concerning stuff while we were running it. Some of them struggled with basic definitions (cauchy sequence/least upper bound), and many couldn't state the definitions precisely without reminders. Those of us running the review figured the "collaborate on challenging programs" would work well so that they could discuss and identify points of weakness, but many of them had the same mistakes, and so they weren't catching as many of their classmates mistakes as we hoped.

We're gonna run something else before the final. I'm not sure if we should again just give them cool problems to collaborate on, or if we should lecture on core concepts from the course, or if we should do something else entirely. We figured that lectures wouldn't be effective since they already had a bunch from the prof/office hours, but at the midterm session, there seemed to be enough missing fundamental knowledge. I feel like we don't just want to do more lectures because they have enough, but at the same time, I don't know how to fill in missing fundamentals like knowledge of definitions in another way. Any suggestions/insight would be appreciated

[charred silo]

May be give them time to look at things individually first. In a group setting, if someone else thinks of a solution quickly and then tells you about it, the answer is much less likely to stick.

[stark pine]

the problems were hard enough that none of them thought of it immediately

they did all genuinely need to collaborate

And everyone was able to contribute

[charred silo]

Ah, okay. Then, unless they got lost and tuned out the discussions, they should remember the definitions afterwards? Assuming they do some amount of review themselves within the week. Otherwise, explanations would have been kinda hard to follow.

[long pelican]

I feel like the best medium for people who have gaping holes in fundamentals is homework questions designed specifically so that you need to have good fundamentals to solve them

Half of the entire time spent in the class is homework, and an hour of discussion isn't really enough

[tepid smelt]

Grant himself has said in general watching videos is not how you learn math. He has always said you primarily learn by doing. His videos are entertainment. That said they are high quality for entertainment purposes. I think they are best for someone who already has a good understanding of the material but wants to see it presented it in a visually appealing way.

[long pelican]

I always took the truth to be somewhere in the middle, especially given the loads of comments saying things like "I learned more watching this 20 minute video than in a whole year of math class"

I don't think even Grant expected that they're absolutely useless for new learners

[tepid smelt]

I always thought he wanted to generate interest in a topic with his videos along with share interesting ways to visualize certain concepts. His goal was never to teach it. I think in that regard he has been successful not to mention his open source with manim had really upped the video production for many content creators.

[molten urchin]

I think visuals and intuitive explanations can lull many viewers into believing they "get it", when they may not have developed enough faculty with the nitty-gritty details.

That said I think Grant's videos are great as a supplement for someone who is learning the subject. I've found them extremely useful when learning myself.

For me, these videos act as the bridge between intuition and concrete details.

[tepid smelt]

I think videos being easy to digest do give the wrong impression with how difficult real learning is. I know my students lack grit and patience it takes to learn math. They want everything to happen quickly and for it to always be this entertaining process. Where in reality sometimes its just not and being frustrated and lost is a feeling that is normal when learning.

Lack of patience in general is a real problem I struggle with my students

[molten urchin]

Precisely, I agree.

[real mauve]

do note that them saying "i learned more..." means they are already acquinted with the topic

[austere inlet]

I don't think anyone writing these comments has actually learnt more from a demonstrative video than from their classes, no matter how awful and unmotivated these can be (depending on the instructor and other contextual factors). They're connecting what they see with previous experiences, at best. Sure the video can help you get the overall idea or provide some visuals for what you're doing but it isn't teaching you how to solve mathematical problems or how to actually work with the concepts — I agree that most students underestimate or simply aren't willing to put the effort required to learn mathematics and have little to no patience nor tolerance for failure and frustration. Overall, these comments are a very subjective assessment of the effectiveness of these videos (assuming that their purpose is for learning and not entertainment, which might is a wrong assumption to start anyway), and in fact a great part of it might just be the illusion of understanding that one gets when the speaker knows their stuff and how to communicate clearly.

[fossil flower]

what software do you guys recommend for tutoring math online? I mean something in which I could write and draw things, some additional gadgets would be very welcome too (currently I'm just using MS paint lmao)

[austere inlet]

do you have a graphics tablet?

I use Xournal++, it's open source and multiplatform

[fossil flower]

nope, somehow I can write with my mouse very clearly, but I've been thinking about getting one anyway

and alright I'll check that one out

[real mauve]

since we get the MS suite at my uni, i use one note

as for tools, graphic tablets are indeed very helpful. i got myself one of those huion tablets last year and it has been great

i can link you to the tablet i got, if you're interested. one of those cheap ones that have no screen, it's just a large pad. you get used to it pretty quickly

[austere inlet]

huion is good for the price I'll shill it anytime

[real mauve]

seconded

this is the one i got https://www.amazon.com/Graphics-Supported-Function-Battery-Free-Multimedia/dp/B082MX126F/ref=sr_1_8?crid=15HIWD562YIIF

though in my experience, you can get away with something smaller, especially if you use a large screen

[fossil flower]

thanks for the link, but I'll look it up on other sites cuz I don't live in the US

[real mauve]

yeah me neither, it was just for reference 😛

[astral laurel]

Any specific reason for use of a specific software? Would a tablet-proprietary software not work?

[real mauve]

i'd wager the best reason is "free but still good"

[astral laurel]

To be a bit more comprehensive, do you have a table of competing tablets? I'm probably TAing next semester and I think I will get a tablet because the last time I tried Zoom+HTML output, it wasn't very good, though a part of it is that I can go too fast with my highlighting and that HTML should be read on user hardware instead. Though TBH I would rather want to have the speed to TeX-on-the-fly but I know I am not there yet

[astral laurel]

[real mauve]

what i usually do is tex some key stuff up and freehand the rest

that's a nice thing about powerpoint, for example. but also if you just text something up and open the pdf in a browser to draw on it there

stuff like one note and xournal++ let you easily type, draw, and copy paste images, but you can't use tex in them

as for tablets, i've only used huion, but i know the uni uses wacom and many people swear by that brand. it's pretty pricey tho

you could also get a proper tablet instead of just a peripheral, but that's more expensive

[stark pine]

ok thanks - this is what my intuition said too, which is why I went for that the first time, I was just worried it was the wrong choice

My favourite problem was showing the following are equivalent

1. Bolzano-Weierstrass
2. Finite Intersection Property
3. Least Upper Bound property
4. Cauchy Completeness of R
5. Monotone convergence theorem

[pastel horizon]

So graded a mock exam paper and came across a very interesting misconception:

"Prove 68 isn't a square number"

"1, 4, 9, 16, 25, 36, 49, 64, 81, 100 are square numbers, 68 isn't in the list so it's not a square"

What do you think the best way would be to show why it isn't rigorous enough?

[real mauve]

they didn't say why those numbers in the list are squares either. with no definition anywhere, they can't check whether or not the number satisfies that definition

[pastel horizon]

For some context, something like
8² = 64 9² = 81 and 64<68<81 so the square root is between 8 and 9 would've gotten full marks (out of 2) as well

[pastel horizon]

Yup that would be the misconception, knowing square numbers doesn't prove a different number isn't square if you don't know the reason why that list exists

Plus there's a more general misconception about not realising that proving something goes beyond just stating facts you need reasoning too

[real mauve]

you could tell them they needed to say somwthing about consecutive integers to justify checking that list. it's a bit of a tough one cuz they might just argue it's obvious

[austere inlet]

you can write TeX equations within Xournal++, it's a built-in feature

[real mauve]

damn i didn't know, that's nice

i guess one note should probably allow it then, too, since word and ppt do so as well

[plain valve]

I wondered if also pointing that e.g. 144 isn't on that list could show the need for more explanation

[long pelican]

Just say his friend looks at his list and tells him “ah you forgot 68”

And ask how he’d argue his case

[real mauve]

that's pretty good, ngl

[pastel horizon]

I'd argue they didn't understand it because they carried on after 81 as well personally

Clearly you don't need to show square numbers past 81 since 68 is smaller

[pastel horizon]

That's a good idea

[pastel sundial]

idk, I feel like I learned quite a bit from his essence of linear algebra series

like I watched them just before taking linear, and I feel like they were a huge help for that class.

His videos provide excellent intuition imo, but intuition is all they provide. Plus videos like his are very easy to watch passively, which is not the best way to learn math.

[pastel horizon]

Definitely. Maths is not a spectator sport that's for sure

Videos are only useful in my opinion if you've already committed it to long term memory but you just need a quick refresher

The type of thing that makes you go "ah yeah"

[long pelican]

We can consider a pretty interesting optimization problem though: given the constraint of a 20 minute video, what is the maximum possible useful learning you can possibly induce about, say, functions, to someone who has (a) never heard of them before, or (b) learned them the wrong way in school

[pastel horizon]

Define learning

I mean you can definitely make progress that's for sure. Would you say they've learnt it though

[long pelican]

maximum amount of progress, then

[pastel horizon]

Then for a), I really doubt it

[long pelican]

I mean, it's an optimization problem. The space of videos is like $(\bR^3)^{\bR^2\times\bR}$

[pastel horizon]

b is kind of up in the air. Maybe it's just one misconception that will go away after watching and everything falls in place

[burnt vessel]

Icy001

[long pelican]

In that space the amount of progress of learning reaches a maximum somewhere

[pastel horizon]

Sure but it's just one problem, you can learn a lot from that example but I don't think it's enough to give a foundation to apply it for similar problems

[long pelican]

I don't think the actual optimal point is findable in reasonable time but the thought of it is useful to guide some discussion

[pastel horizon]

I mean yeah there's a lot to be gained from discussion for sure

[long pelican]

Already I came up with a nice question

Do you think Grant make his videos better for the goal of inducing learning?

And if so, in what way

[pastel horizon]

I think an important part is a variation of examples and non examples

So maybe in your discussion you could say "ok, x=3 and y=4 is a solution but is it the best solution? How do we know?"

[long pelican]

One of the problems is viewers getting a false sense of "I get it"

[pastel horizon]

Then maybe a follow up, "can you find a problem where 3 and 4 are optimal solutions?"

[long pelican]

One of my upcoming homework problems is like that >:)

[pastel horizon]

That's the trap of showing only one problem imo they could have a misconception that works for that situation but not in general

[long pelican]

Give infinitely many examples of differentiable functions $f\colon \bR^2\to \bR$ satisfying:
[f(0,0)=2,\quad \frac{\partial f}{\partial x}(0,0)=7,\quad\frac{\partial f}{\partial y}(0,0)=-2021.]
Make sure to check that all of your infinitely many examples satisfy the above conditions!

[burnt vessel]

Icy001

[pastel horizon]

That's a good question actually

The prompt at the end is generous but they'd definitely make that error without it

[long pelican]

Yep it's there because the students tend to just do minimal work without it (and get it wrong, and learn very little from the experience)

[fossil flower]

quick question, how do you even save xournal files? Ik what buttons to press but when I click on "save" it just doesn't work

[fossil flower]

nvm I fixed that issue

[civic tree]

based

xournal++ good

[grim spindle]

Imma ask here too cuz why not

So i'm doing this supplementary instruction course (It's a student association thing, not officially sanctioned by the uni) for linear algebra 1 that starts in about 2 weeks, about a month before the midterm, and lasts 6 weeks. I'm not sure if I should start from the very basics (The course covers general fields, finite fields and complex numbers very briefly and then moves on to vector spaces), or just refresh the definitions and jump into like, linear independence, bases, transformations etc.

On one hand, the people coming to this course are the ones that are maybe struggling with the basics, but on the other hand what they should mostly take away from the fields portion is being able to manipulate the field axioms fluently in the context of VS and perform basic modular arithmetic.

I'm considering writing notes for a more extensive covering of the topic and adjust according to the students, but that might be too much work

[pastel horizon]

What is your question exactly? Also yeah I have a good discussion point too.

I'm going to be teaching a relatively weak class area this upcoming week. I could very easily just give them a formula and a worksheet but I think they need to be really brought back to basics conceptually before we can move forward to harder area calcs like triangles, trapezia, etc.

My plan was originally to start introducing it by counting the squares, using variation and a spot the mistake exercise so they don't get the misconception that they have to be whole squares. Just had a realisation I could go into an even deeper mastery approach by moving squares around to show the area hasn't changed, or doubling the amount of squares to see if they can spot that the area doubles. What do you think? What approach would you guys take?

It's not that I think that a simple area calc is beyond them as they are, I just think they could benefit highly if I built the concept of area up from scratch to help with future lessons

[grim spindle]

My question is should I go more in depth into the perhaps 'less important' topics at the start of the course or jump right into the meat. Keeping in mind their midterm is soon after the course starts and they might be tested on those

[pastel horizon]

I would use a couple of diagnostic questions, see what their understanding is and go from there. Be flexible either way

By diagnostic, I mean multiple choice, each wrong answer targets a specific misconception

[grim spindle]

Problem is I gotta prepare my class notes in advance and I'll only know who signed up a few days in advance

[pastel horizon]

Class notes online? Surely it wouldn't hurt to submit a more in depth one and just adapt on the day

[grim spindle]

That's probably what I'll do. I just don't wanna put in a lot of effort for nothing as I'm already fairly busy

[pastel horizon]

Better to be flexible though if you're unsure where they are

[grim spindle]

True

[long pelican]

+1 for going back to conceptual foundations. My two cents: a number one sin that happens often in math classes is teaching in such a way that students remember (temporarily) processes of how to find something but never actually learned its definition. e.g.

• Students know how to graph f(x)=x^2, but what is the graph of a function, i.e. is it a set of points? Is it a relationship? Is it itself a function? (Answer: a set of points in R^2)
• Students know how to plug in numbers to a function, but what is a function? Is it the same thing as an equation? Do you solve functions? (No, and no)
• Students know to do the same thing to both sides of an equation, but what is an equation? So many algebra questions in this discord ask whether you're "allowed" to do something to an equation, and I can't help but think that the fundamental issue is they don't even know the meaning of what they are doing.

So my second cent is to really try to come up with a learnable, precise, and age-appropriate definition of area first, so they don't become students who know how to do something but not what it means!

[pastel horizon]

They are like 13-14. But really weak, probably because of COVID

I want to set them up to understand why the area of a triangle is what it is visually so they will remember it easier

[long pelican]

So why the area of a triangle is 1/2 bh?

The nice proof of that will require knowing that area is a quantity associated to 2D shapes that is additive and invariant under cutting up, rotations, and translations

[pastel horizon]

Yup, I've got some good questions planned so they can see rectangles at different rotations and the same techniques still work

And obviously recognising that they are also rectangles, it's not just something that's horizontal or vertically oriented

Translation would be the idea of dividing into centimeter squares and moving pieces around to show it's still conserved. This is a vital concept if I'm going to show other shapes visually

If they can't understand that then it will be really hard explaining how we get from a triangle to a rectangle with double the area

[long pelican]

After these, try just writing down "Area is invariant under rotations and translations" on the board and see the response. Assuming they have the language skills to read that, of course

[pastel horizon]

Arguably with that class you could probably spend a full lesson explaining what is and isn't a trapezium as well that's a really hard concept for that age

[pastel horizon]

I think I'd lose them tbh, but I could probably write something equivalent

[long pelican]

Yeah, anything that emphasizes that that, along with additivity over disjoint unions (in age-appropriate language of course) is the defining property of area

[pastel horizon]

Tell you what as well, I bet you any money they will confuse a parallelogram with a trapezium at some point

The definition is similar but there is a subtle difference

[long pelican]

Trapezium, describing a geometric shape, has two contradictory meanings: Outside the US and Canada: a quadrilateral with at least one pair of parallel sides (known in the US as a trapezoid) In the US and Canada: a quadrilateral with no parallel sides (known elsewhere as a general irregular quadrilateral)

I'm already confusing trapezium with itself 👀

[pastel horizon]

Wait what

Well ok in the UK it only has one pair of parallel sides

Exactly one in fact

A parallelogram would have two pairs of parallel sides

So you could see how kids would confuse the two

It matters because the area of both shapes are completely different

[long pelican]

Arguably pure vocabulary issues are less of a deal than mathematical meaning-making issues

[pastel horizon]

It's more that conceptually they understand one shape only has one set of parallel sides, the other has two

[long pelican]

can they make those observations given the pictures?

[pastel horizon]

They should be able to

[long pelican]

The issue then lies with them remembering what trapezium and parallelogram mean then?

[pastel horizon]

If they are confusing the two they would probably also confuse the formulas. Or say if I showed the visual proof they would confuse the two

Like say if for some reason you accidentally called a square a "circle", obviously intuitively you know what the area should be but then you'd second guess yourself

[long pelican]

But if given the pictures without the words trapezium or parallelogram, they can come up with the right area formulas?

[pastel horizon]

Hmm that's interesting, so almost like building it up conceptually before giving a name?

[long pelican]

I'm picturing the following concept maps:

[pastel horizon]

Yeah actually it could be a fun investigation lesson, we have a look at the facts of the mysterious shape before revealing its identity

E.g. our mystery shape always has one set of parallel sides regardless of orientation. We can get them identifying which sides are parallel

We know it's a quadrilateral it always has 4 sides, etc

[long pelican]

They are trying go directly from the word "trapezium" to formula

because they've been trained to link words with formulas

And then they get mixed up because they aren't good at memorizing (which is why they're considered weak)

[pastel horizon]

Nah I like the idea of going backwards come to think about it

[long pelican]

No that picture wasn't a suggestion on what to do lol

It's a diagnosis of their problem

The correct path way is like this, at least for the first 10+ times they think about it

[pastel horizon]

So it's like saying regardless of what I'd do they'd just take the path of least resistance at the end of the day

[long pelican]

Something like that!

[pastel horizon]

Unless I'm forcing them down that path

I still like the idea of an investigation based approach though

Obviously I'm leading that and not letting them discover new "theories" 😂

[long pelican]

Do they tend to discover wrong theories or...? 👀

[pastel horizon]

It's a big risk

I feel like they could discover something completely wrong then that's stuck in their brain since they came up with it

[long pelican]

You have to teach them the concept of disproving a conjecture then!

Not necessarily proving yet, just disproving

[pastel horizon]

That could maybe be asked as an open question, get another student to convince me why the other's idea doesn't quite work, or to find a counterexample

[long pelican]

yep, that could work very well

[pastel horizon]

When you ask them to convince the teacher, it kind of implies you're expecting a strong mathematical argument without saying explicitly

[long pelican]

After you've presented the axiomatic properties of area, you could get some interesting theories and questions indeed. For example what is the area of a circle? How do you even go about verifying that the area of a unit circle is bigger than 3.14 and less than 3.15?

More importantly how do you define the area of curved shapes to begin with

[pastel horizon]

Yeah with that I want to try and really encourage exact form and get them to understand the decimal is only an approximation

[long pelican]

actually what I was getting at is that the area of a curved shape can be defined as a limit of a sequence of areas of approximations of it by tiny squares

To show that the area of a circle is bigger than 3.14, you demonstrate that you can fit 314 squares of side length 1/10 inside it

and the idea is that pi itself is defined as a limit

[pastel horizon]

Ahh see I was gonna go the other way around and define pi as a ratio

[long pelican]

That too is a limit!

How do you define length of a curved shape?

(circumference of a circle)

Again you approximate it by polygons of increasing side length like Archimedes did and take the limit as n goes to infinity

In fact if you do that you have discovered arc length as a definite integral

Very good connection to calculus

[pastel horizon]

I think they could actually understand that if I showed it that way

[long pelican]

😄 glad to hear

[pastel horizon]

They could quite easily work out the perimeter of a polygon, plus when you visualise it it's easy to see you're getting closer but not quite there

[winged urchin]

I personally thought parallelogram was a great name

It's as parallel as you can get with a four sided figure!

I also like to comment on shapes with a constant measure (length or area typically) extending to give you the next higher dimensional measure

Like a square is just a line of length say, L, and if move that line a distance of L in a perpendicular direction we create a square. And the area of that square is the constant measure, L, times to distance we moved it, so to speak

And if you remember that idea you don't really need to 'remember' some formula

A cylinder? Just a circle of area 2pi*r stretched out over the height of the cylinder

Boom, volume is 2pi r h

[long pelican]

https://cdn.discordapp.com/attachments/694677515645747201/908611613458845716/unknown.png Using this poll as part of a plan to improve math literacy in calc 2 students, anyone got any other tips?

My definition of math literacy is the fundamental understanding that mathematical expressions and statements have meaning (note: this isn't how the word is commonly used)
Includes ability to make sense of, read, and write, mathematical expressions, statements, and notation

[strange bronze]

hmm

do you consider using "solve a function" as a synonym for "find when a function is 0" an abuse of terminology?

i'd avoid it around learners but i think its fine in a vacuum

i mean, we use that jargon for polynomial functions at least

idk, "make sense" is too vague to me in general

like do i disqualify the last one because f(x, y) is not a function rigorously?

its the value of a function at point (x, y)

[long pelican]

"Solve a function" is straight up meaningless nonsense, and notice that f(g(x,y)) is also meaningless because f takes a pair of numbers as argument

[long pelican]

I'm letting this one slide because it's an abuse of notation but one that's used everywhere to the point it's become standard

[strange bronze]

okay sure

ill admit i stopped reading halfway through the last one lmao

[long pelican]

There is nothing in "solve a function" that inherently suggests setting it equal to 0

[strange bronze]

like to be clear

its clearly an abuse of terminology

but i dont think its an unacceptable one?

like what else could it mean

[long pelican]

Set it equal to pi?

Find the minima and maxima?

Find the domain?

[strange bronze]

the word "solve" is never used for that though

[long pelican]

Find its inverse?

[strange bronze]

but its used for "find zeroes" in the context of polynomials

¯_(ツ)_/¯

[long pelican]

I have never seen "solve x^3-x-1" in a real math text anywhere

but I have seen "solve x^3-x-1=0"

"roots of the polynomial" ✅
"roots of the polynomial = 0" ❌
"zeros of the polynomial" ✅
"solutions to the polynomial = 0" ✅

The main reason I'm attacking this particular abuse of terminology is because

I can see it directly adding to the idea that math expressions are to be solved according to the teacher and do not carry meaning in themselves

[strange bronze]

i suppose that makes sense, but idk

id rather reserve "does not make sense" for cases where i cant interpret the prompt in a sensible way

which the last one fits (f(g(x, y)) is nonsense) but the first one doesnt IMO

its clear what the first one means

[long pelican]

It shouldn't lol

When students are given a function (polynomial, say), the first thing they want to do is set it equal to 0 regardless of what the question is asking

Also let's look at what solve means

Solve an equation -> find the values of the free variables for which the statement is true

Let's solve a function -> find the values of the free variables (there are none?) for which the function is true (?????)

[strange bronze]

im not disagreeing that its an abuse of terminology

i just find it an acceptable abuse, in the sense that it wouldnt phase me if i saw it in a paper

from a pedagogical pov i dont disagree with avoiding this

since again, abuse of terminology

but i wouldnt say it doesnt "make sense", since i totally understand what its asking

[long pelican]

Ok I'll take it for granted you understand what it's asking

Since it's calc class though, solving a function means find its derivative

So the actual answer is 2x

Then in the optimization unit, "solve this function" means "solve for when derivative equals zero" and students plug and chug by taking its derivative and set it to zero

ezpz lemon squeezy

[strange bronze]

idk, it just feels like youre occupying a weird space where "the function f(x, y)" is a fine abuse of terminology, but "solve the function f(x) [which is a polynomial]" is not

[long pelican]

Apparently solve also means to plug in a value

[strange bronze]

i guess your point is that the former phrasing doesnt really introduce misconceptions

whereas the latter does

[long pelican]

ya

[strange bronze]

idk, maybe just add "(without ambiguity)" or whatever to the end of the question at the top

and ill be less nitpicky

[long pelican]

Don't think it's a weird space I'm occupying at all because you see declarations of functions like f(x,y) everywhere in papers, wikipedia, etc...

But I have not seen the "solve" abuse of terminology except in K-12 teacher-written stuff, and in student questions

[strange bronze]

i still feel like i see it for polynomials now and then

in fact, im pretty sure the only reason i stopped seeing it was because we had to distinguish between formal polynomials (as elements of R[x]) and their evaluation functions

which i dont think a calc class bothers with

[long pelican]

That distinction seems totally orthogonal to the "function" and "function = 0" mixup

[strange bronze]

but eh, thats polynomials and not functions in any case

so i guess i get your argument

well yeah, thats my point

[long pelican]

Hehe more examples

I'm also pretty sure students aren't inventing this abuse, they pick this up from teachers

https://cdn.discordapp.com/attachments/326138737606262786/908737252232945674/20211112_231653.jpg yikes

I guarantee if a student is asked "solve this function" and given that rational function, they will
(a) set it equal to 0 and solve the equation in algebra 2
(b) find its derivative in the derivative unit
(c) find its derivative and set it to 0 in the optimization unit
(d) split it up into partial fractions in the partial fraction unit
All contributing to the idea that a math problem statement does not have inherent meaning and is just code for figuring out what the teacher wants you to do

[strange bronze]

okay thats terrible

[austere inlet]

big yikes indeed

have we become collectively illiterate as humanity or what, like use some words

[pastel horizon]

Or maybe they will try to simplify it

Wait what that's not even a partial fraction

[long pelican]

😁 exactly, this teachers’s poor students are being taught that words in mathematics are code for what the teacher wants you to do

[astral laurel]

https://docs.sympy.org/latest/modules/solvers/solvers.html

[long pelican]

I should try that in sage

[astral laurel]

(a) and (d) are fine, and reasonable interpretations for 'solve'

[long pelican]

And Mathematica

[long pelican]

My position is that the student should do what they think the teacher wants, that will probably get full marks

But that that’s not how you should approach reading math in general

[astral laurel]

But why would normal people need to 'read math'?

[long pelican]

My students aren’t normal people, they’re majors in science fields

Or economics

[astral laurel]

This is Theorem 1.1 of https://www.sciencedirect.com/science/article/pii/S0019357721000458

Most majors in science fields don't need to read math as much as they need to understand relevant math

[long pelican]

Pretty sure relevant math is math is math

[astral laurel]

Are you telling me the usual science major needs to know what this is saying?

[long pelican]

What site did that link come from

[astral laurel]

It's Indagationes Mathematicae

[long pelican]

Ok pull an example from a science paper

[astral laurel]

https://arxiv.org/abs/2109.00680

None of the notation shown is adversarial

[long pelican]

The language though is pretty mathy

There’s also set notation

[astral laurel]

This (the arxiv) is simple stuff, and while I do think you should fault students for getting the technical questions wrong here, I don't think 'solve' is the worst offender essentially

[long pelican]

It’s not graded and the only point I made about a is that you should not try to make sense of things that don’t make sense

[astral laurel]

Unfortunately that's not how language works

[long pelican]

Or rather not try to assume what the teacher wants if something doesn’t make sense

I’m not really saying anything controversial here so if I appear to, it’s a miscommunication

[astral laurel]

Actually, I literally showed you an example of 'solve' with a default context

[long pelican]

You’re making an orthogonal point I feel

“Solve the function” is up there as only an archetype of K-12 unlearnable math teaching

Not as a singular thing

There are loads of other symptoms I could have equally used

[long pelican]

I’m gonna change this to: if you learned math based on treating math problems as instructions to follow a learned recipe instead of making meaning of the sentences and expressions in front of you, being asked on a test to “solve the function” is something that often further entrenches that viewpoint

But if you insist, this wouldn’t apply for a sympy based course since there is a well defined notion of solving a function there (but it does for typical math classes, and if you disagree, just substitute another of the numerous examples instead)

[astral laurel]

Essentially your take boils down to: math is not algorithms
But actually, to about 99.99% of the world, just about all science is algorithms - though I admit I could be one doing the hot take with my proportion.

Unfortunately, I'd say test taking is algorithmic. You could argue that standardised testing should not be, but to me I think treating problem solving as an algorithm is not wrong though I think it becomes very rigid due to the rigid structure of curricula. There's almost nothing you can do about the rigidity of curricula

[long pelican]

I expected that hot take :P

[astral laurel]

You'd need something like a revolution in education where the performance metrics are a little 'freer' to achieve that. On this I think the usual suspects that do well are nordic countries

[long pelican]

It’s possible to view all of human thought as an algorithm as well

But in that viewpoint you can still distinguish understanding a language and not understanding a language up to a point

[astral laurel]

Oh, and one pervasive problem in math education is the prevalence of calculus (i.e. literally the sense of calculation). Math's use as 'just math' is not shown at a young age to much of the world. You'll notice this if you look at global syllabus. I'd say French education among all I know starts early with some pure math.

There's like no section on 'proof reading/writing' or abstract topics e.g.
https://timssandpirls.bc.edu/timss2015/encyclopedia/countries/germany/the-mathematics-curriculum-in-primary-and-lower-secondary-grades/

Point on calculus: it does take away abstraction and provides concrete example of treating math as calculation/solving

[astral laurel]

On this yeah, I agree with your point on teaching the technical definitions properly, but I feel like it's fine to have students make mistakes. The students probably never got the technical definitions down

[long pelican]

I don’t even mind never teaching the technical definition of solve, just that you don’t teach students misconceptions

Directly or indirectly

[astral laurel]

I just realised for your last sentence, a really loose notation is to take f(g(x, y)) as taking g(x,y) as the first parameter of "f(., .)" where the unknown parameter is assumed to be required later on so you have "f(g(x, y))(z)" actually. Perhaps my notation is too loose

[long pelican]

You’re thinking curried functions in functional programming languages!

Fortunately no one in my class today brought that up 😅

[astral laurel]

But yeah, it's not 'uninterpretable' so to speak

[astral laurel]

In a sense I'd applaud that student for making sense

But I agree it is loose/bad notation

[long pelican]

Ye, another thing I asked is if any of them knew any programming languages and no one did

[astral laurel]

Alternatively g(x, y) does not map to a real number, but maps to a 2-ple

[long pelican]

Which surprised me, I thought coding would be more popular now than when I was in college

[astral laurel]

Coding is more popular now, but could be that your class has no exposure, which I find weird

[long pelican]

Yep

They were interested when I showed them JavaScript in the dev tools console in my browser

[astral laurel]

Did no one have this take

[long pelican]

Nah

[astral laurel]

Can I ask the age-range

Is this uni/pre-uni

[long pelican]

18-19 mostly

Freshman in university

[astral laurel]

Oo pre-uni

Oo uni

omg lol you should shore them up

I see now

[long pelican]

😄

[astral laurel]

Yeah unfortunately uni is typically the first brush

I see why you made the comments you did, I personally do as well

I'm considering teaching (as TA) stats next sem to business school students, no clue how it will turn out. I'm hoping to at least show R

[long pelican]

Whatever your expectations of their math literacy are, divide that by like 10

[astral laurel]

lol when I think about how stats is so butchered in pre-uni, I just get shivers

[long pelican]

Would be the advice I give to myself if I were to time travel back

[astral laurel]

It's not that, it's really that the syllabus isn't helping

Stats to pre-uni is like, everything can be Z/T-tested

That's 'fine' as a take by assuming CLT applies everywhere, since whether they learn more tests can vary, but what I'm not fine is when they can't do the (X_1+\dots+X_n)/n as a first-principles manner to thinking about it, which is just weird

But w/e, gotta start again from 0 in uni

[long pelican]

Yep

[analog token]

In French education, proof reading/writing mostly start at the first year of university
Is it different in others countries ?

You see some "proofs" before but not in a systematic way

[astral laurel]

I must say I don't exactly know, but I would expect proofs in classe préparatoire stuff

[stiff apex]

Yeah, classe préparatoire is all about proof. But it's the equivalent of 1st year uni

[analog token]

Yes, I'm asking about how is it done in others countries than France

[astral laurel]

I'd say classe prep is like pre-uni, but if you equate to 1st-year then I guess whatever. It's not uni in the sense it's not a course to a degree, it's for entering a degree-awarding program as far as I know.

In post-secondary (but not uni) education worldwide I don't think anyone does much proofs. The typical is to show trigonometric or inductive proofs, but they are rather boring and mostly boil down to algebraic manipulation.

[long pelican]

It just occurred to me that algebra teachers have been teaching manipulating equations all wrong

Current lesson: If an equation is true and you do the same thing to both sides of an equation, it remains true

That's only half of it... the other half is that If an equation is untrue and you do the same thing to both sides of an equation, it remains untrue

In fact, that's the more important half; that's exactly the logic that tells you there are no other solutions

(and it immediately explains extraneous solutions when you apply a non-injective function to both sides)

[tepid smelt]

This is a nice subtle point I have actually not taught. I think the issue is we tend to focus on representing situations with equations and solving it. I spend little time on just basic solving but more on word problems where setting up the correct equation can solve the problem in the scenario.

I think I will mention this on monday as just just finished a unit on equations/inequalities.

[winged urchin]

Sometime I like to emphasize that each line in an equation is technically like

If line N is true then line N+1 is true

And show how this is fine when we're talking basic operations like addition, subtraction, etc... But when you apply functions to both sides or you 'cancel' a function on both sides those statements from one line to the next are the conditions for something to be a function or for something to have an inverse

And I do talk about the truthiness of a statement sometimes too aha

One thing that has always bothered me kinda..

We teach kids to never start with the equation or inequality or whatever that you're trying to prove, if that's the goak

But really... If you start with an inequality that is true and break it down to a tautology like 1>=1, again being careful of when we apply functions and all that, then that is a proof

You can actually start with what you want to prove and if you can reduce that to something true, the statement was true, and if you can reduce it to something false then it was false

But its very much the convention to teach kids to only work on one side

[long pelican]

My take: Always talking about "doing things" to equations contributes to the misconception that there is no inherent meaning in mathematical statements, and what one is learning is arbitrary rules. How to fix? Teach them to read mathematical equations, and logic, and teach them to write down that this implies this, which implies this. Or that this if and only if this. Or that this follows from this. (corresponding to P => Q, P <=> Q, and P <= Q respectively)

And teach equation manipulations as theorems instead of procedures

Out: If you do the same thing to both sides of an equation, it remains true
In: If a = b, and f is any function, then f(a) = f(b)

In: If f is an injective function, then for any real numbers a, b, a = b if and only if f(a) = f(b)

Also teach equations and functions together in 3rd - 5th grade, instead of equations in 3rd - 5th grade and functions in 9th (????????) grade

[winged urchin]

I definitely agree with introducing some of these concepts earlier

Sometimes though, I don't necessarily think precise logical statements are the best for students... I'm not sure

I think I would be happy with more...colloquial statements with the understanding from the students that it means a more technical thing

Mainly because as a tutor I find if I repeat something as we're doing it then it sinks in their brain more

[long pelican]

Precise thinking should be the solution to these kids' problems

[winged urchin]

But it's difficult to 'say' the more precise statements and still be able to effectively communicate

[long pelican]

Imprecise thinking is why 80% of students couldn't solve a probability problem on the last exam

because they defined a function to be a divergent integral and didn't think precisely about what they were writing

Also all these "am I allowed to X" questions

[winged urchin]

I would attribute that more to the student's care taken with the question. A student should always look at their answer and take deeply about each step, which does involve thinking precisely about definitions

[long pelican]

come from not having precise enough working definitions of everything involved to think about it themselves

[winged urchin]

I do encourage my students to make 'cheat sheets' of definitions or the more precise statements, though

And I try to get them to refer to them if they really want to make sure what they have is accurate

[long pelican]

I have an interesting analogy I came up with

If you have an 80% precise notion of idea A, and an 80% precise notion of idea B which depends on idea A, and so on until idea Z

Your precision of the chain from ideas A to Z is almost 0

What does this mean? Means you should not accept less than 100% precision on fundamental ideas like A and B

[winged urchin]

I guess though, I just mean, that I am not... confident that if teachers used 100% precise terminology, whether that would be better or worse for their understanding. It could be the case that they need those... simpler (yet imprecise) statements to be able to bridge the gap from no knowledge to working knowledge

[long pelican]

You can be precise without using jargon by the way

[winged urchin]

Here, what is your best attempt at stating the intermediate value theorem as simpler but 100% precisely as possible?

[long pelican]

If $f\colon [a,b]\to \bR$ is continuous, and $c$ is between $f(a)$ and $f(b)$, then $f$ reaches the value $c$ somewhere

[burnt vessel]

Icy001

[long pelican]

Could also say

$f$ attains every value between $f(a)$ and $f(b)$

[burnt vessel]

Icy001

[winged urchin]

I like that better, albeit it loses some precision

[long pelican]

Don't think so!

Side note: this trips people up if they don't read "f(a)" and "f(b)" for their meaning as numbers, but read them as "plug in a and plug in b"

[winged urchin]

I like these as written statements, more than verbal ones, certainly

I think that's part of my issue with some precise statements spoken, the mathematical bits can easily get lost in the students head I find, unless it's written down

[long pelican]

That's why mathematicians don't give math "speeches" :^)

[winged urchin]

So in the verbal statements of things I like to try to avoid f(a) or f: [a,b] -> R, etc..

[long pelican]

(without a blackboard)

I think calculus students are developmentally capable of learning how to read the statement of the intermediate value theorem and understand it without verbal assistance

[winged urchin]

Yeah, it's definitely better to have something written but then that feels.... somehow more restrictive. Sometimes as a student is working I just want a quick sentence I can say to remind them of the core idea of something without having to break out a proper definition

[long pelican]

The requirement is that the teacher can do it too

Which isn't a trivial assumption 🍭

[winged urchin]

I've always liked IVT as a name for a theorem because what it's about is in the name clearly. It concerns the intermediate values, =p

And that, is kind of like my 'simple sentence' explanation of IVT

It's imprecise, sure

But I can say that without really any fear that the student has lost the plot as I was speaking

[long pelican]

In this semester which is really my first semester teaching "normal" non-mathy students I've seen so many examples of imprecise thinking leading to horrible conceptual disasters

https://cdn.discordapp.com/attachments/694677515645747201/909174914483900436/unknown.png

[winged urchin]

Whereas if I 'say' mathematical statements without it being written down I fear students lose the plots in the symbols

[long pelican]

Here's a student trying to answer why this statement is not true and/or has notational errors

So "it will always be 1/3" is correct, which is nice

but then she says y must be < 1/3

which gets a ??? from me

[winged urchin]

Was the total question like... true/false?

[winged urchin]

Ah

So they could've just recognized f(x) is a constant function and boom, the inequality doesn't make sense

=p

Nothing to do with integrals, or FTC, or none of that

[long pelican]

Or rather it's false

But

She clearly did not parse "f(x) > f(y) for all real numbers x>y" correctly

because of this "-> y must be < 1/3"

and that's a serious issue to me

and this isn't an isolated example

I'd say over 50% of students have such serious issues

these are university freshmen by the way

actually I amend my percentage to over 80%

[winged urchin]

What could they have been thinking there...

You know

It's a stretch

But, maybe...

Since y is less than x, and x^2 is positive. If they somehow confused the y in f(y) with the... limits of the integral? Somehow?

[long pelican]

I've tried to make sense of it lol

[winged urchin]

Basically, if we don't integrate as far as the bounds were for f(x), then we integrate less area and so clearly our area will be less than 1/3

There's a couple problems of course, but that's my best interpretation of what they might've wanted to say? =p

[long pelican]

I'm not convinced that's what she is thinking lol

[frosty flame]

Question is tho, to use x in the integral and in f(x). Should have used a different variable.

[long pelican]

That was intentional!

[winged urchin]

Sometimes students just write stuff as well right. Time pressure and all that. Not all a student writes are things they would necessarily be able to defend at all afterwards

[long pelican]

Oh this is homework, not exam

[winged urchin]

Ah, a little better but still. Even in homework there are times students just... go with something and cease spending time thinking on it =p

The variable names are funny ya. ahah. Not technically incorrect of course but you could definitely see students mixing them up

[long pelican]

I wanted them to actually notice that putting x as a variable of integration is confusing and/or a syntax error too

This particular student didn't, which is fine since it could with a stretch be interpreted as a new scope for 'x' inside the integral

[astral laurel]

I personally like using $\iff$ and $\implies$ properly on equation-line-to-equation-line interaction

[burnt vessel]

ShatteredSunlight

[long pelican]

Ya, I do that too

[astral laurel]

For example $x=0 \iff x - 1 = -1$
But $x = -2 \implies x^2 = 4$, and one can ignore the other case ($2^2=4$) since presumably the $\implies$ is the important part in this particular argument

[burnt vessel]

ShatteredSunlight

[astral laurel]

Indeed, when one only requires a string of implies, rather than a string of if and only ifs, I think it shows good understanding of material to know if converses are not easy, not proven, not required, etc.

[long pelican]

I feel like, math would be so much easier for every struggling student if they learned precise thinking

[astral laurel]

Lol the 2nd part got me, so I suppose they are suppose to say that's not true

I think this kind of symbol-handling is great - they need to know to get comfortable with algebra like this, but I think it will trip a lot of people and that's part of the process

[long pelican]

Cool analogy: Learning piano with a mute piano all your life, vs. learning piano and being able to hear what you're playing

[austere inlet]

I think "precise thinking", as in actually knowing some logic, set theory and the structure of proofs is what really got me into math and helped me understand it

before settling for a math major I had taken some engi calc courses, without the proofs of course and it all seemed really arbitrary

barely passed calc 1 and failed calc 2

[long pelican]

I should try to get other teachers to put homework or test questions asking an easy question that requires precise thinking -- I think there's a total lack of that in typical homework and exams, it's all about solving problems similar to ones they've done

If that happens, they will see just deep the rabbit hole of misconceptions goes

And then we'll all want to do something about that

[real mauve]

i wonder, did you do many examples of this in class?

[long pelican]

@real mauve I'm starting to, but tbh I should have noticed it right away after Exam 1 after grading the hardest problem

e.g.:

https://cdn.discordapp.com/attachments/694677515645747201/909242918605312020/unknown.png

https://cdn.discordapp.com/attachments/694677515645747201/909243509062656011/unknown.png

https://cdn.discordapp.com/attachments/694677515645747201/909244084680523867/unknown.png

I really should have picked up on the fact that 95% of university freshmen are rote learning because they don't understand a single thing about math language or precise thinking, and solved that problem 2 months ago

Instead of mathematical misconceptions, they write things that don't even "compile"

[astral laurel]

Have you showed them even/odd functions before

[long pelican]

yea

[astral laurel]

So did anyone write that

[long pelican]

I think 3 people, but at this point I'm not even mad they can't make the connection to even/odd functions

I'm mad that they say things like "shows that x = 2"

[astral laurel]

I feel like your class isn't studying hard enough but I suppose that is the norm

Yeah hmmmmmm

[long pelican]

It's like if it's a coding class and someone submits code that doesn't even compile

for homework, even though they are supposed to run and test their code

[winged urchin]

That's a question that might benefit from negative points for nonsensical answers, but yeah... I bet if you asked them they would probably say they ran out of time and just wrote something

Sometimes to I'll tell students that if you fess up to a part of your solution that doesn't look right to you then you might get treated more leniently

Like instead of just saying "This implies x=2"

If the student wrote "This implies x=2... But the integral concerns x=1, x=0, all values between 0 and 4... so I think this doesn't make sense but it's all I can come up with"

[long pelican]

I was thinking about negative points for nonsensical answers, but at this point I'm glad I didn't, because I realized it's a widespread problem rather than something particular to certain students. Also isn't even the students' fault, if it's so widespread. You have to blame K-12 education for teaching such poor foundations in math language across the board

[winged urchin]

What would you think of an answer like... "This implies that the graph of (x-... (blah)) has negative and positive areas that are equal to each other"?

The fact that it's an odd function not centered at 0 probably also makes some students uncomfortable

[long pelican]

At the time, I'd have thought nothing of it, just a trivial observation from the definitions

But now I would be super impressed lol

[winged urchin]

Yeah... I wonder if any restatement of the question could be better...

Like

Instead of what does this imply

Maybe if it just asked

"Tell me something that's true based on this.."

Or idk something like that

[long pelican]

It might help, but I actually think the responses as they are reveal exactly what I wanted them to reveal

[real mauve]

this is a bit of a different take but, the reason i asked was that i think such open ended questions will inevitably end in failure so early in a program

they've never learned this before

and from their pov you might just be driving in the last nail in the coffin that math is random made up stuff and it's difficult to figure out what the teacher is trying to convey with random symbols

mind you, what you're evaluating is good and important. just evidently not suited for the scenario. if everyone is doing poorly, it also reflects on the decisions you're making regarding teaching and evaluation techniques

this sort of stuff requires a lot of practice to internalize, and maybe you're at a point where you're so far removed from that stage in learning that you no longer understand that

[long pelican]

They haven't learned the skill of mathematical sense-making before?

[real mauve]

well, you're having to teach it for a reason

usual school systems won't exactly do a great job of fostering that, although i'd expect the experience to vary greatly by country

[long pelican]

Even more surprising when you consider this is a selective university with the average student in this class having been top 5 in their high school

[astral laurel]

Surprisingly, no. Let me put it to you this way. I'd say many students don't have the skill of physics sense-making in the sense of statics and dynamics. Everything becomes a jumble of formulae in physics even though there are simple experiments showing what equations say

And mathematics is an additional abstraction on top of things that can be seen

[real mauve]

on top of that, if students are doing poorly, getting angry and failing them more isn't gonna help 😛

[long pelican]

Nah we sort of curved them accounting for this information

I do wish if I did this over again, I'd incorporate sense-making from day 1 instead of assuming they have the skill and/or can figure it out on their own

The syllabus and content from previous semesters have none of that

[real mauve]

was it also not included in previous semesters' evaluations?

[long pelican]

Ehhh

This class as previously taught has an extremely weird definition of sense making

It's the ability to answer questions like "Interpret f(10) in terms of the real life context"

The thing is, if that type of question is the only sense-making question and it's on homework, practice, and exams, it's just another type of question they study for and practice

[real mauve]

another hot take is that real life examples can help them understand better, rather than making stuff even more abstract

like i can guarantee that the way it's going, people are reacting with "wtf is this random shit"

again, it's important, but perhaps in a more guided fashion

[long pelican]

The weirdness is not that those questions exist

the weirdness is that that's the entirety of it

The rest is routine problem solving

[astral laurel]

This sounds way too elementary for uni

[long pelican]

I'm imagining people feeling good about themselves about good results in those "interpret f(10)" type questions and concluding that they are successful in teaching sense-making

[astral laurel]

It's like secondary education or worse

[long pelican]

Yeah I suppose so

[real mauve]

still, i find it too big a jump from "what do you do if you run into f(10) in a back alley" to "yeet an expression: what do your elven eyes see, legolas?"

and as much as it will pain every evaluator out there, having to curve grades is an indicator of BS. either grade inflation, or poor evaluation/teaching

[long pelican]

Not necessarily

Depends on what you want the exams to represent

If you expect the good student to get 100, that's one thing, but you can also design one where the good student gets 80

and the over-and-above student gets over 80

In that scenario making 80 an A sounds extremely fine

[real mauve]

sure, but that's a rather minor swing, in the scope of things. i was more referring to cases where everyone gets under 60 or 50 and that gets curved up to 100

[astral laurel]

But this should not be decided by you. I don't think you have independent power to decide the curriculum of a course

[long pelican]

Nah there was a pretty good spread in both cases. The class has 3-4 students that "get it"

[astral laurel]

Grading is typically very administrative too, and you would need to answer if your grading is not within the norm

[real mauve]

but anyway if you design an exam, you should sit down and solve it yourself and have your TAs also take a look at it, then readjust the difficulty as needed

[astral laurel]

Unless you got to one of those unis with extreme academic freedom I guess

[real mauve]

like in germany

[long pelican]

Ya, there's 2 professors and 2 TAs and we all time ourselves doing the exam the week before the exam

and aim for 1/4 of the time allotted

[astral laurel]

3-4 doesn't seem that good honestly. Like locally we have 'learning outcomes/objectives' and I don't think the administrators would like having only 10% (presumably) 'get it'

[long pelican]

Depends on how high the normal expectations are

I know expectations are probably gonna be that if you can repeat the procedures without mistakes, you "get it" whereas I have something higher (but it doesn't interfere)

A lot more students get it under normal expectations

by the way the other professor concurrently teaching, uses the same exams (we make them together), and she taught it once before last fall

She didn't expect that problem (and the probability one on the second exam) to be as hard as it was either

So I think this revelation (that students understand way less than they appear to, based on their responses on routine problems) is at least somewhat of surprise for everyone involved

[real mauve]

even/odd functions and orthogonality will trip up students all throughout their undergrad, and a bit into grad school. this is well known 😛

if you tell them "check that one is even and the other is odd" they'll immediately understand, provided it's already been taught

[wispy slate]

Hey can i ask something ?

[real mauve]

seeing it immediately takes more experience than you'll amass taking 4~10 courses at the same time over 3 months

[long pelican]

Ahh I think I said this before but I'm not even talking about not solving the problem

I'm talking about something way worse than not solving the problem

[wispy slate]

Can you tell me the difference between orthogonality and orthonormality ?

[real mauve]

try asking in one of the help channels, check out #❓how-to-get-help

[long pelican]

Orthonormality has the additional condition that <v,v> = 1 for all vectors v in the set

Ok I indulged oops

[wispy slate]

Yeah and the same condition they both have is of perpendicular right ?

[long pelican]

So back to this, it's like if I was teaching a music school, all the students have taken 12 years of some musical instrument, then I ask them to play a moderately difficult passage in front of me

It's one thing that they make a mistake, it's another that they can't even begin to play their instrument at all

So here, not noticing even/odd is one thing (less serious), but writing things like

$\int_0^4 (x-2)e^{(x-2)^4},dx=0$ implies $x=2$ (notice it's syntactically nonsense)

[burnt vessel]

Icy001

[long pelican]

What am I to make of that

[real mauve]

d2 is kinda cute, ngl

[long pelican]

hehe

have you seen the most common response to the probability problem on exam 2?

Context: I ask them to give a name to the PDF of a distribution and write a relationship between it and a function previously defined in another part

The most common response: they name the PDF as $p(x)$, and they write $p(x)=\int_{-\infty}^\infty f(x),dx$

[burnt vessel]

Icy001

[real mauve]

ah yeah i saw that one

[long pelican]

So what I'm saying is beyond not seeing how to solve a problem

it's like... showing you don't understand the prerequisites for the class almost

How are we going to define p(x) to be a constant function and say with a straight face it's the PDF of a distribution

moreover, the integral actually diverges

[real mauve]

uniform distribution moment

[long pelican]

At least uniform distributions have support [a,b] for some a and b

That function is a constant function over the entire real line

[real mauve]

at any rate, coming from a country where not even precalc is covered by the time 12th grade is over, i can assert i would've failed all of your tests resoundingly in spite of having graduated top of the class

[long pelican]

Is that country America?

[real mauve]

no

[long pelican]

Darn it was a good guess

[real mauve]

and as you noted, in spite of having technically learned these things in HS, they managed to get good grades without understanding the underlying concepts

i think you overestimate the school system and the students both

[long pelican]

mm-hmm

[real mauve]

and/or are so far removed from the students' shoes that you can't understand how they struggle with "something so easy"

[long pelican]

I actually get why they struggle with it

I have a hunch their education from middle school onwards actively got them further from understanding math than if they had zero education at all after learning fractions

[real mauve]

possibly so

[long pelican]

I read things in #prealg-and-algebra and see some awful teacher-written questions

this discord is actually a good window into it

[real mauve]

hs teachers require little in the ways of pedagogy and education themselves, for one thing

[long pelican]

I've always had in my mind that they're experts in classroom management and not experts at all in math

[real mauve]

pretty much. teaching children is a whole separate thing

my biased take on this is that the sort of stuff you are going for is unfortunately not something that can be taught

you can be exposed to it and at some point internalize it

[long pelican]

Not taught via lecture in 1-hour pieces

But maybe with better thought out homework assignments

[real mauve]

yeah, that was my thinking when i mentioned "guided examples"

[long pelican]

The other thing is

the internet is supposed to be a good place to learn math

but I googled "domain of a function" today

and the top results are all like from K-12 math learning sites that all omit or make more complicated the idea that the domain is the set of inputs the function accepts

And they're all about "how to find the domain"

You can see where students get that language from

[real mauve]

presumably because "sets" aren't usually introduced in Hs

(at least to my limited knowledge)

[long pelican]

Oh I didn't even notice I used the word set

I was using it entirely colloquially though

[real mauve]

the closest thing that pops up is intervals

[long pelican]

They do have a colloquial definition of set I hope

just a collection of objects

[real mauve]

not in my experience, but recall my education is poopy

[long pelican]

Hmmm

If the internet had proper math resources for K-12 that would be a big plus

Like if I google PID or module (something in abstract algebra)

the internet is great

I get linked to excellent relevant stackexchange questions

as well as wikipedia which has a very concise definition

you don't get anything that reeks of TSM

[strange bronze]

yeah thats a good point

[long pelican]

Personally my math understanding skyrocketed after discovering the artofproblemsolving website way back in 8th grade

[strange bronze]

the internet is so bad at k-12 stuff, they simultaneously assume kids are idiots who need how to solve questions spoonfed, but dont bother to actually talk about the math

[real mauve]

i had just taken for granted throughout my studies that, whenever i'd move up a level, i'd essentially have to throw most stuff away and start again from scratch, but correctly

you're trying to build up on a shaky foundation, but are realizing that foundation may as well not exist

maybe this was strongly seared into my mind cuz i also suck at math and struggled a lot to learn what little i know now. you seem to have gone through this stuff so long ago or so easily that this is all too trivial for you or can't get into the students' shoes. like just casually dropping "set"

this is borderline copypasta

[long pelican]

I do casually drop "set" damn

Wonder what else I casually drop

This would've been great on the diagnostic test we gave them at the start of the semester

Less how to compute derivatives, more "do you know what a set is"

[real mauve]

that's exactly it

i bet they could all differentiate and integrate simple functions

without knowing what they're even doing

through no fault of their own, might i add

[long pelican]

Yeah, no fault of their own

[strange bronze]

i remember my undergrad gave incoming honours math students like 3-4 super basic questions to make sure you understand, like, basic thinking

i was responsible for grading them one year as part of my TA job

and one of the questions was

Circle all of the integers below:
1, 0, -5/2, pi/3

A number is called "rational" if it can be written as a fraction of two integers.
Circle all the rational numbers below:
1, 0, -5/2, pi/3

something like that, i dont remember the numbers exactly

the most common answer was correct

but the second most common was kids not circling the integers in the second part

[long pelican]

Interesting

[strange bronze]

presumably since they weren't actually written as a ratio of two integers

and they conflated "can be" with "must be"

though come to think of it, i think the question also included a decimal like 0.5? and basically everyone circled that

so maybe they just thought of "rational" and "integer" as mutually exclusive

[long pelican]

same with real and complex!

[strange bronze]

in any case, about 2/3 of the incoming class got all 4 questions right

but literally no one got 3/4

everyone who got at least one wrong got multiple wrong

im sure you could draw some hindsight conclusion from that

[long pelican]

Misconceptions come in packages 😛

[strange bronze]

i remember another question was one of those intro logic class "implication doesnt work like that" examples

so it was like

You are given the following facts:

• My cat is named Fluffy.
• All cats have fur.
  Which of the following statements can you conclude? [check all that apply]
• All cats named Fluffy have fur.
• My cat has fur.
• All cats are named Fluffy.
• There is some cat that is not named Fluffy.

again i dont remember the exact question but

along those lines

[real mauve]

-> -> <-

[long pelican]

how'd they do on that?

[strange bronze]

i think it was the least commonly answered correctly

again most of the class got it

[long pelican]

Dang

[strange bronze]

but a large amount answered that

• There is some cat that is not named Fluffy.

(or equivalent) was correct

again idr the exact wording

but basically they assumed that \exists is incompatible with \forall somehow

the other 2 questions were uh

one was solving a quadratic equation which i think only 1 or 2 students messed up

and the other was some inequality question but i forget the specifics

[winged urchin]

I'm of the mind that if they had to use the quadratic formula for that quadratic equation one, that that doesn't constitute 'solving' to anything but the most basic idea of solving =p

In fact I think I'd rather ask them to complete the square than make them use the equation

[long pelican]

I am not surprised at the high performance on the quadratics question

[winged urchin]

It can just be rote memorization certainly

I will say where I am I've tutored highschool kids where they talk about sets in class

One weird practice test material thing even talked about depressed cubics, but I think that was a kid in a better class with a possibly quirky teacher who liked to add that in

Ahah

There was also the calculus test question that said that dV/dx was in units of cm^3

V being volume and x being the length of a regular side

I thinkit was just a cube iirc

[strange bronze]

hold on i might have the pdf downloaded still

might be able to find the inequality question

not technically allowed to share but its been over half a decade so i doubt theyll care as long as i dont name any students or anything

You are given the following inequality:

4 < |x| < x + y < 12.

As usual, |x| denotes the standard absolute value function: |x| = x if x ≥ 0, or |x| = -x if x < 0.

Which of the following possibilities makes the inequality true? (check all that apply)

• x = 0, y = 0
• x = 5, y = 3
• x = -5, y = 3
• x = -5, y = x
• x = 10, y = 1
• x = -10, y = 1
• x = -10, y = 3
• x = 10, y = x

wait

oops lmao

sorry i cant copy-paste since its a png, not a pdf

so might have a typo

but thats the question

[winged urchin]

I get the idea at least

The y=x cause some trouble? =p

[strange bronze]

bleh there we go

i think thats right now

students managed fine on the question

[long pelican]

Interesting

[strange bronze]

i dont remember what they got right or wrong exactly

[winged urchin]

The only other thing I'd possibly expect confusion in is like... the student wondering 'which' inequality to use kinda

[strange bronze]

but only a few kids got it incorrect

[winged urchin]

But you can reason out it has to be talking about the whole thing

[strange bronze]

yeah thats fair, you could probably nitpick the wording

but i dont remember that being an issue

anyway students were given ~40 minutes for these 4 questions

it was the first class of the semester, which was about 15 minutes of:

welcome to the honors calculus course. be warned that this is a hard, proof-based course, and you can transfer out at any time. some resources call it "real analysis" instead of "calculus". we will be issuing a quiz to assess where you're at with mathematical thinking. this quiz is not for marks, but if you do poorly, we will recommend you transfer to the main calculus stream.
and then the students got the remainder of the 50 minute block to finish the quiz

they were allowed to leave early, and most did

(naturally, these questions barely take 5 minutes altogether)

(idk why the prof gave that much time, maybe didnt wanna stress them out day 1?)

again about 2/3 of the class did perfect on it, the remaining 1/3 all got an email that gave "extra resources" and warned that staying in the honors course would probably be a significant time and effort investment if they don't get caught up quickly

[long pelican]

At least this course is proof-based!

[strange bronze]

interestingly, that 2/3 included all of the engineering and CS students that took it because they wanted a harder math course

the 1/3 that got it wrong were all math majors

[long pelican]

Whoa

Math major the classic "easy" major?

[strange bronze]

(but about 90% of the course was math majors, so you could argue thats just a statistical coincidence)

[long pelican]

oh maybe

[strange bronze]

i mean engineering students were allowed to sign up for it but specifically recommended against unless they were really interested in math

since you know, time intensive course

CS students were recommended it iff they were interested in graduate research

so i guess we just ended up with the more keen engineering/cs kids

as a result of this

the course started with about 110 people and ended up with like 40 by the end of the semester

i didnt keep track of the major distribution of the students who transferred out

[long pelican]

Oh man

That's pretty intense

[strange bronze]

it honestly wasnt a very intense course at all LMAO

it was really slow paced relative to an actual analysis course (like, it was a 2-semester course and we only defined derivatives at the end of the first semester!)

but it was a first semester uni course so

[long pelican]

Damn

[strange bronze]

a lot of students just werent prepared

which is fine

thats why we allowed them to transfer at any time

[long pelican]

Makes sense given this class has proofs

and given what I talked about today 🤣

[strange bronze]

the students who stuck in it did really well

consistently like 3.6 gpa averages

which in a first year course is like

unheard of

(the calc courses, for comparison, aimed for 2.4-2.6 average)

(one year, a calc course had a 1.8 average...)