math-pedagogy

[tawny slate]

i would personally avoid a book that does this

its also why i dont play the witness and consider it to be an awfully designed puzzle game

[true anchor]

I feel that as long as it’s a very small number per chapter (1 or 2), then it won’t necessarily be hard to figure out. As for the demeaning part, I think this can just be worded well, no?

[true anchor]

See I love that game lol

What about Outer Wilds?

[tawny slate]

no its not about that, its about the fact that if ANY solution could be wrong, and you won't tell me, why do i bother putting in the time to learn anything?

ill just find a different source that respects my time more

if i wanted to do this kind of hunting of what is true and what isnt, id just read politics

[true anchor]

You say “respects your time” as if critical thinking is a waste of time

[tawny slate]

not poisoning my intuition is also respecting my time

[true anchor]

The point is to encourage the reader to actively engage in the process of figuring out valid reasoning steps

[true anchor]

Good point

[tawny slate]

you dont have to go THAT far to challenge someone's intuition and critical thinking

[true anchor]

I wouldn’t want to mess with the intuition for sure

[tawny slate]

just say "this solution is wrong" and write the solution

that already teaches them to be critical

[true anchor]

That’s certainly a possibility, a good middle ground

[tawny slate]

if the solution is obviously wrong, its not a good exercise, and if its not obviously wrong, then pre-warning doesn't hurt because they still need to nontrivially parse why the wrong solution is wrong

[true anchor]

Alternatively I could tell them somewhere easily locatable in the book, so they know they can always check but can try to puzzle it out for themselves first

[tawny slate]

youre adding additional difficulty for little value

this is what makes the witness so annoying

not only do you have to figure out the rules

you also hide rules and interactions where youre not really expected to find them

and to execute the solution requires a lot of tedious travelling and waiting

[true anchor]

Okay here’s my issue with warning them: it means they don’t need to be critically engaged with the other solutions

[tawny slate]

i dont think "highly motivated" readers are going to have that issue

[true anchor]

I think they will! As you say, they will likely work to make things make sense rather than to check if they make sense

[tawny slate]

i think this is precisely what i mean by demeaning

[true anchor]

I grant that might be a good thing, for correcting intuition, but I think they’ll do a mix as long as they know the vast majority of the solutions are correct

They will notice common patterns across the solutions and infer that those must be valid

[tawny slate]

if you have laid out the idea that we need to be careful and critical, and youve made your point with an example, youre now assuming that im not going to apply precisely what you just taught on the other problems

[true anchor]

Well, idk I think it’s reasonable for the reader to trust the author of their textbook.

[tawny slate]

i dont

i dont trust any book, that book has to earn my trust by the virtue of its ideas

[true anchor]

Interesting point

[tawny slate]

thats what being skeptical and applying critical thinking is about

see? im applying what you taught

[true anchor]

I hope that’s the case, that would make this a lot easier

But what about just having some “solutions to the solutions” section?

[tawny slate]

again, that is undoubtedly better, but why make things more difficult than they need to be

[tawny slate]

here

im going in circles and ive given my opinion

i gotta go back to work now so maybe someone else can pick up this if they want

[true anchor]

Yeah I guess this topic is sorted basically

Thanks 🙂

[turbid zenith]

I'm highly wary of the idea of trying to "hide" the incorrect solutions. Much better to have a "what's wrong with this argument?" or "some of these are faulty, which ones?" header.

It's like having "prove or disprove" for a math text. Those questions still make you think. But you're not left wondering where your author may be leading you astray.

Pedagogically, getting rid of wrong intuitions is hard enough as it is. It's a constant battle.

[true anchor]

Well, my approach is basically the same “some of these are wrong, which ones” for each chapter’s solutions

[turbid zenith]

But importantly the students need to know when to be "on guard" for those

Rather than having to constantly be on guard.

[true anchor]

Of course it’s a matter if how many, right? How many they have to look between

[turbid zenith]

Wait, hold on let me make sure I'm understanding right

[true anchor]

I take your point about intuition, I would have to find some way of ensuring they don’t stay wrong for long

[turbid zenith]

So you're giving problems in each chapter, say

And then at the end of the chapter, you give solutions

But some of those solutions are deliberately incorrect?

[true anchor]

Yes

[midnight scarab]

Yes, it's just exhausting. Also I need to first see examples of correct arguments, try to understand why they're correct etc.

[turbid zenith]

Okay ... that makes me EXTREMELY uneasy. I do not like it.

Have that be a separate question perhaps.

[true anchor]

And I’ve told them at the start of the book that the solutions are sometimes incorrect

[turbid zenith]

Then, to be honest, this is a book I would not buy nor recommend my students.

[true anchor]

Yes, I’m thinking of only introducing the incorrect ones late in any given sequence of similar exercises, to ensure they have sufficient knowledge to detect the wrong ones

[true anchor]

Fair enough, of course

[turbid zenith]

I'm not saying "find the error" is a bad type of prompt, FAR from it

[true anchor]

Right, your issue is with all of them being suspect?

[turbid zenith]

But I don't like the sense that the book I'm trying to learn from is trying to trick me.

Like daring me to be as clever as them.

[true anchor]

See, I love books that do that haha

But it’s really just a matter of ensuring students are very actively engaged in the validity of reasoning, rather than just following demonstrations of correct reasoning.

Being involved in the development of the reasoning process, not merely a recipient

[turbid zenith]

So why do you think students often err toward the latter?

What do you think is the root cause?

[true anchor]

Let me think for a moment on that

Well, on first blush, my thought is that they do not initially know how to reason mathematically, because it’s not something one has to do outside mathematics.

So when they start doing mathematics, they have to follow demonstrations to see how it is done. Otherwise they would have to like, reinvent thousands of years of philosophy from scratch.

So they trust the instructor’s word on this, because they don’t have any standard to compare it to.

Something like that, I would think.

I really appreciate this discussion by the way, this is very helpful

I do think there is a substantial difference in how difficult this is based on how many of the solutions are wrong. The higher the proportion, the more on guard they need to be, and the harder it is to detect the wrong ones because of the lack of usable training examples

I think if there are only one or two, out of perhaps 100, that are wrong, it may not “poison their intuition” as Cozmo put it. They will be able to discern which reasoning steps aren’t consistent with the others.

I will have to test it, of course

I recognize this idea is very risky and could go badly, but I think there is a lot of scope for mitigating the downsides and still getting the benefits

[tawny slate]

more mind reading

which people generally dont like

[turbid zenith]

I would say that a reason students err toward "mimicking" is honestly because that's how they've played the game of school up to that point.

This is what I see from my own students a lot.

And at least some of that comes from the focus on abstract symbol-shuffling before the students have really internalized what it is they're talking about.

[true anchor]

Oh I thought you were joking. For pedagogy it’s useful to think about how students think, so that you can teach them more effectively. I was just answering Solid’s question and doing my best to come up with an explanation. I could easily be wrong. I think this is fairly different from what I was taking issue with, especially considering there isn’t a student in front of me to ask

[true anchor]

Great point, yeah that seems reasonable too

[turbid zenith]

To some extent there has to be some attempt at mind reading in pedagogy.

I can never fully know what's going on my students' heads. I have to try my best to figure that out through assessment.

[true anchor]

@turbid zenith Were you asking this as an aside, or do you consider it to bear on the “incorrect solutions” question? (Of course I see that it’s relevant, I just wondered if you were going somewhere with it)

[tawny slate]

i was half joking, i am aware you have to do some level of mind-reading for any form of communication, period

but i was using your own point that you were personally upset about to point out how this was demeaning

which was one of my previous criticisms

the designer of the witness is not smarter than me or clever for making his puzzles unfair and annoying, it just makes him egotistical

it doesn't test me, it doesn't feel satisfying to solve, and its annoying

[true anchor]

Sure, I take your point on that; putting yourself in the position of intentionally deceiving the reader in order to test them is condescending.

Just personally, however, I genuinely love that sort of dynamic. I find it very respectful and intellectually stimulating, as a reader. I’m sure this is uncommon, of course.

[turbid zenith]

A bit of both.

[true anchor]

as long as they actually can trick me. Maybe this is your point. If they underestimate me, that pisses me off.

^ @tawny slate

[true anchor]

I will say that I don't know that it makes him egotistical. A lot of people love that game, and he may have been making it for them and for his own artistic pleasure (which I also wouldn't consider egotistical).

He may be egotistical, and you may have other good reasons for thinking so, to be clear

But there does exist a market for that kind of game, there are people who like it, and for them it doesn't feel demeaning.

[tawny slate]

look, you came here asking for pedagogy advice, not how to cater to a niche audience

im not here to have a debate

if you want to make your work less accessible thats on you

everyone seems to be in agreement here

[true anchor]

We largely are, yes! Doesn't feel like a debate to me, but of course you're welcome to stop at any time. I appreciate everything you've brought to the discussion 🙂

[vocal phoenix]

Apologies if this already came up, but are you basing this on your previous experiences in teaching mathematics to others?

[true anchor]

Eh, not really. I’ve tutored a few people, but it’s more based on my experiences as a student (i.e., sample size 1).

It’s an experiment, rather than the result of previous experiments

[vocal phoenix]

Fair enough

[thorny raft]

I'm writing my math IA currently, and it's on differential equations. I felt confident in my abilities but realized how difficult some of these concepts are to explain to people who haven't taken any calculus. How do I explain the relationship between area and slope to a reader who has just discovered the existence of differentiation and integration? I bit off more than I could chew and during my integrsting factor explaination realized how many concepts I see as simple are difficult to explain to someone who hasn't taken the calculus to know what product rule is, or what the +c is ect ect

[elfin fog]

what level of math

AA SL or AA HL or AI?

[thorny raft]

I think sl

Yeah

It has to be able to be read by something who hasn't taken any calculus

[elfin fog]

are you doing the formal IB diploma?

/the full IB Diploma

if so, you can assume the reader knows calculus

since they cover calculus in IB

showing the relation between slopes and areas also doesnt need to be explicitly stated but u can state the FTC and use it in ur IA

[thorny raft]

Uhm I'm not sure

[thorny raft]

Wait really?

[elfin fog]

yes

it is topic 5 iirc

[thorny raft]

Huh my teacher always said that you have to assume they dont

How different is it internationallly

[elfin fog]

what curriculum are you exactly following

[thorny raft]

Uhm I'm in math 31 ib

[elfin fog]

is this in canada

[thorny raft]

Yeahh

I hate how it's called something different I wish it was all the same

[elfin fog]

as far as i know, math 31 ib is not an official ib course. maybe your school is fusing math 31 and ib together

but if you want to see the ib course outline, i can send it to you

[thorny raft]

That would be nice!

Wait

Are all math ib exams on the same day?

Internationally

[elfin fog]

theres different timezones so Europe and asia do it at different times of the day

[thorny raft]

Cause I do take an ib exam on May 1st/2nd

[elfin fog]

yes that is the general time for math exams

in ib

[thorny raft]

Okay

Wait how familiar are you with the hl curriculum

Don't you cover stuff like IBP and partial fractions for integrals in it

[elfin fog]

yes i did the HL curriculum

you do everything up till taylor series

[vestal tangle]

This might be useful to you: https://danielvoconnor.github.io/MathNotes/quickCalc.pdf
It's a collection of intuitive calculus explanations.

For example:

Another resource is a little book by Mark Levi called "ordinary differential equations, an intuitive introduction" with some simple explanations in it too

For example:

chain rule via stretching:

a "proof" of fundamental theorem of calculus without words:

another simple explanation of the FTC is given here: https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#Geometric_meaning/Proof

[turbid zenith]

This has none of those pictures when I downloaded it

[vestal tangle]

those are from "ordinary differential equations, an intuitive introduction"

which you can read here: https://anton-petrunin.github.io/417/lib/ODE.pdf

[turbid zenith]

Ah there we go thank you!

[strange lynx]

How can we intuitively explain to i think 15-18old students, mix between those that need the extra help and those that are interested, that 3^(3^2)=/=(3^3)^2. I mean sure you can just do the calculation but many of them would basically fall back on the rule a^b^c=a^(bc) and no more thought would be had or just start at one end and do it. I was thinking to use a series argument like (33*...33....33....)()() but nothing satisfying. Do I just tell them to go from the right/top and leave it at that?

[midnight scarab]

I mean it's really just about following the parentheses

Plus perhaps a convention for which one a^b^c is, though I wouldn't write this

[strange lynx]

I'm afraid they will just drop them

[midnight scarab]

Tell them parentheses matter

Like commas

[tawny slate]

so there are two separate things to discuss here

one is that (a^b)^c is not equal to a^(b^c)

the other is the convention that in a^b^c, why is it assumed we do b^c first

it sounds like youre asking about the first explicitly but also probably wanting an answer for the second as well?

[tawny slate]

the short answer to this is you get them to think about "how many a's are being multiplied?"

which is how they understand and intuit the rule that (a^b)^c = a^(bc)

once they intuit that rule, it should be pretty obvious why they arent the same

another more long winded direction to take is to come from prime factorizations

treat prime factorizations as collections of "blocks"

like imagine 2 is a small yellow block and 3 is a slightly larger red block and 5 is an even bigger purple block

so two yellow blocks is 2^2 = 4, a yellow and red block is 2*3 = 6

get them to get a feel for how any combination of these blocks forms a unique positive integer (even an empty set is 1)

then show how even though this is a simple way to understand prime factorizations in terms of multiplication, actually finding the base-10 representation is "hard", you have to compute it

now if you re-ask the question of "how many a's are being multiplied?" you can see that the (a^b)^c case is easily understood entirely using the blocks, the prime factorization

but to understand a^(b^c), you need to do that "hard" step of unpacking the block combination to its base-10, so these things are very obviously not the same

the block combination represents some number, but its not so easy to tell how big it is, whereas a base-10 representation like 5231, you can easily get some sense of its magnitude

as for the convention when you remove parentheses, the second exponent c is "lifted" twice, the superscript of the superscript, so it must be the exponent of some exponent

if you did a^b first and suppose that's equal to k

you would effectively have a^b^c = k^c

but this c is on the "second" superscript floor, not the "first floor"

if we wanted it on the "first floor", we would need parentheses

$$a^{b^c} \ne (a^b)^c$$

[burnt vessel]

Cozmogrgdfschkipkhrshtensi

[tawny slate]

you can even see how this works in the latex rendering

thats how i would explain it, let me know if it works for you

[strange lynx]

I'm thinking maybe I can short circuit this with your formula there saying that we only take b to the power of c in the other case we take the result of a^b. Maybe

[strange lynx]

I like the grouping example of understanding the difference between (a^b)^c and a^(b^c). Need to preface a bit about prime numbers and how all numbers are uniquely broken down into them. In the block example is the idea to show them visually the differences between number and also get the different setup of factors, so yellow yellow red has the set {y, yy, yr, r, yyr} as factors and how that builds the set of blocks to understand (a^b)^c. But a^(b^c) means you first have to figure out b^c. Doesn't that always mean we get singular a's in each actual example. How would one unpack the block combinations in the graphical format? @tawny slate

[tawny slate]

literally just convert them back into products of prime powers and calculate

im not aware of anything that makes this any simpler, and i think this type of question relates to deep number theory problems

the block colors were really just helping to make the visualization here more concrete, but you really should label each of these blocks with the exact prime it represents

minimizing the back and forth conversion pain

[midnight scarab]

Really the point is just that bc factors of a != b^c factors of a

[near oriole]

Importance of necessary parentheses is intuitive to me from coding. If a computer can't understand clearly, then the code will break. But I'm not sure if you can convey that.

[rough agate]

hiii, one question

for those of you who teach maths with technology rather than a white/black board, what specific tech do you use for writing ?

like do you use apple pencil + iPad, the Microsoft equivalent, or some external plug in writing surface ?

or even mouse/trackpad only

[turbid zenith]

I use PowerPoint in some classes.

It depends what course I’m teaching.

[tawny slate]

when im using zoom i just use my mouse

works well enough but i also mostly teach up to high school

[halcyon glade]

same (for holding office hours for a uni class)

[timid shard]

I got a $50 writing surface thingy a few years ago

its great

but I only taught on zoom

i was curious and it would seem prices have gone down, you can get ones that seem good for $25

[quasi musk]

I use a wacom pad a lot when I'm working remotely on zoom

[rough agate]

so do you just not write anything at all in the lesson ?

like the latex is all there already or something

[turbid zenith]

Yeah, I use the built in equation editor and animate it in, although I can also write the LaTeX on the fly if necessary

[rapid tusk]

the worst feeling is when you’re in a lecture where they spend more time writing stuff out than covering substantial content

I think having more complicated latex expressions pre rendered helps save time and maintain pace

[gray shard]

Tbh I prefer hand written lectures a lot more, bc some ppl (ofc not meant as a front) just read their LaTeX/ PowerPoint and call it a lecture

[turbid zenith]

Not taken as a front, don't worry

Yeah it can't just be talking at them, of course

But I hate that writing feels like such a waste of time sometimes

[tepid smelt]

I think lazy lectures are really common sadly. Often they might just be writing word for word from the book.

[rapid tusk]

balancing the two mediums is probably the way to go?

[quiet glade]

can anyone explain why we taught greatest integer function and fractional integer function

[junior junco]

I'm with you. I didn't learn about the fractional integer function in hs but we did briefly cover greatest intieger back in Algebra I. It makes sense when you model things like a tiered flat rate shipping cost based off of weight ranges, but I really struggle to understand why anyone would make a graph or model this with a function.

[tawny slate]

making it a function simply allows you to embed it inside a formula instead of needing to write a program-like procedure

graphing it, like almost any function you graph, is really just useful in that it gives you intuition for the behavior of the function

if you can easily understand what the function is doing, graphing it should be trivial

in some cases when youre writing computer programs/algorithms, you may need to rewrite floor functions in terms of ceil and round functions, and so having a graphical visualization helps you better understand how each of these functions are translates of one another

without seeing the graphical representation, I can't even begin to tell you how many times ive seen students mess up these translates, its not as trivial as it seems

it might be kinda niche, but programming is becoming a more and more ubiquitous tool/skillset to have

whether you have artistic applications or you wanna automate some personal process like downloading and archiving your files or developing a game or actually having a job in it, its all useful

rounding is especially important in finance driven applications because even small rounding errors can be skimmed off and it can be a lot in accumulation

https://www.minich.com/education/wyo/java/lecture_notes/SalamiFraud.pdf

this is not only very common, but in modern days the amount people steal is increasing exponentially

[quiet glade]

Yeah i got the first point because everyone is explaining the same point about the use in finance field but,
this adjoint i really cant understand because i dont have niche in linear algebra. But thnx man u explained well!!!!

[quiet glade]

g.i.f saves the day from financial scams?

[tawny slate]

well, it doesnt do anything on its own, the point is that these are ideas we should care about

math itself is neutral, but it can be wielded for good or bad, appropriately or inappropriately

[quiet glade]

i mean it helps is round off the decimals and my granda always used to tell me

One single penny, can become a ocean then u can buy ur games.

[tawny slate]

i dont want to assume anything about your teachers or your school system, i dont want you to think that you cant criticize your teachers or that they cant be bad or uninformed people, but depending on circumstances, sometimes teachers are overwhelmed, they have limited resources and a bunch of students and the system isnt supporting their needs

maybe thats not your case, maybe your teacher just sucks, but i cant know that

[quiet glade]

i mean i do respect but they will never even i asked they will just move on by saying it is not in our syllabus, it will not asked in exams
I do respect your point

[quiet glade]

i belive no teachers sucks, but they are just bounded by the school systems to teach what is important for exam and also low income can also become reason because at the end i study in govt school uwu

btw thank you explaning guys, igtg studies SEE ya!

[halcyon glade]

I'm not sure how in-depth of an answer you're looking for, but there are tons of situations where you want to turn a real number into an integer. The greatest integer function is just one way to do that (round the number down to the nearest whole number).

It also provides an example of something which is not continuous, which is nice to understand.

[turbid zenith]

My main gripe is in calling it the greatest integer function 😛

And writing it [x]

The floor name and notation is right there

[halcyon glade]

In general if you have a number like 3.786, it's pretty natural to want to split it into 3 and 0.786; the greatest integer function and fractional function are just the notation for that

Yeah the name "greatest integer" is a bit weird.

[turbid zenith]

S: "Why does the number go down if it's the greatest integer?"
T: "Well no, you see, it's the greatest integer that's less than..."
S: *eyes have already glazed over*

Versus

T: It's the floor function. You round down on the number line.
S: Ohhhhhhhhhh

[rapid tusk]

seems like some weird holdover perpetuated by shitty high school algebra textbooks or smth

i first learned it as “floor” as well 😭

[tawny slate]

dont get me wrong, i also agree that it should just be called floor, way less confusing

that being said, i think its good and important to go over that as a definition though, and explicitly point out that these two definitions are seemingly different but actually equivalent. i think from a pedagogical view, these kinds of alternate definitions are very important

many problems require you approach a problem from different perspectives, and this demonstrates that even definitions are subject to some kind of arbitrary choice and equivalence

reminds me of the 3blue1brown video on different definitions of the ellipse

[quiet glade]

yep i'll also use this "floor" word to describe this fxn. it really does not make sense if the number is 2.99 why it will call 2 the greatest integer fxn in fact u can say that the floor fxn is [2,3).

[quiet glade]

BEST!

[stiff oriole]

what is a good way to grow your customer base as a tutor?

e.g. some good, local networking platforms like nextdoor?

[marsh thistle]

what is the second definition you're referring to? I can only think of one way to define the floor function...regardless of how many names you give it

[tawny slate]

floor(n) can also be defined as the greatest integer smaller than n

rather than rounding down to the nearest integer

[halcyon glade]

is "rounding down" a definition? I thought it was the thing we were trying to define

[tawny slate]

sure, that is a fair point

but i dont think im looking at this from a like, rigorous first order definition sort of way

im looking at it more in a pedagogical manner, its a different perspective, in the abstract

in this sense i think these are distinct

rounding down is starting at n on the real number line and travelling left until you hit an integer

to find the largest integer smaller than n, you either come from -infinity and go right, logging the highest integer until you hit n, or take the entire set and pick the highest

intuitively these feel very very different, almost complementary

and its ideas in the abstract like this that i think are important in developing creative thinking in students

[turbid zenith]

(Though how receptive the students are to that creative thinking is an issue of its own)

[tawny slate]

and thats where the problem composition bit comes in

construct problems that prove to students the importance of considering the problem from different perspectives

this floor function one is hard though, no idea where id begin for this one

[rapid tusk]

I think a natural starting point is to consider what happens when you floor a negative number

show that the naive “round down” approach can fail if applied blindly

ie why is the floor of -pi -4 and not -3?

[turbid zenith]

I would call that rounding up or rounding toward zero.

[timid shard]

https://davidbessis.substack.com/p/the-magic-of-mathematical-intuition

[long pelican]

I really liked that article!

[tawny slate]

same

i was just mentioning in another channel actually that the idea that the difference between a number and its successor is 1 is a learned concept

humans do not innately understand this idea without being taught it, and we have studies that support this

idk about anyone else but that was shocking to me. i always knew that there was a lot of math i took for granted, but i didnt expect something that basic to be something that had to explicitly learned

i also use the couch analogy for understanding

to have a couch to sit on, for anyone who doesn't have a couch, they have to actually take time to gather materials and build it, laboriously, by hand, an absurd amount of work just to have somewhere to sit

but once you have a couch, its always there for you to sit on. sometimes it breaks or needs a little maintenance, but it will always be far easier than rebuilding from scratch

for many of us, we forget what it even felt like to not have a couch to sit on

nowadays, to assist with my teaching, whenever i understand a new idea, i always log the emotional process along with the actual math so that I don't forget the feeling of not knowing, making it a little bit easier to relate to my students

[halcyon glade]

I think that makes sense, a number line is a nice visualization for that idea

The idea of counting is already a learned concept right

So it makes sense that differences between counted numbers is learned too

[tawny slate]

in fact the big reason why kids can even learn to count small numbers so easily is because we have particular brain circuits for all of the small numbers

whenever you think of the quantity 1, a part of your brain lights up, whenever you think of 2, a different part lights up

but at some point beyond like 7 ish you dont have dedicated brain circuits for it, but brain circuits for like "some", "more", "a lot", not the literal words here but the notion of approximate quantity

so kids know how to associate those small quantities with the numbers

but once the numbers get big, our brains tend to believe they are closer together

for instance, we innately want to believe that the gap between 10 and 10000 is larger than the gap between 1000000 and 10000000

its only through our learned reasoning that we correct this and know thats not true

[halcyon glade]

this is a bit of an oversimplification but yeah, we (and other animals) have an approximate number system which is good enough for small numbers

(roughly speaking, the approximate number system operates on a log scale, so 10 and 100 are around as far apart as 100 and 1000 in terms of ability to discriminate that number of objects)

[turbid zenith]

This is the future of math education

Using the power of clickbait to engagement bait students. Imagine.

😛

[strange lynx]

This is why we need mandatory chess time

[dapper flume]

It is funny sometimes to use that kind of engagement as a hook. What i like even more about that example is that it's a planned opportunity for students to struggle with a task that they can access. You get inquiry out of tasks like that, and if they can be aligned to learning objectives, you're stumbling across some profound learning opportunities

[midnight scarab]

Is the trick that ||one can step back into the starting room since there was no bug there||?

Also what's the math idea it's meant to teach?

Well unless you want to ||prove it can't be done without the step back. Though I'm not sure how I'd go about it||

[turbid zenith]

Well the solution I had was that ||if you color it as a checkerboard with your starting square being black, then to squash all the bugs you'd need to exit on a white square, but you're exiting on a black square||

But then someone in another chat pointed out that ||nobody said you have to stomp on a bug as soon as you enter a room||

[halcyon glade]

Yeah

[halcyon glade]

Yeah I think this is the cute idea

It's a neat demonstration of parity

[midnight scarab]

I also had that thought but maybe the more natural "solution" is to go into say the room below (and squash the bug) but then move back to the original room

Since there was no bug there

[rapid tusk]

oh i think my prob/stats prof showed us a similar puzzle last year lol

he would start every class with some quick brainteaser it was fun

[austere delta]

But it says no re-entering rooms after you squash a bug. In this case you re-enter the first room after you've squashed a bug

[midnight scarab]

I understood no re-entering the room where you squashed the bug, and there's no bug in the first room

[austere delta]

It doesn't say "the room" though. It says "rooms"

[midnight scarab]

Well if that's really what was meant there was no point in saying after stepping on bug

[austere delta]

It also doesn't say you have to leave out the green arrow door anywhere

[halcyon glade]

I've seen the same puzzle under many different framings, I'm p sure it's supposed to be solved by reentering

[long pelican]

https://liorsinai.github.io/mathematics/2020/08/27/secant-mercator.html
Interesting read for anyone teaching calculus. I could never have predicted that the integral of sec(x) was first found by a teacher by looking at a table of raw numbers, rather than symbolically

[zinc umbra]

Oh my Jesus😂
Still thinking tho 🥲

[turbid zenith]

Send good vibes to my Applied Calculus class y’all … their final is tomorrow

[halcyon glade]

Good luck!!!

[quasi musk]

Break a piece of led

[zinc umbra]

Done🥲
Just tried again

[thin frigate]

Just teleport :0

[gentle slate]

that is technically a valid answer

you didn’t step on any bugs in the first cell

[turbid zenith]

Man ... how do you keep from getting utterly depressed when some students do terribly on their tests all semester no matter what you do

(Also, gotta love when students are completing the auto-graded WeBWorK homework problems in less than a minute but have no idea what to do on exams...)

[tawny slate]

https://i.pinimg.com/736x/2d/84/43/2d8443ec6b68719e22edd3c05259d7f2.jpg

[mint lark]

I think that sometimes for passionate/driven/good teachers it's hard to give up control in this way tbh

Like ultimately students are in control of their own learning. Your role is to help ¯_(ツ)_/¯

(and crucially to enable them to be in control of their learning -- because that isn't automatic)

[vestal tangle]

just realize that learning isn't really a function of the teacher it's a function of the student

pedagogy matters but only a tiny bit, what matters is how smart/motivated the students are

[halcyon glade]

Wait I don't agree with that, teaching can matter a lot

I agree though that no matter how good of a teacher you are, there's plenty of things out of your control

[heavy trail]

I think you have to separate the question of what the modal learning experience looks like from what the ideal (or even achieveably ideal) experience could look like.

If you want to know what difference pedagogy could make, that's one thing. If you want to know what difference you should expect pedagogy to make, that's another.

The "pedagogical variation" in high school and university is pretty small relative to what it could be, IMO.

Folks learning sports or music don't doubt the value of a trainer or coach, for example. And that relationship is certainly pedagogical, even if you want to debate the relationship/distinction between teacher, coach, trainer, etc.

[quasi musk]

I feel this with my high school calculus students that I teach on the weekends

I asked a student what is the anti-derivative of 1/x and he said $\ln(x) + C$. I replied 'good'. Then I asked him the derivative of $\ln(x)$ and he said he didn't know

[burnt vessel]

MoonBears-C-

[quasi musk]

This bit went on for about 7 minutes

[midnight scarab]

How did it end?

[quasi musk]

Not well

[midnight scarab]

Did you kill him?

[quasi musk]

I have two types of struggling students: diligent note takers that use taking notes as an excuse to not think

And those that don't take any notes at all, but then quickly get overwhelmed when the problem isn't one or two lines long

[midnight scarab]

Was this one of the first type?

[quasi musk]

Yep

Interestingly, their literal twin is the thinker that doesn't take notes

[midnight scarab]

I feel like the second type is easier to deal with because you just need to do some hand-holding to get them to write things down step by step

Of course it's not easy to make it be a habit so that they can do things independently

[quasi musk]

I'm not sure if one is better or worse than the other. Eventually they both fail

[midnight scarab]

Well yeah, unless you manage to correct the habit

But I meant that while you tutor, the notes type is less receptive to your attempts and guidance

[quasi musk]

It's more of a teaching scenario. I teach about 10 students at an after school program

[midnight scarab]

Oh I see

Yeah it's much harder to do in such a context

Tbf I feel I'd be very bad at this

Like I can do 1 on 1-3, I imagine I could do lecturing (at a uni not a hs can't handle a class), but helping a group seems really hard

[turbid zenith]

That makes sense to some extent ... I know the students have to actually come to the table and try.

But it's hard to unlearn the idea that, if you'd just done this or that, they would have done so

[midnight scarab]

Blame the teachers they had before you

[next sinew]

also sometimes people who take math classes might be business students instead of aspiring mathematicians 😛

[turbid zenith]

This was indeed a business calculus class

[rapid tusk]

what

what do they even teach in “business calculus” or “applied calculus”

[quasi maple]

You know the Black-Scholes equation, which essentially models the prices of stocks in a frictionless market, is essentially a massive PDE?

Not something I'd expect to appear in an elementary course on business calculus but still

[halcyon glade]

Huh, this seems pretty straightforward, calculus has tons of applications to business, economics, etc.

[midnight scarab]

Rates, maxima/minima/optimisation etc I guess

Something like that

[vestal tangle]

exponential function for continuous compound interest

[heavy trail]

It's introductory calculus but everything is a word problem about BUSINESS

[turbid zenith]

This is not far from the truth lol

Essentially we did derivatives and some basic integrals, basic differential equations (separable), and partial derivatives up to Lagrange multipliers. No limits.

Applications were things like marginal revenue/cost/profit, elasticity, inflation, Cobb-Douglas production model, consumer/producer surplus, Gini coefficient, compound interest

Honestly relative rates of change and elasticity were interesting enough that I plan on teaching them the next time I teach the regular calculus sequence. In fact I will probably include all these topics, but those two in particular are great for motivating certain concepts.

[heavy trail]

Yeah, I think it's a bit of a fib since it's (usually) meant to be calculus for "business majors" or whatever.

There'd be a way to do calculus for them, but it wouldn't just be like..."Ok, we're going to learn about exp(x) because of something you'll never encounter: continuously compounding interest!"

I always wanted to see a business calc type class that made good use of, say, Excel.

But most of the concepts do have a reasonable role, even if they'll never be manually calculating derivatives or integrals in a business context.

[turbid zenith]

Well our school officially calls it Applied Calculus, which I don’t like

Because I feel like that raises the obvious question of “So the main calculus sequence isn’t applied?”

But all my students were economics, business, and accounting majors

They need it as a prerequisite for intermediate microeconomics

And I had thought of doing more Excel based stuff, but the micro prof who gets them after me essentially said “you’re welcome to do that if it brings you more fulfillment personally, but I need students to be able to do the algebraic manipulations because when they get to me they can’t solve basic equations or systems”

That’s also why we have to include Lagrange, they need it in that class

[midnight scarab]

So in short the problem is that they have classes on economic theory, otherwise they wouldn't need any of that calculus nonsense

[turbid zenith]

I guess lol

So I just tried to hit the “real world applications” button harder than usual

But I’ve pretty much accepted that for most of them it’s just a hurdle

[heavy trail]

Yeah, so it goes.

[quasi maple]

I've been posed questions about why you'd need quadratic equations (anything to do with them - solving for roots, finding turning points, simultaneous eqn.s with them, etc.) in real life

And I've often referred to physics simulations in video games

[turbid zenith]

I use a revenue model for hand sanitizer as my motivating example

[rapid tusk]

my uni has an undergrad “applied combinatorics” class that is not “applied” at all and is instead just standard introductory combo lmao

[tall bolt]

We all seen you come here discussing it all semester, I think its very clear you did everything you could and at the end of the day its introductory calc, if the students dont want to hold up their end of the bargin, thats on them

Still incredibly demotivating though im sure 🫂

[turbid zenith]

We'll see how it goes once the tests get graded I guess lol. It wasn't ENTIRELY a lost cause I think.

It is crazy how often bad algebra ends up being the problem though

[turbid zenith]

Also need to figure out what to do about WeBWorK. It's very clear that a number of students just cheated.

[tall bolt]

Do you have any inclination as to how? Was it like answer sharing or something? My university uses STACK (https://maths.ed.ac.uk/research/tech-enhanced-mathematical-sciences-education/stack) which allows you to add a whole bunch of logic to your problems and also randomise them which could be helpful

It also allows you to create graphs of certain shapes or with certain properties (injective etc.) and I believe it’s open source. Potentially allows you to come up with questions that are harder to cheat

Though it’s obviously a whole other system, and potentially not even what the issue is. Still, an option to consider

[rapid tusk]

from above they apparently completed the assignments extremely quickly and then crashed and burned on exams?

[turbid zenith]

Yup pretty much @rapid tusk

I can see how long students spent on problems

Have a student who completed these problems perfectly on the first try, spending less than 1 minute on each problem for example. But, yes, crashed and burned on exams.

[tall bolt]

My question was more, how did they cheat, not how do you know the cheated, apologies if that was unclear

[turbid zenith]

My guess is screenshotting the problem and putting it into ChatGPT.

[tall bolt]

Crazy, I don’t think I’ll ever really understand willingly pursuing further education and then doing everything in your power to not learn

[turbid zenith]

If my class is viewed as a hurdle to doing the stuff they actually want to do, then that explains it.

[tall bolt]

I suppose yeah, I guess a lot of people who do business are doing it for the sake of having a degree.

Is there any scope pivoting even harder into business ideas? Perhaps bringing in stuff from statistics like linear regression, using least squares, frame it as predicting sales or something?

[turbid zenith]

I originally wanted to do that actually!

But apparently they already have a class that does that

[tall bolt]

Just by plugging things into R or excel, iirc you need derivatives for least squares (but I’ve not done any statistics in minute, could be mistaken)

But yeah in any case that’s unfortunate

[turbid zenith]

Right now I'm treating it as "back to the drawing board" and trying to find ways that I can assess students more often (for lower stakes) in class.

[tall bolt]

Would an idea like a short presentation work perhaps? Get students into groups and they essentially have to pitch you, say you give them some data, they predict a regression, work out some average (some sort of business stuff from data you get the idea), but they need to explain the maths to you?

Potentially too large a class for something like that and it’s always possible they’ll just learn a script, but idk, just as an idea that’s more “business-y”, possibly more engaging for them

[gray shard]

Not every part of a major is super cool

Maybe they are super invested in learning other stuff

[zealous kettle]

I hate how pronounced the difference is between students who are there to learn, vs students who are there to get through school.

[turbid zenith]

Am now grading it, and yeah, it's atrocious so far

Just finished grading #2.

• A bunch of people couldn't even find the areas.
• On the practice test a very similar question gave a table instead of a graph, so some students tried to hopelessly mimic creating a table.
• Actually a number of people got c right and recognized that it should be at t=5. But of course there were a number who mistook the graph as being p instead of p', and said the max price was between t=2 and t=4 or something.
• For part d, where they should have used the Product Rule, like half the students left it blank, and a number of others concerningly claimed that R'(t) = p'(t) q'(t).

Oops that's the wrong version

This is the right version.

[midnight scarab]

How did people do on 1?

[turbid zenith]

Okay I guess?

A number of people have the right idea even if they’re not the best at wording it

Though there was at least some floundering

[rapid tusk]

not to invoke business student stereotypes but...

is this entirely unexpected lmao

[turbid zenith]

UPDATE — starting to tally scores, and I have a student who's gonna earn an A- and another who's gonna earn an A so far!

So it's not just me! 😄

The student who earned an A struggled at the beginning but was in my office hours a bunch leading up to the final 😄

[quasi musk]

My disdain for AP calculus grows each time I have to help students with it

The amount of time we spend doing calculator things instead of actually doing Calculus is insane

[rapid tusk]

AP is such a joke in general

high schools make such a big deal of oMfG coLlEGE lEVElL1!1!11!!11!!!

when

1. it stretches barely a semester of material to full year length
2. it's a terribly bastardized dumbed down version of the most basic 100 level classes full of "stuff you should've learned in high school anyway"
3. they've nerfed the hell out of it as student quality continues to plummet

4. it is disgustingly easy to game these tests

when a 65% raw score receives the same score as a 100% raw score

what message does that send?

[quasi musk]

Calc BC is two semesters of calc

More or less

[halcyon glade]

Eh it depends on how much you go past the official College Board curriculum for BC calculus

I would say AB calculus is one semester and BC calculus, if you stick to just the topic written, is a relatively small addition onto AB calculus, maybe half a semester

I remember being surprised by some things I thought were covered in BC calculus but aren't actually listed among the topics (like integration via trigonometric substitution)

https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf Like AB calculus is units 1 to 8 of this and BC calculus is units 9 to 10 (plus a handful of extra topics attached to the ends of 6 to 8), I never really understood how that worked

[halcyon glade]

Out of curiosity, what calculator things? I don't remember it being that calculator heavy

[rapid tusk]

they'll occasionally put questions that lead to equations that you have to solve numerically (ie with calculator)

or just answers that are to be expressed numerically

example from some solutions i wrote up when i was still in high school

there's no depth or sophistication involved here at all

it's just "plug shit into calculator"

[rapid tusk]

no trig sub in BC always struck me as absurd

and an indictment of how fucking terrible the syllabus was

[halcyon glade]

Oh I see

That is annoying

[rapid tusk]

also the official "solutions" are atrocious to read

it's as barebones and unimaginative as the exam itself

[halcyon glade]

I have a feeling the BC syllabus the college board puts out is meant to be like a minimum requirements thing

[rapid tusk]

even in these solutions i wrote up myself

there's ... nothing to explain?

[rapid tusk]

i have my doubts about how far teachers actually go beyond it if at all

i'm sure the qusetion quality has only gotten worse since '22

[turbid zenith]

I don’t have a problem with calculator stuff in general. In principle “most functions” done have nice antiderivatives.

Where I start to have an issue is when you have to start learning how to finagle the TI calculators in particular to do what Desmos does in one click so you can get back to the math itself

[turbid zenith]

Update. Not as bad as I thought.

And if they turn in some last stuff, one of the D’s will be a C+, one of the C’s will be a B, and one of the B’s will be an A.

So… relieved that I’m not just a bad teacher 😂

[tall bolt]

I think it’s been rather clear from how much you’ve come here for advice you’re not a bad teacher but you can take a horse to water and all that

It’s good to see it wasn’t a complete disaster though

[turbid zenith]

Yeah thanks for listening to me vent as the semester has gone on

[rapid tusk]

why does TI even still have a monopoly on the k-12 calculator market

their hardware is outdated as hell and there are plenty of better competitors out there now

[rapid tusk]

the majority of the class at C or below

[quiet jackal]

crazy how the grade representing “average” is the average grade

[gentle slate]

it’s almost as if…

[turbid zenith]

Yeah I wish there were more above C

But I don't think I'd ever seen algebra this bad 😛

I think I learned like 5 new wrong ways to do algebra

[rapid tusk]

https://mathwithbaddrawings.com/2021/07/19/the-seven-deadly-sines/amp/

the classic

[turbid zenith]

And even on the final I still had only like a handful of students who realized that the derivative of p(t) q(t) is p(t) q'(t) + p'(t) q(t)

Half the students just left that question blank, half of the remaining students did p'(t) q'(t)

The hint literally said "If R(t) = p(t) · q(t), how do you find R'(t)?"

[halcyon glade]

Is this a pedagogy question?

[rapid tusk]

what is that even supposed to mean

[rapid tusk]

a good teacher/textbook will explain them clearly, it’s partially on the student to use their own mathematical maturity to make sense of those explanations

[heavy trail]

A Mathematician's Lament: https://worrydream.com/refs/Lockhart_2002_-_A_Mathematician's_Lament.pdf

I think you have to show, not tell. Give them a mathematical question or consideration that interests them and doesn't involve formulas.

There are a lot of historical problems you can describe to a layperson without getting into the technical details, e.g., Poincare's conjecture, the bridges of Königsberg, etc.

Or show them the more mathematical way of thinking about a formula they're very familiar with, like proving the Pythagorean theorem with them (a la Meno's slave).

[rapid tusk]

the problem is when the foundations are weak it’s hard to build on them

this describes how poorly math is often taught at the elementary levels

[hoary pier]

I'm not a teacher of any kind, but I'm currently taking a multivariable calc course and I came into it with a fair bit of knowledge of diffgeo (long story short, I was a very curious slacker) and I can't help but wonder--why not just teach with differential forms from the outset? You can teach generalized Stokes' perfectly well whilst remaining in the realm of R^2 and R^3, and the gradient and curl (via the cross product) can be rather easily replaced with the exterior derivative without much hassle--hell, you could just as easily replicate the same abuse of notation used to define the cross product curl to define it in terms of the wedge product instead.

If we're going to bring up electrodynamics, that's already been done with differential forms, and continuum mechanics requires tensors anyways.

I dunno, for the case of R^2 and R^3 respectively, you can kind of just teach it from the alternativity requirement. Sure, Hodge duals might pose a bit of an issue but that can be handwaved a little bit (like so much else already is--at least in my course).

Cross products and inner products are both bilinear but I've never learned them as such in classroom settings.

I guess there is the issue of teaching linear duals in order to talk about covectors and forms, which I admit would require a non-trivial bit of linear algebra.

[halcyon glade]

Many students come into MVC without any knowledge of linear algebra, which is a shame IMO but just how it is based on physics class sequences

[hoary pier]

Yeah, this I think is a dumb decision in curriculum design, because linear algebra is vastly important to physics, engineering, and STEM as a whole.

Ehh I think that one could reasonably argue that this is a problem of appropriate restrictions on scope.

[halcyon glade]

I also don't think there's any practical advantage to knowing the generalized Stokes theorem unless you go pretty deep into math/physics

[hoary pier]

Okay, that's...painful but fair.

Entirely fair.

[halcyon glade]

That being said, my school's multivariable calculus teaches differential forms and the generalized Stokes theorem at the end

[hoary pier]

Damn where do I apply

[halcyon glade]

https://www.youtube.com/watch?v=cFscZ9c0AIk&list=PL8erL0pXF3JYCn8Xukv0DqVIXtXJbOqdo

These are the lecture videos

[hoary pier]

Thank you!

[hoary pier]

Actually, I take this back, I can't think of any place in high school courses or below where categorical reasoning would have helped me with anything.

I will, however, die a battered corpse on the hill that cross products are gross and unintuitive compared to wedge products.

To my eternal chagrin.

I don't get why the concept of oriented areas being a new object didn't make sense to them.

Ah

How'd Grassmann even introduce his algebra?

[halcyon glade]

You would also have to change all the physics courses to be in terms of wedge products and I can't imagine the physics professors would want to do that

[hoary pier]

It's not that hard... (I say, having 0 teaching experience)

Yeah, I've seen exterior products, coupled with the inner product, applied to a lot of vastly different fields that I wouldn't have immediately ever thought of applying them to myself.

Namely, projective and conformal geometry.

To say nothing of the (categorified) exterior & symmetric power constructions in representation theory and whatnot.

[halcyon glade]

@steep seal has much to say about exterior products :<

[hoary pier]

Oh, I know.

[steep seal]

Hey nice to meet you

[hoary pier]

We've met

😂

[steep seal]

I know

I just thought It'd be funny to act like we didn't

@halcyon glade knows he can keep me happy by periodically pinging me with GA

[quasi musk]

Most students have trouble with Green's, Gauss', and Stokes' Theorem as it is

It's certainly doable, but you'd have to be teaching an honors vector calc class to get away with things like that

[hoary pier]

I honestly sincerely believe part of the issue is the use of normal vectors that pop seemingly out of nowhere and have no real value other than to be dual to some plane of rotation. That, and from what I've seen in my own class, a poor grasp of single variable calculus.

Well that and there's very little time spent (again, this is specific to my class) motivating the theorems and analyzing why they work and how.

My class has, regrettably, been almost entirely calculation-based at the complete sacrifice of geometric and mathematical intuition.

[quasi musk]

The more I teach, the more I appreciate those calculation based approaches. They contain a lot more than you'd think

[halcyon glade]

Calculations are really important. I see math majors not being able to calculate things sometimes, and it worries me.

[quasi musk]

There are very few examples/calculations in the work that I do, and it's very frustrating

[hoary pier]

I'm not knocking calculations, as much as I hate doing them.

I just wish we got more proofs and intuition to go with it.

[lavish yoke]

I don't remember using generalised Stokes at all when I did my physics degree

[turbid zenith]

Mised a bunch of this conversation but will say, getting too abstract too quickly can alienate a lot of students

If they don't have an intuitive idea of what it is they're looking at

[hoary pier]

I wholeheartedly agree but I don't believe that's much of an issue here as long as we ensure the discussion is restricted to Euclidean 2- and 3-space

No need to discuss diffeomorphisms or manifolds or anything of the sort.

[hoary pier]

I would argue that bivectors are more intuitive than whatever the fuck is going on with cross products.

||Can you tell I despise the cross product yet?||

[halcyon glade]

I agree that bivectors are a pretty intuitive notion (more intuitive than cross products)

[turbid zenith]

I don’t like the cross product either XD

I have been considering whether bivectors could be used instead of the cross product for 2D

And maybe 3D

Is there a good way to talk about the TNB frame without cross products though?

[rain hawk]

I'm tutoring a high school student, and we're doing sinusoids

I am finding it pretty hard to motivate why we care about sinusoids with the knowledge they have

I'm thinking maybe a little bit of history, talking about astronomy and chords, would work

but it's kinda out of the way

The real reason we care about sinusoids has to do with fourier series and their connection to the complex exponential, but neither of those are something I can use here

is there a better way to do that

[midnight scarab]

Isn't trig quite natural?

[rain hawk]

in what regard is it quite natural

I think that chord lengths are a pretty natural thing to wonder about, maybe

and sine is half-chords, which is basically the same

[midnight scarab]

Well how do you find the side of a triangle given 2 sides and an angle

Also useful way to talk about similar triangles

And well obviously right-angled ones

And from there talking about the unit circle is quite natural

[halcyon glade]

It is? I feel there there are many obvious reasons to care about sinusoidals (modeling any sort of wavelike or periodic behavior)

Sinusoidals came about a long time before Fourier theory existed

And also yeah, what afqt said about trigonometry being very natural and practical

[rain hawk]

lots of wavelike or periodic behavior is not well modeled by sinusoids tho

[halcyon glade]

As a first approximation it's pretty good

[rain hawk]

I guess I can go through law of cosines, yeah

[halcyon glade]

(Considering it's the only periodic function you learn in high school)

[rain hawk]

it's not something they're doing until later, but it is a pretty natural reason to care about sinusoids

[midnight scarab]

So they're learning about sinusoids before trig??

What's the context?

As in how were they introduced in class?

But yeah, periodic phenomena, going around on a circle, maybe recommend them a video on epicycles for some excitement

[turbid zenith]

https://youtu.be/Lr9Epp5eKUQ?si=Oj4LKVV80OlGWSgV

I like this explanation.

The driving question of “how high up is the sun” is a very natural one

[heavy trail]

"Where da sun at?", a question we've been asking for as long as we've had eyes

[turbid zenith]

What’s y’all’s take on a student saying they’re “not a good test taker”?

[heavy trail]

I think you've gotta dig in. Test anxiety is real, but also lots of students cop to it who don't have it, and lots of students who have it use it as a blanket excuse without taking active measures to manage it.

And if we believe that tests are there to measure understanding then a student w/ actual test anxiety is only adding noise to the measurement, so it's in the instructor's interest to control for it as economically as they can.

[timid shard]

as a student, I am just slow. which means I can do all the homework and I think even deeply understand it because I spent perhaps 50% more time than my classmates, but during an exam it's just impossible for me to finish everything.

[undergraduate proof exams, usually with some choice of questions, say 5/6]

[halcyon glade]

Depends on what kind of test I think. There are ways to make tests more/less stressful.

I think math exams where you have to race to solve every problem are evil

[turbid zenith]

I can absolutely see the time issues

And I feel like it would be clear if a student was doing good work but ran out of time

I’m more concerned about students who leave a bunch of stuff blank or only minimally answered

[timid shard]

Well you leave stuff blank if you don't have time to answer...

I guess if everything is blank then yeah the student should be encouraged to focus on one question

instead of stressing over different questions and spreading themselves thin

[turbid zenith]

The amount that the student I’m thinking of wrote down was nowhere close to three hours worth

What gets to me a bit is the idea that they ”really know it” because they did it on the homework etc

[timid shard]

it all comes down to if you think the student is saying the truth then

[turbid zenith]

I don’t think they’re lying

But I think it’s more like, on the homework, they had unlimited time and there was a way for them to get the solutions to similar problems

So my gut feeling is that the student figured out how to do each problem at the time

And then flushed :/

[timid shard]

so telling the truth about doing homework "the right way"

[turbid zenith]

I definitely have seen other cases where students were clearly just using ChatGPT. This was not one of them.

It was like randomized WeBWorK stuff.

[timid shard]

so I genuinely think some people are just slower, no matter how much homework they do and I don't think it reflects their understanding.

yes, even much slower

[turbid zenith]

That sounds like a good time to get accommodations

I wish I didn’t have to pivot back to giving in class exams :/

[timid shard]

What's the difference?

What's a non in class exam?

[turbid zenith]

I used to do take home problem sets and let students revise them based on feedback etc

[midnight scarab]

There's also the more subtle possibility of them looking up similar problems and then applying it to the problem at hand

So you get some understanding but not much actual problem solving

[turbid zenith]

Yeah, that makes sense

So like... at that point you still can't do it on your own :/

[midnight scarab]

Yeah exactly

Some people just give up if they can't immediately come up with the solution

I feel like learning how to make progress on a problem is one of the hardest things

Though tbh I feel like for the problems you gave the approach should be reasonably straightforward assuming a solid understanding of the content

As opposed to an exercise like, idk, prove Hölder's ineq

[halcyon glade]

Yeah I would probably refer them to your uni's student learning center if there's something like that (I assume there's some center that helps students mamage stuff like test anxiety?)

You can be supportive but I think it's outside the scope of your job to expect that you'll upend your course structure for that student. Idk how many questions you have on the test, but three hours sounds sufficient for any reasonable calculus exam.

[strange lynx]

I was always assuming everyone was already sitting with 5 books and first checking each section that was relevant. Then check if previous students have had similar problems on their handins or exams if they are available. It's super rare to go from a 15min section of a lecture and 4 paragraphs to a solution.

[ember horizon]

I couldn't preform basic arithmetic until I had foundations in real-complex-analysis, boolean logic--And even then because the examples were always of applying but never of the operations in abstracted isolation, it becomes a reflex instead of a generative domain.

[tall bolt]

I think it’s a real thing. I’m a pretty horrid test taker, I work pretty slowly and I struggle with the stress, but like you’ve got to do what you’ve got to do

[quasi musk]

I don't think they're lying; but I think what they mean is "I understand it better than what I demonstrate on the test". This is hard to measure in feelings, but easy to measure in terms of homework completion, quizzes, and tests. If you sat them down & asked them to solve the problems, then they either can or can't do it

I believe it reflects a gap of knowledge; they aren't aware of how much they don't know. They think that knowing is being able to follow along calculations as the instructor does them, or understanding the book as they read it

They never interpret mathematics as something that they can do when things are covered up. Even basic things like a definition, simple examples or counter-examples often lie beyond their ability

Ability of recollection at a minumum

[halcyon glade]

Do you think this only applies to written exams or also to oral ones?

[zealous kettle]

There's kinda a philosophical question: How much of the material needs to be strictly in the student's brain at the time of the test, in order to consider them ready to move on from the unit?

I have directly told my students "You're probably going to forget some of this between this class and the next time you'll use the material. But, you will know what you don't know, and it will be easier to relearn it the next time you need it."

[pastel sundial]

Idk I feel like there is such a thing as genuinely being a bad test taker

Being able to think well under pressure and under a heavy time constraint is a very real skill that I think some people are genuinely worse at.

Learning disabilities are a real thing

[pastel sundial]

I unfortunately feel like this is most exams to some extent

And I say this all as someone who is (I think) a good test taker

[rapid tusk]

i remember taking a computational linalg course my first year

where all the exams were sprints and the prof was well known for writing exams like that

they weren't hard per se but on no other math exam i've taken in undergrad so far have i felt that same kinda time pressure

(i have no clue how the rest of the class did)

(probably not well bc this was also an introductory lower div that engineering/non-math STEM students were also taking)

[tall bolt]

Yeah I should also point that out for my comment about being bad at tests, I have adhd and dyslexia which 100% don’t help

But I mean as I said, what’re you gonna do, it’s just how schooling is and you need just get on with it

[zealous kettle]

No, actually, it is the teacher's responsibility to make sure they have an accurate understanding of the students' knowledge, and if the test doesn't do that, then it needs a redesign.

[tall bolt]

I do agree, I’m pretty anti written exams especially for maths, I think they’re a pretty terrible way to assess understanding in maths but I get why they’re used. This is a reason I’m much more pro oral exam, but unfortunately that’s just not how it is.

At least not in the UK, they love big final exams here.

[zealous kettle]

perks of being a one-on-one tutor for homeschooled kids

[quasi musk]

Sure, but I think a lot of people that say they're a bad test taker don't fall into this category. I would describe it as fundamental gaps of knowledge they don't even recognize

I have yet to work with a student that could answer my questions on a subject one on one 100% correctly and fail a test

[tall bolt]

Yeah that’s another good point, when I say I’m a bad test taker, I mean I average 90~ on homework’s and 60-70 on exams (where in the UK anything 70> is an A), I’m not like failing

Some people do 100% use it as an excuse but I don’t think it can be dismissed entirely

[quasi musk]

Sure, it can't be dismissed entirely. There are people with severe anxiety issues, and that can be amplified by an exam

In almost all cases of 'test anxiety made me do worse' I've heard from students, they had severe fundamental gaps of knowledge that they weren't even aware of

But instead, claimed to know the very material they couldn't get correct on 1. homework, 2. oral questions in a tutoring session, 3. test

So my personal opinion was to try and empathize with the student, but slowly lead them to the conclusion they don't understand it as well as they thought they did

[cosmic ibex]

But, I understand it perfectly when I'm looking at the textbook page! It's only when I have to use it in tests, or homework, or proofs, or an oral cross-examination, that anxiety gets the better of me!

[quasi maple]

This is the thing I really don't like about education systems

When the only thing to show for what you've learnt and whether you've learnt it is a series of high-stakes exams in tense environments also somehow set to feel isolationist, it doesn't really feel representative of how far you've reached

Worse when it's a paid course you've taken, and it gets even worse the more it costs

(source - have almost been there at the cost of my mental health)

[quasi musk]

My freshman physics instructor got fed up with students claiming that they read the book & did every homework problem, but the exams were too difficult. So, after a while, each exam had one problem that was directly from the books examples (which he didn't do in lecture), and one problem from our assigned homework. This was 2 out of 5 questions.

When students inevitably complained that they knew everything, but your tests are too hard he opened up the book to show them the problems from their source that they claimed to know

[quasi musk]

This can be true, but I'm not sure there are great alternatives to written in class exams. Would weekly homework, weekly quiz, and 2 midterms and a final necessarily be a better experience than midterm & final?

[pastel sundial]

Yeah I think for math at least, exams suck but there isn't a great alternative

[cosmic ibex]

I think the underlying problem is really with the system that demands that a course must produce a final ultimate grade that ends up in the student's permanent record at all. Most students learn math not for the purpose of being able to brag to a future employer that "I'm good at math, I passed all these courses", but just because knowing that math is necessary for learning the skills they're actually going to depend on later. So for a student who doesn't get it, that should be adequately reflected in their subsequent failure to learn the things that matter. (And if they do manage to learn the things that matter, then who should care whether they learned some math that -- as it turned out -- they could do without anyway).
Of course, students should be provided with ways to measure their learning performance against some standard, for their own sake. I don't just see there's a deep reason to elevate one of those ways (or any particular combiation of them) to a mythological status of "final grade".

[zealous kettle]

Most students learn math not for the purpose of being able to brag to a future employer that "I'm good at math, I passed all these courses", but just because knowing that math is necessary for learning the skills they're actually going to depend on later.
Most students learn math because their parents and/or government are forcing them to.

(this is a simplification) Parents force kids to learn math out of fear of embarassment and hoping the kid will get a high-paying job. Governments force kids to learn math in the hopes that it will make them more useful to the government.

[gray shard]

Don’t think this is a true statement at least not in Germany

[vagrant meadow]

I think in some sense it's useful to teach in lower education. arithmetic is indeed a little useful, and it does kind of develop their thinking and reasoning skills (though this aspect could be better emphasized).
however, at the college level (and even arguably by high school) I have no idea why the hell I have bio majors in my calc class. by the time they maybe even use it, they're going to have forgotten it because it's not a continuously honed skill (and that's a problem in general with general ed math requirements).

and, yeah, most of them have no passion for their major. no interest in learning the course content for their major, let alone this dumb math class they have to take. "you're going to need this for your major later" just means nothing to them. so... what is the point. id rather just teach the handful of students that actually want to be there. smaller professor to student ratio would be nice.

one of the problems I graded today, for a midterm, had 30% (of 122 students) get 0% because a lot of them just showed absolutely no understanding of the material. and I'm just wondering "who are these kids? why are they here? why does this class have 130 students?"

ok that just ended up being some venting at the end there but I'm gonna pretend that it was productive for the convo

I just really hate this societal/parental pressure in the US for everyone to go to college. it's bad for them, a waste of money, and, at least for me, it's just kind of frustrating. it's just not really enjoyable or fulfilling to teach when it's like pulling teeth, and then they fail because they didn't put in effort. and then there's pressure to pass them anyway because you want your NUMBERS to look good. let it be another teacher's problem.

[lyric fractal]

This is my case too, i just dont do well on exams, that i would later solve easily when im not under the time pressure. Having the recourses and time to work affects me so much. This really sucked when multiple professors i respect would ask me why i did badly on their exams, when they expected me to ace it given my performance in class, and i just wish they would understand that some of us just operate slowly 🙂 luckily now that i started research i feel much more comfortable that i dont have to deal with that sort of pressure again.

I definitely still had gaps in my knowledge, so its not like the assessment is entirely false, but the difference between a exam and a hw is that if you identify a problem during the hw, you can use the recourses at hand to fix it, while in the exam you just either know how to do it or you dont. (for example you missed a key proof idea)

I like open book exams for this reason, and i wish it was more common

I allow my students to use notes they wrote.

Well i should mention that all my exams determined 100% of my grade, so stress was definitely a factor. Imo a good way to evaluate students is through HW, oral exams and open book exams.

I dont see a difference between a student who knows how to apply theorem A immediately and a student who is able to realize it during a open book exam.

[quasi musk]

Is there not a case where an oral exam is more stressful than a written exam?

I think that there's a lack of record and accountability when it comes to oral exams en masse. Moreover, it introduces more biases

But insofar as a student that immediately recognizes which theorem applies, versus one that has to look things up; it's clear to me that the student that recognizes it immediately either got lucky or has a deeper mastery of the material at that moment.

In large part the written exams are just to make sure something was learned, and you didn't just copy it down from your neighbor, the internet, or your AI output

After a certain point you need to know the basic axioms, basic results of your field, and how calculations go. Exams are there to help facilitate that

[zealous kettle]

Exams measure it. They don't intrinsically facilitate it. And later units often reinforce the skills better than the drills for the skills too.

[quasi musk]

However, Exams also serve the dual purpose of saying whether or not someone is qualified. Many students in calculus classes go on to be Engineers. I'd like to think that the people designing our bridges, dams, aqueducts, electrical circuits, etc. have some basic competency in quantitative skils

[quasi musk]

Ah yes, but students study for the exams; which is really what is facilitating the learning

[zealous kettle]

It's facilitating cramming and dumping

[quasi musk]

Then they go on, in their field, to see the material again. This was an interesting article, relevant to the discussion

https://edworkingpapers.com/sites/default/files/ai22-644.pdf

Why even bother teaching calculus if it's just going to facilitate cramming & dumping? Most people don't even use calculus!

Computers do most of the work anyway, so what's the point? AI can answer almost all freshman level math questions

[zealous kettle]

Memorizing sin and cosine of common angles is not strictly necessary in order to apply trigonometric identities, and applying trigonometric identities reinforces the knowledge of the common angles. When we test a student on the sin and cosine before we teach them about trigonometric identities, and that test result ends up as part of their final grade, we are not accurately portraying the student's knowledge of trigonometry.

[quasi musk]

No, but we are accurately portraying how diligent the student was at time T in learning trigonometric angles in the term. My question is, where do you draw the line on 'exams primarily facilitate cramming and dumping of information'

[zealous kettle]

How much time have you got?

'cause the whole system is kinda fucked.

[quasi musk]

Let's say a 16 semester calculus class with 2 exams & 1 final. Exam 1 will be at about week 6, Exam 2 about week 12, and the Final in week 16. To make it more fair, there'll be quizzes on weeks 2, 4, 8, 10, and 14. One question pulled directly from the homework

16 week* semester calculus

We'll follow Stewart's Calculus as the standard, and stop at about u-sub or volumtes of revolution

[zealous kettle]

In a world that truly prioritizes student education over cost-cutting, every student can retake any exam as many times as they demonstrate updated understanding.

[quasi musk]

This is 6 chapters of material over 16 weeks. So just over 15/6 = 2.5 chapters/week with one week to prepare for the final

[zealous kettle]

This is only infeasible today because of modern student:teacher ratios.

[quasi musk]

The first chapter is largely review, and you can cut here and there as you go along in stewart's

But the goal isn't just student learning. It's also making sure students are qualified to go be professionals where people's lives could be on the line

It's the dual service of education. Not everyone should graduate med school

[zealous kettle]

Cramming is not a useful skill for most jobs that require a Bachelor's degree.

[quasi musk]

I agree, but I also think that exams/college classes facilitate much more than just cramming

Maybe freshman levels, and some sophomore level classes can be passed by cramming in the US (where I'm based); but upper divisions are harder to cram

Especially in STEM

[midnight scarab]

I don't think students want to spend that much time in uni. Especially if they just want the degree to get a job (which is an issue in itself but a different issue)

[zealous kettle]

Wherever possible, projects are better than exams.

[quasi musk]

"Ok uhh so like we took a derivative which is like the slope of this umm... function and we uhhh graphed it and got like uhhhh"

[midnight scarab]

Why?

[zealous kettle]

They necessitate and facilitate a deeper understanding of the material. Also there's a level of intrinsic motivation - I have never seen a student excited about an exam, but students often get excited about projects.

[quasi musk]

I'm not a big fan of projects in place of finals or exams, although I think projects can be a great learning experience in elective courses

[zealous kettle]

Personally, I'm pretty sure I've forgotten more from courses with final exams compared to courses with final projects.

[turbid zenith]

Honestly I’m pretty happy if a student thinks “slope” when they hear “derivative”

Rather than “bring the power down and subtract one”

[zealous kettle]

That's so sad

[quasi musk]

I think my students know that derivative means slope of the tangent line

I'll give them a test soon to find out

[midnight scarab]

Ah yes, the famous d/dx e^x = x e^(x-1)

[turbid zenith]

So I'm doing some professional development meetings and stuff at my university right now, and I wanna ask y'all's thoughts on something...
What do you think the grades of A, B, C, D, F mean? Or even, what DO they mean vs what SHOULD they mean?

[austere delta]

It's so cursed to me some countries don't have a grade E

[turbid zenith]

Yeah I know that this is mostly a USA thing, but that's what I'm asking about because I'm in the USA 😛

(And think there's a lot of problematic stuff with our grading system)

[austere delta]

But so are you asking for
A for amazing, F for fail?

Or like a formal description of understanding required for the grades?

[timid shard]

it's so cured to me some countries have A through F

[turbid zenith]

I'm leaving it open ended for how people decide to interpret it

In general I'm asking for what kind of performance should earn an A etc

[austere delta]

Why?

You prefer numbers?

[timid shard]

idk what I prefer, I've only ever had %

[turbid zenith]

If you're not in a country that uses ABCDF I'd be interested to hear what you do instead and how your performance levels are determined

[austere delta]

In my country we have grades A through F, but you know without a confusingly missing grade

[midnight scarab]

In France it's just a number out of 20

[vocal phoenix]

In Poland in pre-university education it's a numeric scale, from 1 (fail) to 6 (exceeds expectation; expected to be fairly rare with the "standard" scale topping out at 5)

[midnight scarab]

Some profs adjust it so there are some 20/20, some don't

[timid shard]

you get awarded x points out of 100, if the grade is really bad across the board they might add / normalise or something

[vocal phoenix]

In university education it's from 2.0 (fail) to 5.5 in .5 increments, except for 2.5 which doesn't exist

So the lowest passing grade is 3.0

And the highest is again normally 5.0, with 5.5 only given in outstanding cases

Rarely more than one, two people per class

Very often none

[hollow musk]

we have both letters ABCDR and numbers 1 through 4 here

the standard that the provincial government wants students to attain is a level 3/B letter grade

[vocal phoenix]

Also typically those grades are only used for entering in the final system, a lot of the courses use some kind of broader numeric scale like 0-100 with an official conversion table from that to the official grade.

Which is only applied at the end of the semester when you have to register the actual course grade.

[hollow musk]

oh but, we still also get a number between 0 and 100

[pure light]

the official designation of grades at my university is A ('excellent'), B ('good'), C ('fair'), D ('barely passing'), and F (fail), which can be modified by + or - (which i think is supposed to make it increments of approximately 1/3 of a grade)

[midnight scarab]

Your uni gets an F on knowing the alphabet

[hollow musk]

the way things are set up here, a B is supposed to indicate that the student understands the material well, while an A is supposed to be exceptional

[pure light]

well the idea is that it's A--D for going through the alphabet and F because it stands for fail

[hollow musk]

but in practice, it ends up being the case that students who consistently get Bs end up having some gaps in knowledge

[midnight scarab]

Unless they're in a specific book by Perec

[hollow musk]

I always found F for fail a bit harsh

[midnight scarab]

Yeah I was just joking

[hollow musk]

I prefer R for remedial, or retry

[midnight scarab]

Yeah that does sound better

[hollow musk]

here, it’s nearly universal for R to be used in elementary schools

while high schools can have either R and F, iirc

mine still used R

[midnight scarab]

Here in elementary it's 2-4 (with 2 being a "fail")

But you can fail one of the subjects and still move on to the next year

[midnight scarab]

Where here actually doesn't mean in France but where I went to elementary

[midnight scarab]

I think?

Or maybe they just magically conjure a 3 as your final grade

I'm not 100% sure

[tall bolt]

My uni has 3 different As and the grades go down to H, unsure if that’s better or worse

But letter grades are somewhat useless here, it’s your final numeric average that’s more important, which puts you into one of the four grade boundaries between 1 and 3

I do prefer the UKs system to like a GPA system somewhat, I have a 75 average which would work out to be a 4.0 GPA if I converted that, but I think my actual GPA is a 3.7 because I have a couple Bs

Though all of that is kinda thrown out the window since upon leaving your average just gets boiled down to a 1, 2:1, 2:2 or 3

[quasi musk]

An A reflects the student has performed well enough in the class that they learned the bulk of everything I was trying to teach; a B means that they learned most of the material I was trying to teach, but missed some important stuff; a C means that they learned some of the material they were supposed to learn, but missed many important details and concepts; a D means that they learned little material they were supposed to learn; a F means that they didn't learn the material at all.

This is measured by homework, quizzes, and tests

[tawny slate]

So to me, the most natural first response to this question was "D and F indicate failing grades, so it should reason that students who perform so poorly that they demonstrate that they do not understand the core ideas and lessons of the course, that they should not get credit for passing, receive these grades."

Which makes sense to me, because I do not want unqualified engineers or medical professionals working on anything or anyone if they don't have basic necessary skills and knowledge, and their degree should be proof of this, and the only real gatekeeper in some cases is the professor

But I haven't seen anyone make this point yet, or maybe I skimmed and missed something. Is there any reason not to adopt this standard? Or is the issue "What should count as competence/qualification?" which is more nuanced? But then it feels like it should be reasoned on a per course context rather than being asked in a general form like this

[turbid zenith]

D is usually considered passing

My school defines it as "minimal passing"

A question I'm interested in is what an "A" means, since that seems to be a big point of tension

Does an "A" mean:

1. The student met all the standards of the course
2. The student went "above and beyond" somehow
3. Some combination of the above
4. Something else

[quasi musk]

I'd go with 1. They met all the standards of the course

an A+ would be above and beyond

I know some departments resereve A for something like the top x% of the class

[vestal tangle]

i remember an A being 90 or above and an A+ being 95 or above

so i'd agree that A should map to 1 and A+ to 2

it really depends on how you're mapping the letters to their score and if you include the +s

[tall bolt]

For my uni that would be considered a C

C is supposed to be met the standards, B is they mastered it and A is "an excellent grasp of the material and have demonstrated some fluency with further material". Thats the description for the lowest level of A at least, we have 3 bands for As. I think assessment varies quite a lot my country, or at least continent

[hollow musk]

on paper, a student here is supposed to get an A for 2

but in practice, it's more like 1

[zealous kettle]

The grade SHOULD answer this question: "What portion of the skills and knowledge taught in this class can this student use?" I don't have super-strong opinions on what the intervals should be.
Typically it DOES answer this question: "How well-suited is this student to this class and its structures?"

[zealous kettle]

I think it should be either 1 or 2, depending on what your standards are. Do you expect all of your students to be able to use all of the skills you taught?

[zealous kettle]

Are grades tied to exam performance as a % score in your system?

How do you distinguish "meeting" vs "mastering" ?

[tall bolt]

Yes, 40 is a pass, 40-50 is a D, 50-60 a C, 60-70 a B, then A3 to A1 for the grades above that

The actual words used there were specific to the school of informatics, the central marking scheme just uses like satisfactory, good, great, excellent or something like that

[rapid tusk]

at my uni i'm not even sure 1 applies 😭

grades are so fucking inflated

i BSed half of my real analysis final last semester and still came out with an A that i absolutely did not deserve

[quasi musk]

You can fine tooth comb what standards of the course mean. What I mean for standards of the course is they learned what was taught, in the book, and in the homework. You can ask them to do basic calculations, or apply basic theorems and the student can do that on an exam

I don't mean that they will fuck up basic calculations, and it'd be left to how generously the grader gives partial credit for them to pass

[tall bolt]

I mean at my uni that just wouldn’t be enough for an A though, getting an A typically does require some level of success with unseen problems which aren’t entirely straight forward

2/3 of exams I’ve sat this semester have had at least half the marks dedicated to things which were introduced in the exam

[strange lynx]

How do people balance "what the students have seen before"? Is it more deepdive into the material, novel new problems, combination of different techniques in chain and/or manipulation of the principles we have learned. I remember prof that tried things like this was very hit or miss since if you couldn't figure out the "trick" you are stuck at 0 and no other path was viable so it also gave 0.

[zealous kettle]

Extrapolation like that isn't strictly correlated to understanding what the student has already seen. Sometimes a student can understand how to add fractions and how to add negative integers, but will need extra guidance before they can add negative fractions.

I think this stems from a fear of failure - like, a student who tries to combine the concepts in that way will feel like they're guessing until they receive feedback, and that's a terrible feeling in an exam setting.

[tall bolt]

I don’t know how I feel about it to be honest. On the one hand I think you do demonstrate a solid grasp of the learning objectives of the course if you can be introduced to new material and use the skills you’ve learned to succeed with new ideas, but it can go too far.

Like I think doing problems you’ve not seen before in class is perfectly fine, I have no issues with that. But exams which define new concepts I think run the risk of pushing it too far, as an example, my topology exam had a question about the dyadic rationals which were not something we looked at in class, but I just genuinely didn’t understand the notation used in the question so I was unsure of what I was being asked to do and I was kinda just locked out of that question. I don’t think that’s a great test of my knowledge of topology.

But in principle I don’t hate the idea, like I said I think being able to quickly get up to speed with new related concepts is an important skill and demonstrates a certain level of mastery. The topology exam I’m not too bothered about because it wasn’t so many marks, but I have other good and bad examples of the same idea.

My curves and surfaces class had a question on the exam about the contraction and got us to prove some basic properties of it before we proved Cartans magic formula. We hadn’t seen any of that in the class, the interior product was defined in the exam and none of us knew about duality of homology etc.

I think that question is fair though, it uses the same ideas as the exterior derivative but in reverse which is a reasonable generalisation to make I think

On the other hand my quantum programming exam last week was entirely based on a new idea introduced in the exam with rather little to do with what we did cover (beyond it being prerequisites to the definition), I think that finds its self on the wrong side of things

The amount of exam these problems comprise is important imo. Going by the system here where above a 70 is an A, you should have like 30%ish of the exam being problems like that.

There should really just be enough straight forward routine calculations to pass, so ~40%. That gives you another 30% for problems which require a bit more thought but should still be pretty achievable

[midnight scarab]

Yeah I think it's good to have part of the test be of this kind, but it must be done well

[tall bolt]

Yeah I definitely see where the idea comes from, and I think it’s a good idea, but quite hard to execute properly

[midnight scarab]

Yeah, I had a prof who taught PDEs and func analysis and his exams were great precisely because they took the problems really exploited the ideas presented in lectures and exercises, and that applied both to the more bookwork problems and those that required some insight (in adapting these ideas!) and finally there were a few questions at the end that required some new insight

[tall bolt]

I remeber in that curves and surfaces exam I was stumped on a 10 mark question to prove the Poincaré lemma, I’d somehow just never actually looked at the proof, and in the last like 5 minutes of the exam I thought I had an amazing idea which cracked it. Got the exam back and the entire proof just had “no, 0/10” next to it

[long pelican]

The format I've settled on and really like is 50% straightforward and 50% new-ish (not too novel, but not straightforward either), with 75% being the de facto threshold for an A. Works for all levels of classes, from calculus to complex analysis

I've not been brave enough to put a really novel question on an exam

I'm in the US too, where this isn't standard. But this format is pretty standard in UK and Europe as far as I know

[midnight scarab]

The one exam I hated was my AT midterm. Most of the class was about homology, Mayer-Vietoris, excision etc. But then the exam had 5 problems which all only used homotopy, + degree of a mapping for the last one

So like everything I studied for the exam was completely irrelevant

And I struggled trying to apply homology techniques to the problems

(Which were not really obvious in any way)

So I think it's essential that studying the material be rewarded in some way

VS have something new in the sense that it's orthogonal to most of the methods in class/hw

[zealous kettle]

I think if you're thinking about grades as a rewards, you're setting your students up for anxiety.

[midnight scarab]

I didn't even talk about grades per se

[zealous kettle]

I'm also mostly talking about primary and secondary education, not university-level

[midnight scarab]

The point is that if you study for an exam you'd expect what you studied to actually show up on the exam

[zealous kettle]

Students shouldn't need to study for an exam. If a student is studying for an exam, it means they're preparing to forget the material after the exam.

[midnight scarab]

I definitely learned things by studying for an exam, and these things stuck not less than what I had learned earlier during the term

On the contrary, it's a great way to consolidate the knowledge and also to form a more holistic view of the material covered during term

Of course, that's also because I study with the objective of learning

Not just passing the exam

Just being key here

[zealous kettle]

Right. The expectations of a student who's there to learn math are different from the expectations of a student who's there because their parents or the state are forcing them to.

In an ideal world, it wouldn't be a problem if the latter got Cs and Ds, but jobs that pay a living wage have stopped hiring people who lack college degrees, and colleges have stopped accepting people who got Cs and Ds, so we get grade inflation.

[midnight scarab]

Of course the hs context is different from uni (which was the original context or the convo).

Specifically ugrad

[zealous kettle]

Was it? I'm trying to go back and find it, but I think the OP just asked what the letters do/should mean.

[zealous kettle]

here's the original question

[quasi musk]

Yes, it was in the context as they teach at a college/university

But not strictly limited to that

[turbid zenith]

(But all the talk that's come out of it has been really helpful)

[zealous kettle]

Ah, I didn't catch that

[tall bolt]

I think studying is where most of my learning actually happens

Through the semester reading the notes and doing the homework’s sews the seeds and I feel like studying is where I reap the rewards if that makes sense

[turbid zenith]

Especially the role of "novel" problems, that's been another thing I've been thinking about what their role should be

[tall bolt]

It’s going back through everything with the broader perspective that allows me to see the area more clearly

I do think bad exams encourage cramming though, and I agree that is orthogonal to learning

[turbid zenith]

BTW was one of those "homotopy" supposed to be "homology"?

[quasi musk]

Studying for my qualifying exams both in my MS and my PhD program were where I learned/mastered the most material. Learning to synthesize a year's worth of graduate study was one of the better skills I've learned.

It's also showed me how shallow problem solving in the k-12 system, undergrad, and even some grad courses are

Like a bag of tricks/problems without a through line guiding it

e.g. Calculus problems to random word problems to other random techniques. It all feels loosely connected

[zealous kettle]

Right!?

[quasi musk]

Which is fine, but what I took for as learning/understanding 10 years ago is nowhere near what I consider having learned/understood something nowadays

[zealous kettle]

Some of my homeschooled students use a workbook called "Beast Academy", which does focus more on problem solving.

[quasi musk]

I'm not railing against the current system or saying it needs to be reformed. I think it's actually quite good, with lots of internal flexibility for people. It's just an individual class/experience can suck.

But man oh man, grad school is illuminating

[zealous kettle]

I am railing against the current system and saying it needs to be reformed - extrinsic motivation sucks.

https://www.youtube.com/watch?v=fe-SZ_FPZew

[quasi musk]

I used to think this way. Then I spent a few years teaching. Really opened my eyes on to why things are in certain ways. It is incredibly difficult to teach the way you think math should be taught outside of a one-on-one session with an extremely motivated/talent

[zealous kettle]

What portion of kids do you think are "extremely motivated"?

[quasi musk]

I'm teaching at an after school program Calculus & Geometry on Saturday mornings currently. This is supplementary to a public, private, or charter education some of my students are in.

Within my 10 students in Calculus, I have 3 that are motivated. Within my 10 students in geometry that I started with, I have about four students that really enjoy geometry

When I TA at my public R1, in a class of about 90 students differential equations I had, there were about 20 people regularly showing up to optional discussion sections

About half of them were motivated to engage with the material

It's strange: because when you begin grading, you can see a clear difference in most cases what A work looks like, B work looks like, and C work looks like. Sure there are fuzzy boundaries between A-/B+.

Yet there are still clear differences in quality of work that you can tell. Grading rubrics often fall along those lines

A small amount of subjectivity doesn't mean that the whole thing is subjective/open to interpretation

[midnight scarab]

Woops yes, the first one. Fixed, thanks

[turbid zenith]

Okay, makes sense!

[turbid zenith]

Grading by its nature is subjective.

Sure you can make it be out of "points" and say "well you only got this many points out of that many points", but the instructor still had to decide how many points each assignment or item is worth and what responses get how many of those points.

Points are objectivity theater, to borrow a term from David Clark.

[zealous kettle]

I think the primary/secondary school system in the U.S. fails at all of these goals:

• Encourage motivated students to be able to combine different skills together in creative ways
• Help unwilling students retain skills after the test
• Produce metrics that accurately reflect students' skills

[turbid zenith]

https://gradingforgrowth.com/p/traditional-grades-are-not-objective

[zealous kettle]

Maybe I'm just still salty that I got a B- in precalc 10 years ago because of the number of arithmetic mistakes I made.

[quasi musk]

To what degree is it subjective in your mind? Is it 100% subjective? 50% subjective? 25% subjective?

How do you even give anyone a score on an exam if your belief is that grading is entirely subjective

If an exam is blank, and receives a 0%. How is that subjective? Or if every problem is solved correctly (as outlined in the class). The student receives 100%. How is that entirely subjective?

[turbid zenith]

You decide what grades mean, set standards, and you give grades based on whether students reach those standards. But you don't pretend that what you're doing is "objective" just because there are numbers attached to it.

[zealous kettle]

It's subjective whether the exam measures the skills that were taught.

[quasi musk]

If it doesn't have an objective meaning, then why give out a score at all?

[turbid zenith]

(Also the idea of giving a numerical percentage to how subjective it is is funny)

[turbid zenith]

Because I'm required to do so. But also welcome to at least one reason why I brought up this question in the first place!

For example, you can say "well 90% or higher is an A, bing bang boom, end of story"

But each individual professor still has a LOT of control over what it takes to earn that 90%

So "90%" is more like "what the professor has decided is 90%"

[quasi musk]

My point in asking how subjective your grading is pertains to: how do you assign grades when you have to? Do you just acknowledge 'yep I'm wrong on what I'm assigning, but I'm assigning it anyway'

[turbid zenith]

I don't acknowledge that I'm "wrong"

[zealous kettle]

Subjective doens't mean wrong

[turbid zenith]

I acknowledge that the grading scale I've set up is by its nature subjective

But I assign grades according to it anyway, because I've made the scale and such clear to my students

[quasi musk]

So it's subject to your interpretation. How do you know your interpretation is what you should assign?

[turbid zenith]

And then I try to use it as a calibration point the next semester.

[turbid zenith]

I feel like you're trying to point out an inconsistency in what I'm saying but if so I'm not seeing it

[zealous kettle]

I mean like, how does anyone know anything?

[quasi musk]

The inconsistency that exists here is that the letter, grades once assigned are objective. Objective in the sense that student X received letter B. Grades/GPA make a material difference in student's lives.
For instance:
Grades that I assign are inherently subjective. Grades that I assign have an objective impact on my students lives. I have a duty to assign grades fairly to all of my students.

The inconsistency I see is this: I cannot assign grades fairly to all of my students as all of my grades are subjective. This subjective grade will make an objective, and in some cases, unfair impact on my student's lives. Therefore I ought not assign grades

[zealous kettle]

I agree that grades oughtn't be assigned wherever possible, but for kinda different reasons. But of course, it's not always possible to not assign grades.

[turbid zenith]

The underlying issues you’re mentioning are precisely why I’m trying to get my university to start having a conversation about precisely what A/B/C/D/F should mean and what it should take to get it.

[zealous kettle]

Good luck!

[turbid zenith]

Because at that point we’ve at least laid out our terms and agreed upon them.

[quasi musk]

Yeah, I 100% agree the terms should be laid out and agree upon. It is very frustrating to have TAs or Professors make up exceptions. The subjectivity of grading that does inherently exist, should be kept to a minimum

To keep this issue/inconsistency at bay

[turbid zenith]

Subjectivity is not in itself bad. It just needs to be recognized and used correctly.

[tardy ember]

i think this is kind of... assuming more of a boundary between subjective and objective than there really is? or something?

[turbid zenith]

Professional judgment is a thing.

[tardy ember]

i don't really know how to phrase the thing correctly but like

[turbid zenith]

So I’m trying to design my grading system so that a student who has reached a particular level of understanding in my professional judgment ends up with a particular grade.

[tardy ember]

there are cases where a "subjective" assessment does, objectively, perform better (by some metric) than any more "objective" method that we actually know how to use

[quasi musk]

This is probably the best example in the paper: the incongruency of a grade and things learned

[turbid zenith]

Yeah. It makes you wonder what would happen if Alice got to take a new exam on the topics of Exam 1

[quasi musk]

I had an instructor in CC, where for borderline students he would let them retake their worst exam. The catch was they could only improve no better than their average across all exams

So if their worst exam was a 45%, for instance, and their average was an 89% (100% on 4 exams, 45% on exam 1). Then their make up to bump them from the A-/B+ range to an A range

[turbid zenith]

Why have that restriction?

[tall bolt]

This is one of the things that’s always seemed really weird to me about US unis, at least through what I’ve perceived through media etc. It seems professors have a great deal of say in how grading works and individual students grades are assigned which is just not at all the case here

[turbid zenith]

If the student would get a 100% on a retake why shouldn’t they get 100% of the credit?

That’s what I’d want to ask

[turbid zenith]

How is it different where you are? (And where is that if you’re comfortable with sharing?)

[quasi musk]

For him, it was because their was a contractual agreement in the class to learn A material by X date. It was the penalty for not meeting that agreement

[turbid zenith]

Ahhh I see

[zealous kettle]

If we're presupposing exams, then the easiest thing to do would be for every exam to come with a retake of every previous exam.

[quiet jackal]

why allow retakes at all then

[turbid zenith]

What I did this past semester is if your final exam shows better understanding of a topic, that topic’s score from earlier gets replaced

[quasi musk]

His reasoning was: it's more cost effective for the college & student if they just redo this exam rather than redoing an entire semester

(For students that failed). For students that weren't failing, it was a token of forgiveness

[quiet jackal]

i understand why retakes are good but this seems incompatible with his reasoning

like if the goal is to pass people why not remove the cap

[quasi musk]

I guess it's what he settled on

I can ask him when I go see him again

[tall bolt]

Im in Scotland. Here all exams and all marking is both internally and externally checked by other academics for fairness of marking, and of questions etc. Curves are a centrally decided thing, the university looks at the average of each class, and compares them to how the students in those classes preformed in other courses before deciding to apply a curve (and often they wont, if they ever do its generally downwards). Essentially every class's grade is either 80% or 100% determined by the final exam, and all of these are marked completely anonymously.

Essentially all exams go through 3 rounds of checking, and beyond deciding what questions are on the exam, lecturers have very little say

[quiet jackal]

its still demonstrating understanding by the end of the semester

[tall bolt]

I believe this is the case in the UK generally

[quasi musk]

Many times the make ups were even after the semester

[quiet jackal]

i think a more compelling argument is fairness to the other students? maybe?

[quiet jackal]

oh idk then

that is very generous

[tall bolt]

And like what these grade boundaries are and should be is a central university decided criteria, with possibly some more fine tuned guidance per school

[turbid zenith]

Wow.

I can't imagine having the time to set all that up XD

[quiet jackal]

what is the point of all that lol

[quasi musk]

To minimize unfairness and subjectivity in grading P:

The question is: for all that effort does it get better outcomes

[tall bolt]

It takes a while to get our grades back, typically around a month.

[quasi musk]

A month to get grades back

[quiet jackal]

lol

[quasi musk]

In the US people would be freaking out. People already email when it's been a day or two after the final and grades aren't out yet

[quiet jackal]

pretty sure profs arent allowrd to take that long at most US schools

[quasi musk]

I think our deadline for grades is Tuesday after finals week

[quiet jackal]

who is volunteering to do this triple blind grading

[tall bolt]

Everything about exams is different here though, its taken much more seriously.

[quasi musk]

I was even pressured by my primary to finish grading finals before my own finals were over with. He wanted me to stop studying for my finals to grade his students to meet the academic counselors demands

They got a strongly worded email & a threat that I'd go to the union if they held me to that standard

[tall bolt]

I believe the system is one round of grading by the lecturer and tutors, then it gets passed to the exam board for review, then the full thing gets passed to the external board for a final review.

[quasi musk]

That's almost a dream come true

[tall bolt]

But again, degrees are just wildly different things in NA and elsehwere, they dont really compare very well I dont think

Even just the goal of exams seems to be quite different, not to mention their weighting

[gray dagger]

Yeah here (US, Cali) if a single exam was 80-100% of a grade, that would be multiple complaints to the dean and I would expect a long conversation with the professor about how insane that choice is

In undergrad I mainly saw ~40% homework, ~60% exams and quizzes depending on the professor. In grad Im seeing the exam scores inching closer to 75-80% of the grade, but thats still 3 exams generally.

[zealous kettle]

In grad school for CS, most of my classes were 100% projects.

[gray dagger]

That makes sense, and from what ive heard of non-probabilistic applied math courses its similar. Math Stats courses (in a math program specfically) are more theory focused, so exams are still appropriate

Though im gunning for a direct applications course before I graduate

[tall bolt]

FWIW I also think 80-100% final exams are insane but the UK has a real thing for them

And we’re flat out not allowed to do resits, so if you have a bad day, unlucky bozo

It’s certainly far from a perfect system, but there are some merits

[gray dagger]

You're also right though that in the US, grades are partially a function of the professor and that can lead to wild swings in course difficulty

My current linear alg professor has stated with his full chest that hes trying to pass as many people as he can, while ive met professors who are... not as generous

So far I haven't met a "my plan is to fail 90% of my students" professor, but I've heard tales

[tepid smelt]

I like this. At the highschool level every school I have been at puts a lot of pressure to allow retakes. Ultimately the teacher can decide. I have adopted you can retake any test once but it caps at like a mid C.

I hate grades in general. I wish students had to pass a comprehensive exam to move on and then we might actually get kids to listen to us.

That way we can also set a benchmark across the board on what we expect kids to learn. The level any class even in college is presented at differs so drastically based on the teacher.

[zealous kettle]

If the student demonstrates that they know all the material when they take the retake, then why must the score for it be a C?

[quasi musk]

Because it was a retake. It's the price of not getting it the first time. An education isn't about learning all of the material. I now know all of my high school pre-calculus. Should I go back and replace my grade? No!

Even in the same class, I do think that retakes completely overwriting lower scores is a little too generous

[fierce rain]

Hey everyone — hope this is okay to share here. If you tutor maths privately then this might be of interest to you.

I’ve been working with a couple of tutors lately to try out a new way of getting students without relying on agencies or platforms.

It’s super early stage and we’re still testing, but it’s been going surprisingly well so far. If any of you are tutoring and curious about how it works, feel free to DM — happy to share what we’re doing and get feedback.

No pressure at all — just trying to make something actually helpful for tutors.

[zealous kettle]

Is it generous to report a grade that accurately represents the student's skills and knowledge. You said earlier that you think the grade should reflect whether the student met the standards of the course - is that not the skills and knowledge that are taught in the course? And if the student has that at the end of the grading period, why should the grade report anything different?

My only caveat is logistical: It would be kinda a nightmare to facilitate unlimited retakes for every student for every test. That's why I propose projects, and embedding a retake of previous exams into newer exams.

[tall bolt]

I do think there’s a middle ground, I think a retake should be allowed and whatever that grade is should stand, but it should be noted somewhere that it was the second sitting of the exam

I do think that a student who got a 90 the first time around probably looks better than one who got it on the second, but equally if they put in the work to improve and succeeded, that should be displayed on their transcript

I also think resits are fair, especially coming from somewhere with such heavy exam weighting and no resists allowed, it just kinda sucks. My algebraic topology exam tomorrow is 100% of the grade for that course, if I have a bad day that just kinda stands forever and I’m not sure that’s very fair.

[zealous kettle]

Grades these days are so divorced from skills and knowledge that tech jobs actually give CS exams as part of the interviewing process, even for people who just graduated with a CS degree.

And if grades are meant to report how hard a student is willing to work, they do that in a super fucking classist way.

[tall bolt]

Not something I disagree with

[quasi musk]

Standards of the course can mean by X date you're examed on A amount of material. Failure to meet that agreement is the penalty

The skills and knowledge of the course are tested at various points in the term. The retake is an act of generosity, but within bounds. And it should be fair to everyone. This isn't my policy.

When I teach a university class in the future, I'll be very stringent on retakes

[zealous kettle]

Why? Why require performance at a specific steady pace, demonstrated at a specific time?

[quasi musk]

Because when you teach each 100+ students at a time, you can't have 100+ different schedules.

But even in smaller class settings, it's easy for people to get out of the loop. Accountability matters in education. If the issue is the time frame of learning, then maybe school isn't right for you (or that specific person)

[zealous kettle]

Then why aren't you on team "Fuck the system" ?

[quasi musk]

Because I think the system is functioning relatively well. I've worked in a lot of different settings, and I don't think there's one system that will be uniquely great. It has a decent amount of flexibility in it.

Going to university/college isn't something everyone should do. There are people that are fundamentally not suited for college. That's ok. I'm fundamentally not suited for all kinds of work

The point of college is to both educate and qualify you. I think the lens you have is primarily the educate part. But I think the qualification part is just as important

[zealous kettle]

Going to university/college isn't something everyone should do.
In today's world, lacking postsecondary education is a great way to die in credit card debt before you can retire.

[quasi musk]

I don't think that's true. You can also say "In today's world, getting a university education is a great way to die in student loan debt before you can retire"

[zealous kettle]

Just because the school system doesn't work for someone, doens't mean they should be denied education

[quasi musk]

Nobody, in the US, is denying anyone to go to college in the US. There is a college that will take you

[gentle slate]

everyone should be suited for university and college, but not everyone must go

we still need to function as a society after all

[zealous kettle]

I mean, if you can't learn in a lecture hall, you should still be able to learn and show employers that you've learned.

[quasi musk]

Sure, is a college class the only way to learn? There are plenty of people that learn trades, work at restaurants, learn management, etc.

[zealous kettle]

I mean learn college material.

Job apps will filter you out if your resume says "Read these textbooks" instead of "B.S. from University"

[quasi musk]

Sure, but there are plenty of jobs that don't require degrees

[zealous kettle]

And they generally pay much worse, and they're not suitable for everyone.

[quasi musk]

Sure, but nothing is exactly suitable for every person. Each person has to decide what is important to them, and what they will do with their lives. But in the US, there is nothing stopping someone from attneidng college/university in general. Some people's specific circumstances can make college difficult.

But everyone must trade their time somehow. It's not even clear nowadays that college degrees are worth the time

[zealous kettle]

But there are lots of factors which can stop someone from learning at a university they attend, and from demonstrating that knowledge to employers

[quasi musk]

At the end of the day, your education must be up to you. I can no longer claim that my 3rd grade teacher didn't teach me well so I can't add fractions when I'm 30

Nobody can learn something for you. In fact many students attend class, pay tuition, and decide not to learn