math-pedagogy

[long pelican]

This method can backfire if the exam turns out easier or harder than you expect

[real mauve]

that was an example, i guess a better match is to simply say there are "more than 100 points" but not specify how many

but the difficulty in formulating it that way is directly connected to the reality that the numeric values assigned to questions that will anyway be curved is meaningless

i might be mistaken, but my reads so far usually (not always) point to grading on a curve being a bad thing for students

but papers like this one limit their study to very specific kinds of curving where the grades are forced to spread out over a range and only a small % of students can land in the top range

[long pelican]

That's not applicable to me then

[real mauve]

that's what i would've thought 😛

[astral laurel]

It depends.

[real mauve]

in europe (or at least germany), i think you'd have to slaughter a student in order to get reprimanded. you otherwise have freedom to do anything, including grading in whatever way you see fit

[astral laurel]

Locally (South east asia) grade proportions are strictly academic-board decided. Administrators basically have a large say although these should also be mostly (all?) academic staff as well

[pastel horizon]

When I was at uni my mentor said he only ever awarded one 80+ mark and that was a dissertation worthy of being published in a journal

So you can see that 10% gap is huge between a strong undergrad student and a student that is already capable of publishing papers to a journal that would have rigourous scrutiny

70% I think is a good system as well. It shows that scientists, engineers or mathematicians don't always have to know everything. That's what a 90% A mark would imply

[cosmic ibex]

I thought that was what "curving" means.

[real mauve]

not necessarily

the rescaling need not force the students to fall into specific grade ranges

something simple like determining a new 100% value will just scale all grades up by some factor

whereas you can do some fancier obscure stuff to make it so that, even if all of the students got grades between 90 and 100%, they well end up spread over D - A

with only 5 or however many you pick in the A range

[cosmic ibex]

Wouldn't that be grading by a "line" rather than a "curve"? I always thought the metaphor behind the "curve" terminology was that the transfer function will be as non-straight as necessary to get the desired final distribution.

[real mauve]

this latter one is the one being criticized in the paper snippet i posted above, but i surmised that wasn't what icy was doing, which is why i mentioned it

well

that's a good observation

i'm honestly not acquainted with the naming scheme for these things

but throughout all of my discussion above, by curving i meant any rescaling done ex post facto

regardless of whether it forces the students to conform to your desired grade distribution or simply changes the 100% line

[cosmic ibex]

Hmm, or perhaps "curving" refers to the "bell curve" shape of the desired final grade histogram ...

(Though my impression is that the desired shapes in US tradition tend to be flatter).

[real mauve]

i don't think that kind of curving makes much sense at all

[cosmic ibex]

Each time I think too hard about it, I end up concluding I have no coherent idea of what grades are ultimately for ...

[pastel horizon]

Oh yeah normal curve basically just flies in the face of what that text says

You'd basically be saying only the top 10% can achieve an A

[cosmic ibex]

I believe there are several institutions and/or systems that do enforce such proportions, though.

[pastel horizon]

Well first you need to define what you are assessing for

If everyone has met your assessment criteria for an A then they should all get an A that's what he was proposing

Of course reality is that some students are stronger than others and you end up with a normal distribution, but that's not curving your grades to fit the distribution that's just the central limit theorem

[real mauve]

there is no reason a small, randomly selected sample of the population should exhibit the behavior of the total pop

[pastel horizon]

For sure

[real mauve]

you could get one tail and be forced to fit them into that curve

makes no sense

[cosmic ibex]

Non-randomly selected, you mean?

[real mauve]

no

that's why i also added small

[pastel horizon]

In fact they are randomly selected. Sure you might argue universities only accept straight A students but that pool of A grade students is random

And in that case you're assessing for the difference between an upper second or a first class student

Those universities have much stricter standards

[cosmic ibex]

I'm sure there is at least a correlation between innate mathematical ability (whatever that is) and which people end up attending university programs where they have to learn additional math.

Of course if we want to be pedantic, I shouldn't have said "non-random", but "not independently random". :-)

[pastel horizon]

Well again in this case it's a random sample from the population of straight A students

Reason why I emphasise that is your standards at university for assessment are stricter, you expect them to have a baseline knowledge. To get good qualifications they need to develop that further.

And in order to protect their status as a prestigious university they have higher standards than some less prestigious universities for qualifications

[long pelican]

Funny thing about innate math ability: the top scorers on my Calc II final all seemed to be international students who got their K-12 math education in China

Some people believe this shows innate ability, I just see this as huge evidence that math teachers in China actually teach math that makes sense, which makes students able to learn it properly

[pastel horizon]

Yup, the "Asian good at maths" stereotype should be more like "Asian countries are good at teaching maths"

[long pelican]

One thing that’s been discussed a lot about Asian style education is that people call it rote, because they don’t teach by discovery learning or alternative methods of addition and multiplication. What they missed is that while here students practice meaningless algebraic manipulations, Chinese students practice the same thing but actually know what they’re doing.
I’ve since concluded that rote vs discovery learning is the wrong dichotomy. For example, here we’re doing common core type stuff but a lot of students are still lost because it doesn’t make sense to them. So discovery learning falls flat if the math doesn’t make sense, which happens if teachers teach TSM

[pastel horizon]

The other thing that helps a lot is their insanely good behaviour for learning

Good behaviour for learning in the US would be poor behaviour for learning in China

[long pelican]

Having students contribute ideas instead of passively watching should be superior with equal content, what’s missing is to teach math that makes sense mathematically (I.e. real world analogies don’t help. Need to teach the language and syntax and logic)

[pastel horizon]

I don't even mean that. Chinese students are basically working twice as much as US students since they have cram schools and strict parents

I also heard Shanghai have a system where they make kids solve problems on the board when they finish their work so other kids can follow their example

I think that can work in the west I've tried it. He loved coming up to the front and I'm sure the other kids were watching curious about what he was up to

So I guess the question is, what do you think is more effective in the Chinese system? Their curriculum or their expectations?

There's no doubt both play a part

[long pelican]

I think… the curriculum is more effective

[real mauve]

it's a bit difficult to observe the effect of these things separately

the competitiveness in the chinese system cannot be circumvented

[pastel horizon]

I think it's more of a fifty fifty split.

No matter how much the west improves the curriculum we can't match their work ethic

[long pelican]

If the subgoal is for math to make sense, the gatekeeper for that right now is improving the low quality textbooks and teacher materials

[pastel horizon]

That's fair.

[long pelican]

Everyone I’ve met for whom math makes sense either learned a lot of math outside school (e.g competitions) or had an excellent teacher

[pastel horizon]

I just think that's half the battle

[long pelican]

Well, judging by the performance of the students who get it, the other half is self-sustaining

[pastel horizon]

I guess you can say curriculum is the battle and work ethic/expectations is the war

With a solid curriculum behaviour for learning naturally improves since kids start making sense of it

The opposite is also true

Even with all that we might never catch up to China

And by we I'm including the UK and US

[long pelican]

Hmm

[pastel horizon]

Not sure about Europe because I know some countries like Poland have a great work ethic

Our stats show something like EAL students are actually better performing than native UK students despite the clear language barrier

So they are students coming in from Poland with little knowledge of English and outperforming people born and raised in the UK

I imagine there's some similar with the US

Don't students immigrating from Mexico outclass US born students?

[long pelican]

Don’t think so

[quasi musk]

https://en.wikipedia.org/wiki/Educational_attainment_in_the_United_States

Not really

"A trend becomes visible when comparing the foreign-born to the native-born populace of some races. Foreign-born Asian, European, and African immigrants had a higher educational attainment in terms of having earned a four-year college degree than their native-born counterparts. According to the U.S census about 43.8 percent of African immigrants achieved the most college degrees, compared to 42.5 percent of Asian-Americans, 28.9 percent for immigrants from Europe, Russia and Canada and 23.1 percent of the U.S. population as a whole.[15][16][17]

The opposite is true on the high school level and among Hispanics, where the dramatically lower educational attainment of the foreign-born population decreased the educational attainment of the general Hispanophone populace, statistically.[3]" - from the wiki

In summary, there are large achievement gaps in education between hispanic/latino people, and non-hispanic/non-latino people

More so than other demographics within the US

[cosmic ibex]

If those numbers count people who immigrated as adults, a significant confounding factor would be that it's easier to get permission to immigrate the more highly educated you are.

[tepid smelt]

Yeah that is a real issue especially since parents in the west don't care if there child is disrespectful to teachers. I would say the majority of k-12 teachers complain most about student behavior.

I don't see a eqdy fix to this as culturally we have lost this battle.

[tepid smelt]

I agree but as a teacher we can make improvement by changing the curriculum in the short term even if it wont make as much difference as changing students behavior. Most teachers quit this job due to the daily disrespect you get from students with zero consequences.

[astral laurel]

I'm in Asia and like uhhh no....

[quasi musk]

Asian countries aren't somehow magical at teaching kids math

There's a higher cultural value, especially within the family, on educational attainment

[quasi musk]

I'm referring to this statement

Which is for the most part, not true in the US

[quasi musk]

The US has more lax laws on uneducated immigrants coming into the US than other developed nations. You've probably heard about chain migration, it's because once one person in a family attains citizenship, they can then sponsor their family members

(Mother, Father, Brother, Sister, etc.)

Which doesn't take into account educational achievement of anyone. Although for the initial person getting into the US it would help to be educated (as in an international student gets a job and sponsored by their company)

Canada and Australia, I believe, have a points based system to encourage highly educated and technically skilled works to go there

With regards to education, this tends to lead to children of immigrants in the US having parents that don't speak English that well, further hindering opportunities in school

I do support the move towards a point-based immigration system to entice educated & technically skilled workers to work within the US

but neither of the two main political parties will ever support it

[real mauve]

europe also has this same system though

where through one faimily member, the rest can follow

[quasi musk]

I can speak to immigration in the US, CA, and AUS

[quasi musk]

From what I've heard, that initial family member getting in can be very difficult

I've heard people trying to get into Germany, and it taking a decade or so

But that's not good evidence to go on

[real mauve]

i got into germany fairly easily, they're begging for "highly skilled workers" in a handful of areas

this includes all STEM and medicine, among others

all you need is a bsc and a job or education acceptance letter

with a foreign bsc it's fairly easy, easier even than getting accepted into undergrad

the declining european population has made the procedure fairly easy

[quasi musk]

I'm talking about low skilled workers immigrating

The US is better at incorporating lower skill workers

Most countries have incentives for high skill workers coming in

[real mauve]

only for the first person, though

[quasi musk]

The US doesn't really have that, even if you're high skill it can be difficult to get an h1b

[real mauve]

if you have a familiy member in the country, you already have a way in

and then there are free cultural integration programs

[long pelican]

Something to think about is that American students at selective colleges who get A's in their classes still have trouble making sense of math... comparing the excelling hard-working American students with the typical (hard-working) Chinese student there's still a difference. In fact the top scorers on my final exam seem to be Chinese international students

[real mauve]

they teach the language and skills for artisanal jobs and the like

[pastel horizon]

That's quite curious then. What's the reason why immigrants from Hispanic countries are worse when immigrants from Europe, Africa or Asia do better?

[real mauve]

my guess would be that many of them immigrate explicitly to do manual labor and send money back to their families, not necessarily to study

[quasi musk]

This is a big one

Another reason is that Northwest of Mexico is the sonoran desert, where people tend to live very spread out, without a lot of access to education

There's very few roads there in comparison to central or southern mexico

Lots of ranchers

They also tend to be pretty conservative in comparison to people living in central or southern mexico, e.g. more catholic

Going to public HS in CA, a lot of my classmates were DACA or kids of illegal immigrants. They said their parents didn't really care too much about their grades, they just wanted them to try their best during school days

Then on weekends go help the family business (like mowing lawns, landscaping, etc.)

At 15, they make good money getting like $200-300/weekend, so it feels more important than schooling

But then they turn 25 and they're maybe making $1,200/week, which isn't enough for a family of four

[long pelican]

Is it possible to help them without intruding on their home life?

[pastel horizon]

Jesus how expensive is the cost of living in the us haha

[quasi musk]

For my area the median home cost is 869k

[pastel horizon]

That's insane! In the UK that would be somewhere like inner city London

[quasi musk]

Well I'm in CA

CA is the most expensive state to live in so 🤷‍♂️

[pastel horizon]

1200 a week would be a good income here too

[quasi musk]

In my area, if you make 100k/year, you can qualify for low income housing

[real mauve]

that's insane lol

[quasi musk]

yUh, welcome to CA

[earnest trail]

yeah it's crazy

I lived there for a while

super expensive

now I live in florida

still expensive but nowhere near as much

[civic tree]

florida

[earnest trail]

bruh

[wispy slate]

WTF
That's actually a livable amount in NYC if you are in the outer boroughs

[earnest trail]

that's a livable amount anywhere in the world except for california

[wise onyx]

Thats fucking amazing XD

[wispy slate]

this channel is no longer math pedagogy, we the "dunk on California" gang now officially claim this land as our property

[long pelican]

California bad, Massachusetts good

[earnest trail]

Florida goodest

[abstract jungle]

what would be the simplest way to explain why the quintic couldn't be solved to laymen or non-math majors

let's say we want to make it interesting and digestible

[pastel sundial]

Wasn't there a youtube video about this?

https://youtu.be/BSHv9Elk1MU

Haven't watched it but I should

[earnest trail]

same

I like the title

"galois free guarantee"

or your money back

[abstract jungle]

interesting

though I guess it does take a long time

if you only have 5 minutes of a person's attention

which is quite a lot in a conversation

I wonder what's the best we could do

[cosmic ibex]

I think the most important part of such an endeavor would be to avoid implying that quintics don't have roots, or that the roots are mysterious things that are impossible to compute in practice. The impossibility result is only about choosing a particular set of tools to do it with, i.e. basic arithmetic and radicals exclusively.

[earnest trail]

so there are ways to compute quintics?

(not depressed ones or anything)

[cosmic ibex]

Sure, numeric techniques work well for them, just as for computing radicals.

There's nothing magical about x^5-a that makes it "more possible" to evaluate its roots (say, to decimal fraction approximations) than any other quintic.

[earnest trail]

that's true

but aren't depressed quintics easier to solve?

[analog token]

Well, this depends
You can't express the roots of the general equation, but for some equations you can indeed express the roots with radical and coefficients of the equation

[earnest trail]

yeah makes sense

[pastel sundial]

@long pelican Wait so who is Hung-Hsi Wu again?

I know he's a math education researcher but like what else

[long pelican]

He’s an algebraic geometer too

I’m not sure what you’re looking for

(Unrelated) Found another good article today though! Very good, too good not to share: https://graduatenyc.org/wp-content/uploads/2019/03/Parker-Math-as-Language-May28.pdf

[tawdry venture]

sometimes students are capable of solving problems that directly require them to evaluate a function at a point (such as "let f(x) = 3x^2 + 5, find the value of f(11)") but not problems which involve the same thing a little indirectly ("let f(x) = ax^2 + 3x. for what value of a does the graph of y=f(x) pass through (4, -5)?")

how come?

[real mauve]

my best guess would be that the introduction of an extra parameter makes the problem seem daunting, even though you give them specific values for f(x) and x. so maybe f(x) = ax^2 + 3x looks too impressive at first glance to focus on the extra info that simplifies the problem to your usual "find x" flavor. at its core, also, it's more like "find a value of x for which f(x) = something" rather than "find f(something)". for a beginner, these two things could make it challenging

the combo can be confounding because one wouldn't really care about x in the first place here, too. so maybe being too used to "x" being the quintessential variable is a problem.

[tawdry venture]

right

[long pelican]

Are you saying the goal that’s unmet is not being able to approach this problem when they’ve not solved a similar type of problem before, but they know how to plug in values?

[tawdry venture]

come again?

i can't parse that, sorry

[long pelican]

Hmm

Do you want them ideally to be able to successfully approach the problem as a new problem?

(As opposed to a “type” of problem they’ve seen before, which they are good at doing)

[tawdry venture]

hm.

okay, let me come clean: the reason i asked this question is bc of someone who came into the help channels with a problem like that

and i wondered what the root cause of someone having trouble w/ such a problem might be

cause i know i personally had almost no trouble w/ such questions. at least i don't remember them being a hurdle for me.

[long pelican]

In extremely general terms my take on the root cause is math illiteracy, combined with approaching problems as keywords for performing recipes

#math-pedagogy message this article says it in better words than I can

Personally I’m in the same boat as you, but taking the experience of me visiting math classrooms, interacting with help seekers here, and seeing what my own freshmen expected to get out of my lecture all points to this

[real mauve]

among the things i wrote above, i would say the main offender is related to what icy said rn. seeing only the first part of the problem, not recognizing it, and getting scared before reading the rest of the info

[tawdry venture]

ah.

okay, so intimidation by large (>1) variable count

[real mauve]

that's at least my impression. or maybe also too much visualization

depending strongly on needing to imagine a parabola, but not being able to do so without knowing what a is

(or not realizing that it was a line in a, alternatively)

[long pelican]

I think you’ll maybe find they can’t understand what the problem is asking or saying. In the article I linked this is mirrored by the author saying that a B student will likely not state a theorem correctly in their own words even having it in front of them

Instead of large variable count I’d just say they don’t know how to read quantified variables in general

[wise onyx]

Whats ur opinion on long take home final exam?

[long pelican]

Good in a class of people who care about the subject

[topaz scarab]

https://www.youtube.com/watch?v=IgF3OX8nT0w
I think I really like veritaserum late videos, and their approach to teach math / sciences
they give the actual historical motivation, and the original techniques
Like, I would love if there are few pages in my textbook dedicated to simply explaining about tides, and why predicting tides was a big deal, and that's one of the main motivation and application of doing fourier in the 1st place (on top of heat equations). AND THEN also a bunch of comtemporary applications.
Otherwise, math was too dry for my taste back then when I was younger...

[long pelican]

https://www.carnegiefoundation.org/wp-content/uploads/2013/05/stigler_dev-math.pdf

[cosmic ibex]

It is true that some students figure out on their own that mathematics makes sense
😭

[cosmic ibex]

The conclusion of the case study on pp. 25-30 is interesting:

[The interviewee] clearly has some understanding and the ability to reason, but one wonders if his shallow knowledge of procedures has been his downfall. What may at first appear to be gaps in understanding eventually reveal themselves to be gaps in procedural knowledge and notation, exacerbated by the disconnect between his often correct reasoning and his often incorrect procedures
The usual and expected conclusion in this genre would be something like "schools spend too much effort on teaching procedures and too little on understanding". However, the interpretation here appears to belie that -- it says the interviewee does have understanding but his procedures were too shaky. In and of itself, that sounds like a reason to double down on procedural learning, doesn't it? It's like the authors wanted to conclude that the student's procedural skills were weak because schools had tried to teach him procedures without connecting them to his existing understanding, but they can't quite find foolproof support for that thesis in their material, so they just leave that matter hanging ...

[long pelican]

Very interesting, don't you think! What do you make of it?

it's almost like American math education teaches people that there's 2 distinct kinds of math: procedural math and conceptual math, and they don't mesh. When a problem has math notation or asks you to solve something (like a word problem), you do the procedural math, and when the problem asks to explain something, you do the conceptual math

[cosmic ibex]

So the underlying problem is not that there's too much of one and too little of the other -- but the underlying assumption that there is a dichotomy between understanding and algorithms in the first place.
(I could add that this dichotomy tends to be the paradigm of most "math education in country X sucks" discourse, no matter which side of the supposed dichotomy the speaker positions himself on).

[long pelican]

Yeah, basically the whole global conversation right now is wrong... strong way of putting it

[cosmic ibex]

And, of course Goodhart's law, applied to the fact that procedural knowledge is a lot easier and cheaper to test (in a defensibly objective and "fair" manner) than understanding. Everybody knows that, but so much of the discourse seems to deal with it by hoping piously it will go away, one way or the other, if only everyone could have pure intentions.

[long pelican]

The fact that most students cannot tell when their procedurally generated answer is nonsense is one of the first things I noticed when grading exams

In fact they don't even detect that their answer is syntactically nonsense (unreadable)

especially when they're asked to write an integral or expression to represent something oh my goodness

[native hemlock]

i regularly get answers like 1 < x < -10 when i do math

[long pelican]

Oh yeah

[earnest trail]

lmfao

people just don't check their work

they trust their gut too much

and they don't care so they move on

[long pelican]

In this final exam, there was the following question:

Let $f(x)=\int_3^x \left(\frac 1{|t|}+e^{t^2}\right),dt$. For which $x$ is $f(x)$ defined?

A good percentage of people were able to say that $f(x)$ isn't defined when $x=0$ but zero people then said that this means that $f(x)$ is undefined when $x<0$ since in that case the integral is still improper, passing through 0

[burnt vessel]

Icy001

[cosmic ibex]

(And thanks to those absolute-value signs they cannot even hide behind a Cauchy principal value).

[long pelican]

Good point

[native hemlock]

i am also incapable of telling when my answer to a word problem is obviously wrong. like if im asked where someone ended up, given their velocity and starting position, i can conclude that they moved forwards even if they had negative velocity

and i wont be able to realize my mistake

[long pelican]

I suspect when people practice and get things wrong like this, all they learn from it is that they "did the procedure wrong" and need to practice the procedure more

[native hemlock]

what i learn is that word problems suck

[cosmic ibex]

Heh. I make software for a living -- perhaps when my product doesn't work, I ought to conclude I need to practice programming more, rather than writing some tests to run before releasing.

[long pelican]

I wouldn't be surprised if there was a study saying that programmers are 10x more likely to "learn math correctly"

[cosmic ibex]

There's a huge overlap in required attitude, if perhaps not in necessary factual knowledge.

[long pelican]

They would have to do the math learning after being a competent programmer, not before

[cosmic ibex]

In order to test your hypothesis, you mean?

[long pelican]

Mm-hmm

The hypothesis isn't about an innate ability common to programming and math

[cosmic ibex]

It'll be hard to find test subjects for it, though.

[long pelican]

hmm

[cosmic ibex]

I have a hypothesis that being a programmer contributed to me never managing to learn differential geometry properly. I simply couldn't get past how a whole bunch of notation was introduced without ever (as far as I remember) being explicit about the types of the things being defined.

[long pelican]

Hmm differential geometry, all the notation probably has to do with charts and stuff

I think of charts and the specific transition functions as an implementation detail

😏

[cosmic ibex]

I think I remember that charts seemed fairly natural to me. The rage came when I reached pages of definitions going something like "the Lie derivative is denoted by such-and-such symbol, and defined by this equation" -- without stating outright which kind of thing the Lie derivative has to be in order for the equation to make sense (and it was something like a mapping that takes functions from function to functions as input and produces a similar thing as output), and half of the time it was to be understood that everything is secretly (and invisibly) a function of a point on the manifold, or perhaps that everything is implicitly lifted into a "functions from points on the manifold" monad, except when (just as invisibly) it isn't.

[long pelican]

Hmmm...

[cosmic ibex]

For technical reasons, tangent vectors were defined as the algebraic duals of germs of scalar fields around each point -- which I'm cool with considering an implementation detail too, except every so often the text would suddently remember that implementation detail and apply a vector to a scalar field instead of writing an honest inner product.

[long pelican]

like: X(f)?

[cosmic ibex]

Yes.

[long pelican]

Hmm

What is a tangent vector axiomatically

[cosmic ibex]

The details decades back, so don't ask me ...

[long pelican]

I'm also asking myself that

[cosmic ibex]

Ah, wondering if you were trying to be Socratic in response to my ranting.

[long pelican]

Let's define the tangent bundle as the natural vector bundle where transition functions are derivatives of the original transition functions

And the cotangent bundle as the (fiberwise) dual of that

X(f) sounds like it would be <X, df> under that interpretation

[cosmic ibex]

Yes, that sounds right.

[long pelican]

So many implementations of the tangent bundle interface

[cosmic ibex]

That definition of tangent bundle sounds pretty physics-flavored: defining what a thing is by describing how it transforms.

[long pelican]

yep

[cosmic ibex]

There's a passage in Folland's QFT book about how mathematicians tend to want to see a concrete ZFC implementation of the things they're working with, whereas physicists are more comfortable with an axiomatic approach where they take on faith that there's something that behaves like the definitions say it will.

[long pelican]

I must be a mathematician that thinks like a physicist

[cosmic ibex]

It's an interesting observation, given how mathematics is supposed to be the heartland of abstract axiomatic approaches.

[earnest trail]

how much more technology do you guys think would be used in the classroom in 5 years? 10? 20?

obviously it's unpredictable but what are your thoughts

[tepid smelt]

I found the opposite true where someone who is competent in math picks up programming much quicker.

[tepid smelt]

I find they are often not equitable to students who are also learning English. I have begun to adapt icys approach of students being more comfortable with mathematical literacy. It sucks that so many modern curriculums have overly wordy "real life" examples that just complicate the math involved for many students.

[pastel horizon]

Ironically real life examples are rarely "real life"

Usually something that's far fetched or something that's been grossly oversimplified for it to make sense with their current knowledge

And the kids will see through those anyday

In fact one thing I think people are starting to realize is that language should not be a barrier to being able to assess someone's mathematical ability. You test their understanding of language on an English test not a maths test

[long pelican]

Indeed, a lot of genuine applications of the math they learn in school are actually to build more math, e.g. second order differential equations using quadratics. Instead of treating terminal “fake” real life scenarios as the true applications of this stuff, teachers should continuously emphasize how current math is built from from previous math using logic and reasoning

I also ran this semester on the hypothesis that reading comprehension of math notation (i.e. math literacy) is the missing bridge between procedural fluency and conceptual understanding. It didn’t start that way (I just thought I could do whatever and students would pick it up), but the mid-semester feedback clearly showed students were getting the wrong things from my lectures. So it’s still too early to tell how true that hypothesis is. What are your thoughts on what the missing bridge is? (See the article I linked for background)

[cosmic ibex]

Another thing that struck me as I read the article was the apparent similarities between bad math education and full-immersion language programs. You throw students into a world where they're only allowed to speak in formulas and symbols, and slipping back into their native language (i.e. intuitive number and magnitude understanding) is viewed as a failure to be avoided.
I'm not sure where that analogy leads. For natural languages, immersive approaches apparently work, or at least well enough that they're fairly popular. The situation with mathematics seems to be somewhat bleaker.

[long pelican]

I had a very different analogy to immersion language! I think current math instruction very closely mirrors current school foreign language instruction where they focus on the procedural aspects instead of using math (and language) communicate meaning. And immersion foreign language instruction would be basically college math where the professor's entire lecture communicates profound meaning (to those who can understand it)

[cosmic ibex]

Interesting. (All analogies are false, of course).

My school days are too far back for me to recall how foreign-language instruction actually went. I do have a vivid memory of my very first English lesson, in grade 5, where the teacher entered and started rattling at us in English and got each of us to respond in kind (something like "my name is X") -- except me who balked completely at the absence of an explicit explanation, and deliberately refused to infer anything from context. 🤣

[mint lark]

I often compare the way that honors math is done at michigan as immersion-based mathematics

[twin lichen]

i don't understand how we teach kids about diff geo without monads

[native hemlock]

so tru

[cosmic ibex]

(FWIW, I was probably wrong to suggest there was exactly a monad involved in my description yesterday.)

[mint lark]

lolwat?

[long pelican]

I also couldn’t find the connection between vector field notation and monads but there could be one

[twin lichen]

I was just messing around. Troposphere's point was that careful disambiguation in notation is helpful in diff geo and they mentioned that speaking formally we could model some of this stuff as lifting into a monad

i was sarcastically taking this to the extreme and suggesting we teach kids functional programming right out of the gate

[wispy slate]

Hey y'all just wanna say thank you for working your hardest to share the grandest, purest art of them all

Merry crimbas

[earnest trail]

Thank you, merry criss cross

[long pelican]

Merry 🎄 Christmas

[pastel horizon]

Merry Christmas all you educators

You sure earned it

[winged urchin]

Yes happy holidays everyone ^^

[boreal yacht]

Is there online resources/games y’all would recommend to help teach kids their addition/multiplication tables?

[pastel horizon]

Time table rockstars

[tepid smelt]

I like gimkit for boring drill problems. They have a fun open world game fishtopia where you answer questions to earn bait to fish. You can then sell the fish to but upgrades/travel to different world to find rare fish and the winner is the one who earns the most money. The kids really enjoy the game and are motivated to do drill problems this way.

Whats nice about the program is it has lots of game modes and you get a detailed breakdown of what each student got wrong.

If your doing it for your own kid I would just quiz my daughter as we were driving until she had it down then moved onto using distributive property and prime factorization to do larger multiplication in your head. Like 18×5 could be (20-2)*5 or (10+8)*5 or 9(2x5) etc key showing how you can do multiplication in lots of ways I then enjoyed doing arithmetic games like the 4 4s challenge. I think it helps to make it a social activity as its pretty boring.

[tepid smelt]

https://erich-friedman.github.io/puzzle/plustimes/ https://mathequalslove.net/make-30-puzzles/ erich friedman has a lot of fun arithmetic games those are two of my favorites I have done with my students

[boreal yacht]

thanks stephen, I'll look into these!

[topaz scarab]

https://www.explodingdots.org/
This is kinda a catch all

[long pelican]

http://academic.sun.ac.za/mathed/174/MollyNotation.pdf
Not sure if I shared this before, but the conclusions are pretty striking and consistent with my observations

[winged urchin]

Just skimmed the conclusion, sounds pretty neat I'll have to give it a more thorough look through

[topaz scarab]

I find table 1 and 2 to be very fascinating. I don't think I ever encounter any of these. I think all of my students understood the concept of variable and I was taking it for granted

[long pelican]

There was the following probability question on the final:

[Context: a probability density function p(x) for elevator waiting time] Let x >= 0. What's the probability that you wait x seconds or less for the elevator?
I thought this was a giveaway problem because questions like
Write an integral for the probability you wait [less than 10 seconds] / [between 10 and 15 seconds] / etc. for the elevator
are exactly like the very first ones they practiced

Turns out the top 2 most common answers were:

1. $\int_0^x p(x),dx$

2. 1, because $\int_0^{\infty}p(x),dx=1$

[burnt vessel]

Icy001

[long pelican]

Another common response was something like:

$1$, because $P(x\leq x)=1$ because $x$ is always less than or equal to $x$

[meager bronze]

what's the issue with answer 1 other than the variable issue

[burnt vessel]

Icy001

[long pelican]

1 is "mostly" correct but 2 points taken off for integral notation issue

[meager bronze]

oh, okay

i mean like, i agree it's nonsense as written but like

if students write that down i think they understand what the question is asking

[topaz scarab]

I can totally see me writing something like that when I'm in an exam panic mode lol

[meager bronze]

they just dont have a full grasp on what the notation itself means

yeah

[long pelican]

It's definitely a plausible explanation until you see how did they did on part b

part b was about a song cycle playing on repeat, the cycle being 25 seconds of song and 5 seconds of silence. What's the probability the elevator arrives during silence?

1/6 was an extremely common response

[meager bronze]

and you were looking for a sum of integrals instead?

[long pelican]

Ye

[meager bronze]

i dont think that mistake has much correlation with the notational issue that some students had in part a

[long pelican]

It's basically showing something lacking in understanding being a likely common reason for both

instead of: most people understand probability but just didn't realize their variable of integration is in the bounds, AND most people understand probability but don't realize p(x) isn't a uniform distribution

Like that's a pretty far-fetched explanation

Also, uniform distribution wasn't even discussed much as an example lol

Hmm one student wrote:

[\frac 5{30}=\frac 16]
[P(0\leq X\leq x)=\frac 16\int_0^x p(x),dx]
for part (b)

[burnt vessel]

Icy001

[long pelican]

Everything about the second line is super weird :/

I just think issues with the meaning of algebra are at the root of like 90% of all this

which is what the paper is saying too

@meager bronze you're an American grad student right? Is the college you teach at pretty good? Did you see a pattern in algebra meaning errors in exams or homework?

[meager bronze]

i finished my phd in 2020

i was a grad student at uchicago

where i mostly taught calculus courses

[long pelican]

Ahh uchicago is a pretty poggers school

[meager bronze]

i'm now a teaching professor at another institution though

yeah i mean i think lots of students at all levels just dont understand what basic algebra means

even like, what equality means

i do see lots of algebra mistakes

that I have to correct

[long pelican]

Have you taught a semester already?

[meager bronze]

yeah just finished my third semester at my new uni

but also i taught for 4 years as a standalone instructor at uchicago

[long pelican]

Yep, I visited uchicago in 2016 and I learned grad students teach full courses there as part of my visit

[meager bronze]

i think the biggest issue i see is like, students who don't understand that facts of algebra are universal, like

when we do basic integraiotn stuff, someone writes sqrt(a+b) = sqrt(a) + sqrt(b)

and i correct it

then 2 months later we're doing series and they make the same mistake and I'm like "didnt we go over this when we did integrals?"

and they say "oh yea i know that sqrt(a+b) \neq sqrt(a) + sqrt(b) for integrals but i didnt realize you also can't do that for series too"

[meager bronze]

that's right

[long pelican]

Surprisingly I only saw one serious case of sqrt(a+b)=sqrt(a)+sqrt(b) in this semester

Weakness isn't really algebraic facts, it's more like

hmm how do I say

Well before I say it, I can mention some basic algebra mistakes

Thinking that 1/|x| is defined for all x because the absolute value takes care of x=0 somehow

Thinking that ln |x| is defined for all x (same reason)

Those 2 probably show that they don't really think about defined-ness in terms of function evaluation

I'd say generally they think of math notation as a 1-way street rather than a 2-way street

They write things that a mathematician can't parse and don't think there's anything wrong

[meager bronze]

yeah

it makes me sad sometimes

[long pelican]

I kind of wonder how to handle it effectively in future semesters

This semester was kind of a wash because I only realized how deep the issues go mid-way through

Probably notation building, error finding, example reading exercises

[real mauve]

i also wonder, if you give a random person in your dept the same exam with the same time limit, how many would make the same mistakes due to simply being in a rush

the 1/6 part was really... weird, but the notation part seems like a very minor issue, and as buncho says, largely unrelated to the second point

some of your questions really strike me as "write T for false and F for true"

not that there's anything wrong with that, it's just that this seems much more a verbal reasoning kind of thing. if that's the goal of the course and that is evident from the course description when students are choosing what to enroll it, then fair game

[tawdry venture]

wym

[real mauve]

like they'll trip a lot of people up even when they understand the problem, only because of how it's presented

before even getting to what the problem is about

[cosmic ibex]

You noticed that tables 1 and 2 were answers from students who had not yet been taught anything about symbolic algebra, right?

[topaz scarab]

Yes I know. It is just really show how much I take things for granted.

[cosmic ibex]

Alright.

[cosmic ibex]

These students seem to have thought they'd identified a trick question -- they'll know by now that you tend to ask some of those. I suspect that describing the density function as "p(x)" led them to think you were declaring X to be the random variable that gives the actual waiting time. This requires them (a) not to care about the difference between upper and lower case letters; (b) not to have internalized the jargon enough to know that "Let x >= 0" implicitly creates the variable x, rather than stating an assumption about an already existing x.

I see lots of people writing things like

Let n be a natural number. (Bla bla bla). Does there exist an n such that (bla bla bla)?"
which seems to indicate they don't understand "Let n" to actually bind the variable, but merely to declare up front what it's going to range over once they do bind it.

This has an interesting correspondence to the situation in typed programming languages (C, Java, C# ...) where it is generally allowed to declare a variable with just a type and only later assign a value to it. In those languages it does makes sense to speak of "creating a variable" as a separate mental operation from specifying how the variable you declared gets a value. Which is not how the jargon or formalism of mathematics work.

On the other hand, that parallel may be spurious, since I don't think most people who make that error can program anyway.

[astral laurel]

I wonder if the scores could be improved if the question was

Let p(x) denote the probability density function of the waiting time. Let y >=0. Write the probability of waiting y seconds or less.

I feel like obvious context and being used to symbols can let us adapt well to write the answer, but devils advocate is that reusing symbol 'x' is notational abuse which invited the students' notational abuse

[astral laurel]

On this, I'd be more like thinking along sympy.Symbol stuff instead. I feel like variables, binding, etc. are very CS-ey and about memory management

[cosmic ibex]

I don't know sympy.Symbol. Can you share some of the insights you get from thinking about it here?

[astral laurel]

As in like, just defining it to use it. So you could write like

>>> import sympy
>>> x = sympy.Symbol("x")
>>> def p(inp_x):
...     return 2*inp_x + inp_x**2 + 3
...
>>> p(6)
51
>>> p(x)
x**2 + 2*x + 3
>>> type(p(x))
<class 'sympy.core.add.Add'>

So writing the first x = sympy.Symbol("x") defines a symbol, for any subsequent use of x while function parameters should be called something else, but if they use the same x then they should be defined in relation to that x. x can be used for substitution later on, it's essentially treated as a real variable

I think it's a bit confusing to write x in both as a symbol, just assuming it is real, and then to write just x in functions, despite this being obviously common and if you are careful you should know what is going on

In other words, humans have complex scoping, and it could be nice to just simplify that scoping

But of course, people are not computers, and it is reasonable to expect some obviousness in knowing what symbols mean in their contexts so....I can see it going both ways

[long pelican]

Interestingly, the people who did the P(x<=x) kind of thing strictly came from the other class, whose notes had a giant box which says “P(a<=X<=b) = \int_a^b f(x) dx”, without explaining what X is. I used Pr[…] notation instead and explained more about X, and the errors my students made were either saying 1/6 or only writing an integral for the first time interval (25 to 30 seconds)

[long pelican]

Good to note that if you declare an anonymous function in C#, the variables corresponding to its arguments are not accessible in the same scope as the declaration of the anonymous function, only inside the anonymous function.

This is in contrast with how sympy works but is a lot more in line with how arguments to functions are treated in math literature and even in the more abstract parts of physics literature (like tensors as multi linear maps)

[cosmic ibex]

I was more referering to how you can declare a local variable first and initialize it much later -- nothing about functions, even.

[long pelican]

Isn’t that in the context of when the problem said “let p(x) be the pdf”, which the students interpreted as making x visible in scope afterwards?

Ah ha I thought about it some more

I can now imagine the student thought “ok so we’re saying we ran the experiment once and found x is the waiting time. This question is now asking what the probability is the waiting time is less than or equal to the waiting time?”

[cosmic ibex]

My point was that in the actual question you and I know that "Let x >= 0" means that x is the name for a new thing with no necessary connection to anything that came before -- but the students may be missing that and simply understand it as putting forth a "type" restriction on the usage that x either already has or will eventually get.

[long pelican]

Right, except that properly interpreted, there was no x in scope at all before

[cosmic ibex]

Right, so both misunderstandings must have been present.

[long pelican]

Yeah

btw I can show you guys the final exam now

I know someone asked for it a couple weeks ago

[cosmic ibex]

I think the only relevant point I was trying to make with the programming-language analogy is that it's not conceptually obvious that telling what a variable ranges over also constitutes a binding of that variable.

[long pelican]

True, you kind of get that experience from reading similar stuff. Another thing to make sure to make explicit at some point!

Do you think any of this is ideally high school's responsibility?

I know realistically speaking there's like 1097123 issues to fix at the same time

[cosmic ibex]

That's a good question; I'm not sure whether or not it ideally should be -- but I believe high school textbooks and tests use similar wordings, and as long as they do that, they certainly also ought to make sure their students understand them ...

[long pelican]

I do try to avoid writing trick questions. In fact I don't think the reusing x aspect consciously occurred to me while writing and reviewing the exam.
But I also more importantly try to avoid writing questions that students can solve by pattern matching and regurgitating a procedure. Success on those types of problems tell me little to no information about what they understand

[cosmic ibex]

"Trick question" in the student's intuitive sense of "question I can answer without any calculation once I spot how simple it is".

[long pelican]

I'd define trick question as something a large amount of completely understanding students will get wrong because of falling for the trick

I think that falls in line with Edd's suggestion of showing the exam to some random people in the math department

[cosmic ibex]

Hmm, so replace "trick questions" with "one of those typical Icy001 questions where beginning to calculate is an error".

This is from the perspective of a student brought up to think success is about following the right procedure, of course.

[long pelican]

Mm-hmm

[cosmic ibex]

In which case my hypothesis would imply you've successfully disabused them of that notion, to the degree that they're looking for conceptual shortcuts even in questions when there aren't any to find. In a larger perspective, that is good!

[long pelican]

Interesting, so the perspective is analogous to overshooting a goal (and eventually correcting for the overshoot)

[pastel sundial]

Icy did you get course evals in yet?

[long pelican]

Yea

One section gave useless evaluations (5/14 response ratio), the other section was a lot more interesting (9/9 response ratio)

Heuristically I think my approach worked very well for about half of the first section and 8/9 of the second section

The other portion is probably people who had too weak fundamentals and didn't care enough

I really appreciated those detailed comments

from the second section

A very interesting aspect is that there were zero comments about exams (complaints or otherwise), which I completely didn't expect

So it seems that hard conceptual exams with a generous curve were a fine thing to do

The homework complaints are understandable; For the find the error exercises, I 100% intended that if they didn't get it they'd come to office hours to learn how to read such things

But they didn't :/

[earnest trail]

but one of them said that since they were weekly assignments, it was hard to get help

maybe make the homework due after 2 weeks but with more problems? this allows people to have more time to ask for help

and why do you think people didn't comment about exams?

[long pelican]

I just think they didn't comment about exams because they were okay with them

Is there something else you're thinking?

[earnest trail]

nah I was just curious

but yeah make the hw less challenging and more similar to the problems done in class, that seems to be the primary complaint

[earnest trail]

and this maybe as well?

or maybe with the difficulty level, make most of them similar to the ones in class but the last couple are more challenging

[long pelican]

With biweekly homework the people who start last minute are still going to start last minute

[earnest trail]

that's true

but then u got the people with busy schedules who struggle to find time to get extra help

[long pelican]

Also interesting thought: with easier homeworks, their exam performance will drop significantly, and they'll complain about exams instead

[earnest trail]

so that's why I suggested this

[long pelican]

Honestly.... that's how the homeworks were

[earnest trail]

oh

[long pelican]

Only 33% of the homework problems were new

67% of them were from old homework assignments

[earnest trail]

did people do poorly with the easier questions

[long pelican]

Not at all

[earnest trail]

so why are they complaining

[long pelican]

Low scores on their homework I think

[earnest trail]

dat shit make no sense

[long pelican]

Mainly the error finding portions

[earnest trail]

hmm

idk then

[long pelican]

Here's statistics for one of the homeworks

Q1 is error finding
Q2/Q3 are textbook

Q4 is a mini-involved problem with COVID-19 modeling

[earnest trail]

is this the % of people who got them right?

[long pelican]

yes

or more accurately, average score on each problem

[earnest trail]

oh okay

welp

maybe they need some more mathematical maturity

what class is this?

[pearl gate]

icy, are u a TA or something?

[long pelican]

Integrals, diffeq, multivariable calculus

I'm one of the 2 main instructors

[earnest trail]

ah interesting

[pearl gate]

all in one?

[long pelican]

yeah

[earnest trail]

is an instructor not the same as a professor?

[long pelican]

It's the same

[earnest trail]

damn seems like a lot

[pearl gate]

i mean diffeq. + multivariable calculus in one?

[earnest trail]

so there are 2 instructors for a class?

never heard of that before

[long pelican]

Yes

Just the very basics of each

[pearl gate]

what classes have the students already taken

[long pelican]

AP calculus mostly

[earnest trail]

oh

[pearl gate]

at least single variable calculus, through differential and integral right?

[long pelican]

Yep

[earnest trail]

I see the problem

[pearl gate]

oh yeah that sucks

yeah lmao

[earnest trail]

lol

[long pelican]

I take it AP calculus is a joke in 2021

[earnest trail]

not a joke

[long pelican]

When I took AP calculus myself in the 1800s, it was actually pretty good

[earnest trail]

but just not enough

[pearl gate]

1800s huh

lmao

[earnest trail]

1800s LOL

[long pelican]

I kid

[earnest trail]

duh

but yeah ap calc is good but just not enough

[pearl gate]

AP Calculus is just not a very mathematically rigorous class

[earnest trail]

are ur students then mostly freshies?

[pearl gate]

very computational

[long pelican]

Yep freshies

[earnest trail]

that's the issue

[pearl gate]

well is your class computational as well?

or are u covering proofs and going in depth on individual concepts?

[long pelican]

No proofs 😉

[earnest trail]

they go straight from the equivalent of calc 1 to calc 3

[long pelican]

It's a mix of computational and conceptual

[earnest trail]

im sure they're doing a lot better on computational questions

I would suggest reinforcing the conceptual ideas more and spending more time on those

but don't take too much time away from the computational stuff

[pearl gate]

computation is important, but not as much if its a class for maths students

in my opinion

[long pelican]

That's the in-class approach I used-- the challenging part is that they're poorly equipped to understand the concepts

And there was a lot of diagnostic work to do

[pearl gate]

review single variable for like 2 or 3 days?

is that possible

provide visuals and intuition

actually well this all sounds nice in writing but its probably much harder in practice

teaching math is quite difficult

[earnest trail]

yeah

how about this

delete the course

easy fix

[long pelican]

The structure of the course was out of my control and it was frankly unideal

[earnest trail]

but yeah reviewing single var might not be useful if what they're struggling with is the conceptual aspect of things

maybe they should take calc 2 as a prereq before this course

or smth else to prevent freshies from taking it

so that they can gain more mathematical maturity

[long pelican]

How much mathematical maturity do you expect from someone with an A in AP Calculus BC?

You've both taken AP calculus recently, right?

[earnest trail]

im currently taking ap calc ab

but yeah for students who took ap calc bc I imagine they should be a lot better off, especially if they did well

but idk high school teachers don't typically teach at the same level as professors do, at least from what ive seen

so someone who took calc 2 should probably be better off than someone who took ap calc bc

[long pelican]

HW corrections might be a good idea when people bomb homework problems

[earnest trail]

that's true

and encourage people to get together to work on the hw

[long pelican]

That's already encouraged

[cosmic ibex]

There's a fundamental ambiguity in what homework is for. a) Doing this on your own time (and, where applicable, considering the feedback you will get on your hand-ins) is one way to improve and refine your understanding on the subject. b) This is how we decide whether you're good enough or not. c) Seeing the results will enable the teacher to find out before it's too late which parts of their teaching are or are not working.
Student expectations will be all over the place between (a) and (b), but generally skewing strongly towards the latter if they are used to homework grades contributing to their final scores. A student getting more problems designed for (a) but thinking of them as purely (b) will naturally tend to feel unfairly treated.

[long pelican]

With regards to c), I got a lot of valuable results in the error-finding exercises but I only thought of them for the last 3 homeworks, so there wasn't anything I could do about them. Curse of first-timers...

It's a tough dilemma. Ungraded homework is the best but only if students are motivated to do it, which these students categorically are not. Homework graded on completion will just lead to low effort homework submissions

[earnest trail]

mm yeah they're not motivated

that's the issue

and they're not math majors or anything, right?

[long pelican]

They are not

[earnest trail]

so yeah there's honestly not too much you can do

[cosmic ibex]

The long "additional comment" seems to be someone who is instinctively thinking in terms of (b) -- but they know the ideal of (a), so they're tacking an "oh, it didn't work from that perspective either" on at the end.

[earnest trail]

like I said, maybe the class should have more prereqs

[long pelican]

That's something I'd support

Then again: the majority of incoming freshmen who enter take Calc I to satisfy their math requirement. They opt to take Calc II if they're up for the challenge

So they were already self-selecting for being good (in their view) at math

So it's kind of like, it's not practical to set a prerequisite if the set of people who meet it is the empty set

[earnest trail]

yeah but they need more mathematical maturity/experience

so really any other math class would probably also help them

[long pelican]

I'm thinking it's probably better to design the class to remove some of the material and replace it with mathematical maturity building instead

Without the requisite mathematical maturity, you'll just forget what you learned anyway

[cosmic ibex]

Yeah, you can't really teach mathematical maturity in a vacuum. Every subject would demand that they students they get have already passed M.M.

[earnest trail]

I would say that's a good idea, the question is, do you know how you would actually do that?

[earnest trail]

yeah that's true also, you cant really teach it, but im sure there are ways to build it

[long pelican]

The error-finding exercises (i.e. the homework problems which students found extremely hard for some reason), as well as the simple questions on exams that only test ability to read notation, both have something to do with understanding notation as containing meaning, which could possibly be the most important ingredient to begin learning mathematical maturity

[cosmic ibex]

Right.

[earnest trail]

yeah I agree

so work on that and your students should hopefully improve

[long pelican]

Doing proof-based linear algebra this coming semester 🙃

I was warned that the students although being top math freshmen in this school still lack mathematical maturity

A verbatim quote from the professor who last taught it: "The hardest part of the course is to get the students to write in complete f______ sentences"

[earnest trail]

LOL

I hate writing complete sentences in math

but yeah do the students take computational lin alg before?

[long pelican]

I think it's just the conceptual leap from writing math as scratch work to writing math to communicate a sequence of logical ideas

No they don't

[earnest trail]

damnnnnn

it's gonna be a rough ride lol

good luck

[long pelican]

🍀

[earnest trail]

yeah that's true, people perceive math as this abstract thing and when ur asked to explain it, students just decompose on the spot

[long pelican]

Decompose on the spot damn

[earnest trail]

LOL

[long pelican]

There's a couple of students (racist comment redacted ) who actually understand how to write math as communication and consistently do well on exams. A thing both of them have in common is that they use $\therefore$ in their work

[burnt vessel]

Icy001

[long pelican]

Ok I can be un-racist and just say they're international students

[cosmic ibex]

Hmm, interesting. I've always understood \therefore and its ilk as the result of a misguided ambition to avoid tainting your mathematics with natural language and complete sentences.

[long pelican]

One thing I see American students completely unable to do is to write an explanation with a quantified variable in a proper manner

This is a typical style of explanation I see a lot: explaining something in terms of "a function cannot be zero"

[cosmic ibex]

Perhaps they don't know the word "argument"?

[long pelican]

I don't think that's necessary

A grammatically correct version of the explanation above would be:

$f(x)$ is defined for $x>3$ because $\ln x$ is defined for $x>0$, [...?? I have no idea how to reach the conclusion now]

[burnt vessel]

Icy001

[cosmic ibex]

It looks to me they are attempting to say "the argument of ln cannot be 0 or negative", but lacking the terminology to do that.

[long pelican]

My version uses a quantified variable x instead of the word argument

[cosmic ibex]

I could see wanting more reasoning after the observation that the argument to ln must be positive, and how to tie that fact together with that argument being (I assume) x-3 in this case. But that defect seems to be orthogonal to the language problem in apparently claiming that ln cannot itself be 0 or negative.

[long pelican]

Well, in the actual problem, this was f(x). Turns out the student is completely wrong

But I wasn't pointing to that, I was just pointing to the style in one example

[cosmic ibex]

Okay, that kills my explanation.

Or does it? I think I stand by the student probably meaning "the argument of ln cannot be 0 or negative", even though they somehow reach an incorrect conclusion from that correct fact.

[long pelican]

ye, I understood what the student meant

No points were taken off for style

It's just an observation in passing

My conjecture is that seeing math with symbols and variables to communicate and explain something instead of solely in scratch work for worked examples or procedures to follow is something missing in American math education

[pastel horizon]

Pedantically the integral of a reciprocal is ln |x| isn't it? So it's just the ln0 case that's undefined

They've missed the point of defining the function this way anyway

[long pelican]

Actually the integral of 1/|x| is pretty weird. But you don’t need to calculate it. The integral in the problem is improper and divergent when x is 0 or negative

[earnest trail]

it's simple, just use ftc

[long pelican]

Share solution and I’ll grade it :D?

[earnest trail]

im currently not in the best position for that

laying in bed with chest pain lol

[long pelican]

That sounds serious

[earnest trail]

minor pneumothorax, if it gets worse im going to the hospital

I've had a pneumothorax before

doctor says if it's minor just be careful but if it gets bad go to the ER

[long pelican]

I’ve had crippling stomach pain 2 months every year since 10 years old and it took until 4th year of grad school for a doctor to suggest it’s gastritis and prescribe a simple PPI which worked like magic

[earnest trail]

damn that's uhhhh

interesting

lol

[long pelican]

Yeah

[astral laurel]

This sentence is #lifeofateacher right there

[quasi musk]

I can confirm this is a big issue with linear algebra or first proof class

What I always do is grade leniently but give lots of feedback for the first two or so weeks

And expect them to improve on that feedback, and then I start taking off points for the things I would

For instance, I'd be super nitpicky about every vector having a vector sign

This is to train students to know what is and isn't a vector, so when the vector sign is no longer written

They aren't as lost

[grim spindle]

What if a student consistently doesn't write vector signs, but it's clear they understand when something is a vector

Would u take off points for that

[quasi musk]

Yes

If I clearly communicate that this is something that you should do in terms of a rubric

And you knowingly do not follow that rubric, then it's not fair to the other students that they get away with it

You have to be uniformly fair

[pastel sundial]

Icy I wonder to what extent a lot of the complaints your students had about the homework could be addressed in the future by just clearly and directly explaining your thought process.

[pastel sundial]

Like

"one of the most common and critical to address issues at this level of math is a tendency to only be able to approach math as rote computation. As a result, I've designed many of my homework problems to require more than just a straightforward application of the things I go over in class.
Unfortunately, problems like this are by their nature going to be more difficult than what you're used to and so I plan to be very generous grading them. I considered grading on completion but this tends to lead to a lower quality of work which would ultimately hurt you since these problems are also designed to help prepare you for exams. So instead I will be grading your homeworks on a rather heavy curve and also giving you the opportunity to do homework corrections. In addition I am always available in my office hours and am happy to walk you through any problem you're struggling with"

Not sure how applicable this would be to proof based linear though

[quasi musk]

Not a huge fan of corrections, but maybe that's just me

Some of my profs would grade 5 pts completion, pick 3 problems and grade those out of 5

Then your score is out of 20

[cosmic ibex]

I think Icy's homework complaints of that sort might be from the tail end of students who didn't pick up when he did in fact give an explanation of that sort.

[pastel sundial]

Well didn't Icy say that he didn't really figure out his homework strategy until like halfway through the semester?

[long pelican]

Ya

Also interesting to note: the section that complained about homework also ended up doing the best on average on the later exams by far

[pastel sundial]

So he was never able to give this sort of an explanation up front

[long pelican]

They were also easily the people who followed the lectures the best towards the end

[cosmic ibex]

Interesting. Don't mind me, then. :-D

[pastel sundial]

And consistently getting 70% on homework is arguably more frustrating for a good student

[long pelican]

Yeah it might just be the grades should be adjusted somehow

[pastel sundial]

Especially when the professor is realizing at the same time as the student that the homeworks are on the hard side

Like as someone who cares about their grade, that would be mega tilting to me

[earnest trail]

same lol

[pastel sundial]

I'm imagining having a similar experience with an intro class in another department.

[earnest trail]

yeah would be shitty

[pastel sundial]

And it would be nice at the end when I get my grade back and find out it was curved and I got an A, but it would be tilting as hell in the meantime

And I imagine students submit course evals before they get their final grades back

Honestly one thing I'm learning in college is just how important having clear grading criteria is for a student's (my) peace of mind.

[earnest trail]

^^^

bc it's like wtf am I supposed to do

[pastel sundial]

Which kinda surprised me since my highschool didn't assign grades, so I went into college with the mindset of "focus on learning the material and the grade will follow." But confronted with the inescapable reality that grades do exist and do matter, I find it really important to know what the professor expects for an A and how my work measures up.

[long pelican]

"focus on learning the material and the grade will follow" worked for me in college

[pastel sundial]

Well that's still my primary approach.

I just like to also have clear feedback on how I'm doing in terms of grade so I can correct in the off chance that I miscalculate what leanring the material means or the professor has some weird grading criteria.

Especially as I've somehow managed to stumble into a straight A average that I'm now trying to maintain.

[long pelican]

Do you think it's weird this section complained about homeworks but said "homeworks were much harder than exams", and didn't complain about exams, while the other professor told me their sections complained about homeworks and exams equally?

My section also spent on average 8 hours a week outside of class while the other professor's sections spent on average like 6

We use the same homeworks, labs, and exams

[quasi musk]

I don't think it's weird in a one term thing

but if it consistently happens, then it's weird

I think it's normal for students to just complain about everything

[long pelican]

Maybe

The other professor also mentioned this batch of students overall might be weaker than usual due to learning precalculus/calculus during COVID-19

That's likely, right?

[meager bronze]

100%

i noticed that too

in my courses last semester

[long pelican]

That's good to hear

[earnest trail]

yeah I was talking to my former chem teacher a few months ago (who teaches mostly high school sophomores) and she said her students are struggling because they've never had high school in-person

so im sure it's the same for college

since im sure the jump from middle school to high school is much less severe than the jump from high school to college

correct me if im wrong

[long pelican]

My college experience was actually such that I found it easier to get an A in college than in high school, because the professors taught the real thing instead of dumbed down things to memorize :~D

AP biology vs. biology with Eric Lander was night and day

[earnest trail]

damn that's reassuring

[shut saddle]

is it bad to ask someone to sum from 1 to 1000

[long pelican]

More context?

[cosmic ibex]

To see if there's a Gauss in the class?

[shut saddle]

was just hoping they would get tired and try to find a pattern

they did 1 to 100 by hand/calculator

[cosmic ibex]

Brian Hayes in https://www.americanscientist.org/article/gausss-day-of-reckoning argues that it would be hard not to notice at least some useful patterns if you actually set out to do the sum by pencil and paper.

[earnest trail]

seems more like a research experiment than a fun activity

[elfin valve]

https://tenor.com/view/the-office-pam-beesly-theyre-the-same-picture-the-same-picture-they-are-the-same-picture-gif-20830121

[earnest trail]

LOL

[elfin valve]

XD

[tepid smelt]

This is so true especially with math. Secondary level in particular its really bad since us math teachers seem to be the only ones who actually test students or give hw. We get complaints from students/parents/admin constantly.

[pastel horizon]

Homework I'm kind of skeptical about. But tests are just part of maths, I don't see another way of reliably assessing whether a student truly understands something conceptually

[pastel horizon]

Maybe I should explain my view on homework more. What does it actually achieve?

If it's used as assessment, you won't get a reliable picture compared to what you would achieve in class.

If used to reinforce learning, only a small percentage of students would engage in the intended way. Most would just rush without putting much thought into it. This ties in with my first argument.

Is there even proof that homework in maths raises achievement compared to not giving homework? I'd love to see if anyone has a study.

[pastel sundial]

I feel like this is a terrible take

Math is not a spectator sport. If you never do practice problems you're never going to properly learn math. And homework forces students to do practice problems.

[grand laurel]

not a study and not technically math but i used to TA algorithms and datastructures which features voluntary homework that is nevertheless corrected and commented
we used to tell students that doing homework is important for exam success etc but didnt actually have any data to support it, so one semester we just wrote a script that checked every students homework success against exam success and there was pretty big correlation

[pastel sundial]

To be fair that could also be explained by "better students did better on both homeworks and exams" rather than "doing good on homeworks taught students to do good on exams" but still a good data point.

[grand laurel]

yes, but my experience is that the better students don't do (every) homework since there is no reason to

getting back homework with 100% correct and 0 comments might be a nice ego boost but if you get nothing else for it and its not compulsory, you might spend your time elsewhere

(but this is just personal experience/bias)

[long pelican]

This semester I did something similar to Lochverstärker and did an item analysis comparing individual homework problems to combined exam performance. (I might have alluded to this before)
A clear pattern emerged and I saw 3 types of homework problems

1. Problems that basically everyone got right. Low correlation by fiat, and I can’t say much about how helpful they were from this analysis

2. “Hard” but unhelpful problems. Less than 90% average, high standard deviation, but low correlation. These included wordy problems, confusing problems, etc.

3. “Hard” and helpful problems. Less than 90% average, high standard deviation, high correlation. By far the highest correlations were attained by proof-based problems or similar problems (also includes the error checking exercises) despite this class and exams not being proof based in the slightest

The existence of type 2 should rule out the null hypothesis “better students do better on both homeworks and exams”

At least, "better" in the sense of hard-working or studious

At a minimum we need to invoke mathematical literacy to explain it -- "Students higher in mathematical literacy did better in homework, and also did better in exams (even though the things they did better at on the homework didn't directly relate to the exam questions)"

[pastel horizon]

Should they not practice problems in class instead?

I'm not disputing that practising problems is important for learning of course, just the fact that extra work is given outside class.

[long pelican]

In university, 4 credit classes are allotted as: 6-8 hours a week outside of class and 3 hours a week in class

[pastel sundial]

Yeah the issue with practicing problems in class is that there's only so much class time

[pastel horizon]

Maybe I should also mention I'm on about secondary school. University is a lot different, students who attend university should, generally, be expected to know how to take full advantage of homework

An average kid in secondary might not fully appreciate this and just rush it for the sake of completing it

[pastel sundial]

Secondary school is like highscool right?

[pastel horizon]

Kind of

11-16

In year 11, the final year homework pretty much just becomes "do some revision for exams"

[pastel sundial]

Because even then there's no way a student is going to learn AP calc without doing problems outside of class.

Or even like algebra 2, probably

[pastel horizon]

Yeah, like I said with that higher class of students there is merit in setting them some practice. Only because they understand how to gain the most information from practice once they've reached that level of maturity

[long pelican]

Something just occurred to me: In high school, homework is typically practicing routine skills, yet the majority of students forget them in the following year or when they enter university. But with proof-based math classes, students almost never forget how to write a proof. Shouldn't we be paying attention to the fact that the knowledge we supposedly teach students in algebra 2 is very fragile, and realize that teaching math as practicing routine skills is a horrible way to teach it?

Like, math knowledge is not supposed to be fragile, given that math is an interconnected web of logic and intution

[pastel horizon]

Just think as well of all the middle aged people that would see something simple like adding fractions and comment "I forgot how to do those it's been a while"

Would you forget something like how to drive after twenty years off the road?

[grand laurel]

my comment was also about university, i am more skeptical about homework in highschool (especially math homework)

i think the goal in highschool is to prepare for exams

[pastel horizon]

For sure

[grand laurel]

by practicing routine skills

which isnt great but a necessary evil (?)

[long pelican]

Not a necessary evil; exams can be made better and un-teach-to-the-test-able

[pastel horizon]

Exactly

That's what some exam boards in the UK are now doing

Which in turn means teachers can no longer rely on teaching methods, they need to teach concepts

[grand laurel]

i will believe it when i see it

[stark pine]

how do you feel about larger projects? Like something where the students get to do exploration or discovery and it's done over the course of a couple weeks not a couple days

[grand laurel]

any education change that requires vastly better teachers i am skeptical about

[pastel horizon]

They are forced to. If they don't then students fail exams simple as that

[long pelican]

Does anyone else agree that we can vastly speed up the rate of improvement of teachers by publishing excellent textbooks? (Kind of like how Singapore does it; their textbooks even teach teachers on the job about the intuition and logic behind the math)

More specifically textbooks that present math logically and don't routinely ask students to assume or infer unnecessary things

[pastel horizon]

If teachers are relying on textbooks then yes that should be a given

[grand laurel]

better and continuous education of teachers is always good

[long pelican]

Hmm another thought occurred to me: for most jobs, you can become a master by doing the job a lot. Master craftsman, master doctor, etc. Why is it that being a master secondary math teacher requires outside assistance?

Maybe it's because the job itself does not practice the skill of delivering/teaching real math

[grand laurel]

kind of unrelated but anecdote: there is a german guy who used to be a professional mathematician and professor and then decided for some reason to switch to teaching highschool
i think he is a very good teacher and he sometimes writes about his struggles with the education system but also other teachers and damn some teachers in germany are really, really bad (they mark stuff students do as wrong because of learned dogma that is often also objectively wrong from a mathematical pov)

[long pelican]

Marking insightful observations from students as wrong must be one of the most harmful things you can do to a student's math education

[earnest trail]

damn I never thought of this, I agree. how do you think we should teach algebra to high school students then?

[earnest trail]

also yeah the textbooks can help teachers understand, bc ik a lot of teachers know what to teach but not the why behind everything

[mint lark]

“Students almost never forget how to write a proof” is just bullshit

Having taught many students proof based math. Many of them forget

[long pelican]

How do you know they forget? I can't think of examples like that in my experience

[mint lark]

I have watched students solve such a problem, checked their proof was correct, then given them the same problem on a quiz the same day

And they cannot replicate their work

Have done it with time delay too

You overestimate proof based techniques bc those who get to proof based math generally already have math ability

[long pelican]

is your claim that students forget specific proofs?

[mint lark]

This is the majority of students in the intro to proofs class

[long pelican]

Mine is more that students generally remember what a proof means

[mint lark]

Ehh

[long pelican]

and what makes one valid or invalid

[mint lark]

They forget that too

Like they can’t read them

Or will replicate errors and need them reexplained

Again I think you are overestimating

[long pelican]

Maybe it's difference in levels of students we're experienced with

There's definitely people at both ends of the spectrum no matter what trait we choose

[quasi musk]

I think the issue with this is that writing proofs is a much more complicated skill than just skill & drill problem sets

Like even math majors at university have trouble with learning how to write proofs

It was so difficult that in the "Geometry" class taught in HS, they had to simply proofs to two column lines of reasoning

Which isn't how anyone proves anything

[long pelican]

Yeah proof for sure is a more integrated skill and that's maybe why it's less forgettable

[quasi musk]

Even getting students to write down algebraic steps without skipping things in their head

Can be difficult

Let alone requiring complete sentences

I guess a lot of my work has been in remediation, and those types of "fun" things for advanced students don't do anything to help remedial students

[long pelican]

Generally would you agree the more interconnected their knowledge is, the easier it is for them to not forget it?

[quasi musk]

It's tricky: the more interconnected the knowledge is, the harder it is to set up

But if you put the puzzle piece together correctly, you end up with something that is hard to forget

So yes, in general I'd agree, but that doesn't mean that has to be writing proofs

[long pelican]

Mm, I might have written something misleading, making people think I think writing proofs is the only way to have interconnected knowledge

I was just more contrasting two examples

Puzzle pieces is a good analogy

Struggling students are missing some or many puzzle pieces

[quasi musk]

And it's not necessarily easy to identify what is missing or where they go

[pastel horizon]

Arguably the whole of maths can be thought of as puzzle pieces

[tepid smelt]

I think hurting a students grade for not doing hw is wrong at the secondary level because its not equitable in that some students simply can't do work at home. That said many students want hw to practice more at home to solidify understanding. I have tried icys approach of giving more open middle problems as hw and only a few drill problems. I just don't really count them against a student.

I think students who struggle really need focused support. There should be a support math class for low students to be able to catch up on skills they lost as they were simply passed along prior to hs. My toughest students are those at a elementary level in HS because they were never required to learn anything.

[pastel horizon]

If they want to do it, that's no problem. Question is should you force people who don't want to do it at secondary? My answer is no.

It definitely shouldn't count against them. Sometimes you get lazy but talented students who can (rightfully or wrongfully) get an A on the exam with minimal effort. Others need to put in the extra effort for the same outcome

[earnest trail]

I think homework should only be mandatory if the student is struggling, i.e. if their grade is below 70 or smth, depending on what the teacher sets it at

if they have a satisfactory grade, the homework should be optional

[patent beacon]

I'm starting to give homework in my classes. It doesn't go towards grades, but I still expect them to do it. I think it's important for my students to spend time outside the classroom thinking about the topic. It's not a pile of homework, always less then 10 questions that cover content from the previous week mostly, so that they're seeing that content again.

As a teacher, it's evidence that they're spending that time studying, and when I call home because of a student's behaviour or performance in class, I have something to point to. I'll also just get in touch throughout the term if i student isn't handing in homework, and also on the other end, if a student is consistently handing in good work and performing well.

For context, I'm teaching 13-18 y/o's

[wispy slate]

https://mathweb.ucsd.edu/~fan/teach/add.htm

[quasi musk]

Gonna be teaching again at Russian School of Math

[tidal mica]

but then how do you measure their grade without a consistent metric like homework

also is this an appropriate place to ask about self-study tactics (personal pedagogy if you will)

[real mauve]

i think math discussion and advanced lounge are better suited for that

[pastel horizon]

I disagree with using hw as assessment

Especially if you're using it to track grades. Maybe they are strong in one topic but weak in another. So they might get an A for one HW and a D in the other. Does that mean they are a weak or strong student overall? Not necessarily

[long pelican]

I think of homework as the time where you (the student) really takes the time reading the text line by line and learning the math deeply, in order to solve the problem(s). So you're supposed to spend as much time as necessary to get everything right, asking for help as needed. If you get a D on it, then you just didn't do what was asked of you

Of course, that's the college viewpoint. For high school, if you want students to learn real math, wouldn't this also be the necessary approach?

[pastel horizon]

They are talking about using it to get a grade

I think it's fair to say practicing problems is the way to learn but attaching a grade to the problems isn't the right way of assessing them

[long pelican]

I saw one aspect of your response that stood out though: Maybe they are strong in one topic but weak in another

If they did homework properly they shouldn't be weak in it

(assuming sufficiently difficult homework)

Also, homework is not meant to be merely practicing problems, but there's a significant amount of learning done by doing homework

Would you offer re-writes of homework?

Until they manage to solve every problem

[pastel horizon]

I think if it was optional and non graded they would engage with it more in the intended way

[long pelican]

They just won't do it

[pastel horizon]

Feedback would need to be meaningful as well of course not just "14/20"

[long pelican]

What do you think of the model used in graduate classes where there are no exams and your grade is based completely on problem sets (and maybe attendance?)?

I guess first instinct is that people will cheat?

[pastel horizon]

At my uni they based exams off the tutorial problems

So ok, attending tutorials is optional but if you don't then you're less likely to be able to do the same problems that are on the exam

That would basically be a series of formative assessments then with that model?

[long pelican]

I kind of dislike when exams are based off homework or tutorial problems. It kind of gives students the false impression that they understand more than they do

also emphasizes a problem-memorization mode of studying

[long pelican]

Yeah I'd say that's accurate

[pastel horizon]

It's interesting. Definitely has its flaws even if you assume students are honest

They might be stronger with one problem set later in the year for example, are you capturing that?

[long pelican]

I should mention the problem sets are extremely challenging by the way. Proofs you've never seen before are typical

You're saying like a take-home exam

[pastel horizon]

The traditional end of year test has flaws too though as we all know

[long pelican]

These problem sets could possibly be seen as take-home exams which measure how good you are at learning the material on your own

[pastel horizon]

I'd probably say the one thing that kinda bugged me with engineering is all the problems are ideal circumstances you can solve on paper

They should've done a final exam on computer with real problems

[astral laurel]

Is this a moral or policy stance? Using homework as assessment is sadly one of the more effective ways of encouraging people to do homework in grade-focussed climates

[cosmic ibex]

The most problematic unintended consequence of using homework for formalized assessment is probably that it lets some students remain stuck with a mindset where learning is supposed to take place while the teacher is talking, and the homework is just a take-home test after the fact to verify that they've been attentive in class. Then whenever they don't already know how the solution to a homework problem goes, they'll instinctively conclude that their learning has already failed. Actually figuring something out for yourself feels like a mild form of cheating, a desperate last-ditch remedy for either their own or the teacher's inadequacy, with the goal of being able to pretend they learned the problem in in class. And thus we get them showing up in places like this server with "I have no idea where to start".

[astral laurel]

I think this is a comment on a wider topic on how much lectures should cover and how relevant lectures should be compared to homework.

In many cases I would expect at least that students who listen in class can begin on homework. Other than that in a certain sense homework difficulty is up to academic freedom.

In terms of learning, just about no subject has learning as a spectator sport and this is not restricted to mathematics despite what some comments on mathematics would suggest. In no subject can one just go to a lecture and not do anything (i.e. not think about what they are hearing at all) and yet be considered to learn at all. For these non-learners I think not knowing where to start is expected and appropriate. These people basically put in effort levels below expectations.

[pastel horizon]

I genuinely think it's not a good way to assess pupils

You're putting the focus on what students are getting correct. I think the opposite should take place, teachers diagnosing and addressing misconceptions

[cosmic ibex]

Yeah, I think my point is that the idea of homework-as-assessment is what enables those people to believe they are doing it right by merely showing up to lectures (or classes). They are the people who look at a homework problem for three seconds, don't immediately see a solution, and then conclude "I was present and awake in class, yet I still don't know how to pass this test; I must be inherently bad at math", rather than "okay, I'll need to think about this".

[earnest trail]

through exams

and in-class assignments/projects

[tepid smelt]

At the high school level I have moved more towards standards based grading. So students are evaluated primarily on traditional exams and alternative assessments like projects to demonstrate they have learned a particular standard.

Students who engage in a meaningful way with homework assigned do better on exams. Yet I do have a few students who do no homework but listen well in class and ask good questions and still crush the exams.

I do think the high school vs college approach is quite different because I am still dealing with children who often don't have home lives where learning can take place. Not to mention in high school I have way more time with the students in class compared to a college student. So the value of hw is generally greater for a college student

[tidal mica]

i remember hearing once "you dont have to do homework, it's just an excuse to interact with the material" so that's pretty much my pov on it

so as long as theyre getting excuses to interact w material i think it can be substituted yeah

[earnest trail]

I think the few students you mentioned in your second paragraph are fine. those types of students should be allowed to not do the work if they don't need to

[tidal mica]

but then allowing some people to not do homework and making others do it will lead to division in the class and probably resentment

[earnest trail]

that's true

but like I was suggesting, the teacher could set a benchmark

if your grade for that class is, say 80% or higher, then you don't need to do the hw

but if it's under that cutoff then you need to do it

[astral laurel]

This is a good thing to strive for, but it is hard. I'll also add that it is easy for administrators to set KPIs directly in conflict with this.

[pastel horizon]

That's right, I do forget about admin in US. Luckily in the UK the main KPI is GCSE results which only Y11s sit

Don't get me wrong, OFSTED still expect to see assessment and "progress" being made but at least this doesn't have to just be a meaningless grade

Progress is in quotes because it's a big buzzword that's easy to throw around but not actually trivial to define

Just an unfortunate case where the people who hold educators in account actually interfere with what I personally think is best for kids

Already a long post but one final point. One advantage of only using Y11 exams for grading means that education becomes a marathon not a sprint. Say if a Y9 doesn't understand a topic fully. That's perfectly fine - there's still another two years to fill in that gap. Last thing you want to do is drag their confidence down for not knowing something at that stage of their learning when it doesn't really matter that much.

[long pelican]

No class grades then? Just GCSE/A levels?

[pastel horizon]

Pretty much

We use pathways instead to differentiate ability

That's exactly right, feedback is crucial though

[cosmic ibex]

It makes administrators jittery if they don't have measurements along the way -- something could have gone wrong for the whole marathon and they'll only learn about it at the finish line, a disaster!

[pastel horizon]

There are ways to measure progress without giving a grade

[cosmic ibex]

But it's the easiest thing to imagine.

[pastel horizon]

Problem is then they'd have to trust the teacher judgment

[cosmic ibex]

And that's not good if their worry is that the teacher might not be competent.

[pastel horizon]

An educator can clearly see a student's progress in mathematical understanding over a year for example

What freaks them out is having to rely on that abstract understanding rather than data

[long pelican]

Any ways to data-ize understanding?

[cosmic ibex]

I'm not disagreeing with you, just pointing out there are social-dynamics reasons when that ideal doesn't become reality, and it doesn't all boil down to evil men secretly wanting education to be bad.

[pastel horizon]

Oh yeah I'm aware

[long pelican]

If I think of humanities classes where writing is involved, there's none of this data obsession

[pastel horizon]

Not every teacher has good assessment skills either

[long pelican]

And math should be approached like a humanities class

[pastel horizon]

I think there's a misconception that getting 100% on a test means someone is perfect at maths

Not necessarily

Because they might've shown some weaknesses in understanding in some correct answers, but the mark scheme allows them to get away with it. Or of course the question is poorly designed

[cosmic ibex]

Nobody are much worried that kids won't become masters of English compositions. But it worries them a lot that kids end up bad at math, and they're scared shitless that it will become even worse on their watch, and their defensive reactions actually prevents the education from becoming better.

[pastel horizon]

Actually, I think the opposite is true

Have you noticed most people are willing to accept "oh I was never good at maths"? But hardly anyone says "Yeah I couldn't read when I was at school"

[cosmic ibex]

That's on a personal level -- in public administration where you view things in aggregate, the prevailing view is that a society needs a lot of math-strong citizens, but who cares how many or how good poets we have?

[pastel horizon]

Another thing to add, what does making progress in a subject like English look like? It's totally different to maths

They judge progress based on the quality of their work

[long pelican]

To be more like humanities, we probably need to revamp what we think "basics" are. Right now it's a whole list of things like "find vertex of a parabola in vertex-slope-intercept-blah-blah form"
Should be primarily "how to read mathematical statements and arguments" and "how to write a coherent mathematical argument" instead

[pastel horizon]

There should be more emphasis on fluency too with that

As a really basic example, take something like this:

123 + 99

Say if a fictional person just used the column method to solve this. You could ask your student to suggest a more efficient strategy

[pastel horizon]

That's a good point too. I think that again boils down to people accepting maths as a hard subject, not everyone can do it so many don't even attempt to end up on those careers

[cosmic ibex]

Part of the tragedy is that so many people end up with a vague feeling that "just add 100 and then take 1 away" is a less mathematical way to go about it, which they should feel vaguely inadequate for resorting to.

[pastel horizon]

Yup, because it's not the method they were shown in school

I mean neither way is the "wrong" way they both give the correct answer. One is clearly more efficient than the other though

[tepid smelt]

It is interesting how students will just blindly go into a computational method they know the stop to think about the problem. I was doing some basic systems with my students last week and had the following question system 5x-6y=1 and 15x-18y=3. The question was is it possible to find a pair (x, y) that satisfies the first equation but does not satisfy the second. Just about all immediately just jumped into trying to solve the system with either substitution method or elimination.(most doing substitution). Instead of quickly seeing they are the same exact line.

I have seen this blind algorithmic thinking constantly and its been hard to break them away. I think icy has the right approach though with really focusing on mathematical literacy

[long pelican]

Very related! This is one of the pre-course survey questions I'm asking my incoming honors linear algebra students

[cosmic ibex]

Do you award partial points for correctly answering "no" to the first question? :-)

[long pelican]

Here's another one. It's said in a certain pre-calculus textbook that you haven't really learned precalculus if you don't know how to prove that sqrt(2) is irrational

[long pelican]

Everyone who completes the survey will get 20/20

[tidal mica]

not to play devils advocate but how tempted do you think students would be to google it

it's a hellish circle, bc if you tell them they get the automatic 20/20 there'll be those who just submit bullshit and if you dont some will undoubtedly google it

[long pelican]

Well I'm telling them straight up both that it's graded on completion and the purpose is for the class as a group to let me know their rough math literacy level, that way I avoid the chance of very misaligned assumptions about their math level like last semester

[cosmic ibex]

If someone can write a convincingly short and complete answer based on googling, that would still indicate they're pretty up to speed anyway.

[silent coral]

What do you think about Moore's Method or Inquiry Base Learning to any math class? I'm not a teacher, I'm just a math major student, personally, I would prefer receive each class with that methodology. But I don't know if it's really fits in any math course. Have you try on some or do you have any opinion about the method?

[earnest trail]

idk what moore's method is but inquiry based learning is only helpful if you have the right set of students

if you're having them fo their own explorations and all that, they have to be motivated to do that

my business teacher does this because these students chose to take the class because they have some form of interest in business, so we are willing to do research and learn that way instead of him lecturing all the time

but a teacher of a required class probably wouldn't have the same results because most students are taking the class just for the credit

so in summary, inquiry based learning and stuff like that is likely only to be successful in elective classes unless the teacher of a required class makes the class interesting and engaging enough that the students enjoy it and are willing to think independently and deeply

kinda like eddie woo

actually, exactly like eddie woo

[wispy slate]

iirc Moore's method is independent workshopping hard problems + student presentations each class that purposefully yields a sink or swim kind of framework

I would imagine it is not great for education generally since is literally meant to cut some people out and strengthen others rather than bring the entire group of students up, but could be useful in small classrooms of already-expert students like a grad class

Because if you do well, you will really really do well and learn a huge amount/strengthen your abilities greatly
If you don't, you will be so far behind that it's hopeless

[earnest trail]

that's stupid

you gotta strengthen everyone, especially the people that are struggling

[wispy slate]

yeah apparently the guy it was named after was not exactly Mother Theresa either lmao

[pastel sundial]

Small personal anecdote, but I remember getting a pretty comfortable A in complex analysis while feeling like I learned nothing.

The issue was covid hit halfway through the semester

Honestly I am wondering how much merit there is to the pedagogical approach of making tests a lot harder but making the threshold for an A (or any grade) a lot lower.

[lyric fractal]

thats not the case here
if someone gets a high mark thats a good indication they genuinely understand the material pretty ok
considering the difficulty and understanding needed to even pass let alone get a A

[pastel sundial]

Hm?

No I'm saying the tests were easy for that class, but I'm wondering what it would be like if they were harder.

Probably wouldn't have made much of a difference in that case since it was mostly covid holding me back.

But I do wonder in a broader sense. Like what's a better test of a student's mastery? Being able to perfectly answer a set of easy questions or being able to answer 70% of a set of hard questions.

[long pelican]

I mean, it's kind of obvious, right? Look everywhere in life. Prepared answers give you no information about competence, unprepared answers give you a lot

[tepid smelt]

This is true for every class. I especially feel this once I teach a topic realizing how much I didn't fully understand. That said a student getting an A would have sufficient knowledge to go onto the next class and be prepared. Also once going onto further classes previous knowledge is solidified also. I remember going through calculus having a much better grasp of algebra/trig then when going through those classes

[tepid smelt]

The difficulty of a class is something I do struggle with. There is a sweet spot where the class is sufficiently difficult but not overwhelming so where you just frustrate and lose the majority. I think this is something I know many of my colleagues struggle with also. I don't have a good answer for what level a class should be at. I am getting better(honestly thanks to discussion here) on the types of questions I want to ask

[pastel sundial]

I also wonder what people here think of the approach for exams where you ask n questions but only require students to answer m of them.

It's an approach many of my professors have used, and while it certainly is nice as a student, I wonder about its merits or lack thereof pedagogically

(Autocorrect at its finest)

[long pelican]

I think the following are pretty equivalent in quality:

1. curving but not based on a set percentage of students getting an A
2. points sum to over max points
3. what you suggested

Well 1 is the most flexible, but also has the most uncertainty

[tepid smelt]

I think its ok to have several questions which all students should respond the same.(kind of like drill problems) That said I do think its important to have several questions where students can arrive at a solution in different ways. Also questions where students are able to mathematically communicate a particular solution. It is hard to come up with the later but the discussion here especially the approach icy has used has helped me. I think many teachers(at the secondary level at least)sadly don't have a strong enough grasp of the material to come up with good questions. This is all the more reason we need better textbooks at the secondary level

[long pelican]

both 2 and 3 require you to be pretty damn sure you are good at predicting the difficulty of the exam

[long pelican]

Where are the people writing mathematically valid algebra textbooks anyway

They're out there... setting up camp...

Alone...

[tepid smelt]

AOPS has a good one. Honestly older textbooks were higher quality then the newer ones being produced which is worrying. Israel Gelfand has a set of books for the secondary level that were quite good. We definitely need strong mathematicians writing textbooks rather than for profit companies just producing pretty looking books with low quality content.

[long pelican]

I wonder if there's any teacher who experimented with using exclusively AoPS textbooks and has written about it

[quasi musk]

At RSM, we have our own in house curriculum that's pretty good

Very little fluff, lots of focus on problem solving skills

I taught 7th grade yesterday as a demonstration of how I run a classroom. There was a girl that forgot that d=rt. I gave her the equation and she said she didn't know how to solve the problem

I said keep working at it, 5 minutes later she had a full solution

Mind boggling what students can do if they don't just give up right away

[long pelican]

Sounds like a completely different experience from your previous place!

[quasi musk]

Yeah, it is lmao

I'm also getting paid double

What I was getting paid, in terms of hourly

[long pelican]

Damn that's amazing

[quasi musk]

I talked to my principals about doing my PhD while I teach here and they said it's totally fine

Since it's an after school program, and all the University classes are in the morning early afternoon, and all my teaching will start at like 3 or 4

So it seems to be a good gig, probably better than grad TA

[pastel horizon]

It gives the false impression that some of the material is optional, that they can just choose to answer questions they feel comfortable with and not worry about weak areas

[tidal mica]

asking as a student: how much of the homework do you expect your students to google?

obviously going to vary between classes and honors students but still

[long pelican]

I'm planning to be fine with (and may even encourage) any internet googling as long as they don't get into the habit of looking up the exact problem and copying from it

[silent coral]

Hmm. I would say they don't need to be hard problems. They can be guided problems that helps you understand how the concepts come up.

For example, see Steps into Analysis of R.P. Burn or the Journal of Inquiry Base Learning. They have full courses on math subjects with Inquiry Base Learning method (All free)

http://jiblm.org/guides/index.php?category=jiblmjournal

[tidal mica]

im just wondering to what extent professors might think googling is a good thing - especially for something like analysis i had to google proofs i couldnt come up with, though i always made sure i understood as well as possible before copying

[long pelican]

I definitely made good use of the internet (especially stackexchange) when I was stuck on something like a subpart of a problem. Sometimes I'd run across the exact problem but in this scenario I'd only peek up to the part I got stuck on, inevitably go "D'oh that's so obvious" then finish the problem on my own

I think that's a very good outcome for a problem set

[pastel horizon]

I mean, isn't the point of graduate level maths that they learn how to look up what they don't quite understand and figure out how to apply it?

[long pelican]

Yes, but not just graduate level math. In fact, high school and undergraduate students are already kind of doing it, just getting pulled into sources that are harmful in the long term (e.g. Khan Academy style youtube videos that teach shortcuts)

[pastel horizon]

Shortcuts never work for sure

[cosmic ibex]

Depends on what you mean by shortcut? In your 123+99 example from last week, being numerate enough to see the shortcut is surely a good thing?

[pastel horizon]

A short cut in pedagogy

Using shortcuts should be encouraged if they understand the mathematical reason why it works

The short cut in pedagogy for that example would just be teaching column method and saying you can apply that for any sum

[pastel sundial]

This is why I don't use the residue theorem for contour integration.

Actually I probably shouldn't shitpost here

[cosmic ibex]

Ah, okay then.

[lethal light]

What's a good calculus 1 and 2 textbook that's not a billion pages long that would make a good reference for teaching?

[turbid zenith]

Kinda depends what you want out of it.

What makes something a good reference for teaching in your opinion?

[lethal light]

covers the fundamentals, has a good mix of theory and application, has problems that are more focused on the understanding rather than do the same variation of the same problem 30 times until you memorize it.

[turbid zenith]

Gotcha. I mean my go-to is Garner's book

https://www.lulu.com/en/en/shop/chuck-garner/calculus-dynamic-mathematics-single-and-multivariable/ebook/product-rk7jq5.html?page=1&pageSize=4

Totally not biased because he's a friend of mine

But it's legit my favorite calculus book, and its problems are nicely selected

[lethal light]

ty! I'll check it out

[turbid zenith]

It's also really affordable in PDF form

Now I've got a question for anyone who's up for thinking about it.

So like ... I get why proof is important. Really.

But why does it seem that as soon as mathematics curricula move toward proof, they simultaneously move away from computation?

At least for me, I have a much easier time understanding something if I can sink my teeth into particular examples and see how they tick, and then I have something to build the foundation for abstraction on. But the usual curriculum seems to often move away from that.

Reason I'm considering this is that I'm about to teach undergrad abstract algebra starting in Feburary. (Our semester starts late because we have an accelerated January term.) And obviously I want to have my students doing proofs, but ... I feel like having them do computations would help them grasp things better and make sure they're not just juggling definitions without understanding what anything means.

[meager bronze]

what kinds of computations do you have in mind for group theory?

just curious how you might approach that class computationally

[turbid zenith]

Things like modular arithmetic, permutation notations, orbits under transformations, generated subgroups, etc. I see the computations coming in a little more when it comes to rings and fields because I definitely want to have an emphasis on polynomials and why students learned to do what they did in high school.

Obviously still with an emphasis on proof, but I want to try to make this class as "active" as I can, in that students play around with things to notice the theorems and properties before they're formally introduced.

[twin lichen]

imo proof and computation are not distinct per se, it's more that computation is a very simple and limited form of reasoning that involves chaining a sequence of equalities or isomorphisms together. The transition from computation to proof is more about expanding the reasoning tools they have available and making them more sophisticated.
I'm sure you're right about computations helping to avoid juggling definitions. In my option a proof which proceeds just by unfolding definitions is also not very sophisticated so there's no reason to prefer such a proof pedagogically over a computational proof.

anyway you should definitely be able to incorporate lots of computation into your class and i support that strategy

[cosmic ibex]

Hmm, perhaps it tends to give students a wrong impression of what proof is about, that so many of their first encounters with proof writing tend to be in "build familiarity with the definitions" exercises that just use "prove this or that" as a pretext for thinking about those definitions, and really lack the creative elements of proving things.

To the teacher, the mechanics of proof is second nature, and then it's actually a good way of exploring a set of definitions to sit down and see what you can easily prove from them. But a student who is still struggling can easily feel that "prove such-and-such" must in general just be another step-by-step procedure (which they just haven't understood the precise algorithm for yet), when so many of the examples really do follow fairly linearly from the definitions and the statement of the conclusion. In particular, when they seek out homework help. the well-meaning helper might put a lot of focus on showing how each of the steps in the obvious proof follows almost by itself from what's already on the paper.

[turbid zenith]

That makes a lot of sense. And that's another aspect of proof I have to figure out how to deal with, to be honest -- how to get students better at the kinds of proofs that aren't "just follow your nose and the definitions", the kind that require a leap of insight

Because that's a kind that I struggle with personally as well

So my natural inclination is to try to find a way to take the trail that I've blazed with much difficulty and somehow give more guidance to them so they're better prepared to tackle it

[pastel horizon]

I think that's kind of understating computation. My argument would be the computation involves much more reasoning: you need to draw on your own experience, select the most appropriate and efficient method, evaluate and check your answer makes sense in the context of the problem

If you have a look at the UK curriculum for example. Fluency, problem solving and reasoning are the key focal points. Reasoning would cover proofs, fluency would cover computation and problem solving also overlaps with computation. Of course reasoning and problem solving also have a lot of overlap

[turbid zenith]

https://twitter.com/solidangles/status/1481286185620287494

Bill Shillito (@solidangles)

Anyone teach a one-semester Abstract Algebra course (even better if out of Judson's book) who'd be up for sharing their syllabus/assignments/etc? Trying to get a good idea of pacing of both topics and workload. (Still figuring out how to calibrate the latter...)

Twitter · Jan 12, 2022, 3:25 PM

[twin lichen]

I meant this in a fairly logical sense of like, computation being only an "equational" branch of logic based on rewriting and reduction (or rewriting a group based on a chain of isomorphisms), as opposed to involving like existential reasoning, i.e. constructing an element having a desired property. I don't mean to generalize too much from this narrow perspective, I don't really mean to say that computation doesn't require any kind of creativity. Certainly the idea of drawing on your past experiences with what works is something which is really not well addressed by the analogy I'm using.

[stark pine]

@turbid zenith My group theory class had some nice kind of computational questions that were in the form "show G and H are isomorphic" where G and H were groups given in terms of generators and relations. You had to like, actually just try sending different generators to different places and seeing what happened. I don't have any examples since I don't have access anymore, but I really liked those questions for making me work with abstract groups in a way that I was still doing computations and manipulating identities

One assignment question we had that I did remember that was in a similar vein was to find a ring R and elements x and y such that (x) = (y) but x and y are not associates. I came up with a gross example in a polynomial ring, but that question was also really nice for making me get my hands dirty with the definition of quotient rings and ideals and such

[turbid zenith]

I like this a lot

I view it kind of like, if you can prove a bunch of things about integrals, but can't actually take one, then something is missing

[stark pine]

Yeah! And in fact, I had already taught myself group theory and taken galois theory when I took this intro course, but I still learned stuff because that was the first time I had ever done this kind of crunchy exercise