#math-pedagogy

1 messages · Page 55 of 1

long pelican
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57% correct, and I bet half of those got it correct in a roundabout way rather than thinking about definitions

real mauve
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i guess you probably covered it that way in class

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i had to flex like 5 neurons together to come up with it lol

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maybe mentioning the fundamental theorem of calc as a hint might make it less obscure

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but again, idk how this was approached in class, this is just me in a vacuum

long pelican
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Fundamental theorem of calculus would only make this more obscure

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I consider this "roundabout"

unknown.png
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This is short and sweet

unknown.png
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This is pretty much perfect

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real mauve
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but you see that 2 out of those 3 thought about it in terms if the relationship between integration and differentiation

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i would personally put those under "usage of the fundamental theorem of calc"

long pelican
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FTC connects definite and indefinite integrals

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an indefinite integral is defined to be the set of all functions whose derivative is the integrand a priori

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This is reflected in what you are doing when you calculate an indefinite integral, and also why you add +C at the end

real mauve
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oh you're right, that was just me not actually knowing what the FTC says haha

earnest trail
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interesting

next relic
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I totally agree; some KS3 SAT questions are WAY harder than GCSE Foundation. Even KS2 has some pretty nice questions for Foundation students. c;

spice snow
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How do you guys deal with students who are lacking in prerequisite knowledge for a course in the beginning?

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Even if they already took the prerequisite courses

winged urchin
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As what? Are you a tutor or a teacher?

spice snow
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ta

winged urchin
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Or that, okay

spice snow
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but as a teacher is also useful

long pelican
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This is exactly the situation I'm in lol

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And probably what every other instructor of college freshman courses faces

winged urchin
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Yeah as a tutor (my experience) you have enough time to work through whatever misunderstandings they might have, typically. Unless it's like exam crunch students who just want help right before the test

spice snow
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this happens every semester for calculus 1 with a good portion of the class

winged urchin
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Icy probably has the best tips as far as teaching or taing it goes

long pelican
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I'm also learning as I go

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This is my first semester dealing with it 🙂

winged urchin
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Also depends how uh.. generous you are with your time

spice snow
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in the past, we have had old videos from the professor to help students reference the relevant prerequisite knowledge

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but not very effective

winged urchin
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The thing I think, is most students nowadays don't learn as well from just reading or hearing something

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They typically need to do it, in my experience

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Which requires more... personalized teaching

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But you can kinda give out tests for a tutorial or class or whatever, just on prerequisite stuff and whenever someone has a question, answer to the whole class

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Because more often than not, someone else has that same question

spice snow
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On a separate note, because this is traditionally a first year course, students also tend to struggle with more challenging problems that require critical thinking

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the homework problems are done with the stnadard homework systems

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but those are usually fairly similar to each other

winged urchin
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I remember once I was a TA for a course requiring MATLAB except they basically just give them a week to pick it up right and just jet off from there.

I was given a little flexibility in how to run the tutorial so I made a worksheet that kinda... worked through all the most simple of basic things in MATLAB and didn't really 'teach' infront of the class so much. I just made them follow this fairly guided work and would answer questions to the whole class and demonstrate whatever I needed to

And that seemed to help them pick up at least somee basic skills? There were still some silly mistakes from some students but I think it went well?

long pelican
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I think that in order to even start learning to solve problems properly, students need to learn how to read math problems

winged urchin
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A little different but I could imagine doing something similar with pre-requisite math knowledge. Factoring, expanding, solving simple things, etc etc...

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But it all depends on how much leeway you have as a TA

long pelican
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And also, in order to even start learning math correctly, they have to learn how to understand the language that the instructor uses

winged urchin
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Mhmm, use consistent notation for sure

long pelican
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Without either of those things, it's just more rote learning, which they are good at but it only helps up to a point

spice snow
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which isnt always bad, but it limits exactly how much you can apply what you have learned when you come across a different problem

long pelican
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Exactly!

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Good illustration of this actually

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Another one

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(For both of these, the prompt is to identify why the sentences are wrong)

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So I also believe the underlying reason they use the template-matching strategy is that they don't have the mathematical literacy skills to actually read the problems in front of them

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And the core issues with mathematical literacy in particular are reading sentences with variables and functions

spice snow
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yes

long pelican
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If I do an intro course again I'm going to play with dedicating the first week or 2 (or 3) to reading variables and functions in various scenarios

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before the actual content

winged urchin
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Ya, I like some problems in K-12 where it's something like

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"The sum of two numbers equals twice the first number and their difference equals twice the second number. What are these numbers?"

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It's not applied so much but it still shows whether they can understand how to use variables and interpret english to math

long pelican
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That could be one example

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The more varied the examples the better

earnest trail
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when he says "sin times x" instead of "sin of x"

long pelican
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What?

earnest trail
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one of my friends

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he said "sin times x"

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when reading sin(x)

long pelican
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Dang

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Sounds like a misconception not properly treated by well-designed exam or homework problems

earnest trail
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yeahhhh

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he also said something else that's similar but forgot

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oh right

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he said e^x as "e of x"

long pelican
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lol

wispy slate
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there's a guy who comes to my math club only to interject with "why is this useful" and demands to be taught basic statistics and doesn't seem to realize no matter how much I try to gently explain that this is a recreational club and not a class
I even tried to do some fun probability paradox stuff to satisfy him (which genuinely contains and helps nurture fundamental intuition needed for prob/stats) and he replied with "who cares about this stuff? only crazy people and geniuses, doesn't seem useful"

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dude looked like he was in a super depressed mood the whole time too
any ideas of how to help a person enjoy a club session when they are like this?

quasi musk
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Just put it back on them

wispy slate
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I don't have the heart to tell him to not show up because there is basically 0 interest, even among the 20 or so people who signed up for the email list announcements for meetings, and I get like 0-2 people attending every meeting

quasi musk
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Just say "Is this a productive attitude for a math meeting"

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"What did you hope to gain by attending"

long pelican
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0-2 people attending, is this usual? Is there less interest in math than 10 years ago?

quasi musk
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"What applications are you interested in?"|

wispy slate
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postered all over campus early in the semester

long pelican
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Oh, I had the impression you were hosting math club at high school or middle school

wispy slate
long pelican
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So he's not a kid

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Is he .... mature?

wispy slate
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I mean
he wasn't acting immaturely

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just the living antithesis of everything I believe about math tho LOL

long pelican
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It's weird he attends math club

real mauve
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maybe to try and rekindle the flame

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but motivation to do stuff comes from within, sadly

earnest trail
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wtf

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when I was reading that I thought it was some high schooler

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but from an adult?

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that's... strange

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and if the math club isn't succeeding maybe you either need to make it more appealing to people or end it

quasi musk
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It's really not that strange, adults have little patience for things that aren't going to directly affect their income or well-being within a reasonable time frame

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most adults aren't that mature

pastel horizon
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If he is 50 you pretty much won't be able to change his view

wispy jolt
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Nothing is worse than a like 50+ year old man who aggressively doesnt know math and just wants to question everything

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I had a class with a dude like that tho he was prolly more like 35 to 45 or so and he would just devour the TA's time in discussion

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He wasnt so much hating on math like this dude but he just could not accept anything

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And really gave the TA a hard time like it was a philosophical debate

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Seems like a bad attitude tbh

long pelican
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Interesting, does he even understand what he’s debating about?

pastel horizon
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I think at that point in his life he's already decided what he believes in

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At least at 18-22 they're a bit more malleable. Which can be a bad thing too

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I think it's something like at 25 you reach full maturity

tawdry venture
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any ideas for how i should eradicate this

strange bronze
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give examples?

tawdry venture
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of the commutative law?

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i guess i could try to do that

real mauve
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maybe something simple like 3*4 = 4*3 and then expand 3 = 1+1+1

trim violet
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draw dots in a rectangle ?

brittle crag
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wow.
like bring the analogy back to physical objects.
"if I have 3 apples, and I double that, that is, multiply by two, how many apples do I end with? 6, thats correct. Now lets put the apples back and start over. If it take two apples, and then take two more apples, and then I take two more apples, that is, I multiply my original two apples by three ..."

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then remind them multiplication is just repeated addition. Doesnt matter in what order you add numbers, does it?

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but I think if you have physical objects, and he physically saw that you got the same number regardless of the order

dire igloo
real mauve
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what kind of target audience did you have in mind for this

long pelican
tawdry venture
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common core is not a thing in russia

long pelican
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Wait you’re in Russia?

tawdry venture
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yes i am in russia

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why could you possibly think otherwise thonkzoom

long pelican
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I had no reason to think you were in Russia…

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Ok take my statement minus common core; the approach isn’t limited to common core

tawdry venture
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right ok but like

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hm

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i dont imagine anything really gets PROVED like

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formally proved

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in primary school here

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the student seemed not to deny the commutative law outright, just objecting to like

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the process being different or somesuch

long pelican
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I mean if he forgot that it was proved, then he has no reason to believe it’s true?

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Might not have been exposed to the arguments like the array argument or the area argument OR might have not understood them and their consequences fully

tawdry venture
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hrm

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i guess that would be a possible explanation

long pelican
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Another possible thing is that he’s convinced of the commutativity for concrete numbers

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But with a complex expression with symbols he doesn’t intuitively think the argument extends to “symbols”

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The missing link is obviously that the symbols represent numbers (and not noncommutative objects)

tawdry venture
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i have only had these kids for several months, so i probably did not have time to address such issues individually

long pelican
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Same with my calc 2 students, they come with a lot of background issues — yesterday I gave someone who came for extra help the exercise to find all functions f from R to R such that f(a)=f(b) for all real a and b, practicing reading notation and understanding their consequences. He found it challenging

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He interpreted “f(a) = f(b)” in like a million incorrect ways first

analog token
tawdry venture
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just so i can be wary of such misconceptions should they show up later

long pelican
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  1. So the set of outputs is the same between the functions a and b
  2. So the set of outputs of f are the same
  3. A function of a is the same as a function of b
  4. f of the set of all real numbers is the same as f of another set of all real numbers
  5. Two functions f and (he imagined another function g I guess?) have the same output for every input
dire igloo
tidal mica
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im grading homeworks for a calc class made for business students (who wont do calc again) - should i go easy or is it too harsh to grade as i were grading for a math major?

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not looking for a definite ans ofc, just opinion

real mauve
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depending on how much time you can/are willing to sink into it, maybe grade homework going easy but giving lots of comments/feedback, then grade quizzes and exams as usual

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but anyway the content of the evaluations shouldn't be quite the same

tidal mica
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i mean it's calc 2

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theyre doing normal calc 2 stuff

tidal mica
quasi musk
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then as weeks go by, I start docking points for things I told students to be careful about or to watch for

earnest trail
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and I doubt that students at that age (or even my age) have a concrete understanding of numbers

earnest trail
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there was someone in #discussion I believe like 3 weeks ago who was saying that he wanted to dock points for someone saying "Domain =" instead of "Domain:" lmfao

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like some people are way too harsh with grading, all u gotta do is tell a student that that's not necessarily correct, and they will probably keep that in mind in the future

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no need to punish them for something minuscule that they didn't know

long pelican
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Also, isn't the goal of elementary math education to have a concrete understanding of numbers

pastel horizon
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I would certainly hope so

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In the UK they're teaching them to the point where they decide for themselves the most efficient route for an arithmetic calculation

earnest trail
long pelican
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By "proof" I mean arguments like the area argument: rotate a rectangle 90 degrees, same area

earnest trail
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oh is this like

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group theory esque

earnest trail
long pelican
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ab is the area of a rectangle of width a and height b, and ba is the area of a rectangle of width b and height a. The rectangles are congruent therefore have the same area, so ab = ba

earnest trail
earnest trail
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it is just "here's commutativity now do some problems"

long pelican
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Well that's pretty bad

earnest trail
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and of course it's different for each teacher and country and whatnot so ya

earnest trail
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american education amirite

long pelican
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Unfortunately

earnest trail
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yeah

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I wish it wasn't structured like "cram as much at a time"

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and more "help students understand as much as possible"

pastel horizon
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Split it up into squares. n rows of m is clearly the same sun as m rows of n

long pelican
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Well, the rotation establishes congruence in the real number case, while summing in two different ways works for the positive integer case

pastel horizon
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Ah yeah I see what you mean. The commutative law can be a great tool to use for decimals actually because you could say for example 4.9 lots of 9 is the same as 9 lots of 4.9

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Also the famous percentage trick, x% of y = y% of x

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Definitely something that needs emphasising (and I think it is in the UK) for 5-11 education

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Now here's an interesting one, what's your opinion about teaching scientific notation early?

long pelican
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Shrug, it's not foundational to anything so there's no real need for any particular place for it

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Back when calculators overflowed and displayed things in scientific notation, it might have been more necessary

pastel horizon
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You don't think it's useful across STEM?

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The other benefit is you would be surprised how many kids can't convert centimetres into metres

long pelican
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Scientific notation: writing a number as $a\times 10^b$ where $b\in\bZ$ and $1\leq a<10$? That's something that can be taught in a heartbeat with correct foundations

burnt vesselBOT
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Icy001

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long pelican
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Would you rather spend 5 months to get students to get it 10 minutes to get students to get it?

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What are some errors in thought process when converting centimeters to meters?

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In particular, errors they cannot detect themselves after a bit of thought

pastel horizon
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Honestly? The error is usually "do I times or divide by 100"

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If they could somehow learn one cm as 10-2 m that would help a lot

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I guess it's not really strictly maths pedagogy but it does remove a lot of barriers in science when it comes to big numbers or small numbers

magic minnow
long pelican
wet sail
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That's really cool

tawdry venture
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okay so like, is there a name for this thing students do when they overuse the word "it" without making it clear what is being referred to?

real mauve
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perhaps ambiguity or recitation, since it usually happens when one learns buzz words but not exactly what they mean or what object they are referring to

tawdry venture
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is there a fix for it

real mauve
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maybe reviewing the definitions with lots of simple but diverse examples

earnest trail
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so essentially there are psychological tricks as well

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kids typically know the definitions and stuff but the "it" comes from laziness or other similar reasons

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so the fix is usually psychology

long pelican
long pelican
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But yes, both things are extremely common in people of all ages and I wouldn't be surprised if it's not entirely their fault

gleaming flame
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This channel needs more discussion of undergraduate maths education. Everything from the lectures and textbooks and online sources to the teachers are worse in quality than their K-12 counterparts.
The only upshot is that students are usually heavily self-motivated and have a decent understanding of high school maths, but it goes all downhill from there unfortunately.

long pelican
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I think I'm going to disagree completely with

Everything from the lectures and textbooks and online sources to the teachers are worse in quality than their K-12 counterparts.

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But the definition of quality for me is probably very different from that for you

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Like if a teacher is extremely engaging and her explanations reach everyone, but teaches test prep instead of math, I don't rate that as high quality at all

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What are your thoughts?

gleaming flame
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Ok well, two things,

  1. Altho most K-12 material sucks, some (if not most) ppl and books are aware about making an effort to appeal to intuition, to be engaging, to hook u with interesting ideas and so forth. Uni professors and books don't seem to care so much about your understanding, droning on about definitions that come from no-where. The sheer lack of resources don't help either.
gleaming flame
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If she's engaging, and her explanations reach everyone, maybe she's not teaching them purely maths but if students are engaged they should be able to fill in the holes, no?

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Also it's sometimes hard to understand preciselywhat one is saying when one talks about pedagogy. It often sounds too abstract to make real connections. Apologies if I didn't answer/get your point.

pastel sundial
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I think the thing is that university math tends to pivot away from appealing to intuition and towards appealing to rigor which can take some getting used to.

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But that's just because past a certain point you need rigor to get anywhere

long pelican
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Tangential (and I will respond later) but I'm currently grading this question and I'm seeing a nontrivial amount of people who have gotten this wrong because they were victims of the y = f(x) test prep teaching

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pastel sundial
long pelican
pastel sundial
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Lmao

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That's painful to look at

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Also Icy I've said this before but holy shit your homework problems are cool

long pelican
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😄 Thanks

pastel sundial
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I'm curious though Icy. What kind of feedback to you get from your students? I'm wondering if the students these problems are intended to help would complain and say they're trick questions or intentionally confusing or whatever.

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Like in office hours and whatnot

long pelican
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I haven't gotten any negative comments in office hours really

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Everyone who's asked for help on these questions (and there's been many) has genuinely wanted to understand

pastel sundial
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Have you gotten positive comments?

long pelican
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Yeah, in fact

pastel sundial
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Like what do they say when you help them understand

long pelican
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This is part f of this question

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pastel sundial
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/how difficult is it to get them to understand

long pelican
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A lot of responses have been heartwarming

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like this:

Calculus class this semester has taught me about the precision of language; and that math itself is a language expressed in symbols and words. Understanding math is not about manipulating examples to get the right answer, but about comprehending concepts, and being able to think about them abstractly.

pastel sundial
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Is this from course evals?

long pelican
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This homework, their answer to part f

pastel sundial
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What that was a homework problem

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Would you have like taken points off if they said "yes"

long pelican
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Nah I'm lenient on that part

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everyone who wrote something, even if it's generic, got full points

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The students who are still unable to read math were the ones who tended to write generic responses

pastel sundial
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Was this just your sneaky way of getting extra feedback

long pelican
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Obviously there's going to be bias in this (because they want points) so there's grains of salt in this, and I'm still patiently waiting for course evaluation results

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But I'm convinced at least some parts of these are genuine

pastel sundial
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Yeah seems like the kind of questions where the best responses would be exaggerations of genuine feelings.

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But still have genuine thoughts behind them

long pelican
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yep

pastel sundial
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I'm very curious to hear how you do on course evals

long pelican
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I'm prepared for a very mixed result

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People who did badly will say "This class is unnecessarily hard, it should just be about how to solve problems"

pastel sundial
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how many of those people are there

long pelican
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There's 3-4 who never come to class in one of my sections, but zero in the other section

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So I'm betting 3-4

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maybe some quiet ones who sometimes come to class

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In terms of grades I'm pretty nice I think

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The later section is getting all A's I think

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That section has just 9 students though

pastel sundial
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ah

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like how many people are in the class (both sections), and roughly how many fall into each of these categories
a). understood how to read math from the getgo and aced the class without really trying.
b). was a strong math student, but still got something out of being taught to approach math in a different way.
c). came into the class with shaky foundations but left having patched things up.
d). came into the class with shaky foundations and left having improved somewhat, but still having significant holes
e). "this class fucking sucks. What's the point of this f(x) notation when you could just write y"

long pelican
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14 in section 3, 9 in section 4 (sections 1 and 2 are taught by someone else and have like 30 people each -- no, it's not because of dropping or switching! The semester started out with this discrepancy)

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a) In the other sections, there's 2 I think, both likely international students (yep I'm racist), in mine there were 0

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what's that sully for 👀

pastel sundial
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just the "yep I'm racist"

long pelican
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Lol

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Honestly pretty much everyone I had had shaky foundations in varying degrees. Think everyone in section 4 fall in c), and probably half of section 3 too. Half of the other half are in d, and the people who don't come to class in e

pastel sundial
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so no one in b)?

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and no one in a) in your sections?

long pelican
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Yeah

pastel sundial
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damn is there like an honors calc class taking all the good students? Or is the proportion really that bad?

long pelican
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This class was for non-math majors who didn't get high enough score on the AP calc BC exam to get credit

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Or like, took AP calc AB or regular calculus in high school and got A in their class

pastel sundial
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ah

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I'm curious what the calc 3 class looked like

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how many people still had shaky foundations

long pelican
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Apparently I heard from the abstract math professor (who's basically teaching the best freshmen math majors) that even they, as freshmen, are so bad at reading and writing mathematical proofs

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most of them have never heard of a set, etc

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So apparently I have not gotten the short end of the stick in any significant way

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I have no idea about calc 3 class unfortunately

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But if calc 2 students and abstract math students have never heard of a set, probably calc 3 students have never heard of a set either (before coming to college)

pastel sundial
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what is covered in "abstract math"

long pelican
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Intro to sets, functions, mathematical proofs basically

pastel sundial
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and is it like the class that you test into if you did calc 3 in hs or just BC cacl

long pelican
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Honestly not sure. I think it's just an "elective" that you can take as a freshman if you're brave enough

pastel sundial
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ah

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I mean admittedly I was not great at proofs in freshman year

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like I could read them, but I was bad at writing them

long pelican
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If you could read them you would definitely be very above the curve here

pastel sundial
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it's crazy how non-representative the active people here are of math students as a whole

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like you'd expect there to be a significant difference due to the self-selecting nature but still

long pelican
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I think the ones who ask for help are very representative tbh

pastel sundial
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yeah but I mean like, Moth

long pelican
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Yeah, if you can communicate a well-formed mathematical idea in this discord you're very above the curve

pastel sundial
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or really any of the highschoolers that aren't just here to ask for help

long pelican
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Browsing the help channels in this discord actually helped me a lot in diagnosing what I was dealing with throughout this semester

pastel sundial
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interesting

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I should do that more tbh

long pelican
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I'd encourage everyone in this channel to do the same from time to time, yeah

winged urchin
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Oh ya just like anything you learn. Having more examples is the key to greater understanding. So the more times you can help students and get experience with that the greater your chance for insights!

long pelican
# gleaming flame Ok well, two things, 1. Altho most K-12 material sucks, some (if not most) ppl ...

OK finally finished grading, so I can respond. Pearson and such try to be engaging with flashy stuff, but at their core they turn the math into test prep, into a digestible series of pattern matching and rules which are easily forgotten, and don't have much logic or definitions connecting them. And they don't teach mathematical maturity or literacy in the slightest.
In the contrast, university professors assume both mathematical maturity and literacy in their students, which can be problematic for students who are neither mathematically mature or literate

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But for students to do have it, the math content is just intrinsically engaging, albeit difficult (math is hard)

long pelican
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See examples of student answers to questions I shared

long pelican
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I've been realizing a lot of my math problems, especially the ones that illuminate misconceptions, are open middle problems

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Although mine are more conceptual rather than putting numbers in boxes

winged urchin
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Hmmm

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Do you think something like solving 2(x+3)=4 is an 'open middle' problem because the steps you could take could vary?

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Like, you can algebraically manipulate in two obvious ways, divide by 2 or expand the brackets

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But even then, you can kinda go more... intuitive I suppose?

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And notice that 2 times 2 is 4 so we want x+3 = 2? (which of course is what the algebra does but I think that kind of approach 'feels' different to a student)

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Though 'if' that is an open middle problem it's a narrow one

long pelican
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Yeah something about it feels very close to a routine problem

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You know what's an excellent class of open middle problems?

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Proofs!

winged urchin
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I think

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If you intended to use an open middle problem

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You should strive to give multiple answers if/when you do give answers

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Show them that there were multiple paths to the end

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And likely even more that you didn't write

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Sometimes I do talk kind of... 'romantically' about algebra or arithmetic in that way

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There's not just one path to the solution, it's like a forested hill and the ways you can move in this 'arena' are governed by the rules of addition, multiplication, etc... but that's as far as they constrain you. You can't walk through the hill unless you have powerful machinery to do it, but you can choose many routes around the hill

long pelican
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Already beat you there! This was a solution to an 'open middle' problem on the 2nd midterm

winged urchin
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Mhmm, lovely!

long pelican
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Also this

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on the same exam

winged urchin
#

Any comments on these from students?

long pelican
#

Hard 😛

winged urchin
#

Any... "Well.. which way should I do these? Which one is easier?"

long pelican
#

Didn't get questions like that interestingly

#

They're also getting better at them

#

Performance on open problems like these was like 0% on exam 1, 4-7% on exam 2, and a whopping 40% on exam 3

winged urchin
#

Hey, small sample size but that's a compliment to your teaching style perhaps ahaha

long pelican
#

Final is in 3 days, so that'll be another data point!

#

There's several "open middle" problems on the final too

winged urchin
#

Do you think you should separate open middle questions from direct skill-checking questions? Perhaps?

#

Like

#

One fear I think students might have, and do run into

#

Is spending 'too much' time on a test trying to think a problem through

#

Or sometimes I hear them say that they got something wrong because they originally thought that they 'thought it through in a clever way' but ultimately had a misconception

#

I wonder if you directly separated the questions so there was a 'formulaic' portion of the test dedicated only to those and then a second portion of the test with open middle questions

#

Where they should 'expect' to need to really chew on the problem

long pelican
#

James Stewart said in the intro to his textbook that on problems like these in the exams he sets for his students, he makes sure to award significant partial credit for trying something even if it doesn't work out

winged urchin
#

It's funny then

#

On a lot of webwork I see

#

Questions are 'very' hand holdy

#

With several answers boxes where the question essentially asks each individual piece

#

These are quite the antithesis to open middle questions, perhaps

long pelican
#

Mm-hmm

#

In the article, it basically exposes traditional problems as having the huge problem of having too many false positives

#

If 92% can get a traditional problem right but only 41% in a slightly open-middle version of it, that indicates about 50% of students who got the first problem right still have misconceptions

#

(if we assume that truly understanding the problem would entail getting both right)

winged urchin
#

Do you think it is bad practice then... that say.. if a student asks for steps to a problem, to give them defined steps? I'm hesitant to say it's bad...

I do admit when tutoring I will comment things like. "Notice this is an optimization problem due to the use of 'most', 'least', 'greatest', 'smallest' etc etc. And in these problems we need to identify an objective function to optimize"

#

And it's common afterwards to then ask about critical points and so forth and so forth

long pelican
#

Hmmmm that example is an approach I explicitly avoid

#

It reminds me of the "key word" approach where you short-cut understanding the problem and just identify the "type" of problem it is

#

Which students love to do

#

That was the de facto standard approach they came into this semester with

#

Which they un-learned 😛

winged urchin
#

Idk know where exactly my opinion lies in that area...

#

Hmm

#

On the one hand, it is kind of like just saying to someone "Oh you were really good!" vs. actually giving them specific compliments or potentially criticism

#

"Your footwork was superb" or "You really held that note for a long time!"

#

In a similar way, you can say "Oh you were really good!" when you're just trying to be nice and didnt actually notice why they might be good

#

But... I'm trying to think of semi-similar examples where the more keyword-ee approach seems preferable...

#

It's not great but my mind cant seem to go to another example, but like saying 'island' instead of describing it more like 'a mass of land above the water that is surrounded on all sides by water'

#

It's really not a great example at all.. but just exploring

long pelican
#

Is the island example illustrating using and understanding definitions?

winged urchin
#

It's trying to find an example where you could potentially... not understand what you meant by a term but it's also unwieldy to describe what you mean instead... kinda

#

Like.. how would you talk about an optimization problem without using keywords

#

No optimize, no maximize/minimize, no objective function, no critical points, not even derivative perhaps

#

Certainly it would take 'more' words to try to describe what you mean to do right?

#

Instead of just saying 'we're looking for critical points'

#

You'd need to say something like.. what?

#

"We're looking for the point where the <whatever quantity we're interested in> is no longer changing if we were to consider small changes in another quantity"

#

Point is perhaps a keyword there too no?

long pelican
#

I'm not against using vocabulary lol

winged urchin
#

As in, we could expand that word into a more labored description as well

long pelican
#

I was cautioning against a particular superficial problem solving approach called the "key word approach"

#

Where you look for key words solely to identify what type of problem it is with the goal of identifying what procedure to apply

#

In place of simply reading the problem and understanding what it's saying

winged urchin
#

So then say you had a problem like... "A company makes circular and square cookie cutters from a strip of metal 15cm long. The company wants to try to make these cutters as big as they can. They plan to cut the strip and bend the two pieces into the desired shapes. Where should they cut?"

#

If a student just tells you "I have no idea what to do"

#

What's your approach instead of using keywords?

long pelican
#

First would be to read the problem with them

#

Slowly

#

Multiple times

#

Then draw the situation with them, starting with asking them to try to draw it

#

So have them draw a strip of metal...

#

Go from there

winged urchin
#

Assuming you guide them to the correct answer, do you think afterwards it's good to talk about the keywords?

long pelican
#

Nah, I always encourage completely understanding the problem anyway

#

I'm not a test prep kind of person!

#

It might be slower but that's not an issue

winged urchin
#

Well, it shouldn't be an issue

#

I think there are some cases where being too slow can be bad, not necessarily for their understanding, just as an approach

#

Students have gotten mmm frustrated if we end up spending half of the time they paid for on one problem out of the seven on their practice exam that they brought to our 'first' session with me with their exam looming

#

But that's an unfortunate reality in some cases

long pelican
#

I've never done paid tutoring but I think it'd be a reasonable self-policy never to do the test prep approach even with an exam looming, but maybe indulging in test prep sometimes is a reasonable policy too

#

I think we had a discussion in here about that exact question actually

#

with someone else

#

I'd categorize the keyword problem solving approach as test prep because I can't think of any long term benefits to taking that approach with all problems in the future

winged urchin
#

Yeah the discussion has been brought up before I think

#

Though I don't think I could just not do test prep like that. There's quite obviously a higher demand around exams and I'm not yet financially sound enough to be able to turn down that income, at least myself

#

I just try my best to hopefully get them to understand something

#

Or if it's their last math course or something then it at least doesn't feel 'too' bad knowing they likely won't retain the knowledge

long pelican
winged urchin
#

Nope

long pelican
#

It's a classic

winged urchin
#

There is definitely an.. assumption made in school that the problems you're working on are well defined and so should have a solution that aligns with how you've been taught

#

And as a student there were definitely times that I did something only because it was the only thing I thought was able to be done and I expected the problem didn't have a 'trick question' as one might say

#

Though I still tried to hand-wave-ily justify the answer after even if I didn't really understand the symbol pushing

#

That didn't happen 'very' often for me but it did for sure

#

I have also encountered.. in tutoring.. that the questions asked 'sometimes' do have some mistake that makes them unintentionally a 'trick question'. And it's just perhaps, unfortunate, that I think I have to tell the student to try to interpret it in the closest way to make sense.

#

Ideally, they would answer that this question as stated doesn't make sense because... idk.. the units don't make sense or whatever error in the question makes it ill-posed, and the marker would see that and give full marks. But I'm afraid that is not the case in general

#

So sometimes students have to betray their understanding of the problem in order to try to get something

long pelican
#

Standard approach is to write their honest answer followed by "but I assume they meant this, and in this case the answer and work is as follows"

winged urchin
#

Mhmm, I do encourage that kind of thing. As well as mentioning when their own answer seems to not make sense

#

If that is the case

real mauve
#

that video :(

winged urchin
#

I really appreciate having this server, and more specifically this channel (I hardly look at others beyond the help ones)

tepid smelt
#

How do you all do review for final? I feel I just don't do review well. I end up helping kids on completely different things and I feel like I am reteaching quickly rather than actually reviewing. I end up giving review problem packets but it feels so lazy and I doubt it helps much but I don't know what else could work.

long pelican
# tepid smelt How do you all do review for final? I feel I just don't do review well. I end up...

I analyzed all the homework item performances and found that the "open middle" problems, including the proof-based problems, were the best predictors of performance on the 3 midterm exams. So I compiled a list of "best" homework problems and told my students to make sure they understand these problems like the back of their hand. I also emphasized the value of being able to read math

During the actual review (only 1 hour allocated for this), they asked me to quickly re-teach probability and I did. Including going over the "open middle" probability problem on Exam 2

Unlike the other review sessions, I didn't do any review practice problems per se.

wispy slate
#

that video is crushing

cosmic ibex
#

The kid who answered 42 should probably have counted as getting it.

tawdry venture
#

does anyone kind of dislike when people say "between AB" or "between [1, 10]" or whatever

real mauve
#

i'm not sure i get it, you mean when people formulate an interval incorrectly?

tawdry venture
#

yeah

real mauve
#

yes. i also get angered that many people can't even say "interval" correctly in spanish

#

this one seems like a minor mistake, but it's anyway important to use notation correctly or define your own notation explicitly when needed, not just use stuff willy-nilly

#

since one would usually like to communicate the idea to others

tawdry venture
real mauve
#

spanish is very clear with its intonation. stressed syllables are clear from the way words are written. the world intervalo has intonation interVAlo. people often read it as inTERvalo, but this would be written as intérvalo, with an explicit accent.

#

i guess in some sense it's a similar problem of disregard for the notation

tawdry venture
#

oh, so when the stress could not be predicted from normal rules, you indicate it with an acute accent?

real mauve
#

hmm more or less, yeah. there's a decision tree for when to use an acute accent when the intonation is on one of the last 2 syllables. whenever the intonation is on the third, 4th, etc (from the right), it always has an acute accent

long pelican
#

(And probably a potpourri of other deeper misconceptions to go along with it)

real mauve
#

i would say it's difficult to tell that much only from that alone though

#

maybe they understand it but just don't want to use the notation for whatever reason, or don't know the notation well

long pelican
#

Honestly the prior probability that a school math student without outside exposure to math understands the nature of sets should be pretty low to begin with

pastel sundial
#

What do you mean "understands the nature of sets"

#

Isnt basic set stuff very intuitive?

long pelican
#

You'd think that

#

I had to spend 20 minutes in an office hour explaining what "set of all triples of real numbers" (i.e. R^3) meant

#

The person came in thinking it's the 3 axes or the 3 planes or something else

#

Basically, if there's something anyone can take away from my experience, it's to never assume someone already understands something in math from their prior classes, even if it's intuitive to you

pastel sundial
#

I wonder if that's a case of math education actively harming intuition

#

Actually probably it's just that "set of all triples of X" is actually a less intuitive notion that it might seem

#

Because a triple is itself a collection so now you're taking a collection of collections

long pelican
#

Hmm yeah, and yet teachers gloss over it like it's just obvious and jump straight into the procedures

pastel sundial
#

So I guess it comes down to being able to reason about sets as objects in their own right and not just collections of other things

#

Like you need to think of triples not just as 3 numbers sitting in a box but the box itself

long pelican
#

that's a good point

pastel sundial
#

I wonder what the best way to motivate sets is

long pelican
#

Honestly it's possible the concept of set is not very hard to teach, it's just that teachers just don't teach it

#

and more importantly

#

They don't base their teaching off of it at all

pastel sundial
#

Yeah but I think you'd still want to motivate them somehow

long pelican
#

There are so many ways teachers can use sets to clarify things yet they don't

#

For example, graphs and function transformations

pastel sundial
#

Like I remember reading a naive set theory book way back in highschool

#

And it was like "a relation is a set of ordered pairs, an ordered pair is {{x},{x,y}}, an equivalence relation is this special kind of relation, etc"

cosmic ibex
#

That doesn't sound very naive.

pastel sundial
#

And I remember thinking "cool why the fuck should I care"

long pelican
#

Lol at the definition of ordered pair

pastel sundial
long pelican
#

I know it's standard in ZFC

#

but for teaching elementary school people... just don't do that

pastel sundial
#

Or any of the underlying proofs

long pelican
#

There is a point where more foundations gives diminishing returns for understanding

#

defining (a,b) as {{a}, {a,b}} is past that point

cosmic ibex
#

Well yeah, but if you're not going to cover axiomatics anyway (good choice!) then it's hard to motivate why one would want the everything-is-sets-all-the-way-down conception of mathematics in the first place.

long pelican
#

Also, no one in the real world in mathematics ever thinks of things as everything-is-sets-all-the-way-down.

#

except people working in foundations

cosmic ibex
#

Right.

pastel sundial
#

I forget if you've said your thoughts on this Icy but I wonder if teaching computer science could help illuminate some of these math concepts.

cosmic ibex
#

It (everything-is-sets, that is) can give the brightest of the bright students a wonderful aha experience, but will just push the rest away.

long pelican
#

Definitely, python list comprehension is basically set notation, and ordered tuples are built in

pastel sundial
#

Like functions in cs are "almost" as general as functions in math

long pelican
#

I took me 5 seconds to explain R^3 to a python programmer

pastel sundial
#

Was there a Python programmer that asked you about that?

long pelican
#

Yea

cosmic ibex
#

Many students seem to come away with the impression that functions in cs are more general than functions in math -- they end up with an impression that even writing a function as a definition by cases is somehow "dirty" or "less mathematical".

long pelican
#

Functions in math are taught in the worst possible way in schools and in youtube videos

pastel sundial
#

oh my god that video you linked here Icy

long pelican
#

Yep

pastel sundial
#

started out good

#

then literally said "f(x) is the same thing as y"

#

how did this way of teaching functions even arise in the first place

long pelican
#

I think it's how functions were conceived in the 1800s, and education moves even slower than politics

cosmic ibex
#

Historically there was a long period in the development of the function concept where people preferred thinking about variables whose values are linked. rather than reifying the link itself as an object to think about. Much of physics, chemistry, engineering is still written that way.

long pelican
#

Ideally the function concept would be taught in such a way that it's absolutely clear how it's distinct from the linked variables concept

#

Start with the sqrt function which, if they use the internet at all, they have seen written like sqrt(3) and such

#

Also

#

Linked variables clashes with the way variables are used in math, to stand in for a quantified object

#

So bleh

cosmic ibex
#

I recall being taught somewhen in middle school from a book that actually had little drawings of machines -- there was a funnel at the top where you drop in the argument, then a handle you could imagine cranking, and f(x) would fall out the bottom.

long pelican
#

Yeah that was also in a book I was gifted in 3rd grade by an awesome math teacher

#

My less-awesome 3rd grade teacher would get annoyed that I'm reading that in reading period instead of the usual reading books

#

and take it away from me

earnest trail
#

don't we love those teachers

earnest trail
cosmic ibex
#

If I remember correctly, the little machine pictures were also a good illustration of what operator precedence means. It's a matter of how to connect up the machines, not about enforcing a rigid order in time for when to do which part of the arithmetic. Many people seem to be taught the latter, and then are horribly confused when they get to algebra and suddenly it's allowed to simplify 3x+5+2 to 3x+7 when they have learned "multiplication before addition" and you can't do the multiplication yet because x doesn't have a value ...

short ivy
quasi musk
#

Unless you're teaching in some sort of advanced class, or somehow students care about it

#

It's going to be a waste of time and cause more confusion

#

Some students struggle to add fractions in calculus

#

I'm not sure spending more time on sets will remedy the root causes of what's going on in math

#

From an educator's perspective

long pelican
quasi musk
#

Each excelling student is alike - they are all (more or less) excelling in the same way; each struggling student is struggling in their own way

long pelican
#

My take agrees with that aspect: each struggling student carries their own set of misconceptions brought by their teachers that they haven't managed to eliminate yet

quasi musk
#

So, if I had to boil it down to the primary issue, it's too large of class sizes to individualize education

#

Which I think we agree on, but it's not going to be a solution to say "Ok we'll just individualize people's education and somehow make the students engaged"

long pelican
#

Is there a mathematical component to that take? Because math is somewhat unique among subjects

quasi musk
#

Yeah, I think this is particularly disastrous in math, because you don't learn how to add fractions

#

Then you go to algebra and you need to add & subtract fractions

#

Clear denominators, etc.

#

But you can't

long pelican
#

I wonder...

#

The people who can't add/subtract fractions in algebra yet passed that unit

quasi musk
#

Yeah, they pass it on short term memory

long pelican
#

Maybe they passed it because they could add it then, but quickly forgot it

#

Mm-hmm

quasi musk
#

So they study for like 3 days

#

Yeah

long pelican
#

What's the cause of that?

quasi musk
#

You have to go to each student ask them why they do that

#

Some common answers might be

long pelican
#

My radical take is this

quasi musk
#

"Oh I just need to pass this class and then I can never take math again"

long pelican
#

For those students, adding and subtracting fractions is no different from a "How old is the shepherd" problem

quasi musk
#

Oh is that the one where they were given a nonsensical problem with numbers

#

That you can't answer

#

But students write down an answer anyway because we trained them to

long pelican
#

Yep, the one where 75% of the students did some operation to get an answer

#

yes

quasi musk
#

I think that's not too big of an issue to fix

#

The one that is more dangerous is the experiment where students were given faulty calculators

#

That did simple arithmetic wrong on purpose

#

and students trusted the calculators more than themselves

#

even if it was like 3*3 = 6

#

https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0223736

When calculators lie: A demonstration of uncritical calculator usag...

Calculators are often unnecessary to solve routine problems, though they are convenient for offloading cognitively effortful processes. However, errors can arise if incorrect procedures are used or when users fail to monitor the output for keystroke mistakes. To investigate the conditions under which people’s attention are captured by errant cal...

long pelican
quasi musk
#

Pretty recent out of texas tech

quasi musk
#

I don't think teaching sets is going to somehow make math less complicated when people can't add fractions

#

I mean, I'm guilty of this when I teach

long pelican
#

Sets won't help with fractions, but they're somewhat necessary for some details in algebra and later

#

Understanding how graphs transform is a big one

quasi musk
#

I go above and beyond to teach these things in short asides

#

like I'll dedicate 10-15 minutes of each lecture I give to extending what we're doing

long pelican
#

Like, seeing that the graph is a collection of points (i.e. a set of points)

#

instead of some mysterious shape

#

graph is defined as such too

#

So you need that concept of a collection of points to do any sort of reasoning with a graph

quasi musk
#

Oh I agree we should work more w/ definitions starting at about the pre-calculus level

long pelican
#

Whenever variables and functions are introduced perhaps

#

I think perhaps using a clear definition of a fraction as a point on the number line (and making it clear that pizza is just an analogy)

quasi musk
#

I guess what I mean by working with them is getting the students to think about them

long pelican
#

helps too

quasi musk
#

I always introduce them when I teach it

#

and say "If you like math, then you should think about this"

long pelican
#

Is your audience college age remedial students or high school age?

quasi musk
#

I just quit my job, but I was managing a tutoring center at a university

#

I'm gonna go back and teach at Russian School of Math

#

Which is k-12 system

long pelican
#

Is that the same place Ann teaches?

wise onyx
#

Oh you quit your job moonbears?

quasi musk
#

I'm going to try to get something in the 8-12 range

wise onyx
quasi musk
long pelican
#

Oh wait, Ann said she's in Russia, so maybe not

quasi musk
#

I worked there for like a year, got management at a university on my resume

#

I was beginning to stagnate, and I don't think I could take another semester of remedial algebra TA duty 3 times a week + managing the center

#

after spending a semester doing remedial algebra TA duty 4 times a week

#

And over the summer it was 5 times a week

#

Same class

long pelican
#

Ouch

#

Did you have a prescribed syllabus for it?

quasi musk
#

I could do whatever I wanted

#

in my TA sections

long pelican
#

If I ever did remedial algebra I'd start with examples of reading what a variable means in a sentence

#

perhaps

quasi musk
#

But I just couldn't do it again, I wanted to do other types of math

#

It was a treat to do calculus

long pelican
#

When you did calculus were there widespread gaps in fundamentals?

quasi musk
#

Like not adding fractions

#

I was working at a school geared towards a population that didn't traditionally like math

long pelican
#

Yea, it seems that way at all but the top universities

quasi musk
#

It was worse than usual

#

I had about 7 years of experience at community colleges before I took on that role

#

and most community college students were more engaged than those university students

long pelican
#

Wow interesting

quasi musk
#

In general, CC students in transfer courses

#

Are very motivated/engaged because they want to transfer out of CC into a good school

#

Usually they are students that are from lower income backgrounds that can't afford a schmancy tuition

#

e.g. my CC had an Honors Calculus curriculum that started with Spivak's Calculus

#

And ended on Spivak's Calculus on Manifolds

#

We finished the entire book of CoM, then did intro to riemannian geometry

long pelican
#

Damn that's sick

quasi musk
#

Yeah, almost everyone got into UCLA/Berkeley for transfer in Math/Physics/Engineering

#

So I definitely know curriculum programs like the one you're suggesting can be done

#

But for the average student it is overkill

#

But the average student doesn't want to have a career in Math heavy stem fields

long pelican
#

I actually also think

#

A proper treatment of math is less of a burden on memory for the average student as well

#

So even the lower-than-average students benefit greatly

#

Like, if math doesn't make sense, but you need to do well on an exam, the only recourse available to you is to memorize lots of problem types and practice them over and over

#

But if it does make sense, you can do well with much less effort

winged urchin
#

Oh boy, lots of discussion hehe

#

On the y is replaceable for f(x) thing... again

#

Do you think it would help to use many different valuable labels?

#

Like if as a teacher, whenever you wanted to draw a graph representing an equation if you just picked a random variable for the horizontal and another random variable for the vertical

#

Because normally when we talk about the normal 2d axes it is relatively common to refer to the horizontal as x and vertical as y

#

Sometimes when I tutor I do try to emphasize the use of just saying the "horizontal" distance versus "x" (or "vertical" distance versus "y")

#

Though it's certainly possible that when we're frustrated in the middle of an explanation with a student whose having a lot of trouble, we might refer to something to the x and y values of a point on a 2d plane

#

Kinda like these naughty common descriptions that are used sometimes but ultimately might create problems down the line

#

Similar to like... 'cancelling' terms or 'moving' terms

#

Oh and even there I made a mislabeling error right? Because in a fraction you don't cancel <terms> since a term is a specific definition

#

If I have 2xy/(4x^2) and 'cancel' 2x, that's not a term of either top or bottom

#

It's a factor, more formally

#

But then do we always want to write it out like

#

2xy/(4x^2) = (y/2x)*(2x/2x)

#

= (y/2x)*1
= y/2x

#

I usually only show that once or twice to try to emphasize what is commonly meant by 'cancel' in these cases

pastel sundial
#

Like I remember telling someone I struggled a lot in Latin because I am terrible at memorizing things, and they were like "but don't you do math?" As though you need a good memory for that.

cosmic ibex
long pelican
#

Well good thing I didn’t get a topology professor that tested us on these names, and for good reason too whew

#

In my experience remembering names for important concepts comes automatically with learning

winged urchin
#

Mhmm! The best concepts/theorems have names that are pretty self-descriptive

#

Intermediate Value Theorem

#

Bisection Method

long pelican
#

Here's a proof from the Stewart calculus textbook

#

Let's try to read it as a math student taught in the ways of TSM would read this

#

"To see why this is true for n = 1, we assume that |f''(x)| <= M". Honestly I think they throw in the towel at this point for two reasons

#

One is that maybe 20% of the students can't parse what "why this is true for n=1" means

#

Of the 80% that do, most or all of them will not be able to figure out why are we are assuming that |f''(x)| <= M out of the blue, unless they ask the professor

#

Also, is n a variable? No, a constant? Ok. Is a a constant? d a constant? What does a <= x <= a+d mean? Isn't x a variable? Is x changing?

#

If x is some mysterious changing quantity what does a <= x <= a+d mean

winged urchin
#

(Not to interrupt you, keep going, but just wanted to say on |f''(x)| <= M (and the other ones for other formulas) the best way I've found to explain where this comes from is drawing graphs and giving the 'hand-wavey' idea that M is some kind of measure or over-approximation to how much the function can change as we move from the centered point.)

long pelican
#

Then they read that integral inequality and they will probably just accept it as true because they have no idea how the author came up with it

#

Because they don’t know that there was logic connecting the previous sentence to the inequality

#

Then on “an antiderivative of f’’ is f’ “ I know only 57% of students understood on exam 3 that an antiderivative of a function is another function whose derivative is the first function

#

So that means 43% of students won’t understand how the textbook made that conclusion

#

So basically to understand up to line 3, you need to understand how to read letters in sentences, how to translate assumptions to special cases, how logic is used in between statements, and a nontrivial conceptual connection about an antiderivative

#

Ok I might have strayed a bit from my point

cosmic ibex
#

Is your point here that the particular textbook is bad, or that "taught in the ways of TSM" (what is that, by the way?) has harmed the students?

long pelican
#

Nah this example isn't saying the textbook it came from is bad

#

This is a college calculus textbook

#

I guess what I'm saying is that high school textbooks anti-prepare students to understand things like this

#

especially when they say variables are changing quantities, that y is another name for f(x), and they don't give examples of how variables are used in sentences in mathematical statements, theorems, or proofs to express an idea

cosmic ibex
#

Apologies, I'm still stuck on "TSM". My brain wants that to mean The Scientific Method, but that doesn't seem to make sense in context.

long pelican
#

Ah

#

Textbook School Mathematics

#

I'm finding some slides right now

#

The term is introduced on slide 13

cosmic ibex
#

I assume that is "school mathematics, as taught by such-and-such textbooks" rather than "mathematics according to the 'textbook' school of thought"?

long pelican
#

Yeah

tepid smelt
# quasi musk And ended on Spivak's Calculus on Manifolds

I know a CC in so cal that does this. At my CC you vot a wide range in classe s based on who taught it. You might get a typical engineering calculus course or see some more proof based calculus course. I am a huge fan of CC as someone who needed it as a second chance as I did awful in HS.

Russian school for math is great I am thinking of enrolling my daughter in it.

long pelican
#

So many good quotes

#

Variables belong to language, not mathematics.

Understanding of the implied quantifiers makes all of this clear, but as many students have discovered, math gets bewilderingly complex when you’re compensating for incorrect fundamentals.

Teachers should model variables as a communication technique. When variables are only introduced as part of new procedural learning such as solving equations, it’s not surprising that students connect their meaning to those procedural skills.

Quantification should be taught explicitly.
Logical quantification is sometimes viewed as an advanced topic in either philosophy or mathematical logic, and reserved for classes like [pre-]calculus or even the university level. But we can see here that quantification is implicit in much of the basic middle school curriculum! Therefore, we cannot get away with not teaching it.

quasi musk
#

It's Orange Coast

strange latch
#

Does anyone have any opinions on what fresh college students are lacking/what skills you wish they learned in high school?

long pelican
#

Yep, basically every post of mine in this channel is about that!

lyric fractal
#

so this is an issue thats been on my mind for a while and i wanted some feedback
im not a tutor just a sophomore in pure math
my professor who is teaching metric spaces and basic topology rn has this system going during the week
2 days--> sends lecture +1 day--> goes live or meet with us
and during the meeting he tries to work with the students as we have to rapidly figure out the questions and how to solve them as he picks a person each question and lets them do the work and a lot of us are put in this awkward spot of "i need some time to think" so is this ultimately beneficial for the students to be able to work on the spot and under pressure or is it perhaps counter productive
i suppose one might argue the student has a responsibility to prepare beforehand

wispy slate
novel kraken
wispy slate
#

@novel kraken well no one came for 2 meetings after that, making nearly half of the meetings this entire semester no-shows except for myself
so I announced that I'm closing out this iteration of the club and that anyone can restart it with the office if they wish

novel kraken
#

ah that's a drag

long pelican
#

Damn this final was brutal for these students. The hardest problem was to give a correct interpretation of the value of a probability density function, and no one got it, which is disappointing but normal

earnest trail
earnest trail
long pelican
#

Well these are non-math majors

#

One of the hardest problems was surprisingly whether $\lim_{a\to\infty}\int_{-a}^a x^3,dx$ converges

burnt vesselBOT
#

Icy001

143174781561339904.png
long pelican
#

This problem was designed to see if people would use shortcuts or rules they've learned (which they would conclude it diverges because improper integrals were emphasized a lot), or actually read the notation

#

Evidently there was some trouble still, which makes sense as this class is the first time in their life they were expected to actually do math via reading notation instead of applying procedures

winged urchin
#

Wait... what do you mean diverges? You mean incorrectly get that?

long pelican
#

yep

winged urchin
#

Okay I was scared for my understanding for a second there ahah

#

How did they manage to get that one wrong? o.O

#

It feels like if you just calculate the it, you should see it cancels and then you get the limit of 0

long pelican
#

Because the notes said $\int_{-\infty}^\infty x^3,dx$ diverges, WATCH OUT, COMMON MISTAKE TO SAY IT CONVERGES even if it's an odd function

burnt vesselBOT
#

Icy001

143174781561339904.png
long pelican
#

(and also because the definition of $\int_{-\infty}^\infty x^3,dx$ is not $\lim_{a\to\infty}\int_{-a}^a x^3,dx$, which converges)

burnt vesselBOT
#

Icy001

143174781561339904.png
long pelican
#

But of course they remember the rule but not why it's true

winged urchin
#

Right of course

long pelican
#

nor how to read the notation for themselves

#

even though it's a goal of the class to be able to read notation for yourself :c

real mauve
winged urchin
#

Is it really? o.O

#

In what field?

#

Whenever I get students making that mistake I try to remind them that those infinities really represent limits and we know they're going to infinity (or -infinity) but we don't know how quickly

#

Or take an example like say

real mauve
#

at least in engineering

winged urchin
#

integral from -a^2 to a of whatever

#

And we can then take limit as a goes to infinity and we observe -infinity to infinity

#

But we can take specific values of a and realize the integral is not symmetric about 0 as we take this limit

real mauve
#

that makes sense, indeed

#

i had never seen that other than explicitly, putting infty as an integration limit always implied lim a->infty of a, or written explicitly some other way if that was not the case

#

very curious. nevertheless, if this was covered in class (and it definitely sounds like it was), it should've been ok

#

about the pdf, i'm guessing it was a continuous one? (qgain, not all resources i've read use different names for the discrete case)

long pelican
#

Yes it was a continuous pdf

#

I can show the most common wrong answers

#

This is something that probably indicates they didn't even read the first sentence of the question...

unknown.png
#

Another one like the previous

unknown.png
real mauve
#

and you were looking for is something like the derivative of the cdf, or?

long pelican
#

$\lim_{h\to 0}\frac 1h\Pr[\text{waiting time is between }10\text{ and }10+h\text{ seconds}]$

burnt vesselBOT
#

Icy001

143174781561339904.png
real mauve
#

right

long pelican
#

That's also the derivative of the CDF, but there are a bunch of ways to get there without using the CDF

#

for example, $\int_{10}^{10+h}p(x),dx\approx p(10)h$ for small $h$

burnt vesselBOT
#

Icy001

143174781561339904.png
long pelican
#

which leads to the same answer as above

#

It was definitely a hard conceptual problem (as opposed to a problem that uses a clever computational trick or whatnot) which uses understanding of derivatives and integrals in a somewhat deep way

astral laurel
long pelican
#

Well as first semester freshmen between you and me they're non-anything

astral laurel
real mauve
#

i'm not sure i followed :x

astral laurel
#

I'm not too sure I follow why it diverges either tbh

#

Anyway I will 'teach'(?) prob/stats-ish applications next semester so wish me luck people... I offered to be 1st line of contact for questions though I wonder how much they will trust me to email me questions

long pelican
#

It's because $\int_{-\infty}^\infty f(x),dx$ is a notation specifically defined by convention to mean [\lim_{a\to\infty}\lim_{b\to-\infty}\int_b^a f(x),dx] or equivalently [\int_{-\infty}^c f(x),dx+\int_c^\infty f(x),dx] for any choice of $c\in\bR$

burnt vesselBOT
#

Icy001

143174781561339904.png
astral laurel
#

So that's -infty + infty I see..., and the single-term case is 0 by odd-functions... interesting

long pelican
#

Ye

real mauve
#

i think if i sat down to take these exams rn i'd probably fail them haha

real mauve
long pelican
#

Funnily enough, like 1 hour ago I went to check wikipedia on improper integrals and it gave as clear a definition of a doubly improper integral as one could imagine

real mauve
#

i have learned a lot of bad notation NervousSweat

#

seems what is often used in eng is the cauchy principal value. good to know

#

were these the only troublesome tasks for the students?

#

if so, there seems to be a common theme

long pelican
#

There were more

#

Let me see

#

Part (b), correctly identifying not defined at x=0 but not realizing this implies not defined for x < 0 either (because the integral in the case x < 0 crosses through 0)

unknown.png
#

For part (c), seeing that f(3) = 0 but not proving that 3 is the only zero

#

In this one, lots of people didn't realize the "room temperature" in Newton's law of cooling is changing with time and wrote either 70 or 450 for it

unknown.png
#

In part (b) here, the responses were very strange. They either wrote a single integral from 25 to 30, or a sum of two integrals, one from 25 to 30 and one from 55 to 60, and stopped there

unknown.png
#

When in reality the answer is an infinite sum

#

I think that's the major issues

#

Top score on this final is looking like 85%. Quite a doozy, really stretched their brains for sure

real mauve
#

aha

#

tbh 85% sounds pretty good for this to me

#

at least out of context (without having taken your course), i assure you most university staff here (phds and profs in engineering stuff) would flunk it

long pelican
#

Ooooh

real mauve
#

i'm honestly pretty upset that checking your exams, hw, etc. always brings up one or more misconceptions i have and am probably perpetuating. something is pretty wrong with the education pipeline i have gone through, myself included

long pelican
#

Damn, the education system really does perpetuate misconceptions to the vast majority of people that go through it

#

Well pre-university that is

real mauve
#

that aside, limits, integration, and probabilities are topics that are classically accepted to be difficult for undergrads, so your findings on this exam are not outlandish

long pelican
#

Was that the common theme you were going to mention?

real mauve
#

ah the common theme is integration

long pelican
#

Makes sense. Math is purely a tool when it comes to engineering

#

For an engineer, whether you understand what you're doing when solving for x or whether you don't, makes no difference

real mauve
#

for whatever reason, integration often seems to get completely separated from the underlying concepts, it turns into some magical sui generis operation

long pelican
#

It only makes a difference if you want to build on that knowledge mathematically-wise

real mauve
#

at every step of the way i have wished i had studied math instead, but it was impossible given where i did my undergrad. it has just gone downhill from there. oh well, more reading to do

long pelican
#

You're doing pretty well I'd say. That improper integral thing is very minor compared to the bigger issues people have such as not knowing mathematical notation is meant to be read (Lol)

real mauve
real mauve
#

maybe as an option to salvage a point or 2

long pelican
#

Honestly when I think about why they truncated it, I think the underlying issue is something like never having learned to generalize in math class

real mauve
#

that may also very will be the case, i wouldn't expect that to be common in HS

tawdry venture
#

i still have that kid in math club insisting that the expressions 4 * x and x * 4 refer to "different processes" and hence cannot be used interchangeably

#

i hate to admit it but i have been unable to convince him

#

and based on his behavior last class it sounds kind of like he's trying to be pedantic on purpose

real mauve
#

what was their argument?

#

also different processes can give the same result, that feel when 2 + 1 \neq 1 + 1 + 1

tawdry venture
#

their argument was that i "wasn't writing things down correctly"

#

or something

#

that and the "different process" thing was all i got from him

real mauve
#

do you think they might be satisfied by showing the commutativity of multiplication of naturals?

real mauve
#

though on second thought, it seems to be more related to a misconception regarding what "equality" means

native hemlock
tawdry venture
#

they acknowledged it gives the same result, to which they agreed.

#

but they argued that the process was different and i must always make sure the things i write correspond 100% to the things i say

real mauve
#

then it seems it's more an issue with equality and equivalence

#

we had some members here that complained 1+1+1 \neq 3*1 because the "operations" or "symbols in both expressions" are not the same

#

so maybe some discussion about equality and different mathematical objects can be useful

#

presumably you're dealing with real numbers, so one would like to establish their equality (though probably with naturals for simplicity)

long pelican
#

Probability with continuous random variables was one of the units

#

Yes I agree with Edd. Some avant-garde people like to separate equality and equivalence and I see some disasters like this one here

#

But good news is that your student doesn’t seem as clueless as I first imagined

cosmic ibex
#

The whole point of algebraic rewriting is to discover when different processes yield the same result and put a = between them.

native hemlock
#

I’m taking calc 1 now and I definitely should be able to answer your question 3, but I can’t. For part b, I know that a function is integrable on an interval if it’s continuous on the interval or has only finitely many removable discontinuities, but idk if the converse holds. And for part c, i don’t even know how to determine if f(x)=0 for any x that’s not 3. Maybe put it in closed form with the FTC and then set it equal to 0 and try to solve for x? I worked so freaking hard at calc this semester and I’m still a complete failure at it

#

I just took my final and it didn’t go great either

long pelican
#

Don’t feel bad, the parts of question 3 were intentionally in increasing order of difficulty

real mauve
#

maybe some argument like the integrand being nonnegative, so f(x) is nondecreasing

cosmic ibex
#

The integrand is actually positive everywhere it is defined.

real mauve
wispy slate
pastel sundial
#

Oh wait is it that you don't go to both sides simultaneously in an improper integral

long pelican
#

Yes

pastel sundial
#

I guess the point is you can get to $\int_\infty^{\infty}$ in multiple different ways

burnt vesselBOT
#

Kanga Gang Mole (sleepy agent)

381174875215364109.png
pastel sundial
#

Like $\int_{-a}^{3a}$

burnt vesselBOT
#

Kanga Gang Mole (sleepy agent)

381174875215364109.png
pastel sundial
#

Pardon me as I forget both TeX and basic analysis

#

Also Icy holy shit that exam looks brutal

#

I assume you've going to be curving the hell out of it since the top score was 85%. Did you warn the students ahead of time that the exam was going to be very difficult and likely curved?

long pelican
#

Hmm I expected it to be similar to the midterms in difficulty lol

pastel sundial
#

How difficult was the midterm?

long pelican
#

Averages on the 3 midterms were 75, 66, 75

#

Average on this final is looking like 63

pastel sundial
#

This looks like a solid calc 2 exam from a universe where math education was good

long pelican
#

Ikr lol

#

I made sure every problem didn't just test a procedure they study and forget in a week, yet is not anywhere approaching Putnam-tier difficult

#

well my problems at least

#

The other professor put 3 problems which are, between us, free points

pastel sundial
#

Meanwhile Putnam this year: use l'hopital's rule

long pelican
#

I'm happy the students got to see what matters in calc 2 rather than procedures that artificially boost appearance of understanding, and even if the average on the substantial problems was 50% I think it's significantly better than what they would have averaged on a similar calc 1 final exam, so it's a good result for now

pastel sundial
#

But yeah I think this being a fair exam would mean you managing to correct 12 years of garbage math education.

long pelican
#

Bring it on

pastel sundial
#

Seems cool if you curve it though

#

I just feel kinda bad for your students

long pelican
#

I'm eyeing an A to be like 68%

pastel sundial
#

Not in terms of the math they learned, but in terms of how scary that exam would be to them.

long pelican
#

Hehe, I did tell them halfway through the semester that getting 70 on an exam is very good

#

It hit them the hardest on the first exam

#

when they had no idea to expect something like that

pastel sundial
#

Ok then that's better

#

Don't want to traumatize your students in the process of teaching them actual math

pastel sundial
long pelican
#

after, unfortunately

#

It influenced mid-semester feedback a lot: I got a lot of complaints that I wasn't teaching problems that would appear on the exam, that the stuff I taught was "relevant but too general to be helpful"

pastel sundial
#

What influenced mid semester feedback? The exam or saying a 70 was good.

long pelican
#

The exam

#

I said 70 was good somewhere before or after the 2nd midterm which was well after the mid-semester feedback

pastel sundial
#

Ah

#

I guess it all works out if you're very generous with the curve

long pelican
#

Yea

#

I don't ever plan to be that guy who fails half the class

pastel sundial
#

I feel like the students who end up with an A will feel very good about it

#

How are grades weighted? Like what percent is homework, exams, attendance, etc

long pelican
#

standard pretty much
15% webwork, 15% written homework, 10% labs (free points), 40% midterms, 20% final

pastel sundial
#

Also Icy I will say this. I wish I could have been in your class

#

I self studied into calc and tested into calc 3. But I think I would have gotten a lot out of calc 2 with you.

#

As in, after my self study so I would technically be "repeating" it

long pelican
#

I wonder if it would've been too easy for you

#

I spent some time practicing reading notation

#

which is like, probably under your pay grade

#

Things like discussing whether |-x| = x for all real numbers x

pastel sundial
#

I definitely would have had an easy time with it, but I had an easy time with all of my classes at that time

#

Took calc 3 junior year of hs, didn't have a difficult math class until my algebra reading course freshman year of college.

#

But I think I would have gotten more out of your class than I got out of calc 3.

#

(The class didn't get to any of the interesting parts and was basically just multiple integrals and partial derivatives all semester)

#

And has basically no actual theory

long pelican
#

One thing I didn't talk about was theoretical aspects of differentiability and such

#

If that's what you're considering theory

#

I wonder if talking about those things is a common mistake new professors would make

pastel sundial
#

I just think my mathematical maturity could have been better at that time

cosmic ibex
#

That's generally an unavoidable effect of maturing: lamenting that it didn't happen earlier.

native hemlock
#

@long pelican would you be ok with posting the exam questions or dming them to me? I just want to use them as personal practice problems

long pelican
#

Makeup exam with the same questions is happening on December 20, so it'll have to wait until then

native hemlock
#

I’m guessing you didn’t make up the hw problems right?

long pelican
#

mmmm about 2/3 were old and 1/3 are new

native hemlock
#

If they’re similar to your exam questions and you’re willing to share them, I’d love to work through some of them

#

But no worries if you’d rather not ofc

long pelican
#

I'd be curious to see how easy or hard you find this from the latest homework

unknown.png
native hemlock
#

Problems with (a):\

  1. $\int_a^b |f(x)|dx$ may not be defined. e.g. take $f = x\mapsto \begin{cases} 1, \quad x \leq 0\ 2, \quad x >0 \end{cases}$, $a = -1$, and $b = 1$. \
  2. $|f(x)|$ does not mean that f(x) is always bigger than or equal to 0. it means that $|f(x)|$ is always bigger than or equal to 0
#

goddamn it

burnt vesselBOT
#

EdgarAlnGrow

407696154034438145.png
long pelican
#

Yeah #2 is the main problem with (a). #1 is also right but "outside the scope of the course"

native hemlock
#

wow beautiful typesetting

long pelican
#

although your example for problem 1 doesn't actually break it

native hemlock
#

is that true if we're talking about so called "riemann" integrals rather than "lebron" integrals, or whatever?

#

lebesque

long pelican
#

Doesn't matter

native hemlock
#

note: i have no idea what the difference is. i just heard that the first aren't defined when there are discontinuities which aren't removable

long pelican
#

$[x\in\bQ]$ is the classic non-Riemann-integrable function

burnt vesselBOT
#

Icy001

143174781561339904.png
long pelican
#

although it is Lebesgue integrable

#

(Definition: $[P(x)]$ is 1 if $P(x)$ is true and 0 if $P(x)$ is false)

burnt vesselBOT
#

Icy001

143174781561339904.png
native hemlock
#

hmm

#

above my pay-grade

#

anyway

cosmic ibex
#

(b) is devious.

native hemlock
#

for (b), saying that f(x) = x^3 + sin x is just shortand for saying f = {(x, x^3 + sin x) | x \in R}. you can swap out the x for a y and you'll still be talking about the same set of points

long pelican
#

Yep that's one way to say it

native hemlock
#

for (c), the domain is a set of pairs of numbers, not a set of numbers

long pelican
#

exactly right!

native hemlock
#

(d) is something that's always confused me. I think the idea is that, for every fixed C, you're defining a function f(x) = x/ (x-C). so since x isn't a fixed number, you can't take C = x

#

it's like backwards

long pelican
#

exactly!

native hemlock
#

idk how to say this precisely

cosmic ibex
#

It's basically that in a function definition we always understand an implicit quantification of the dummy variable, so it doesn't even make sense to ask whether C=x outside the definition -- x doesn't exist there.

native hemlock
#

for (d), g(x, y) is in R, not R^2. so f isn't defined on it

cosmic ibex
#

I hope Icy has mentioned that implicit quantification explicitly in the class -- usually it's something students are expected to sort of absorb by generalizing from usage examples on their own.

long pelican
#

Yeah, implicit quantification needs to be talked about a lot more

#

from when functions are introduced in middle/high school, or possibly right when variables are introduced

#

Students are anti-taught implicit quantification whenever teachers talk about variables as mysterious things that change or co-vary with something else, or something else weird

#

And it makes so much of math unreadable for them

long pelican
#

Dang, that homework was super easy for you

#

It was extremely hard for my students

cosmic ibex
#

I'm not sure it's only visible variables that suffer from it. Even solid, respected textbooks in logic often attempt to convince students that \rightarrow in propositional logic deserves to be pronounced "if ... then ..." in English -- whereas I've never seen an honest prose if-then outside logic teaching whose correct formalization didn't involve an implicit \forall of some sort around the implication.

long pelican
#

Statistics below

unknown.png
native hemlock
#

Well i started learning math by reading Paul’s online notes for hs algebra, and Vellemans “How to Prove It” side by side

#

And Velleman gives super pedantic definitions for things like functions and domains and whatnot

cosmic ibex
#

Interesting how (e) scored lowest, it's the one that has the most syntactically nonsense in it.

native hemlock
#

I don’t think the questions are hard. You just need to know a bunch of pedantic formalism that you don’t normally learn by the time you’re in calc

long pelican
#

And you probably read those and intended to learn them, as opposed to students in schools where the big definitions are just glossed over in favor of the teacher's hand-wavy interpretation and procedural approach

long pelican
#

I'm not sure if I would call it pedantic... these questions specifically might sound pedantic but lack of ability to see what's wrong with these statements kind of makes math unlearnable

cosmic ibex
#

They seem like fair questions to ask, because students are definitely expected not to make those mistakes themselves.

long pelican
#

Yep!

cosmic ibex
#

Hmm, I'm trying to figure out just how I would explain intelligibly that in f(x)=x³+sin(x) there's an implicit quantification around f(x)=..., but if I write

a fixed point of f means a point where f(x)=x
then there obviously isn't. (It would be slightly easier if I had said "a point x", but I'm not always that precise).

long pelican
#

One is defining the function, the other is a statement about the function (after it's already defined)

cosmic ibex
#

Yes, so if one already knows what I'm saying then it's indeed obvious.

real mauve
#

i would really make sure to mention the curving from the start. some people can be entirely driven away from a topic thinking they suck at it

long pelican
#

Honestly we just didn't have any idea what the distribution would look like just looking at the problems on the first exam

real mauve
#

that's true enough

#

just for future ref

#

but also if you expect peoplwle to be good and score 70%, the exam could be made differently. and well, i just happen to dislike curving in general. i know it's useful because one should somehow adapt the grading scheme to the students' level but also immediately knowing everyone will score 30 points below max kinda means (to me) the exam should've been designed differently

#

it's my main complaint with the german system, where 100% of the grade can come from a single exam and you get stuff like passing with 20 and 50 curving up to 90~100

long pelican
#

In a subject that isn't factual recall-based, the kind of exam where the mean is 85% is also the kind of exam that encourages studying by memorizing problem types. So you can't win in this philosophy

real mauve
#

many students explicitly said they avoided doing their research with the staff members responsible for this, even when they got grades of B and over after curving cuz they lacked confidence in the topic

#

i know, that's a fair point

#

i don't have any great solution for it yet

#

but it really crushes some students

long pelican
#

Does it crush confidence unfairly though?

#

I'd expect someone who understands the subject to get A's after curving

#

In this class I can't think of any particularly strong students who have a final grade of B

#

let alone any that would potentially be interested in research in a math-related field

real mauve
#

ah but that is what i mean

#

students that get A, but know they got several answers wrong of left questions empty

long pelican
#

So they're used to solving every question

#

Maybe it's good to rid themselves of the mindset that you're not good unless you can solve every question

real mauve
#

when you consider a context involving international students from several backgrounds, yes

#

and you're also correct in what you just said rn

#

but dealing with these things when you have students from like 10+ different countries clustered together really becomes challenging

long pelican
#

I bet America is one of the tougher countries to handle...

real mauve
#

the countries where the minimum passing grade is 70 or above tend to be tough to deal with

long pelican
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In the UK, 70 and above is first class honors and 80% is extremely rare

real mauve
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yeah

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european and uk students are already used to the system

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but for others it's a huge culture shock

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i think in some asian countries there is little or no curving as well

earnest trail
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curving is ehhhh

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if you feel the need to curve the test ahead of time, the test is too difficult for the students

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so you should make it easier

long pelican
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Why do we need to make numerical grades conform to 70-80-90?

earnest trail
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now if it's after the test and you realize you need to curve that's a different story

earnest trail
long pelican
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There is no need at all except for the arbitrary idea that 90 is A

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If someone decides that getting 60% of problems right is excellent and deserves an A, then what's wrong with that?

earnest trail
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yeah that's true

long pelican
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90% being A sounds appropriate only for a highly recall and procedure-based class

earnest trail
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because in college, professors can decide their grading scales right?

long pelican
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Yes

earnest trail
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yeah so you're right

earnest trail
real mauve
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those are all fine, as long as the students know ahead of time

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but even the wording and formulation of the grading scheme makes an impact

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you could equivalently say that there are 130 points up for grabs in the exam, but you cap out a 100

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and that would probably be received more positively in spite of being equivalent to setting A level way lower