#math-pedagogy

1 messages · Page 49 of 1

grand laurel
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if they get a double check from a peer, they are less likely to be afraid of being exposed by a "dumb" question

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also give positive reinforcement when questions are being asked

round robin
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oo thats a good idea but discussions online are a bit the T_T
sounds quite good in person tho

real mauve
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breakout sessions moment

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i actually enjoy those in language courses, but they only work if students talk

grand laurel
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oh yeah, breakout rooms on zoom are hit or miss in my experience

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when students already know each other, they usually work fine but otherwise its hard

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our approach was to offer non-class social meetings for people to get to know each other beforehand so there are no awkward breakout moments

shadow basalt
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practice, get it wrong, try again

round robin
round robin
dawn walrus
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Thoughts on lying to people when you're tutoring?

civic tree
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Tastefully.

novel kraken
dawn walrus
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This is just a joke from secret club land

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Ok

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A half joke

novel kraken
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oh I keep that channel muted

dawn walrus
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Lmao

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Wise

novel kraken
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secret crib always full of crying

dawn walrus
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Anyway I was talking about how sometimes when I'm helping a friend with math problems, in specific cases, I will sort of pretend to not know the material super well and act like I'm working through it with them collaboratively

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Especially if we're in the same class

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But even if I took the class and can plausibly say I have forgotten the details

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I just think if you make someone feel like an even collaborator it can really boost their confidence

novel kraken
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depends I guess on what you want to get out of it and your mood at the time

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seems fine to me

dawn walrus
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Yeah, it has to be done very enthusiastically I think.

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Like

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If the person realizes you're doing this

novel kraken
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as long as you're not using it as a way to like pretend to get the answers fast your first try lol

dawn walrus
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It's way way more harmful to their confidence than if you just help them outright I think lol

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Well my goal would be to get them to get the answer and feel like they got it themselves mero

novel kraken
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often times when studying with other people I'd just get into a role of explaining things and teaching it to them and they would enjoy it and my enthusiasm would be contagious, and I prefer it that way since I feel like I learn through talking

dawn walrus
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Yeah

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I do that too

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But like, I guess if someone is explicitly asking me for help

novel kraken
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I don't have all the answers though so I'd collaborate too, I mean

dawn walrus
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I try to keep from being too precise and seeming like I know the exact way to do it

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So that I encourage them to help me fill in the gaps

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Or even to get the main idea of the problem, if I sort of hold back from giving the full main idea

novel kraken
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even if I knew the answer I'd be more inclined to just give helpful hints but some people are less motivated like you're saying if they know you're just holding out on them

dawn walrus
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Yeah I think it has to be done pretty carefully. Especially face to face.

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I was thinking about this because a while back Dami and I were looking through some slides and I was not super knowledgeable about them but maybe came across that way, and he commented that he wasnt sure if I was just pretending to not know the material

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And I realized today that I actually do this intentionally with people

novel kraken
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well what do you mean carefully, I guess I don't operate the same way so I'm just not in the same context

dawn walrus
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Just like you can't let the person have any clue you're pretending or it's gonna feel super patronizing

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At that point it's better to just have been honest that you know the answer and to give hints

novel kraken
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I'd probably just shrug it off and say "you caught me" and smile and laugh

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just admit you were wanting to help give hints without being too obvious about it to try to help them, it's well intentioned

dawn walrus
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No no no. The web of lies must grow.

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I will tell them that I'm actually being threatened by a local mafia to pretend I don't know differential geometry of curves and surfaces.

novel kraken
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lol

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I don't think I have the energy to lie that much

frosty flame
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Hmm, I've never thought about lying about not knowing how to do it. It doesn't seem necessary. Like I can give partial hints and ask probing questions from a helping perspective without pretending I don't actually know how to do it. And I would be uncomfortable if I realized someone was pretending with me.

shadow basalt
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I am tempted to mostly agree with luna here

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in particular about being embarrassed if someone was pretending with me

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and also about it being unnecessary. theres nothing wrong with openly not giving someone the answer and expecting them to think about it themself

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(this is arguably what teaching is all about)

dawn walrus
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sure, but especially if you're talking about a classmate who wants help with a problem, they don't always want to be taught

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i could definitely just say "screw what you want, here's what you're getting"

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but if i do the thing i'm talking about, i find that they start to get things themselves over time and after a little while we're totally on the same level anyway

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and if i do the straightforward thing, people just kinda.... don't change? from my experience at least

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whether or not it's necessary in the moment isn't really what i'm worried about

shadow basalt
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oh with classmates im a lot less pedagogical

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maybe thats a bad thing

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i only have the energy to teach people well if they are a few levels behind if that makes sense

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i normally just give answers to classmates idrc

dawn walrus
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the thing is, i kind of was doing this subconsciously and was never putting much thought into it, and i think that helps. and there's always the excuse of "oh, sorry, i've just mostly forgotten this stuff / i just wasn't thinking carefully about it"

dawn walrus
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like i wouldn't do it if i were teaching someone calculus, that would be too obvious

shadow basalt
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i think that like

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in person

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i give off kinda dumb vibes

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people rarely try to get me to help them

dawn walrus
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here's what you do

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you pretend to not know what's going on

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then when they solve it through your gentle, imperceptible nudges

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you say something condescending about how the problem was obvious and how you should have gotten it immediately

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then they think you're coping big time, and feel smugsmug

quasi musk
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As any good play, I have about 3 acts a day

feral vector
# frosty flame Hmm, I've never thought about lying about not knowing how to do it. It doesn't s...

Although, as per my experience, the attitude of the person asking the question changes if you pretend to be dumb as well. For example, when I was helping out a friend, in the topics he was sure that I knew the answers, he was more inclined to give up, so I will just jump in and give hints until I basically lay out the answer. However, at other times, when he felt I didn't know the answer, he would himself get into an investigation mode and suggest possible strategies to approach the problem, which I guess was much better for him.

orchid bane
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I might be a bit late and I don't know if it's relevant, but this year one teacher was doing something I found interesting ; he was often doing small historical "breaks" inside his course (talking about the mathematician's lives, historical fun facts with cool pictures), often after a big proof/result, thus giving us time to digest the material

I don't like when it's too slow because I get easily bored nor when I don't have time to understand what we're talking about

This felt like a great compromise because it allows everyone to follow the course / catch up if they're a bit late / digest and rest / not be too bored

uneven hemlock
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It depends on your audience IME, and on how good the class materials are and whether lecture is recorded

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People are a lot more willing to handle fast paced lectures if they dont feel like they’ll miss something important

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Plus encouraging frequent questions helps keep you from getting in a rhythm and going too fast. In class exercises are good too

uneven hemlock
lofty garden
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Hey so I'm writing this lesson plan thing for my education class and I'm struggling a bit here with this one standard. Both C.D.2.a and c mention logarithmic differentiation. So when I bring up Log differentiation, which standard should I use?

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shadow basalt
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It seems like as far as logarithms are concerned the two standards are identical

lofty garden
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Weird

shadow basalt
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students are expected to understand/know the log derivative and apply it

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so it shouldnt matter ig

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probably just a lazy writeup

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that wasnt proofread well

lofty garden
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That's south carolina for you :p

winged urchin
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Personally I feel like both logarithmic and implicit differentiation are kind of misnomers

shadow basalt
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Although its possible that "logarithmic differentiation" and "derivative of logarithm" are different here

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but i havent taken calculus in years

winged urchin
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As the actual differentiation is the same in those and in normal 'explicit' differentiation

shadow basalt
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Gemini do you know what log differentiation is referring to here

lofty garden
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Maybe but I'm not gonna stress too much on it. It's a minor part of the lesson.

winged urchin
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Yeah

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Or at least.. I'm 99% sure

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Log differentiation is the use of a logarithm applied to an equation before differentiation

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The effect is to either make the differentiation simpler or in some cases possible at all (barring some clever tricks)

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Like with say y = x^x

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Power rule only work when x is just in the base

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And exponential derivatives are for only when x is in exponent

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So you have no rule for x^x

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And no obvious chain rule or whatev fixes it

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There is a trick but by log differentiation what we do is apply logs on both sides to get

shadow basalt
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I think the chain rule does fix it if you are careful enough?

winged urchin
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ln y = x ln x

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If you think chain rule is applicable then tell me what functions compose to make x^x

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I think you will be unable to find functions which do that

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But anywho, so you get ln y = x ln x after applying logairthms

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And then you can differentiate that with elementary rules

shadow basalt
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after reading a bit I guess the issue is that the construction of x^x as exp(x ln x) is what is happening but thats exactly what you see there

winged urchin
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By the way, just to be complete. You are correct in a round about way for chain rule. That's the sorta clever trick

Basically if you take

f(x) = e^x
g(x) = x ln x

Then you could see that f(g(x)) to get x^x

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That's the trick ya

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But that is NOT log differentiation imo

shadow basalt
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But I agree this is not elementary nor a good way to expect a calc student to approach it

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so I see why that trick is handy

winged urchin
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Honestly I try to really focus on how one can manipulate equations to enable techniques we are familiar with

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If we cannot differentiate y=x^x for whatever reason then we must think how can we change the equation to enable us to take a derivative

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To add as well. Like I said log differentiation is also used to make differentiation easier

One might consider

y=((x+3)(x-2))/((x+1)(x-4))

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To approach that derivative normally as written we'd have to use quotient rule and two product rules

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Now one might think. Oh I can change this and expand the top and bottom and avoid the product rules

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But further if one thinks... Well logs turn products and quotients into sums and differences which are much nicer to take derivatives of!

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So they try to apply logs to both sides and simplify and end up with four relatively simple log terms to differentiate

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That is log differentiation

shadow basalt
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cool!

molten urchin
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Has anyone used Manim for math videos before? What has your experience with it been like, and how much background do you think one needs to get it to work (with respect to programming, etc.)?

kindred stag
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I made a small ~30 second Manim video before and it wasn't that technically advanced (I didn't have to use any graphs or anything). But my impression was that as long as you know python, like you'd learn from first semester course, you shouldn't have too much of a problem.

molten urchin
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I see. I'll probably have to pick up some Python. How long did it take you to create that video?

kindred stag
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It took me several hours over the course of a couple days probably, but most of that was just figuring out how Manim actually worked. So the actual creation process was probably only an hour or two.

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Also, its probably important to mention that my video didn't have any narration, adding narration means adding a lot of time where nothing much happens in the animation. So I probably could've doubled or even tripled the length of my video by adding narration, without much additional code.

molten urchin
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Aah

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That makes sense

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Thanks for the information!

vagrant meadow
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any advice for tutoring someone who is taking a class way beyond their level?
someone came in with multivariable calc questions but struggled immensely with basic algebra. it was a frustrating slog for everyone involved. it didn't help that they kept insisting i help them with the calculus part of it which wouldn't do them any good if they couldn't do the algebra (so basically asking me to do the problem for them).
i have no idea how they passed their previous calc courses or if it had just been years since they did any math.
so yeah, any suggestions?

real mauve
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i would be honest and suggest to spend some time refreshing concepts

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same old proverb of building a house on sand

sinful thistle
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also try understanding why they don't want to spend some time on the basics if you tell them that and their answer is just "ok but now help me with mvc" hmmCat

shadow basalt
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oh i mean that part is kinda obvious

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if all you want is to survive calc and never take math again

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makes sense to not want to build a strong foundation, you probably want to learn as little as possible

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but yeah, nix, just be honest w them

sinful thistle
shadow basalt
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I'll put $10 on it

winged urchin
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In my experience as a tutor I have definitely had to deal with students with shaky knowledge bases

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And sometimes they just didnt give themselves enough time to properly prepare

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But they still are paying for some degree of help

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In those cases, you do have to compromise some

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Or you just don't take those students, depends on how principled you are I suppose

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I find practice exams (hopefully made from the current teacher) are the best to look at in those cases

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Those questions have the highest probability of showing up again in some on the real test

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Ideally you can then ask how they would approach certain questions

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But in some especially rushed situations, all I can do is just work through the problems to show them how one could approach the problems

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By no means is this ideal of course, but you work with what you have given or you don't take it at all

small quartz
vagrant meadow
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yeah i guess its a different situation when they come and personally pay me my hourly rate versus when they just come into the tutoring center i work at for free math help for specific questions. with the former im more open to just start going over some basics first since its their time, but with the latter idek. i do just want to be honest and say "we cant help you" but that doesnt feel right i guess

vagrant meadow
uneven hemlock
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personally i have no problem cutting off students who don't want to think for themselves

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5 more minutes spent solving the question for them equals 5 less minutes spent helping other people

wispy slate
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I think you need to do both... show some examples, go through problems step by step and then ask the student to do some on their own.

shadow basalt
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Does anyone have a favorite way of writing blog-post style stuff with latex (and preferably tikz?)

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I tried wordpress but the $latex thing annoyed me too much

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Maybe I should just embed pdfs or something

real mauve
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can whatever you're writing for display jupyter notebooks correctly?

shadow basalt
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it would be for a website but idk what that is

brisk ruin
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I don't think jupyter notebooks can do tikz?

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Jupyter notebooks are a way of having code/latex/writing all in the same document so you can have them interspersed and stuff

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They're pretty useful especially when you want to have math implemented in code and then you can write up the math alongside the code and stuff

real mauve
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i think you can just copy paste the tikz chunks right there, just like you do here

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not very organized

shadow basalt
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im just writing expositoryu stuff

brisk ruin
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Jupyter notebook definitely doesn't seem like the correct tool for this

real mauve
brisk ruin
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If you're on a MacOS you can use LaTeXit to export latex stuff into png and then embed that into your blog post?

kindred stag
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I have no familiarity with it besides knowing that Math SE uses it, but would MathJax work in this context?

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Oh wait you need tikz not just equations

brisk ruin
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Yeah you can't use tikz on MSE

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MathJax/Katex both don't support tikz

kindred stag
round robin
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diagrams urgh

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i personally convert to svg

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and throw it in

shadow basalt
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sad

uneven hemlock
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walking through similar exercises, giving them the first few steps to get started, etc. there is a real effect where people will do less thinking for themselves if they around others who know the answer though, and if i feel like that is happening i tell them to think about it on their own and move on to someone else

limpid dirge
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3rd year in uni (senior next year), planning to start tutoring lower years. i often help/tutor friends that need help with university courses (IT uni, not just mathematics) and i've gotten positive feedback, so i'm thinking of expanding upon that as a side gig alongside studies. any tips on keeping tabs on multiple people and their progress at a time? i've only had to deal with no more than 4 students at a time

along with any experience as a tutor/teacher you may have, it would help out a lot

uneven hemlock
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what country / area do you like in? the types of tutoring that are in demand can be quite dramatically different depending on that

unreal onyx
# shadow basalt Does anyone have a favorite way of writing blog-post style stuff with latex (and...

I've looked into different ways of doing it, and it seems a lot of people opt for tex4ht, which uses the latex compiler itself to convert .tex to .html and should be able to handle everything... but the output looks terrible. It tries to render the math half-way, then let mathjax do the rest, but it's a really unhappy middle.

I've instead opted for pandoc, which is a separate tool that is way less smart when it comes to latex, but it can handle basic equations, amsthm, and that sorta stuff. The nice thing about pandoc is that you can write filters in it (in haskell or python or whatever floats your boat), and you can overcome most of its shortcomings that way. The second answer here mostly works for me https://tex.stackexchange.com/questions/431719/how-to-use-pandoc-to-derive-output-from-latex-and-tikz-to-a-docx-file

I've swapped out ImageMagick's convert for inkscape's command line utility
call(["inkscape", '--export-filename=' + outfile + '.svg', tmpdir+'/tikz.pdf'])

honestly though, it would probably be better to use pandoc to convert markdown to html, and use a filter to allow for tikz code in markdown. It's too finicky to go .tex to .html

TeX - LaTeX Stack Exchange
How to use PanDoc to derive output from LaTeX and TikZ to a DOCX file?

I have a LaTeX document with the following code which generates a picture using TiKZ. (Minimum working example here)

\documentclass{article}
\usepackage{amsmath,stanli}

\begin{document}
Analyse...

limpid dirge
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I have 2 plans: either teach Maths (and maybe IT) to middle school kids (yes that's a thing), or try to tutor fellow university students that struggle with classes (which is a very, very common occurence)

austere inlet
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If you're comfortable doing both you'll eventually end up doing both. Here (Chile) public schools are similarly awful so a lot of people enroll in cram schools or hire private tutors, and the latter pays decently.

austere inlet
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there's probably better tools than spreadsheets e.g. in terms of formatting, I'm thinking stuff like https://tiddlywiki.com/ (which I want to try but haven't bothered to set up)

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but a spreadsheet is readily available and does the job catshrug

uneven hemlock
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college students have specific demands about what they want to work on and don't have much money. high schoolers and middle schoolers are (IME) dragged to tutoring by their parents and their parents have money

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it usually pays decently

limpid dirge
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middle school it is then

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thanks a lot for the insight and info you guys

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i'll start revising the material and start preparing for the upcoming semester

spark thorn
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helping physics student

brisk ruin
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I'm also not sure what this is going to do to help anyone learn math

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Maybe it's nice for math history? But even then, as slime said, there are lots of omissions

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There also doesn't seem to be any correspondence between the fields of study of the listed mathematicians and the periodic structure of the periodic table

marsh thistle
brisk ruin
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Oh yeah I guess if you want to keep some semblance of correspondence between names and elements

marsh thistle
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I like Pascal B though. Like where is Pascal A PikaLUL

brisk ruin
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But there are much better math periodic tables

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As an example

winged urchin
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I've always thought theorems should have descriptive names and not named names

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Like Intermediate Value Theorem is about the... Intermediate values that must be between any two coordinate values in a 2d system

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Stuff like Newtons Method is so just absolutely nondescriptive

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But I don't think it's fruitless remembering names either though. Knowing the sorta story of the discovery of maths can bring some insights

strange bronze
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if you can come up with a descriptive name for grothendieck-riemann-roch

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im all ears

winged urchin
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^

New challenge problem for the server? Ahaha

brisk ruin
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What do people think about this type of teaching?

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If you don't want to read the blog post, it can be roughly summarized as:

  1. Introduce concept
  2. Give a good example problem to work through all the subtleties
  3. Give an open problem for exploration
strange bronze
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Warm up: Suppose that in a group of people, any pair of individuals are either strangers or acquaintances. Show that among three people there must be at either at least two pairs of strangers or else at least two pairs of acquaintances.
dude has a lot of faith in first graders being able to tackle this

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even phrased in a more first-grade-appropriate way

brisk ruin
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Yeah I agree that some of them are not quite age appropriate

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But in a more general sense

strange bronze
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what pedagogical merit does this actually have?

brisk ruin
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I don't know that's why I asked

strange bronze
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and can we trust educators to implement this sort of teaching style with those pedagogical goals in mind?

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i think "get students interested in stuff by talking about open problems" isnt necessarily bad pedagogy but id be very suspicious of writing it into curriculum

brisk ruin
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I can image that an argument can be made that asking these sorts of open questions can try to get people away from rote memorization of formulas

strange bronze
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since im very doubtful of the ability of the, say, 25th percentile educator to be able to handle it well

shadow basalt
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I would say like

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I don't see a reason for it to be an open problem that seems arbitrary

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why not just give a really hard but deep and interesting problem that might have been solved before

small quartz
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All of this depends on the system. In which people are being educated. Are constant tests required? Are unreasonable standards to be hit? So... the best approach is always holistic depending on the student's surrounding. It is the teacher's main job to model and guide them. With guidance, a teacher must be weary of their student's zpd (zone of proximal development) and adjust how they learn from there. There is no 1 good way to rate educators, nor should there be 1 good way to rate student's. This depends more on what society wants and demands.

tawny slate
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i ran into an interesting thought recently while teaching pre-algebra/algebra

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obviously, students should get into good habits when showing their work, such as:

2x+3 = 7
2x = 4
x = 2

like this

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but another way in which students should be organized in my opinion is that whenever they are simplifying an expression, they should [almost] never put it on the other side of an equation, they should try to simplify on the next line

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for example, you want to avoid:
2x + 3x + 4 = 5x + 4

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you want to try and do:
2x + 3x + 4
5x + 4

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I've found that a lot of students will try to simplify on entire lines of equations and then completely forget what they're supposed to do with the equations

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they will like tunnel vision trying to simplify and then forget they were supposed to substitute or something

small quartz
strange bronze
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what do you mean by that

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if an advanced textbook showed me 3 steps to derive x = 2 from 2x + 3 = 7, i'd think there was some reason the arithmetic was nonobvious

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so including that would make it more difficult if anything

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i wouldn't dock a student marks for including it in a proof, if that's what you mean

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but if a textbook or professor showed it, i'd think they're being patronizing

lethal leaf
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No college has ever taught those things are too trivial

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The most I've seen is "include the calculations if you want but we won't dock points"

next relic
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It really depends on the course we're talking about. If we're saying prealgebra, then algebraic steps are definitely important and must be included. In a calculus course, the algebra part isn't the main stuff you're evaluating students, and in a DE course, integration isn't the main focus of the assessment.

nova verge
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Students should only be judged to the clarity of their instructors' own proofs. If a teacher has sloppy proofs it isn't really fair game to go ham on a student when they regurgitate those same lazy proofs.

tawny slate
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i literally mentioned this is something i figured out working with a student doing pre-algebra, this is a 5th grader we're talking about

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there is no "advanced textbook"

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it is just a small detail that I worked out while tutoring the student and trying to figure out exactly where they were getting confused or unable to reach an answer

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this isn't something id put in a textbook, this is just a guideline for work habits to be more organized, a structural form that provides more clarity into what the student is trying to do

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is this like not an appropriate channel to discuss teaching methods for early education/elementary school or something?

brisk ruin
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No discussing math education for younger people is fine

small quartz
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The etiquette of math can become quite pomp depending on where you live. I remember failing a linear algebra test because (even though i got the right answer) I accidentally wrote the wrong matrix transformation value between two of my matrices. It was simply putting the 4x5 matrix in RREF with showing every step, and 1 of the small steps was wrong. It's a cultural thing.

small quartz
lethal leaf
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I knew people in my calc 3 class who didn't know that logarithm = inverse of exponential

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they knew log properties tho

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and how to take the derivative of ln(x)

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but it was just some magic function

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the weird shit slips through in education sometimes idk how it happens

real mauve
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i had someone in uni ask me what the difference between algorithm and logarithm was, or if one didn't exist

unkempt gale
quasi musk
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Students that don't know anything might find it fun, or might get frustrated. You have to come prepared with hints or clues to guide students

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There's no one way or optimal way to present information. There's a lot of ways to do it, and it's good to change it up to keep people on their toes

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When they switch from engineering to applied math

tawny slate
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honestly, i'm not all that surprised by a lot of the things you guys mentioned. yes, a lot of it sounds ridiculous in a sense, but it is merely the consequence of how our education system works

we teach math classes in broad strokes, covering the meat and potatoes of topics for large numbers of students, often times glossing over many details of rigor, such as why sqrt2 is irrational, either because the rigor is dry and difficult or because it's just not worth it in terms of some end goal

by design, there are just going to be lots of students that will miss quite a few details, it takes more individual attention and effort to smooth out those gaps

in addition, a lot of students will focus more on the execution of solving problems and less on the abstract value or nature of the work they are doing, because going through the motions is easier than understanding why these steps are being done in context. this happens if a student just doesn't like abstract high-level thinking or if they just don't care about math or a plethora of other reasons. a lot of these situations described invoke a kind of implicit context that is needed to make sense of the differences, such as the switching from engineering to applied math

we laugh, sure, but i think my point is partially supported by the fact that in a channel meant for pedagogy, a point made about specific teaching techniques devolved into platitudes and anecdotes about strange questions people asked about math

quasi musk
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Uhhh

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Someone in grad school should know what log is

wispy slate
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should

dawn walrus
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Ok but like, if the grad app process is not (in almost every case) doing its job of discerning who knows what log is and who doesn't know what log is, then I feel like it's failing at its job (and I really don't think it's nearly as bad of a process as the undergrad app process is, even if the grad app process is imperfect)

#

Though I didn't need to take GREs.

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So probably most people's experience with it feels much dumber.

tawny slate
# dawn walrus Ok but like, if the grad app process is not (in almost every case) doing its job...

And this is precisely why I said what I said before, because I have issues with this

Obviously, grad students (assuming they are going into the relevant fields) are expected to know what logs are, but are we really going to filter candidates solely by things like this? Is the candidate aware that they don't know what a log is? If it is something as simple as we all say it is, it should be equally quick and easy to explain to the grad student what it is and fill the gap of misunderstanding very quickly. Do we disqualify someone solely by that basis, for something we could fix in the span of an hour or even less?

And for those people thinking "how can you not know you don't know what a log is?" My response is simply that there are [probably] a lot of math concepts out there that are extremely basic that you don't know the origin or motivation or reason or definition for. Why is the order of operations the way that it is? Why do we even define properties of operations like communtivity or associativity, what is the practical value in that? Why do we specifically define square roots to only take the positive value? What are numbers really?

It is perfectly reasonable and normal for people to have these gaps of understanding. We can joke about them that indicates their simplicity and importance, but we should not joke about them as a means of judgment or mockery.

dawn walrus
#

I agree with all of this mostly

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And yet I have trouble with the idea that shows up again and again over several courses, every time showing up naturally in conjunction with the exponential, possessing precisely the opposite properties of it

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And then never asking "hmm, maybe there's something deeper here going on that I should investigate"

#

The reality is, grad schools want students who have shown at least some capacity for seeing the forest instead of studying all the individual trees

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And they aren't wrong in wanting that. That's a huge skill in both research and exposition.

#

Maybe that can happen with other things and just happen to slip through with something like logarithms

tawny slate
#

Ok but grad schools also want students who are motivated and driven and show potential, not someone who missed an observation that was never pointed out to them

Just because I said we shouldn't disqualify them solely on the basis of not knowing what a log is doesn't mean I think they should automatically be accepted

dawn walrus
#

I agree with that

violet onyx
#

is this where actual conversations happen

dawn walrus
#

I'm saying that this ability to zoom out and see the big picture surrounding a mathematical object/concept is one of the many skills that the grad app process should attempt to evaluate. I think that's a huge part of "potential."

#

One of my first semester profs talked a lot about this

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And designed lectures to try to get people to come to the big picture ideas themselves, using careful hints / working proofs in class in specific ways / well placed questions.

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I don't think you run into this issue if you have professors practicing really good pedagogy, the type that forces a student to question the boundaries they've drawn between different mathematical ideas and to make an active effort to blur those boundaries.

tawny slate
#

sure, but my point was simply that we are human and we make mistakes

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one mistake shouldn't ruin someone's potential at a future

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you evaluate exactly that, the ability to see the big picture, not something specific, such as not knowing what a log is. if you focus on the fact that someone doesn't understand what a log is and not on their abilities and potential as a whole, then you are the one who is missing the big picture

dawn walrus
#

Again I agree, but at least in my case if I have a story like this it's not ever going to be featuring someone for whom this was a one time mistake (well, unless it features me, I'll poke fun at myself all I want).
It's more going to be featuring those for whom this might be a symptom of a broader concern (only if I have seen plenty to point to it). It's not my job to make a judgement of someone for a mistake, it becomes more meaningful when it's a consistent thing that impacts my academic relationship with the person (and surely the relationships with others). So I'll share my funny story about someone asking what w is at the end of a lecture about ordinals, and I'll never poke fun at the person, just at the lecturer. At the same time, the story wouldn't have any substance or relevance if the person had just gotten up on the wrong side of bed. It's an example of a pattern. Not a pattern that makes me doubt someone's mathematical potential, just one that needs to be unlearned at some point on the path to doing research mathematics.

#

Maybe other people are just sharing funny stories instead of relevant ones

acoustic ridge
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I'm curious to learn about hearing impaired students' experience in math, and stem in general.

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(my server search yielded only 5 hits on the key words "hearing impaired")

small quartz
# acoustic ridge I'm curious to learn about hearing impaired students' experience in math, and st...

Each case is very different, but it all comes down to the specific diagnosis (a lot of the time.) The most common thing you will come across diagnosed/undiagnosed adhd students. Best approach is to tell them to go to the water fountain every once in a while, or they will become too frustrated to learn. If you're in elementary school teaching adhd and autism symptoms are often very similar. Later on, you can tell the difference by their differing levels of neuroticism and conscientiousness. Also, 10% of individuals with autism are savants. Both adhd kids and autistic students have horrible handwriting.
Sometimes teamwork challenges work best for: A common diagnosis for high school kids: general anxiety disorder (gad); -- kids with opposition defiance disorder (ODD) whom will give you a challenge.
Every once in a while you will get a rare case, I had a student who did not have a corpus callosum in their brain. This hinders their ability to switch from one side of the brain to the other in thinking, and special steps have to be taken to make sure they learn all subjects well, at once. & just like hearing/visual problems, you will need to make larger or seperate copies for these students. 🙂
The label that I've seen scare other teachers the most is "ED" - Emotionally disturbed. Despite what you read on diagnostics from google or anywhere else, there is only one way to get this label. It is by going through some incredibly traumatic event. This includes - Watching a family member commit suicide, murder, or rape. These students will always become psychopaths.
The #1 thing that pissed me off in teaching was seeing foreign language students get labelled as intellectually deficient... Get ready to fight for students 🙂

P.S. - feel free to dm me if you have any questions/comments.

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Sorry for the long message, partially had to vent, haha

acoustic ridge
#

Thank you. That was a good read.

quasi musk
#

But I feel that if too many things pile up in the "should know" category, then you shouldn't go to grad school for Math

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Without at least taking a refresher

lethal leaf
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^

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I do not know how you can get to math grad school without encountering logarithms in your school work tho

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Hell I'm just going to be a sophomore and I had logs show up in my probability theory class all over the place

round robin
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if you spend your entire undergrad computing like pi_23(S^45) and spamming anss or something

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you probably would be quite slow at like say

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some funny problem with log inequalities

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compared to someone who is staring at probabilities with logs appearing all day long

austere inlet
#

one linalg TA I had was doing his thesis in alg geo and in one session forgot what the integral of x^n was

#

it happens

shadow basalt
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although that is trivially 0

round robin
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right LOL

lethal leaf
#

@round robin but not knowing the concept of a logarithm fundamentally?

round robin
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i dont think knowing something fundamentally equates well to doing it decently quickly in a timed test

#

evidently logarithm is just a map from some multiplicative group to some additive group

lethal leaf
#

When people are talking about not knowing what a logarithm

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They mean like not knowing it fundamentally

dawn walrus
#

Yeah we aren't discussing "one somehow missed out on learning about logs throughout their undergrad, but is competent in everything else they need"
The discussion is more about situations exemplified by, say, someone never hearing that log is the inverse of exp, knowing all the log rules, seeing log show up in several situations near exp, etc, and then never questioning "hey, these look really closely related"

quasi musk
#

I think here was the notational difference between log and ln

quasi musk
dawn walrus
#

E.g. if you learned about logarithms and exponentials as power series, proved a ton of rules about them, the first mathematical instinct that IMO indicates the skill of seeing the forest from the trees would be saying "well, these power series look completely unrelated but they both swap addition for multiplication in opposite ways. They're probably closely related, let me try composing them."

dawn walrus
# quasi musk I think here was the notational difference between log and ln

This is the problem with discussing anecdotes in this context lol, obviously no one is disqualified from being a phd if they were taught log means base 10 or something, and honestly I don't really understand why it's funny to ask what base log something is unless you're studying an asymptotic and it shows the person maybe doesn't understand that different base logs are just multiples of one another.

#

This happened in my CS classes all the time

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"wait, is it O(log_10) or O(log_2)?"

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Like you're in CS, no one is paying you to make that connection quickly, it's fine.

#

But... if you're in math, making these quick connections / answering these quick questions in your head as you work is a skill you build over time and which I imagine is gonna be hard to live without later into a phd.

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My perspective is, math is just as much about questioning and exploring as most fields of science are marketed to be, and one of the goals of a PhD admissions committee is to identify how efficiently one has learned to see connections pop up and then question/explore them, without a ton of direction.

dawn walrus
#

That makes sense

limpid dirge
#

How do you go about preparing to teach material for tutoring? I'm preparing to tutor middle school material, while i'm revising through it i can't see any way to devote more attention to the material other than going through theory and a few exercs

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I don't have any students yet, I'm still in the preparation process

grim spindle
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Are you doing 1 on 1 tutoring or class tutoring?

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I think with lower-level maths it's important to constantly keep it grounded with lots of examples and try and encourage a natural process of discovery with guiding questions while going through the theory

#

that's just my experience tho TAing for pre-academic prep school, maybe people here with middle-school teaching experience would beg to differ

limpid dirge
quasi musk
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It's reactionary

lethal leaf
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^

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Unless you're told like "they want to learn more outside of their classes"

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it's reactionary

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you are there to help them with their current classes

limpid dirge
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Sounds like a case to case basis then

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I'm guessing the best thing I can do is grasp the material as best I can so that I can offer simple explanations on broader concepts, and have a few exercs at hand

Other than that everything will be formed around the student I guess

winged urchin
#

You know, I might be speaking from a privileged position since I'm usually exceedingly familiar with the topics I tutor, but I actually kind of like not preparing for sessions

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I know it sounds crazy at the face of it, but it often yields really organic development of the session and they get to sorta see my own sanity checks as I'm working along

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I'll say something like "I think I remember it's this... and then well that makes sense given that this is X and Y"

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Whereas if I prepare material I find myself a little more rushed somehow? I feel like I have less patience?

quasi musk
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I go in to almost all of my math lessons with little to no prep outside of what the company has ordained I cover that lecture right now

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Sometimes I botch a problem, but that's ok

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I want to model what it looks like to get stuck, and how to unstuck yourself

quasi musk
#

Also why is that a place of privelege? And if it is why would that matter?

winged urchin
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Because I know the subject so well typically I don't have to prepare unlike perhaps other aspiring tutors who don't have the experience, I suppose

quasi musk
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That's more of experience versus inexperience

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rather than a place of privelege

lethal leaf
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knowing the subject you're tutoring at a high level is not "privilage" it's just what should be the goal when tutoring

tawny slate
# limpid dirge I'm guessing the best thing I can do is grasp the material as best I can so that...

To give more practical advice, yeah that's about the most you can do realistically. There's generally a limit to how useful your preparation is. Often times I discover new ways to teach a concept when a student struggles in a very particular way, which leads me to think about even basic problems on a more fundamental level or from a different perspective.

For instance, a student is struggling to solve the equation:

4(x-3)+7 = 2x+9

If the student is able to solve equations with a few steps but is overwhelmed by the amount of symbols, your job is now to offer abstract concepts or strategies to help them untie this knot. You could explain that they can substitute, that they can simplify one side first, or to isolate x on one side by "peeling the onion". Each of those explanations serves a different purpose on different equations and each student is receptive to different strategies to different degrees. You need to feel out and navigate which one of these works best or if you need to try something else entirely. This is just as much a process of self-discovery as it is teaching, because you are breaking down the low-level abstract pattern recognition you take for granted (which is part of what makes programming a CAS more challenging than most people expect).

If you want to prepare further, having a grasp of those abstract strategies helps some, and the core concepts from subjects like set theory, logic, abstract algebra, etc. help provide you with the framework to be very very clear and rigorous in your own mind about those foundations and underlying structure. It will help you a lot when students ask you questions like "What does it mean to solve an equation?" or "What's the difference between a solution and a root?" or "Why is the order of operations the way that it is?"

#

Having a range of problems is good, but at least for simple problems, you probably don't need to prepare them, you could probably improvise them on the fly. You should prepare and collect special problems that are designed in a way that test something specific or clearly favor one approach vs another

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If you need to prepare large batches of problems for say homework, then I personally just write a quick javascript/python thing to autogenerate problems super fast, but if that's not an option you could just do it by hand then

modern solstice
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which of these is better

unknown.png
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i was taught the first way but my little brother says they taught him multiplication with rectangles

mint lark
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Both!!!

winged urchin
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I think the left method is for when you understand the concept but just need to do an annoying calculation

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And dont have a calculator

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The right method (supposedly) helps people understand the concept better

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In my tutoring I found that most students move past the right method very quickly

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And in fact, I don't even advise them to do the left method

austere inlet
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visualizations and algorithms are equally important

winged urchin
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I equate the left method with little better than a calculator

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I try to push students to do the calculation in their

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Not 3 digit times 2 digit stuff

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But then I find most highschool and even university problems meant to be done by hand are not calculation intensify

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Or... they don't need to be

#

So so soooo often do I see a student work on a problem like

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4/3 * 9/2 * 5/21

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Or something like that

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And they just multiply all the tops and all the bottoms and get some horrid mess

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Without thinking how they can reduce the calculation complexity

austere inlet
austere inlet
austere inlet
#

but I find the right hand side can be used nicely with e.g. polynomial multiplication and it's actually what I do when I need to multiply e.g. two polys. with ≥3 terms

round robin
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idk if this is relevant here but is there anything that should be done/avoided when writing recc letter for CommonApp for US university undergrad admissions? say like how much do you talk about academics vs nonacademics and like what counts as nonacademics, art, anything outside of school, tutoring?
(is there "too long" as well or like as long as it is cohesive it's ok)

shadow basalt
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There's a word limit isn't there

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wait i read that wrong @round robin why are you writing a rec letter

round robin
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my teacher is q busy haha so he let me write my own and he'll edit it legitly later

shadow basalt
#

uhhhhhhhhhhhhh

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uhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh

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i can't tell if im being a cop of if that is wildly unethical

brisk ruin
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This is a not uncommon practice, it happened a few times at my high school as well

copper violet
#

greetings. I need teachers/tutors feedback.

I've designed 3 math problems which use only + - * /, and I would like to know what is typical grade when kids should be able to solve such problems.

  1. The company XXX spent 300_000 UAH and developed an online MMO game. All the users are predicted to pay cumulatively $10_000 each month. How many months is required for game to pay off?

  2. The company XXX doesn't get all the money users pay it. 30% takes the market place (Apple Store or Google Play), 18% are taxed by govt, it also has to compensate 70_000 UAH/month to each of 2 programmers. Now, when would game pay off?

  3. 30_000 users in-game cumulatively pay $10_000 per month. How many users should you buy additionally, so you have enough money for Lexus RX 350 credit (600_000 UAH)? You can buy users at Instagram for $1 per user.

inner turtle
shadow basalt
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I’d feel so uncomfortable lmao

real mauve
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i thought it was super common for profs to just sign stuff instead of writing it themselves

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i have only once seen it done otherwise

shadow basalt
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Ive never heard of it happening at any institution I’ve attended

stark pine
shadow basalt
#

Jesus

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What the fuck lol

stark pine
#

🤷

real mauve
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are any of you from the same region as max? maybe it's that

shadow basalt
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My region is Chicago or Arizona

real mauve
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germany here, which is arguably THE place you'd expect this to be an issue

dawn walrus
shadow basalt
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I am gonna be a cop about this tbh

dawn walrus
#

It's gonna be fine.

shadow basalt
#

I stand by the take that it’s unethical

real mauve
#

max cop arc

dawn walrus
#

Anyway I did this once and it was very uncomfortable

shadow basalt
#

It puts students in such a wack position

dawn walrus
#

But then I emailed it to him and he was like "thanks, I'm rewriting it totally but this is just good as a reference"

#

So

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I guess it's not super different from just sending a CV in the long run

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At least it wasnt in my case

shadow basalt
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Yeah except the CV is way less uncomfortable lol

dawn walrus
#

I thought he was just gonna put his name on it and send it though, at first

shadow basalt
#

CV is just a list of (slightly embellished) facts

#

A rec letter is like

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Supposed to sell you

dawn walrus
#

This was for an REU, and needless to say I did not ask said person for a letter for grad school

#

Like

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I feel like unfortunately in some ways its code for "I don't want to say no, but I don't think I can write a strong letter"

real mauve
#

idk, here they'd just say no in that case

frosty flame
#

I could also see it happening in institutions where the profs don't really know any of their students. Like their classes are too big to know students individually. So they might just like look at the grade and say "yeah that's good enough for a letter, write your own and I'll sign it". It hasn't happened to me luckily. But I actually thought it was common, so I was surprised when I didn't have to write my own lol.

dawn walrus
#

Well they usually say no if it's actually bad

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Like

#

My backup rec prof was very excited to write me a letter but needed more context from me about my research plans

#

So i gave him a short talk about them

#

But he could also have been like "write a paragraph of the letter about your research plans, and I'll edit and incorporate it"

#

I liked what he actually did much better than if he had done that.

#

And obviously it was better than him saying no.

#

But if he had said nothing and just written the letter, the letter wouldn't have been as informed.

#

Of course if someone tells you right off the bat they're busy, it's not exactly a great idea to come back with "how about I give you a 30 minute presentation instead?"

stark pine
# shadow basalt Supposed to sell you

One of my friends applied for a scholarship that required you be one of your own references, and exclusively refer to yourself as "the student". It was apparently one of the most uncomfortable writing pieces he's ever done. I definitely think making someone write their own reference is not great, but it's not uncommon - at least, I know a few people across a variety of different schools and even countries who had to do it

dawn walrus
#

ok but like

#

this doesn't feel that different from any kind of "accept me for your thing!" piece

#

writing statements of purpose and stuff is uncomfortable, but it's also really ubiquitous

shadow basalt
#

the discomfort comes entirely from the "letter" not being written by its "writer"

#

like you are speaking for someone else about yourself

#

its very strange

#

moreover when you are asked to write a letter like that, you aren't just selling yourself, you are projecting what the professor thinks of you and trying to match their esteem

#

idk the whole thing is very uncomfy

dawn walrus
#

are you doing that

#

i didn't do that

#

i just wrote what i thought, since i figured that would make it harder for him to just use it without changing things

shadow basalt
#

im doing nothing at all this is about ari lol

severe kelp
#

This was also a thing in my highschool, teachers made you write your own letter of recs and then edited lol

twin lichen
#

I take Max's side here

winged urchin
#

I was heavily discouraged from even reading my own rec letters

#

Writing your own is super whack

#

Like it just IS unethical. Would you admit to where you're applying that you wrote that letter?

#

Like if they said "The rec letter was superb!" I have a hard time believing someone would then say "Thanks! I wrote it myself!"

#

And I think that's what makes it unethical

#

If you feel like you have to hide it, then it's probably not 'right'

shadow basalt
#

well i want to clarify that like

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the unethical person in this situation is 100% the professor

#

students are just sort of doing what they have to

tawny slate
#

Also I feel like maybe not every situation here is equal

#

There's I think at least a minor difference between a student writing the letter and the prof just takes it as is

shadow basalt
#

i think even asking the student to do it

tawny slate
#

And the student writing a draft and the professor actually reads through it and puts his own ideas and thoughts into it

shadow basalt
#

is pretty ridiculous

tawny slate
#

Like there's degrees of unethicalness

shadow basalt
#

I think the first answer is missing a big thing

#

i don't think that it is entirely unethical to just sign a letter written by a student if you genuinely agree with it

#

i think asking the student to do it is unethical

wise onyx
#

I think its fine if the prof asks you to provide some info about yourself to include in the letter. But having the student write a letter for themselves is kinda dumb

#

as the student, you should provide as much information as possible to make it as easy as possible for the prof to write your letter

shadow basalt
#

big agree

round robin
wise onyx
civic tree
real mauve
#

another instance of this sort of stuff happening. i'm part of a lab within a larger uni department. yesterday the lab leader tells me "this student is defending their thesis soon and they need 2 reviews of the thesis. the professor is NOT writing one, so i'll do one and you do the other, then the prof signs one of them"

#

not saying i agree with the practice or think it's ethical and whatnot, i'm only pointing out that it does happen rather often, and it might actually be the norm

round robin
#

tho like what so unethical about uni apps as long as you arent lying should be ok righttinktonk

winged urchin
#

Well rec letter are also supposed to be a way for someone to speak about the students faults right?

#

I mean I know most supervisors are trying to give their student the best foot forward but sometimes negative remarks are a purpose of rec letters when necessary, correct?

wispy slate
#

I'd rather have somebody shout my praises from the rooftops

#

I think you'll change your mind on negative remarks being necessary when you have an educator actively trying to hinder your chances of attending an institution because of their own personal problems that they're projecting onto you

wise onyx
shadow basalt
#

@round robin the issue is that you are asking a student to imagine what you think of them, which would be incredibly awkward for many if not most people

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Like I would rather cut my hand off then write what I think my undergrad mentor thought of me and be wrong or something

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Plus students don’t know anything about writing rec letters or what should be in them or how flowery they should be etc

#

There are different rec letter norms in different regions and countries too

#

That students aren’t necessarily familiar with

round robin
#

but in that case yea it's p awk

shadow basalt
#

No it’s supposed to extol the students virtues

#

And sell them

#

Its entirely not just a second resume

#

Like for example your resume might say “wrote blank REU paper”

#

But the rec letter should say like

#

“The students blank reu paper was incredibly insightful and well written”

#

Which is awkward as hell to write about yourself

#

Or should be if ur not a narcissist

winged urchin
#

Hmm, for reference I've only ever had to write a ref letter for a previous supervisor, and I definitely tried to fluff it up haha

limpid dirge
#

i've never thought of having a script generate exercises though, that sounds fun

round robin
#

honestly im prob one of the least suitable to write fluff lol idk how people can turn one sentence to a whole paragraph

severe kelp
#

Just use the best adjectives to describe ur self and ur work

round robin
#

i guess

#

it's time to go on theasourousidkhowtospell

molten urchin
#

Read Mochizuki's response to SS to get a glimpse into exaggeration

#

it is a very challenging task to document the depth of my astonishment...

clever jetty
#

*depths

empty gull
mint lark
#

TFW the implications of using that word as an insult

civic tree
#

i really hope you don’t mean special as in the same vein as “Pepe special forces” sully

molten urchin
#

I thought they meant Mochizuki is someone who stands out as an eccentric genius or something. catThink

empty gull
#

I like how three completely different takes were given

kindred stag
#

I was doing a little armchair thinking, and was wondering why we ask students to prove specific things instead of just setting them up with the theorems and letting them prove whatever they want with few constraints. It seems most of the humanities operate off of the later method, and I don't see much of a reason why math couldn't work that way too.

brisk ruin
#

The later method is a lot slower

winged urchin
#

Yeah for sure

#

I was just thinking in that way Solar haha

#

Basically it's the idea that when a student sees a specific problem posted, they can generally assume that that problem is both possible and that they are somewhat prepared for it

brisk ruin
#

Right now the way things are set up is we have the benefit of hindsight and we can look and see which theorems and definitions are important and come up both in more advanced mathematics as well as in other fields, and then we can tailor a curriculum to rush those

winged urchin
#

Oh that too, tailoring the journey mhmm

brisk ruin
#

Of course, at a research level, you do end up in the situation that you described, where you have existing facts and are exploring for things to prove

#

But this is really really hard

winged urchin
#

And you need buy-in from the students

brisk ruin
#

For a more basic example, if a student has never seen the mean value theorem before, it really isn't obvious

#

There's going to be a lot of stumbling around in the dark before they've worked with enough examples to realize that this is something that's true

#

And then they need to figure out a proof of it

kindred stag
#

That's a good point, so even sans the difficulty aspect something like this is much more time consuming, which makes sense since humanities assignments seem to be on much longer timescales.

brisk ruin
#

Yeah

#

I can imagine a more limited version of this working in a more structured setting but it still seems difficult to pull off

kindred stag
#

But even as a longer term project students would struggle with running into frequent deadends and stuff.

brisk ruin
#

I also imagine that it would work better with more advanced students who have already seen a lot of math

kindred stag
#

And I suppose I actually have seen something similar. On a couple problem sets I've had questions along the lines of "Here's an a specific thing, what's something interesting about it? Prove that this fact extends to a more general case or provide a counterexample."

#

Which is kind of a similar vein

winged urchin
#

Yeah but at least there they direct you still

brisk ruin
#

And even when you're doing research you aren't completely without direction

#

You usually know which direction you should be working towards and what types of results you can prove

winged urchin
#

It's not a bad idea by all means though

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I'd certainly be interested in trying out that kind of free-er exploration

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With my students

kindred stag
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Yeah, I was specifically thinking of questions being of the form "Using these two theorems prove something interesting" and maybe also asking students to explain why it's interesting. But maybe a little more direction would be needed. Something more directed, perhaps specifying what the result has to be about could provide the needed direction.

winged urchin
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Hmmm

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In my head I imagined kind of a process like

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  1. Introduce some kind of function or operator or identity or whatever

  2. Ask the students to just... play around with it. Are there any results that you observe?

  3. Attempt to prove your observation (or disprove)

brisk ruin
#

You'll want to have the students work a concrete example before proving any general facts I should think

winged urchin
#

Just kinda very rough thoughts

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And ya, you should probably demonstrate the approach to some degree

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Although that is kinda hard. Often when you present a result it's not presented as you uh... investigated it

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I find

kindred stag
#

Yeah, that kinda makes sense I think. This seems like something that would require a lot of preparation and you'd either have to lead the student along to make them discover a specific themselves, or have this be a long term project. The former defeats the point a little and already kinda happens and the later seems like it would be incredibly difficult for the student to do well.

tawny slate
#

Man there are a lot of things to say about this topic from different perspectives

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I've found there are a lot of ideas and concepts in math where a dose of "marketing" can go a long way

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In this sense I find math similar to magic

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You want to present the trick, the idea that makes kids go "wait wtf" and then go into the explanation

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You want to spark inspiration and curiosity to maximize the buy-in you get from them, and then present the explanation in ways that first explain the elegance before hitting them with the "well duh" explanation

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How you present the explanation plays a significant factor depending on the topic of how they feel about the idea

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For example, when I teach that the sum of n choose k is 2^n, I have two general ways of presenting this to students:

  • double counting proof
  • every element in the above row is added twice (inductive proof)
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The inductive proof is (curiously) way simpler to explain, but the double counting proof is what is truly elegant

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I always present the double counting proof first, because for about 60-80% of students, the inductive proof almost "disappoints" students by just how plainly obvious it is

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I think this is the role of the teacher in exploratory learning, you want to provide students with the foundation to figure things out themselves obviously, but you also want to be there to put what they discover into perspective

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Sometimes they discover some truly amazing facts but don't realize or notice what they have done

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In addition to that, I think students need a baseline level of mathematical maturity in order to be able to fruitfully pursue this kind of exploration, and that depends heavily on the individual student

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I have students who are really really good at going through the motions of algorithms taught in class but struggle immensely with even answering the most basic questions regarding what they are doing at a higher level, such as "what is a function?"

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As a consequence, they can't even solve problems that require being slightly more conceptual and abstract, so at that point exploration is difficult or even impossible

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One student I had was given:

f(x) = 2x

And I asked them to plug x-2 into the function, and they couldn't because they thought "x = x-2 is impossible"

signal fiber
#

it is impossible 🗿

winged urchin
#

Oh boy understanding variables X X

hexed sluice
#

A method of getting the roots of quadratic equations I hadn't seen before: https://youtu.be/ZBalWWHYFQc?t=106

YouTube
Expii
A Different Way to Solve Quadratic Equations

No more guessing while factoring quadratics! Po-Shen Loh has been thinking about how to explain school math concepts in more thoughtful and interesting ways, while creating lessons for our Daily Challenge (https://daily.poshenloh.com). One night in September 2019, while brainstorming different ways to think about the quadratic formula, he came u...

▶ Play video
oblique bobcat
strange bronze
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any method of solving quadratics is equivalent to the quadratic formula

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(assuming it works in general)

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the point isn't that it gives novel results, the point is that it's a different technique

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unless you think that, once we have 1 technique, we should never ever think about alternate approaches

tawny slate
#

I have 4 different abstract evaluations of any approach to a concept: accessibility, modularity, elegance, and function

when teaching any concept, accessibility is important for getting people to understand the concept, elegance is important for inspiring students and motivating them, and modularity is how you get students to learn how to generalize and extend their concepts to more advanced ones. function just refers to how the method can be applied. for instance, this particular method to solving quadratics is awesome if you need to solve them by hand or in your head, but it may be more direct to use the quadratic formula for a specific problem that requires you to, say, manipulate the discriminant

I recommend evaluating different approaches of a single concept against these 4 ideas to see the difference between them. I find that when you have some structure for how to think about teaching concepts, it's easier to get a feel for the value of different approaches

wispy slate
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100 different methods are just confusing for the students

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What bugged me the most about school math is that the teachers and their methods suck. At university math was a joy.
Maybe you should teach in schools like at uni. Hold a presentation about a certain topic with all kinds of information relevant to it, then let them do exercises. And stop doing the same fucking topic for like 2 months, it’s super annoying and boring.

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Also, the students will understand the material better when it’s defined in a precise way and not in those school bs manners. Nobody understands jackshit when teachers just say “yeah so we introduce this concept, we won’t prove and just assume it fell from the sky” that sucks

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That’s the reason why so many students lack understanding, they are (mostly) just given formulas.

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But yeah 90% of the students don’t even give a shit about math so at the end doesn’t really matter how it’s taught at school😄

wispy slate
#

Possible

mint lark
winged urchin
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Yeah multiple ways to do something is how I become more confident with most results I get

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Do it one way, get result... hmmm maybe I made a mistake, it doesnt feel obviously wrong

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Let me try it another way

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Oh the same result!

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Now I'm getting more confident

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Also for me, personally, the more connecting threads of information the more likely it is to be able to forget one and still be supported by the others

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If there's just one thing I need to remember and nothing else to rest that on, then it's either remember it or nothing

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Like sometimes students get mixed up what the trig derivatives are. Is (sinx)' = cosx or -cosx?

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And if you just think it's remember it or suck then well, not so great

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But if you ALSO know the graphs then you can infer the derivatives from that to a degree as well

#

It's a bit of a departure from different methods/techniques for something but it's in that same vein

tawny slate
# wispy slate What bugged me the most about school math is that the teachers and their methods...

i find it amusing that this response was posted right after my evaluations of teaching approaches, because those evaluations are exactly why I disagree with this sentiment

what makes teaching a math class difficult is that everyone learns differently, and different students are all at different levels, because we segment the curriculum by grade level and not by proficiency (honors/remedial class helps some, but there's not nearly enough granularity or customization). math is especially tricky, because many concepts are built upon earlier ones. it's really difficult to explain quadratics to a student before they have a grasp of basic algebra. it's really difficult to explain trigonometric functions without first explaining geometry and functions

sure, this might be the way you want math taught, but it's not the way everyone else wants to. the reason 90% of students don't give a shit about math is because no one gave them a reason to. math teachers are often times tied by the fact that they need to cater to lots of different students, and that the primary goal of their teaching might not be to inspire or motivate, but to develop skills. of course math would be dry this way. but in these circumstances, it is precisely those new novel methods of understanding problems that will engage students and demonstrate to them that math is either practical or elegant. if you just throw them a bunch of exercises, some students who don't have that foundation and motivation will only fall further behind

if you only teach 1 simple method shared by all other teachers, you are maximizing accessibility. if you teach by explaining a concept and then just throwing exercises at them, you are maximizing function. if you explain a single method in many different ways, you could be maximizing modularity or elegance. these are all varied depending on your students and your goals as a teacher

frigid crest
tawny slate
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i only halfway agree

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I don't think it's "not sensible", but i do think there can be massive improvements

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but yes teachers can't really do much, they have their hands tied

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that much i fully agree with

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i also don't think that's the reason students don't care

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i think the issue of students not caring is much simpler

frigid crest
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The improvements called for would be a shock to the system.

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You could expect resistance from some educators that would insist on keeping things very familiar.

summer falcon
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My math in highschool was enjoyable after geometry

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geometry was rough bc my teacher was old school and i went from an easy middle to a tough HS

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but omg precalc and calc are so much more fun than the rest of math 😂 idk why

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i wonder what could be done to improve geometry pedagogy, because i know I'm not the only one who struggled

turbid zenith
#

New content for the first time in forever. XD

tawny slate
# summer falcon i wonder what could be done to improve geometry pedagogy, because i know I'm not...

bill's quote is super relevant in my response here: "In short, we need to stop conflating logical foundation with pedagogical foundation."

i found that geometry was often times difficult for a lot of students because this is the course where formal proofs (what constitutes a valid proof and how to write one) are introduced. it's not even that the subject material can't be fun and engaging, it's that it involves a LOT of writing, making it feel dry and more like an english class than a math class

however, i think that's a good motivation to build on. you take the geometry concepts that students were already introduced to in pre-algebra, such as that triangle angles add up to 180 degrees, or pythagorean's theorem, and really question the students with "yeah that's what you've been taught, but why is it true? do you just blindly trust your teacher?" I think a lot of geometry teachers don't have time to go over the "why" of the triangle congruence theorems. then find a way for students to express the structure of the formal proof without needing to write a ton

circumcenters/incenters/centroids/orthocenters are also incredibly cool, and I think it's a shame they're not taught in school. i think these are the first/simplest concepts in geometry that I really found to be incredibly elegant

twin lichen
#

In short, we need to stop conflating logical foundation with pedagogical foundation
Great point!

winged urchin
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I'm curious and want to test my own understanding

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Logical foundation would be like...

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What theorems, axioms, knowledge, etc do I need in order to prove this thing?

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Pedagogical foundation would be like...

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What knowledge, understanding, etc do I need in order to be able to grasp this concept?

turbid zenith
#

Exactly, @winged urchin .

tawny slate
#

one thing that sort of blew my mind when i first learned about it, and kind of a personal story related to this topic:

once you learn all of the stuff from school (pre-algebra, algebra, geo, pre-calc, calc, etc.) you go on this track thinking that math only ever gets more abstract and more complicated, rather than simpler or more foundational

then you learn about set theory, abstract algebra, etc and you realize that much of the weird crazy abstract stuff like quaternions and infinite ordinals aren't just starting from an abstracted place, but actually from a perspective that is so foundational that in theory you could explain each individual step to a child, and that same foundation also defines things as basic as natural numbers

and then you realize that that foundational math is also insanely complex if you try to abstract from it to determine a system's consistency, decidability, or any of its other properties in relation to other things, and then you find out about category theory and now you realize that everything in math is both insanely simple and insanely complicated

but strangely enough, it means that young children learning about rudimentary math, even things as basic as counting and arithmetic, are not starting from the very core foundations of math, they're actually starting from the middle. if you go in either direction, up or down, it only gets harder and in certain ways, weirder

that's when it clicked for me that there was a huge difference between the logical and pedagogical, but also developed in me an appreciation for math that no matter how complicated and abstract a certain idea is, there probably exists a very simple accessible way to explain it, if not the exact conclusion at least the motivations and method behind it

and additionally, I find it amusing in a poetic sort of way that math has a parallel to physics in the same way, that we start learning about physics from the middle (classical mechanics). when you go down, quantum physics gets really weird and difficult. when you go up, you find that astrophysics is also incredibly mysterious. just about any subject, you will find similar abstractions in either direction of scale behave similarly, like it's almost some fundamental cosmic property of reality and knowledge

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ok i promise that's the last wall of text for a while

twin lichen
#

I honestly don't think people really think critically about what is meant by "abstract". I think like, when we just don't understand something, we often reach for that word, even when it isn't really accurate. What we're dealing with is novelty and unfamiliarity, not an excess of abstraction

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not commenting on you lol, just something I've observed.

tawny slate
#

ah, fair enough. though i think even my colloquial usage is a bit much in retrospect, so noted anyways for next time

twin lichen
#

one of my students in Calc 3 told me that space curves in R^3 were abstract and i just don't think that's fair, a curve in Euclidean 3-space is a very concrete notion

tawny slate
#

I do agree with your general sentiment about people abusing the word, but I think it's also relative and non-quantifiable so I've never paid it much thought I guess

unreal ledge
#

Bill teaching algebraic geometry to high schoolers I'm in

summer falcon
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Like what happened to all the proofs and pretty sentences :( he taught it completely geometrically and it was near impossible for my visually disinclined smooth brain to follow

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Looking back i kind of want to relearn calc iii sometime but not if it involves that much diagramming and that few proofs / algebra. Haha

turbid zenith
#

But that's supposed to be the power of analytic geometry IMO. Translating between algebra and geometry, so that if you struggle with one for a problem, you can think about it the other way to help you out.

tawny slate
# summer falcon Looking back i kind of want to relearn calc iii sometime but not if it involves ...

Sounds like you just got a professor that didn't really click with you, and sometimes that happens. My calc 3 prof did very little of it geometrically and instead used notational magic to prove and explain everything

Maybe pick up a textbook and try again? There are lots of resources out there and each one approaches the subject a little differently, so with some diligence you can probably find at least one you enjoy and can figure out, and bring any questions to like this discord or something

lethal leaf
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teaching calc 3 without graphical intuition is a disservice to students

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however

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Curves and surfaces in R^3 are not abstract

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it's literally 3D space

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like

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???

turbid zenith
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They can be taught in an abstract way though.

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Like when they're described with a bunch of symbols and there's not enough of an emphasis on getting to see the things you're learning about.

dawn walrus
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i mean they're literally abstractions which exist in most of these students' experiences to model real concrete things like the path an object takes in space, the boundary of some object, etc

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just because they're not abstract compared to other objects in math doesn't make them extraordinarily concrete hmmCat

summer falcon
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The big thing was... Calc II in my university (which i took because my highschool advisor said not to use ap credits even if you made a 5 🤦‍♂️) was basically "for" computer science, things like sequences and series were super useful for us

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And then calc iii was basically "for" mechanical engineers bc fluid mechanics apparently

lethal leaf
winged urchin
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I think students GREATLY under-use graphing software or even simple little code if they've learned a language

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I think maybeee it's a "it kind of seems like cheating" kind of vibe?

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Like if a calculus question asks where is this function's derivative equal to 0

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I nearly never see students graph the function

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Like, that's not all your work of course, you gotta show the calculation but at least to make yourself more confident in your results?

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I remember as an undergrad I abused the heck out of MATLAB to code up whatever I was working on. It was usually pretty hard for me to actually get something wrong because I could check what 'should' happen with the right solution

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"Oh this differential solver has a value of this at t=4.. but my solution is different... hmmm"

kindred stag
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I think providing visual intuition is good, but it's just that, intuition. I definitely think we could do better with providing intuition, and I think visuals and concrete examples are something that's absolutely missing from the curriculum. However, the proofs, formalisms, and symbols are the part we care about. You can't provide just the intuition and stop there, which is kind of what this conversation feels like it's heading to.

brisk ruin
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I mean it depends on the target audience right

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DMAshura originally brought this up in the context of teaching computational algebraic geometry to high schoolers

dawn walrus
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the algebra brain knows no bounds

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imo obviously any mathematician needs to learn how to write rigorous arguments but like

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some undergrads take this way too far

lucid monolith
#

The graphing intuition will also eventually fail for certain functions. For instance, every non-differentiable continuous function is impossible for a graphing calculator to do justice too. If a student tried to find where the derivative was 0 by looking at a graph of a weistrass function, no matter how far they zoomed in they would always be getting it wrong. I think graphing is helpful to work towards building intuitions, but all the interesting problems one can solve are precisely those problems which can't be immediately solved by software. I still think people generally underuse graphing applications and the like, but its a better problem to rely too much on math and rigor than it is to rely too strongly on graphing calculators. You don't want to end up like this guy:

a32169717671cce53617b04b7dc87c94.png
dawn walrus
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at least i couldn't

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sure, it's a sum of cosines, i know those are spiky

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and i can prove for you the derivative doesn't exist by using this

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that doesn't tell me why it's a natural object

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the picture does

lucid monolith
#

The picture could help, but you might still rely on the wrong intuition that if you zoomed in far enough you'd eventually get something that looks like a line, since thats how every other continuous function you've looked at follows that rule. Even if you graph the weistrass function on a calculator, you could zoom far enough in where it becomes a line. Its only when you understand what the object actually is that you understand that the weistrass function simply never turns into a line at any zoom level, its is always an oscillating function.

kindred stag
#

The problem is that your mental image is wrong. You can't imagine all the spikes at once in any concrete way. The graphing calculator tells you what it should look like, but it only truly offers an approximation.

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(in responce to ryc)

lucid monolith
#

A graph can't help with you this fact, because every single graph ever computed has a lowest level of resolution. I.e. its impossible for a graphing calculator to represent the weistrass function so your intuition has to do the work instead

dawn walrus
#

approximations are how people understand complicated things

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when i show you a picture of a sierpinski triangle

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is your first thought "oh, the pattern probably stops when it gets down to pixel size"

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the calculator is "wrong" literally, but it still sends the correct message

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maybe that message needs to be reinforced with a few words

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i don't think you can send the correct message without a picture, even if your picture needs a few words to go with it

winged urchin
#

I just believe there should be multiple pillars to your understanding. Even when we talk about proofs

turbid zenith
#

Just to add something here ... I agree that eventually graphing intuition will fail for particular functions. But I think some people take it too far when they say that using pictures for intuition at all is bad.

winged urchin
#

I believe there should be multiple facts supporting your overall conclusion/result. Hmm the graph looks like this, analytically we can show this, if I take a limit of functions... etc etc

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It's integral to my ability to remember math facts at all really. I might forget whether some identity has a plus or a minus but then if I remember the graph or remember another identity that's related more confidently then I can reason whether it should be plus or minus

kindred stag
winged urchin
#

If I momentarily forget one thing, it's often supported by other threads of knowledge that can help me remember it

#

You should never rely solely on one thing

dawn walrus
#

a crutch in which fields?

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not in all fields imo.

winged urchin
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In my opinion

dawn walrus
#

ok, probably a crutch in most fields

kindred stag
#

can you give an example of a field where it's not a crutch?

winged urchin
#

That's a pretty tough ask ngl

#

Trying to go through the whole of a field and conclude each and every thing cannot use visual aids as a crutch

turbid zenith
winged urchin
#

And it's not like you can't make mistakes with analytical stuff either

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Someone might get reallyyy confident just doing the algebra and getting results that way. Then make a mistake, apply a theorem in a subtle-y incorrect way, a typo whatever it might be and thus extremely confident in their incorrect result because they didn't think to reinforce their understanding with other approaches or supporting evidence

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Science is built on doing things lots of different ways and also doing the same approach with different people of course, thus giving lots and lots of evidence to hold up their result

dawn walrus
winged urchin
#

You shouldn't rely on a graph by itself. You shouldn't rely on a calculation by itself. You shouldn't rely on any one thing to conclude a curious question

dawn walrus
#

where the less visually you're working, the more you're depending on symbols and structures which are kind of "artificial" in a sense

winged urchin
#

Of course in practice sometimes you have to just... go with an answer, like on a test or assignment, or whatever =p

dawn walrus
#

in the sense that they exist purely to describe structures we see instead of to prescribe interesting new structures

winged urchin
#

And even when you work in higher dimensions you can't visualize, you use <= 3-dimensional contexts to better understand the theorems you are going to apply in higher dimensions

shadow basalt
#

Visual intuition isn’t a crutch as long as you’re not wrong

mint lark
twin lichen
#

can you folks not directly perceive truth

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true statements appear to me as green
false statements as red

mint lark
#

Ultimate synesthesia

turbid zenith
#

This statement is red

earnest trail
#

are there good ways to intuitively teach multiplication to students instead of referring to it as repeated addition?

mint lark
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Area of rectangles!!

Even more intuitively, counting things in a grid or in groups

earnest trail
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that's true

#

ty

strange bronze
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does area make sense to young kids?

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seems more conceptually weird than distance

frosty flame
#

I think so. Kids can tell when an area is bigger than another are. So maybe just "how many squarea of this size are needed to cover this area"? If the area is bigger, it needs more squares.

clever jetty
#

But how do you explain irrational length/area

frosty flame
#

Explaining irrational numbers in general is hard. I think the important part is having a general understanding of the concept, but if you really wanted to, I would phrase it in terms of infinite non-recurring decimal expansions and say that if you round off at any point then the distance becomes too big or too small, so this is the only way we can represent that distance.

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But, if you're talking about really young kids, they probably don't even know about irrational numbers, so this wouldn't be relevant.

turbid zenith
#

I agree with the area model, because it's a visual arrangement that lends itself well to other things they'll learn in math (basically anything having to do with the distributive property)

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But I think it's important that they see it as repeated addition as well because that helps motivate the order of operations without having to refer to silly mnemonics about someone's uncle or whatever

kindred stag
#

You can probably accomplish both ideas at once by demonstrating that if I have a rectangle of A times B, then there are B rows of A squares. Some coloring could help get that across to maybe?

earnest trail
shadow basalt
#

Many perspectives > a “best” perspective

earnest trail
#

that's true

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but yeah the rectangle idea is good

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and I think students can grasp the concept of area and think of something they can relate to

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depending on their age, maybe they now the grass or something, and if you have more grass then there is more area to cover

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and I thought of this last night, transitioning to volume could be fairly simple since you can make them think of a pool: sure, a pool might be big in terms of area, but if you really want to know how much water is in your pool, you need to know its depth

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and thus for each unit area you have a certain depth that you have to factor in

turbid zenith
#

Minecraft

earnest trail
#

OH SHIT YEAH

mint lark
#

Minecraft is a good one

earnest trail
#

I didn't even realize that but that can be amazing

mint lark
#

How many blocks do I need to cover this floor

earnest trail
#

^^^^

tawny slate
tawny slate
# earnest trail are there good ways to intuitively teach multiplication to students instead of r...

a third way that i really like to use is the idea that addition is a translation and multiplication is a scaling/stretching

basically take the vector/complex operations but don't use the words vectors or complex, just use arrows and analogies to like sticks or rubber bands

i like this idea because it doesn't require throwing in another dimension and prepares students for when they do actually need to visualize these concepts for things like vectors and complex numbers. makes explaining those concepts much easier

turbid zenith
#

Yeah I get you! Stretching/scaling is another great metaphor to use

#

Multiplication can be all of these things and it's useful for students to see them all together

tawny slate
#

a fourth option i dont especially like pedagogically speaking without contextualizing properly, but technically you could do is the idea of using the idea of number bases and units to explain implicit multiplication in the base definitions, and then extend it to real numbers. this is tedious and annoying to most students and would also probably need some rudimentary intro to infinite sums/series

#

for instance, teaching that 345 = 3 * 100 + 4 * 10 + 5 * 1

#

it's also slightly circular in definition, because in a sense you're also using it to describe itself

#

but where i think this could be helpful is getting students to realize that they are already doing a kind of multiplication implicitly when they are simply describing numbers that they may not realize

twin lichen
burnt vesselBOT
#

diligentClerk

340405629111894016.png
twin lichen
#

this is a conception of multiplication that appears in ancient greek geometry

earnest trail
#

@twin lichen @tawny slate thank you guys so much, these help a lot!

empty gull
turbid zenith
#

So, in case anyone here teaches high school, or knows someone who does

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(I'm becoming chair of this SIGMAA in about a month, and we're trying to collect as many resources as possible that people have come up with)

tawny slate
#

Wait where do I send messages?

#

And how long should the description be? How much detail are you looking for?

earnest trail
#

damn thats cool ashura

#

I wish there were higher level math classes in my high school lol

turbid zenith
#

@tawny slate do you use twitter?

tawny slate
#

Meh

turbid zenith
#

If so a short response to the post would be appreciated, but you could then send me anything more extensive in pm or by email if you want

turbid zenith
humble aspen
#

Hi all! I’m starting a research project on surveying better pedagogical methods and environmental things for secondary school learning. I may ask a few questions here and there for general inquiry purposes!

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For mathematics instruction, is there a need to individualize mathematics learning? If not, why? If so, have any of you implemented anything for this to happen?

#

Additionally, what mathematics topics should be taught more often in 9-12 math curriculum?

quasi musk
#

It's a balancing act in giving every student the personal attention that they need, the social/group assignments that'll create a community of math thinkers, and following along with a generally structured lecture replete with examples and questions

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Some students respond better in other types of systems, like a completely individualized system is great for motivated students who have time

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But not so great for students with attention issues, motivation problems, or not enough time to allocate to their course

#

The social/group assignments that get students involved with their classmates, and thinking together has to be carefully constructed. It shouldn't be "here's a list of 100 problems to solve" but should pull them in with interesting questions for them to explore

#

This is a lot of fun, but takes a lot of experience to pull off correctly. Many students when faced with problems that require creativity get frustrated, uninterested, and ultimately demotivates them. So you have to have to give some scaffolding to students as they go through these

#

This is the ideal way to learn in terms of developing a sense of ownership, involvement, etc. but 1) is very time intensive 2) frustrates most students and 3) doesn't develop procedural fluency as thoroughly as other models (explicit instruction and support structures)

#

The lectures are efficient at getting information through peoples skulls quickly, but the drawbacks are lock of motivation and interest on students part

#

So there's no guarantee that will work at all

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Ideally you do a mix of all 3, and leave some time for supplemental instruction to really help students

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@humble aspen

humble aspen
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THANK YOU SO MUCH!!! So while individualization is needed for mathematics courses, there still needs to be a blend of various teaching strategies? I’m also looking into how to develop motivation intrinsically to learn to see if there is a way to do so.

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I see the benefit for motivated students quite quickly, and there’s always a bit of an issue with students with all unique needs and skills

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thank you for your answer!!

quasi musk
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Yuh

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It's a huge issue, my entire job is thinking about these things and how we can support or improve courses

turbid zenith
brisk ruin
#

Anyone have experiences with teaching mathematics while needing to take into account various disability accommodations?

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(Both formal and informal accommodations)

quasi musk
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It just amounted to talking to them, asking them how they want me to help, and then just doing that

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I had the luxury of being a tutor, not really a teacher/instructor

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So teaching might be different

brisk ruin
wispy slate
meager bronze
clever jetty
tawny slate
strange bronze
#

new edition of the book is the exact same book but with a section added to the introduction saying "my publisher forced me to write a 2nd edition, its exactly the same as the first, buy used"

tawny slate
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Thought this was mildly relevant and cute

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Not sure if I should've posted this in like chill or category theory or what

marsh thistle
#

category theory is a conspiracy theory confirmed

brisk ruin
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It came up during a training I was doing about accessible teaching

meager bronze
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I see, in my institutions i have been explicitly told to not do anything extra because that opens me up to complaints from other students about fairness

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Like, if it’s not documented with the student disability office but I just decide to give one student extra time, if another student finds out, they could complain and I would have to re-give the exam giving everyone extra time

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I would do informal things like “student wants to sit close to me so they can see/hear better” so i always make sure there’s a spot for them

brisk ruin
#

Oh I see

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I think we were encouraged to give informal accommodations of a less impactful nature like alternative ways of completing assignments, not entirely sure though

#

I guess a lot of it will depend on the course coordinator and their specific policies

meager bronze
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oh, that makes sense. yeah I guess like, I'm always flexible with that kind of stuff

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for everyone

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as a default policy

#

so I don't really think of that as "accommodations" but more like "being flexible is good for every student"

brisk ruin
#

Oke

strange river
#

hi there peoples!

I accepted a job offer to teach some math at a high school. the whole thing came at a very short notice, so basically I've had this weekend to cram about the first lessons. I'm pretty excited and happy, but I'm also really scared because I have no pedagogical degree 😄 I've taught some stuff at uni previously, but only in assistive capacity

#

there's a thing that confuses me in one of the textbooks which deals with rudimentary calculus stuff - limits, continuity and derivatives, to be more exact

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at the very beginning of the book, there's an introductory chapter on rational functions. I'm not entirely sure why the authors have decided to put this at the very start of the book, because it doesn't seem to have any super-clear relevance to the later chapters. there are a few explanations that come to mind:

  1. there are rational functions that can be understood or studied more comprehensively by deploying limits or other tools related to calculus

  2. the book later deals with derivatives of rational functions and the authors thought "oh crap, we're not entirely sure if our readers even know what rational functions are, so better put an introductory chapter on that at the very beginning of the book"

  3. I'm dumb and just don't understand how rational functions are fundamentally essential (or at least extremely helpful) in understanding rudimentary calculus

brisk ruin
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The difference quotient used to define the derivative is a rational function of sorts and if f is a polynomial then you can directly compute the derivative by taking the limit of the difference quotient by manipulating this rational function

wispy slate
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hmm I remember at high school functions like at first 1/x, then going into higher degrees, were serving as pretty good examples to how the graph will look like, what's the limit and to learn how to draw graphs by hand in general. Also, for functions like ax^2/(cx+b) you can justify it blowing to infinity by saying "it behaves like x".

next relic
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I think they've been introduced to the idea of asymptotes before and the chapter is simply helping them to recall what they've learned. Usually rational functions are taught prior to calculus and students are usually taught how to find the equations of the asymptotes without much understanding about why they're doing what they do.

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Let's say, for y = (ax+b)/(cx+d), vertical asymptote is x = -d/c and horizontal asymptote is y = a/c. Yeah, fair, but there's a good chance they can't explain why we have the results.

wispy slate
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Has anyone read Mathematics for Human Flourishing by Francis Su? It's excellent so far, picks up on a lot of things I and I'm sure many have thought but could never express as well as Su

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A couple interesting thoughts from reading it that I want to take further into hot take territory than he did and I'm curious to see how people tackle practically:
How do we teach people to explore mathematics without shame, envy, conceit, and any kind of math ego? Maybe this is a question better for a Buddhist discord but I have found the practice of non-attachment to my own abilities or knowledge to be necessary for genuine joyful and flourishing pursuit of mathematics and it informs the way I teach others

limpid dirge
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Sounds like an interesting read, i'll add it to my tbr

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I do want to ask this though

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Is anyone using/Has anyone used these websites where you can post basically an advert for doing private tutoring? In my region there is a not small number of these websites where you can post your studies and contact info so that people can contact you for tutoring.
In your experience (if you have any), do they work? Or are they just there to take your money? (Yes they charge a fee for the service)
Struggling with finding students so any bit of info helps

stuck bramble
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@limpid dirge do you mean classified style sites such as Craigslist?

burnt frost
compact kernel
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I use Kijiji and Facebook Marketplace

pastel horizon
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Starting my maths PGCE this September any advice?

brisk ruin
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PGCE?

pastel horizon
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Teacher training

next relic
#

I'd recommend questioning some of the "best practices" they cover in the course, because some stuff sounds right but has no body of evidence.

pastel horizon
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Luckily I will get a lot of observations in my first month so hopefully I'll get to see what works

meager bronze
next relic
# meager bronze Do you have an example? Like, something you were told is a “best practice” but w...

Please correct me if I made an error anywhere.

Universal Design for Learning is the first thing popped up in my mind. One particular part that struck out to me was the principle that you should use several media to convey the same content to "make it easier for learners to use their strengths and work on their weaknesses." This is quite similar to learning styles, when a child is thought to learn better when the modality the content is delivered matches with their preferences.

The research on learning styles shows that it isn't the case. This is shown in at least two papers I'm aware of, https://journals.sagepub.com/doi/full/10.1111/j.1539-6053.2009.01038.x and https://www.researchgate.net/publication/256537666_Do_Learners_Really_Know_Best_Urban_Legends_in_Education.

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ResearchGate
(PDF) Do Learners Really Know Best? Urban Legends in Education

PDF | This article takes a critical look at three pervasive urban legends in education about the nature of learners, learning, and teaching and looks at... | Find, read and cite all the research you need on ResearchGate

brisk ruin
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Is your critique of learning styles or of universal design

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Universal design, based on your description, is something that has been really emphasized in the course I'm teaching (calculus) where a big thing is the rule of 4 and representing functions in many different ways

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And the teaching coordinators have good data to back this up

next relic
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Let's take this for example.

image0.png
brisk ruin
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Ok I think these are important for reasons other than learning styles

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For example some students might have not great hearing but it might also might not be diagnosed

next relic
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Fair, but the point I'm trying to make is that saying "I'm an audio learner so content should be recorded ahead of time" is kinda meaningless.

brisk ruin
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Also in the thing you linked for 1.1 they have a list of experimental things they link

next relic
winged urchin
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I'm not really up to speed on the discussion of these things but I will say that generally I think that more threads connecting to whatever idea you're trying to teach is better than trying to rest on one super solid pillar

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If my student is having trouble with an explanation I don't restate what I said if I can help it

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I try a different angle

shadow basalt
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What would be an example of an alternative to audio information? Lecture notes?

winged urchin
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Now I'm a tutor myself, so I have the luxury of working one on one usually and it's way different from teaching a class

shadow basalt
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(Of course that is pretty reasonable)

winged urchin
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Sure lecture notes I guess, in less defined ways perhaps...

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Body language?

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like as you're saying something you can do hand or body gestures that emphasize whatever you're saying?

brisk ruin
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Yeah so like if you present information, you should present it in different ways right

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If something is important, you should say it multiple times, write it down, have it in materials that are distributed, draw pictures of it, etc...

shadow basalt
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I guess I am more referring to a student with an undiagnosed hearing issue, say

#

Yeah fair

brisk ruin
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And like

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A lot of things that are good for people with undocumented disabilities are also just good things in general for everyone

shadow basalt
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Maybe it’s a good thing if this is true, but does that slow down lecture quite a bit

brisk ruin
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Slowing down lectures can give students a chance to catch up if they've fallen behind

shadow basalt
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For sure I’m not saying it’s bad I just also wouldn’t want to like get behind of some course with strict coordination

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On*

brisk ruin
#

Oh yeah there is a balance between keeping up with a course schedule and slowing down to let everyone digest

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My teaching this coming semester is a fairly unusual format as far as uni courses go because there's a large emphasis on group work and we aren't supposed to actually spend that much time lecturing

shadow basalt
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Terrifying

winged urchin
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That sounds interesting

shadow basalt
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I’ve really had such negative experiences with group work lol

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Where they just turn into responsibility chicken

winged urchin
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Oh sure, and ideally the students need to get a little extroverted I suppose

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The idea being that they're bouncing ideas off another

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"Oh I think it might be like this..."

brisk ruin
#

So yeah one of the things we learned about during teaching training week was about effective strategies for facilitating group work

winged urchin
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"But then if that's like this then what does this mean?"

shadow basalt
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Voluntary group work (with my friends and people I trust) is great

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Enforced group work has a bad track record for me

brisk ruin
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According to the course coordinators, students are generally enthusiastic about group work

winged urchin
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Hmm

shadow basalt
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You’ll have to let me know how it goes

winged urchin
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I am not... completely without doubt

brisk ruin
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One of the facilitators mentioned a study about this whole teaching structure done by the AMS a few years ago and I am trying to locate it

shadow basalt
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I’d be interested to know if there is a correlation between performance on individual work and how much someone enjoys the group work

brisk ruin
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Here are two studies about group work

shadow basalt
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(For example if you sort your class into groups of 5 and 1/5 people does vastly more work then the others, then 4/5 students could potentially show positive feelings about group work)

brisk ruin
#

Well

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Active learning

shadow basalt
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I liked one profs scheme

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Where every pset had to be done in pairs

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But you got to pick your partner

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The catch was you could only use each partner twice

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Anyway I shall take a look at those studies before bed thanks ange

brisk ruin
#

Oh

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The students also have team homeworks

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Which they need to complete in teams of 4

shadow basalt
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Oh god

brisk ruin
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We pick the teams

shadow basalt
#

Scheduling a meeting with 4 people is harder than publishing a paper

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I shouldn’t be too negative I just had bad experiences maybe

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I’m curious to know what you think as it gets going

brisk ruin
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I am already thankful that I don't need to lecture for 80 minutes three times a week

shadow basalt
#

hahaha that is fair

brisk ruin
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If for nothing else then for the sake of my voice

shadow basalt
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is it like an IBL class where the "learning" is broken into psets

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i honestly love that teaching style

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i want to write a textbook for AT computations in that style

brisk ruin
#

Yeah so it's like we do a short mini lecture on a topic, then break the class into groups to work on some problem, they work at the boards for a bit, reconvene, discuss the problem, move on to the next problem/topic

shadow basalt
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is this like, an experimental class or is it the standard class for whatever subject

brisk ruin
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No this is how all the classes are run for this subject

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1 instructor with 18-24 children

shadow basalt
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Oh cool, has it been going for awhile?

brisk ruin
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Yes it seems that this structure is reasonably established

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We are LPU1

shadow basalt
#

whoever did this graphics design needs to be shot

unknown.png
brisk ruin
#

The discussion about teaching style begins on page 98

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But there is also discussion in some of the other documents

wise onyx
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I'm the 4th member to join this group

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Hope I don't make it hard for them to schedule a meeting with me

shadow basalt
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I think that scheduling is like, exponentially harder in n

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at least at uchicago people had the weirdest commitment schedules

winged urchin
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Hmmm...

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Given sets S_1, S_2, ..., S_n with elements within [0,24) chosen... somehow... What is the behaviour of [0,24)\{S_1,S_2,...,S_n}

meager bronze
#

you're in the wrong channel

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idk what channel you should be in, but it's not this one

wispy slate
winged urchin
#

I was phrasing the problem of scheduling a meeting with n people in a semi-mathematical way ahaha

wispy slate
#

Oh lmao

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I hadn't really read the discussion up to this point

pastel horizon
#

Don't know about you guys but I remember maths being kind of solitary and the group activities where usually the 'fun' lessond

pastel horizon
brisk ruin
#

Sure, this is for uni students

pastel horizon
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Ahhh uni is a lot different then. Guess I can see why group work would be more of a chore

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My lecturer had a strategy where you have individual tasks on a group project and a group task. So even if you don't contribute you still lose marks for not doing the individual part

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Or I suppose you could also have individual parts be sections of a report and have a group mark for presentation

shadow basalt
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The issue is that thing like reports on how the group worked and stuff are extra busywork irrelevant to coursework

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Which is generally avoided at the uni level by competent profs

mint lark
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I ca’d for it last semester

brisk ruin
#

CA = course assistant?

mint lark
#

Ye

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297 if you want to look it up

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Real analysis course

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Taken after linalg

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What course are you teaching that’s heavy group work Ange?

brisk ruin
#

115

mint lark
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Suspected

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Good luck

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Can’t remember who the coordinator is this year but some of the coordinators can be…

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Spicy

brisk ruin
#

Class sizes are bigger this year because of COVID reasons so we’ll see how that works

mint lark
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Bigger? That is inceresting

brisk ruin
#

20-24 instead of 18-20

mint lark
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Wack

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Will be interested to hear how it goes

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The intro calc courses at MICH have a bad reputation rip

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Both among undergrads and grads

brisk ruin
#

To make up for this we have less team homeworks

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I understand why they would have a bad rep among undergrads, but why among grads?

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Just a lot of work?

mint lark
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Yeah. Mostly depends on coordinator

grim spindle
#

what's 115?

brisk ruin
#

Calc 1

mint lark
#

Some of them are not very understanding and were especially not understanding last year with everything going on

grim spindle
#

interesting. The only classes that have group work in our department that I know of is combinatorics for CS majors and sometimes real analysis/intro to measure theory

brisk ruin
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This is not the only group work heavy class

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Precalc and calc 2 are as well

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As well as some others

pastel horizon
#

Feel like I'm the only one doing secondary here

brisk ruin
#

Teaching for secondary school?

pastel horizon
#

It's interesting hearing from a higher ed point of view as well though

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Yes

brisk ruin
#

I definitely don't think you're alone

pastel horizon
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And a level

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Hopefully

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Secondary is more behaviour management than uni for sure 😂

brisk ruin
#

DMAshura has a lot of experience teaching high school

pastel horizon
#

Nice

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Will be interesting to hear their experience

turbid zenith
#

Yo

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What did I miss? 😮

brisk ruin
#

A lot

crystal mortar
#

help, how do I prepare to encourage students to do group work? I am going to be teaching three sections of Calculus 2 (recitations) ?? I start this Tuesday 😦

brisk ruin
#

There are several things that you can do:

  1. Set an expectation for group work beginning from the first day of class
  2. You might present some research about the efficacy of group work and how it makes learning better
  3. Make sure that you are a good facilitator of group work and help it go smoothly
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Depending on what sort of room you're teaching in, another thing that you can do to get students engaged is have them work on whiteboards/blackboards so that they aren't sitting at their desks

crystal mortar
brisk ruin
#

It also helps to know your students so that you can target problems to their interests

strange bronze
#

are your lectures online or in person?

pastel horizon
#

Also I think it's a good Idea to pick smaller groups otherwise some people end up more passive

grave elbow
#

anyone good with set theory