#math-pedagogy

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long pelican
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Yeah, definitely there was more than 1 problem using the area perspective on the homework

pale wasp
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Pretty good you got them to do proofs in calc

long pelican
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🙃

pale wasp
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I know of upper division math classes with essentially no proofs

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Like undergrad pde, probability, cryptography, etc often aren't really proof based

long pelican
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Mm-hmm

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I think a lot of students are liking the proof aspect of it even

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When I reviewed the results with my TA today, she was utterly floored by how few people got the freebie problem too. She even told me that if this was a test given in China in primary school, most people would get it

pale wasp
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They teach random variables in primary school in China?

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I guess also consider that calc 2 students are ones who didn't take it in high school

long pelican
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Are you saying the stumbling block for this problem was the random variable bit?

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I felt like it was the problem solving aspect

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Oh no, my students mostly took AP Calc AB or BC in high school

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I think people who took that and got 4 or 5 get recommended for this class, and the ones who didn't get recommended for a lower level Calc 2

pale wasp
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Oh so it sounds like it is more advanced than normal calc 2

long pelican
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Yeah

pale wasp
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Ok that makes a bit more sense

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Since when I think of calc 2 I think of students just learning what an integral is and how to integrate

pale wasp
long pelican
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Ya, that's part (b) which I fully expected to take some thinking

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... but not part a lol

pale wasp
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Fair enough given they had hw about probability areas. Also I misread earlier and thought you meant the second part b was the freebie

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I do think they might have been more likely to figure it out if it were phrased more in terms of regions in the disk than in rv

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Since rv does add a bit more abstraction

long pelican
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Someone earlier mentioned that interpretation/analysis might have been the issue

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in terms of reading the problem

pale wasp
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Maybe it'll encourage them to study more on random variables for the final

long pelican
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For sure. The time from now to the final allows them to internalize everything too

tawdry venture
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oh the dartboard thing?

long pelican
turbid zenith
long pelican
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He makes no sense in general, borderline crank

unknown.png
turbid zenith
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He is very much in crank territory

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He is convinced that -5 > -2

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Whats even more annoying is he’s taken a social justice bent to it to sneak it in

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Once he started making it all about how India had the right idea instead of dumb Europeans, people eat it up

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And I’m like … I don’t disagree that we under emphasize non-European mathematics, but uh, this isn’t how to fix it

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https://twitter.com/jcrabtree/status/1582243245589155841?s=46&t=9eZiNjSFUeo0xPoBZTMb4w

Example

New Podometic K-8 Math Ed. = Arithmetic Debugged! (@jcrabtree)

#Symmetry

🇨🇳🇮🇳✅

⁻𝔹 ⊜ ⁺𝔹
𝔹 Negatives are the EQUAL & OPPOSITE of 𝔹 Positives

±𝔹 & ∓𝔹 ∑0

⁻5 ▷ ⁺2
5 Negatives are ABSOLUTELY GREATER THAN 2 Positives

Teaching 5 neg's are less than 2 neg's is white western racist non-sense!

#DontBeRacist
https://t.co/bEJ1G3c3Pc

Twitter ·
long pelican
winged urchin
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Well... 5 negatives are absolutely greater than 2 positives

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I'm joking around of course, and the racism accusation is just... yeah I won't touch that.

But it's a fair point that sometimes the notation of what's bigger and smaller is sometimes confusing for students

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We do often talk about the size of numbers with regards to their magnitude

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Like small x being values near 0, not going towards negative infinity

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Of course

turbid zenith
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Situation: Want to make a video on related rates.

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Problem: Most related rates problems are contrived as hell. Literally the two trains problem.

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Solution: Make a video directly lampshading the two trains problem. XD

winged urchin
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The like... shadow problem is phrased in a way of like.. who cares? But it can be pretty easily translated into a problem of like... a camera tracking a person or thing

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How fast do we need to rotate the camera to keep it fixed on the subject?

turbid zenith
cosmic ibex
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Hmm, I'm having some trouble mapping that to what Google tells me "related rates" is about. Is there a simpler way to compute this than writing the distance as a function of time and differentiating?

turbid zenith
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Yes, you don't have to have distance solved for

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Like you could get h = √(x² + y²), and then plug in functions of t, but that's pretty ugly

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$$\begin{align*}
h^2 &= x^2 + y^2\
d(h^2) &= d(x^2 + y^2)\
2h,dh &= 2x,dx + 2y,dy\
2h,\frac{dh}{dt} &= 2x,\frac{dx}{dt} + 2y,\frac{dy}{dt}
\end{align*}

burnt vesselBOT
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DMAshura
Compile Error! Click the errors reaction for more information.
(You may edit your message to recompile.)

209014240848117760.png
turbid zenith
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And then plug in given information and solve

long pelican
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I feel like a student who understands the chain rule should be able to solve a related rates question without being given instruction in related rates, if they think about the problem enough. I always wonder why there is a chapter on related rates problems? Same with "area between curves" for integrals...

turbid zenith
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I think it's a good use case of implicit differentiation

long pelican
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Well I'm agreeing with that, just that it can be a more rewarding experience to "derive" what the chapter says from the foundations than to be given instruction in how to do the problem then practice that type of problem

turbid zenith
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Sure, that's fair

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I'm going to frame it as how to use the same problem-solving process we used for applied optimization

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Making assumptions, drawing a diagram, etc

long pelican
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Lately I've been coming to the conclusion that noticing and observing things is by far a calculus student's biggest weakness in problem solving. Like a problem doesn't become hard unless they have to do that

turbid zenith
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Sort of modeling "what to do when you're faced with an unfamiliar problem"

turbid zenith
long pelican
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The dartboard probability question on my exam got few people successfully solving it not because they were weak in probability but because it was 100% noticing and observing

turbid zenith
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What was the question?

long pelican
cosmic ibex
turbid zenith
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I mean ... yeah?

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So yeah I guess you could solve it the other way

turbid zenith
long pelican
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Correct!

turbid zenith
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Okay good, I was worried I was missing something

long pelican
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It was supposed to lead into part (b) which was actually a calculus question

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Meant part (a) to be a freebie (lol)

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and it wasn't so

turbid zenith
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Oof

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Yeah you can't assume anything it seems

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I have students in my intro stats class who don't know what ≤ means

long pelican
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😱

turbid zenith
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Yes you do end up dividing by the square root etc

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But you have to do the Chain Rule for the square root, AND use Chain Rule again take the derivative of a variable that isn’t already in the expression, and a lot of my students tend to have trouble with keeping track of more than one derivative rule :/

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So if you’re already very comfortable with derivatives then yeah that’s the same thing

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But being able to keep the functions you’re differentiating from “getting too hairy” seems like it could help students who struggle

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You might see that two methods are mathematically equivalent, but pedagogically one could be much easier to grasp

cosmic ibex
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Hmm, I suppose I'll have to bow to practical experience.

tawdry venture
turbid zenith
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...yeah

tawdry venture
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how old are they

turbid zenith
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College freshmen

tawdry venture
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is it just a foreign symbol to them or do they have out of whack ideas as to its meaning

earnest trail
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regardless that's scary

turbid zenith
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I think it was just one or two students but it seemed foreign :/

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But there are a number of students who seem to be unable to think through a problem on their own that isn't "Step 1 do this, Step 2 do that"

long pelican
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0mg that’s like half of my students

turbid zenith
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I need to start making a list of words and phrases that annoy me when teaching math. Two of them are "step(s)" and "the answer".

long pelican
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They’re the students who think that what they’re supposed to get out of me doing an example is the steps

turbid zenith
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Yup

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They know how to get from point A to point B, but put them at point C between them and they're lost

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But seriously I've gotten so annoyed at "So is that the answer?"

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I'm like ... "look at the question, have you answered it?"

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Also it's crazy how some students will get things wrong MORE often than like if they were randomly guessing

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Like ask them the derivative of $2^x$, and you'll get $x\cdot 2^{x-1}$, or $2^x$, or $2^x\ln x$, but pretty much never $2^x\ln 2$

burnt vesselBOT
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DMAshura

209014240848117760.png
long pelican
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Was calculus a prerequisite for intro stats?

turbid zenith
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No

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I teach Calc and Stat separately

long pelican
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Oh so that was a question in calc?

turbid zenith
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Yeah

long pelican
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Meant as a problem solving activity rather than recall of a formula?

turbid zenith
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Like I feel like even if I say "okay so always do A, never do B", they'll respond with "got it, so always do B, never do A"

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Nope, just basic recall

long pelican
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Oh

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I have learned that students retain 10% from lectures but like 95% from problems they struggled through

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When I proved the fundamental theorems of calculus, the TA told me that during discussion the day after, almost no one even remembered that I proved something in class

turbid zenith
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That's what I thought too

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Which is why I flipped this semester and almost all the time in-class is spent with them doing problems

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But it doesn't seem to have made that much difference :/

long pelican
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Hmmm

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Do you observe the weaker students successfully doing problems too?

turbid zenith
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Not really. Nothing seems to help.

long pelican
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Oh I’m guessing in class they always say they are stuck and don’t know how to proceed

turbid zenith
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Oh also, only 75% of the students have signed up for the automated homework system still, and at midterms only 25% of the students had attempted more than half of the assignments

turbid zenith
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No "I tried this, but I don't know how to do that." Just ... "I'm confused."

long pelican
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Yep

turbid zenith
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Here’s another: “So do you always X?”

earnest trail
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yeah anytime something slight changes from what they're used to, they freeze

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if you do integration by substitution with v instead of u, I've heard kids ask if that's allowed

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...

turbid zenith
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Yeesh

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What I’ve been running into this semester actually is confusion on Product, Quotient, Chain Rule

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“How do you know which one is f and which one is g? Does it matter which is which?”

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Like they can’t see it as a template, they see it as “one of the functions must be f and the other g, after which you follow the rule”

earnest trail
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yeahhhh

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it's scary

turbid zenith
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They’ll blank if you write f’(g(x))g’(x), even if you’ve explained outer and inner functions in plain English and had them work through problems, but if you write outer’(inner)*inner’ they’re like “AHA A RULE I CAN FOLLOW”

earnest trail
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bruh

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my friend (college freshman) has to write out each step when solving basic algebraic equations

turbid zenith
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And I feel kinda helpless

earnest trail
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like she will physically write "+5" on both sides or smth

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like wut

earnest trail
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that sounds really infuriating

turbid zenith
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Instead of just guess sometimes

earnest trail
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true

long pelican
winged urchin
# turbid zenith I would rather my students do that

Yeah actually I think students writing out the steps is a good thing. Totally agree with Ashura here.

Most times students will think they are 'cancelling' something but half the time they aren't doing the nitty gritty steps and just think they wave their magic wand

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But totally the way you write the formulas can drastically change how a student understands it.

Writing outer(inner) like you said or first*last for product rule.

Similarly if you write sin^2(x) + cos^2(x) = 1 or sin^2[ ] + cos^2[ ] = 1. They will be able to substitute more easily into the empty box version rather than the x version

earnest trail
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that's true

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students are typically very hyper-focused on the specific letter for the variable

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as soon as you use something other than x they get confused

tribal tiger
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When I was helping him I ended up explaining it like f'(u)u' with some simple example like f(x)=(x+1)^2 let u=x+1 and he seemed to understand it better after that

deep kindle
burnt vesselBOT
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lexitorius

296975504907436032.png
deep kindle
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Writing that out helped me sometimes

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As opposed to f'(g(x))g'(x)

long pelican
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Have you tried to see if using zero notation is understandable?

winged urchin
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I think someone mentioned it earlier and sometimes it's not that one approach is necessarily better than another sometimes, it's just showing a student multiple interpretations can really reinforce their understanding

long pelican
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I'm just thinking they are and should be mature enough to parse and interpret arbitrary math notation without needing any crutches

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(Obviously they aren't)

mint lark
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99% of issues with math learning at the algebra to calc level is not understanding variables and what they mean (including function variables)

It’s tragic that this is very very difficult to correct on

long pelican
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btw today I made an analogy between epsilon definition of convergence and a mate in 2 problem in chess. A mate in 2 has the same logical structure: For all X, there exists Y such that it's checkmate.

Then did a game where people, in order, chose epsilon, then N, then n, to prove or disprove that 1/n converges to 0 (resp. 1) (This should also reinforce the notation that epsilon, N, and n are meant to stand for numbers, not just abstract letters)

mint lark
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I explain combining quantifiers this way to students a lot :)

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There’s a real game theoretic formalism for first order logic I believe to this

long pelican
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Ya, I made an interesting observation that every arbitrarily nested quantified logical statement can still be played with only 2 players

long pelican
cosmic ibex
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It seems the options for explaining nested quantifiers are basically:

  1. Game formalism.
  2. "Look, every subformula has a truth value -- that is, I mean, once we've given values to the free variables -- and this quantifier is part of the subformula that quantifier ranges over ..."
  3. "Shut up and pretend you understand what I'm doing".
spark widget
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If a piecewise function is continuous, what's the consensus on defining the pieces? Like for example, if y = x+3 for all x < -2 and y = 2 for all x >= -2

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Does it matter where we put the ≤ symbols?

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I know at least one of the "branches" of the function needs to cover it, but is there some kind of violation of notation if both pieces cover it, but the function is continuous so it's still well-defined?

cosmic ibex
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In that particular case it definitely matters, since you either get y(-2)=1 or y(-2)=2.

winged urchin
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Yeah unless you're careful with the overlapping regions your piecewise definition won't be a function as Tropo pointed out

turbid zenith
long pelican
turbid zenith
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Any particular part of this?

long pelican
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Intro, chapters 1 and 2 mostly

austere inlet
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that's a long read, will save for later

deep kindle
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I almost want to teach variables with more of a focus on the idea that their symbols can be anything instead of just x which is usually used

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I feel that I would have understood it better in Algebra I if they were introduced to me as empty boxes as opposed to the letter x

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Like saying 4(□) + 2 = 12

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Where the empty box is sorta there to be filled in with correct answers

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With some stress on the fact that there could be more than one answer that works, especially as we move on towards higher degree polynomials where equations often have two or more solutions

lethal leaf
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Alot of the conversations here seems to fall back to "we need to play catch-up but we don't have time"

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Of course the solution is better early level education so this isn't needed

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But is there any research going into how to play catch-up in the most efficient way possible?

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Not sure what keywords I would look for

south raptor
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That reminds me about one of the kids I work with who's only six years old and already hates math class blobcry

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Obviously that's an extreme example but it's not surprising that students go into college missing fundamentals when they're taught to hate math

winged urchin
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My favourite method for catching up students I tutor is having them work through problems and I comment on their steps or query them on slight variations that I know students might have common misconceptions about.

It's a longer session usually but I find it is effective. But that's one on one and again, if it's longer but more effective then it still might not be the most efficient

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I do tutor younger kids too and there especially I am very conscious of not 'taking math too seriously' as to kill their curiosity

vast sun
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Guys, so basically, since I’m good at maths, I wanna be able to explain to my classmates maths when they don’t understand it, BUT, nobody wants me to explain to them cause they don’t understand me :’)
What are the first steps to learning pedagogy ?

turbid zenith
# vast sun Guys, so basically, since I’m good at maths, I wanna be able to explain to my cl...

Here’s a link that may help you get started: https://www.quora.com/I-am-pretty-good-at-math-but-I-am-bad-at-explaining-it-to-others-How-can-I-become-better-at-helping-people-with-math

Quora
I am pretty good at math, but I am bad at explaining it to others. ...

Answer (1 of 10): Thanks for caring, and for caring enough to ask.

It's quite difficult to determine what makes teaching math hard for you, and what you might be able to improve or resolve, but here are a few ideas. Some of those things may apply to you, some may not – I'm afraid I have no way o...

vast sun
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Ohhh thanks youuu

turbid zenith
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And if you have questions you can always ask specific stuff in here too

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You could even try explaining a thing or asking how someone else might do so

vast sun
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So basically, from what I’ve read, I’m really bad at storytelling, but I got a book arriving tomorrow or overmorrow, which is a philosophical book, so nothing about maths right there

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It might help me maybe

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Thanks lol

winged urchin
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One thing too is math is difficult to just... Explain through speech. Indeed it's difficult even for someone to 'get' something in math without doing

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So instead of just explaining you can explain while drawing a picture or somehow giving an idea of what you mean with gestures. Incorporate the visual.

Sometimes writing an equation or something mathy down will be easier to understand than saying it out loud. I think most high schoolers are daunted when someone rattles off the quadratic formula but writing it down is... Less so?

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Even better than just explaining is getting them to do some of learning by asking them specific questions, possibly simpler questions to work up to the material you need to cover

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You get to see what misunderstandings they might have through their working out and you get them to do rather than just listen or watch

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Finally sometimes you have to have different approaches to show the same thing. One explanation might work for someone but not for another. Or someone might need to hear both explanations for the idea to really sink in

long pelican
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You can imagine that the best possible explanation in the world has a 30% chance of resonating with a random student, and every other explanation has a lower chance than that

turbid zenith
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Another big one is numerical examples

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Sometimes when you write down a big equation with a whole bunch of symbols, students eyes will glaze over. But if you show it with Numbers plugged in, that’s some thing they can grasp onto.

lethal leaf
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I was wondering more at the class level

vast sun
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Thank you all by the way you’re so nice

slate shell
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Does anybody have a suggestion for a nice and easy proof by induction for students who just started their first semester in compouter science? By nice I mean a proof by induction such that the students have to guess some formular which they then can proof by induction. Most proofs are either of the form "Proof the following equation" or are to hard for these students. But we want them to come up with a formular to prove on their own.

turbid zenith
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Sum of odd numbers

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Have them calculate 1, 1+3, 1+3+5, etc

wispy slate
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sum of angles in n sided polygon

turbid zenith
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Students are very likely to notice the formula themselves

wispy slate
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This one is a good one too imo.

unknown.png
turbid zenith
wispy slate
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Oops meant something else. Meant this one.

unknown.png
turbid zenith
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Ok that’s much more doable

wispy slate
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I like these geometrical ones as beginner problems

turbid zenith
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Doing it in general you run into issues with convexity that can make it hard to describe

long pelican
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If someone complained in the midterm feedback form that lectures are “not the most engaging”, but that’s because the same 2 people raise their hands when I ask a question and no one asks questions when I pause for questions, what is it that those students want? Is there a chance they complain even more if I start selecting random students to do things like pick a number?

deep kindle
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How do you prompt questions? Do you wait in silence or ask something like "do you have any questions?"

long pelican
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Both

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It’s section dependent

deep kindle
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Try rewording that to "What questions do you have?"

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I had a teacher in high school that managed to get the quiet kids to talk when she just assumed they had questions

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Source: Am quiet kid

long pelican
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What if that doesn’t fix it?

deep kindle
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Hmm I know health classes often prompt questions by giving students index cards and having them write a question at some point during the class if they're not comfortable speaking in class

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Health classes are usually very different in subject from math but I imagine there are enough people that aren't confident in math to think the questions they do have are valid enough to be asked

deep kindle
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I know that many people do not like that at all though, for some it helps and for some it just causes more anxiety

long pelican
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I think I should mention that this section is a hugely mixed skill group and the participation so far has been from the students who understand the most

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Which probably leads to the low skill majority being scared of asking a dumb question compared to the types of questions that have been asked

deep kindle
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Understandable

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In that case I would probably either ask for questions in a way that assumes they have questions: "What questions do you have?" instead of "Do you have any questions?", or give them an index card or something to write questions on that they're not as comfortable asking in front of the whole class

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Or even just emailing you their questions, though people procrastinate

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I do want to note however that I am not currently a teacher so all of this is coming from my own school experience as a student and from talking to lots of teachers about their pedagogies

turbid zenith
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What class is this?

long pelican
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Calc 2, semi honors, freshmen in college

winged urchin
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I always had this vision of teaching where I could have some kind of anonymous chatroom going on with the class and students would be able to ask whatever they want there anonymously and I could read the chat as I was teaching, addressing anything that comes up

turbid zenith
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So first off, about questions ... I've tried moving from "do you have any questions" to "what questions do you have". A lot of people make that suggestion, and yeah the idea is ostensibly that "oh if you assume they have questions, they'll open right up!"

Still doesn't work very well. You get maybe one or two more occasionally but most students are just as silent with that rephrasing.

One thing you could do is give them time to write down two questions that a person might reasonably have about the material, whether it's their personal question or not. The stronger students will come up with questions that they might already have answered themselves but might help others, and the weaker students are more likely to have their actual question in the mix, but having a set number of questions helps. Then ask three random people.

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Here's a thing I wrote up you're welcome to use:

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Second of all, a more global change you might want to consider making is implementing a more active pedagogical approach than traditional lecture in the first place. If you give students time to do problems, then believe me, the questions will start coming up like wildfire when they have to actually put pencil to paper. Because lots of students will just kinda nod along while you're explaining the material, because you're the one doing the math, but all the "wait a minute" moments happen when they have to do it themselves.

deep kindle
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I feel like when I'm in classes I can ask questions without interrupting the rest of the class

deep kindle
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Like giving them an implicit diff problem before teaching implicit diff just to make them use some problem solving skills and see what they can do, even if they get nowhere

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Then at least they're working on stuff together---which inhibits more proficient students to help mentor students that struggle more---and exercising problem solving skills which I see as one of the big things that math teaches you in general

turbid zenith
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Yeah exactly

deep kindle
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Sorry if I'm rambling lol

turbid zenith
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Here's an example of what I have them do in class...

deep kindle
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Definitely saving that helpful questions doc, thanks

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Exactly

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I'm in a data analysis class right now (stats content with calculus & emphasis on how data is analyzed in the real world---we've taken field trips to big companies in my state and such) and my prof very frequently has us do labs as opposed to quizzes

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And she did the same in Calc II, among other things she had us find an object irl that could be modeled by rotating a curve upon an axis and then use integration to find its volume, then test that by filling it with water

lethal leaf
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something to do with trees

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ideally something with a clear recursive structure

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after all, recursion is induction with a funny hat (or induction is recursion with a funny hat, whatever makes you feel better)

earnest trail
long pelican
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I also started class with a 24 puzzle: make 24 using 5, 5, 5, 1 and the 4 operations, to prime them into the mode where you try things in order to solve a problem, rather than "search for what method you're supposed to use"

deep kindle
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I like that

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Make them use their problem solving, that's what math should be about

chrome flax
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What do you guys see as more effective for teaching: Writing down all the theory for a topic on the Blackboard/Tablet and having students copy everything, or passing out the theory already noted down or just to have them write parts of it down, and have the rest on the printed paper already? I'll be starting by first (substitute) teaching in a few weeks, and the person I am substituting for suggested me to do the first option. However, in my school time, my teacher always handed out notes and just had us add some (crucial) points/conclusions... I am a bit worried that noting down everything will take away much valuable time..

turbid zenith
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The latter, for sure.

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I use guided notes myself when possible, and I'm gravitating toward using it more. Less time spent playing "scribe" = more time spent thinking and doing mathematics.

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What class is it for?

chrome flax
chrome flax
chrome flax
turbid zenith
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I do lesson by lesson because at this point I'm still MAKING them lesson by lesson 😛

chrome flax
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Haha - I see! Do you write them in LaTeX or Word? If LaTeX: Do you see an advantage in doing them there?

turbid zenith
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LaTeX unless I'm pressed for time

chrome flax
# turbid zenith

I see, thanks a lot! Planning to do the same. This is what you would pass out, right?

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Is that also LaTeX?

turbid zenith
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Yup yup!

chrome flax
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Wow, very nice! What Font Setup are you using with that, if I may ask 🙂

turbid zenith
#

The main font is Lato

#

I forget which math font it is

#

But I follow the convention of using sans-serif for text and serif for math, to help delineate the two

chrome flax
#

Ok, thanks a lot! I'll try that out 🙂

long pelican
#

I don’t have a good theoretical explanation for this but what I’ve seen empirically is that guided notes increase the feeling of understanding in class but decreases the ability to do nonroutine problems

austere inlet
#

if they don't take notes themselves they partly lose the feel for writing down those routine problems I guess

long pelican
#

You saying it detracts from ability to solve routine problems too?

turbid zenith
#

I imagine it depends on how you do them

long pelican
#

I’m remember a complaint a student made for another class who did guided notes: I understand fine in class then the test is completely different “application” problems

austere inlet
#

also writing one's own notes is a very important skill though I guess there's no need to have students write them while in class

turbid zenith
#

I’ve had students complain about that almost no matter what

#

Anything other than “step 1 do this, step 2 do that” yields that complaint

long pelican
#

You mean anything other than that on the exam?

#

Or during lecture

#

cuz I imagine that in lecture would make students feel like they understand the best, and do worst on the exam unless the exam is also like that

turbid zenith
#

In general

deep kindle
#

I think part of a solution to that could just come from variety of problems

#

I know DMAshura uses some cool examples, like that bitcoin bit in your implicit diff video

#

Like for conceptualizing rates of change maybe talk about more than problems involving velocity and acceleration

earnest trail
#

my stats prof allows students to choose

#

she has a copy of the guided notes and works through it during each lecture and many students print them out and follow along

#

I personally don't bc im too swag for that

#

jk our university limits the amount of free pages we can print and I didn't want use it up lol

lethal leaf
#

Why not just upload them online

#

and let students choose if they want to print them or not

#

I know I'll have exactly 0 use for a print out (since I have a tablet I mark stuff up on)

#

and also some students take their own notes anyways (like me)

#

options are good

#

let students do what they think is best for them (part of learning is learning how you best absorb information)

turbid zenith
tepid smelt
# long pelican I’m remember a complaint a student made for another class who did guided notes: ...

I do some guided notes at the high school mainly because it's easy and students respond well to it. In reality they have to be able to solve routine problems and guided notes help with that. It also just paces the lesson well.

I have been experimenting with group tests where I give non routine problems where an individual test will just be testing basic problems and conceptual understanding. I have found that they are more willing to engage with these problems with a group more so than in class as a challenging problem where it's not worth as much. I don't weight the group test as much and it's mainly to force them to engage with harder problems and be able to talk to each other about it.

vagrant meadow
#

one problem with guided notes, though, is if they get too bloated. you have to make sure you have discretion for when to skip some of the planned examples. my ODEs professor included way too many examples, and went through all of them way too slowly. we had to squeeze multiple chapters into the last two weeks of classes, and make the last exam a take home because there was still a whole chapter we had to cover, and we needed to use the final class meeting to speed through it.

turbid zenith
#

Yeah, you have to be okay with deciding not to do an example or two based on time

winged urchin
#

Hey all, kind of a lower level question in education. More so just looking for general opinions I suppose.

So I've started tutoring a younger student in fifth grade recently. I have tutored kids around this age before but this particular kid is being taught to solve equations right now and they're using that... well.. frankly what I think to be a poor way to teach how you solve equations. Where x - 2 = 5 and 4 - x = 8 are 'different' or taught differently. When they described how to solve for x they were saying stuff like "when x is being subtracted from then we add blah blah" or "when x is subtracted from another number then we do blah blah blah".

Now, they're a pretty bright student and their parents get them to do some like.. math competition problems at their level too. So I figured they didn't need this kind of methodology and instead I've tried to show them how they should be saying what they're doing to both sides or adding zero or multiplying by 1. (Frankly adding zero doesn't really show up much but of course multiply by 1 happens when working with fractions of course) It's only been one session since then and I do think they followed my explanation at the time but we'll see how well it sinks in.

My question is, what do you think of this kind of deviation from what they're learning in class? Do you think it's a good idea to introduce this kind of thing to them or what thoughts might you have in this direction?

long pelican
#

Where x - 2 = 5 and 4 - x = 8 are 'different' or taught differently. When they described how to solve for x they were saying stuff like "when x is being subtracted from then we add blah blah" or "when x is subtracted from another number then we do blah blah blah".

This is quite monkaS . I even think this goes against the common core?

winged urchin
#

I have actually seen this before in tutoring. And... if I recall correctly... it was in a first year university Math for Teachers course

long pelican
winged urchin
#

Yeah it's pretty odd. I remember the student in that course was actually so confused because the book laid out these solving problems as different varieties

#

But maybe it's actually gotten into some schools somehow

#

Your reaction makes me feel better about deviating from what the student is learning though ahah

long pelican
#

I'd even bet the book has some common core verified stamp on it on the outside

quasi musk
#

So I teach 5th graders right now in a specialized math program

#

Just emphasize how you undo adding or subtracting

#

Get variables on one side, numbers on the other. Watch your signs for good luck

#

Anyone have any familiarity with contest math? Like Art of Problem Solving books?

restive apex
# winged urchin Hey all, kind of a lower level question in education. More so just looking for g...

Yeah that is pretty confusing. I agree that this is a poor way of communicating, especially with such a young student, and I'd deviate as well. Unfortunately, some teachers are...not so great, and may not appreciate that the student isn't strictly following the directions. Sometimes tutoring a younger student with a bad teacher is a balancing act between, like, actually teaching, and appeasing the teacher so the student gets good grades and the parents are happy. A good teacher will see that the student's work is logical and be satisfied with that.

tepid smelt
# quasi musk Anyone have any familiarity with contest math? Like Art of Problem Solving books...

Beast academy is good for elementary students. The Berkeley math circle books are good also. I am also a fan of the math kanagroo contest you can buy all the old tests for 5/6 grade level for like 30 bucks. At 5th grade they are pretty close to pre algebra level and you could go right into AOPS or math counts level problems.

I own all the AOPS books but am not really a pro at contest math but my daughter has gotten into it so I keep up at her level which is currently 5th grade.

tepid smelt
# restive apex Yeah that is pretty confusing. I agree that this is a poor way of communicating,...

Sadly most curriculums are bad. It's hard to blame a teacher when they given poor curriculum and trained on it. Not to mention forced to follow it. Yeah sadly for tutoring parents often care more about results than actual knowledge. It doesn't hurt to reach out to the teacher on what they will accept. I think you should absolutely show the proper way but also make sure the student can do well in the class.

cursive barn
# winged urchin Hey all, kind of a lower level question in education. More so just looking for g...

I don't know how closely it might relate to your use case, but when I was teaching assistant for first year students, we had to support maths lessons for other schools (economists, biologists, computer scientists, ...) and in practice, not all students there were thinking "the same", so to speak. The teacher of the theoretical lesson often gave some methods to solve problems, and fortunately they allowed us to show other techniques while monitoring the practical exercises (the teacher was open minded and welcoming any technique, as long as it was "understandable" from the theory in the class)

and often in practice, the methodology sketched in class was not efficient for some students. some were "brain fast" enough to find shortcuts by themselves, and deduced by themselves faster way to solve problems. other were struggling more than average, and needed more guidance and more "rules to learn by heart", because (as much unfortunate it could be, it felt to be quite the reality) they just had trouble in thinking more abstractly and got confused when things were not "super scholar", so to say

and of course it's important in a math class to teach them some abstract way of thinking and generalizing and bla and bla; but in practice it might not work for reasons that are just, like.. "life is life"

#

maybe a way to approaching that kind of pedagogical conflict between different methods, would be to first, acknowledge that they were teached the way they were teached. it can be frustrating for a student to hear "your method is bs and bloated, erase all and now do like this". for some it might be blocking. maybe you can simply approach the problem by showing them the method they've learned is correct, but there exists a more general/more flexible/always working way of solving those equations. for the students that are confident enough, they would infuse the information and make it theirs by themselves. for the others that struggle a bit, it might take more time, or maybe never occur; and "life is life"

(you can also start slowly by providing 2 versions for the solution of the exercise, then quickly get rid of the one you like less; that's more like, a cruisade approach were you implicitly force people that are interested in the solutions, to follow the conventions)

as far as I remember, it's also how my high school teachers were teaching "conflictual" methods; for a lot of students the information was just like "wow wow I don't get any of this?! can I just stick with my previous way"; for some it was "aaah, clever, clever; I'll try to use it"; for some it went "yeah.. that's like, obvious, no?". as long as the teacher is kind and patient, I think transitions go more smoothly than expected, on average

idle estuary
idle estuary
# wispy slate Oops meant something else. Meant this one.

picking 3 consecutive points of the n-gon, then triangulating the n-gon will split it into (n-2) triangles.
the sum of interior angles of each such triangle is 180 always. the result follows by induction on n.
in your version of the problem the n-gon is necessarily convex since it is inscribed in the circle.

r1.png
long pelican
#

I'm assuming you're using induction to show that there are (n-2) triangles in the triangulation. Which raises the question: most of us probably say this is true without using induction, even if under the hood we're making a tiny induction argument (add 1 side, adds 1 triangle). Would you consider this to be an example of a problem that uses induction in that case?

idle estuary
idle estuary
# long pelican I'm assuming you're using induction to show that there are (n-2) triangles in th...

I was reading through the introductory section of [1] which starts out with many motivating examples of premature conclusions.
I think the purpose of that introductory part is to give examples of pitfalls.

The book also contains multiple problems on n-gons designed to be solved by induction.
Some of those are sensibly more suited for induction than the one previously mentioned (even though that one is also a good start).

[1] Induction in Geometry by L. I. Golovina , I. M. Yaglom (published in 1979, Mir Publishers)

Disclaimer: I've only read a small portion of the book.

https://archive.org/details/little-mathematics-library-l.-i.-golovina-and-i.-m.-yaglom-induction-in-geometry

Internet Archive
Induction In Geometry ( Little Mathematics Library) : L. I. Golovin...

The preface says: This little book is intended primarily for high school pupils, teachers of mathematics and students in teachers training colleges majoring...

idle estuary
idle estuary
lethal leaf
#

It may not be interesting

#

But honestly

#

Stuff with summations and products

#

Those are operations people are familiar with

#

But it's really really obvious to see going from the nth term to the n-1st term + a little extra

#

And that's what students need to see first in my experience

#

Before you get to the cool examples like coloring a plane or dominos or recurrences for runtime

tepid smelt
#

When do you allow for students to discover a technique through a series of problems and questions vs just giving them the technique and having them practice?

I find a lot of my current curriculum pushes the discovery based lessons and I feel it's sometimes a waste of time. Where I can just present the technique and get students to develop fluency and work towards harder problems and applications of the technique

lethal leaf
#

Why do you feel it's a waste of time?

#

I feel like those discovery things, if well done, make great homework problems

austere inlet
#

I feel like that's a big "if"

turbid zenith
#

It depends on how much time you have, I think.

#

I understand the need for discovery sometimes, and I'm still trying to figure out a way to make that happen.

#

The issue is that if you do "discovery" in class, different students will have those "a-ha" moments at different times

south raptor
#

I think giving students a moment of discovery is really important to get them to hate doing math a little less. The problem is if they've been conditioned to be spoonfed the methods then they might get frustrated being told to figure it out on their own

#

So it's important that it's done rjght

wispy slate
#

I would almost say if it was possibly I would do exclusively discovery based lessons

#

At least the way my brain worked, I could only understand a topic if I knew why it was that way. If I want to apply techniques to hard questions as you suggested, I would’ve needed to know why the rules are the way they are so I can manipulate them when it gets to loosely related styles of questions

#

And idk how it works where you’re teaching but in the NSW extension 1 and extension 2 papers they will include a tonne of questions that combine logic from different topics, like he most common one was calculus induction

#

I don’t think I would’ve gotten through without the discovery lessons

long pelican
#

You raise a very good point. My experience so far suggests that students who have had zero discovery experience fail to make any headway on non routine problems, and even worse, fail to even see what the value there is in those types of problems

wispy slate
#

for undergraduate exams, should points be subtracted for making a small numerical error that results in a wrong answer to an exam question?

for example the question is to compute the spectral diameter of a matrix and is graded either 0, 0.5 or 1, the student does everything correctly except a numerical error in calculating the roots of a quadratic equation, hence gets wrong eigenvalues. the course obviously doesn't cover how to solve quadratics, and the student knows it from middle school. From what the student has written it's obvious he knows everything covered in this course, but made a numerical error. Should the question be graded 0.5?

#

I'm interested in how these cases are handled in different universities

tawdry venture
turbid zenith
#

If it didn’t trivialize or drastically change the problem I would give full points.

quasi musk
#

So usually strong students will make just a few arithmetic/algebra errors worth 1 or 2 pts here and there

pale wasp
#

the only case where I'd take off further points for using the wrong answer is if it greatly simplifies/changes later answers

#

e.g. if they got the wrong answer 0 instead of say something that depends on x and y and having 0 makes the other parts of the question really easy

#

or e.g. if they're computing probabilities and get a negative or >1 number and don't make any note of that being wrong

deep kindle
#

In high school I started making the distinction between "imaginary" and complex numbers, mostly because I don't like calling any numbers imaginary

#

It's much more recently occured to me that one could do the same to the real numbers by perhaps calling them rudimentary

#

Idk I don't like implying that some numbers are real and others aren't when they are metaphysically the same

tidal mica
#

has anyone done a Budapest Semester of Math? asking here bc the program seems to be very focused on pedagogy

mortal crag
#

Hello all. New to tutoring here. What are some good preliminary questions to ask a student I’m tutoring?

#

/What else should I prepare?

narrow nest
#

This is an extremely simple question that very young students struggle to grasp:

#

4+3 is the number that can be found by counting up 3 from the number 4.
Student's incorrect answer:
4,5,6, so 4+3=6
Correct answer:
5,6,7, so 4+3=7

#

How do you even explain the concept of an off by one error without using too many big words or confusing the student? Surprisingly tricky

long pelican
#

Fun fact: Intervals in music work like the "incorrect" example

turbid zenith
long pelican
#

10:07 😆

turbid zenith
#

XD

tawdry venture
lethal leaf
#

Just help them with that

#

And as holes appear, you'll know what to prepare

amber quail
tribal tiger
mighty terrace
cloud zealot
#

If it were a requirement to construct each number system from the naturals to the complex numbers in a real analysis class, when would be the best time to present this material, and why?

south raptor
#

I'm so confused how I'm supposed to teach this to a student who hasn't learned about variables yet

20221115_1917512.jpg
#

My strategy was to start by having her make a table with different combinations; so the first row would be 25 chickens and 0 cows, the 2nd row would be 24 chickens and 1 cow etc

long pelican
#

Just based on my memory of years ago, it's questions like these (I remember very vividly an age question in particular) that really motivated why variables were useful

south raptor
#

Then from there hopefully she would notice that each time you replace a chicken with a cow the number of legs would increase by two

#

One of many problems is she kept getting confused why adding a cow adds 2 legs even though it has 4 legs

south raptor
long pelican
#

I remember that I grappled with a single age puzzle question for a ridiculously long time, on my own

#

Trying to fit together and understand everything how the technique of variables worked

#

I think that kind of grappling is unavoidable and probably even the best way to do it for the first time

#

Your primary goal would be to motivate the student to want to grapple with it himself in his own time

#

rather than going for any specific explanation

#

Puzzles like age puzzles and this are perfect because you can try the method of variables, get a result, and you can check whether what you did worked by plugging the numbers in

south raptor
#

Right now if a kid doesn't want to do work I usually just ignore them and work with a different kid

turbid zenith
#

(In all honesty I would not mind getting to make videos like this for a living ;P)

turbid zenith
#

Anyone have experience in creating a class Discord?

lethal leaf
#

Haven't created an official one

#

But I've seen them used in multiple courses (some of which I TAed for)

#

@turbid zenith

winged urchin
#

Then the students boost the server for you and demand their math emojis ahah

#

But I like that idea in general, especially with anonymity

lethal leaf
#

So I think Discord is much much better than alternatives like Ed, Campuswire, and Piazza

#

because you can easily have back and forth conversation

#

however it has no ability for "private" posts

#

which makes it really hard to use for some things

cursive elbow
# lethal leaf however it has no ability for "private" posts

I don't manage discord servers so I'm not 100% but I feel like there ought to be a way to do this with the forum system, otherwise if the class is small enough one could set up a channel for each student and set perms so each can only view their own, worst case scenario they can just DM you for private questions

lethal leaf
#

Yea but it's alot harder to setup

#

DMs are a no go

#

It's nice if all instructors can see every private question

#

iirc forums are all public so that may be harder to manage. I'll think about that though

lethal leaf
#

I think maybe a ticket type system could work for private posting

#

but those are hard to track and even worse, ticket bots delete channels but often we want to keep those private posts around

#

if those private posts are something like accommodations for tests, or regrade requests.

pastel sundial
#

Like if you dm modmail a question it'll post it in a private channel that only mods can see. And then any mod can anonymously respond via the bot

#

At least I was under the impression that's how it worked. In any case most things one would want to do on discord can be handled by a bot

lethal leaf
#

I've used modmail for large scale things (CTFs) and I've used Piazza to answer private questions in a course

#

and idk how to describe why I think this

#

but using modmail for that second purpose to me seems quite clunky

still elm
#

for question a) assuming i just wrote linearity of F_A without explicitly writing out the matrix multiplication and jusut wrote |A| = max sum(a_i,j))

unknown.png
#

and this has 10 points

#

how much do i lose

lethal leaf
#

wrong channel, this isn't HW help

still elm
#

did you read

#

what i said

lethal leaf
#

NVM I can't read

still elm
#

np

lethal leaf
#

you're asking what a good way to grade this would be?

still elm
#

if you are a TA or smth please give ur input

#

yes

#

i should have wrote the matrix multiplication explciitly

#

and then took out the sum (a_i,j) as a common factor

#

i literally just wrote norm(ax-ay) <= norm(a)norm(|x-y)

#

<=*

lethal leaf
#

if this is a HW/exam you took

still elm
#

and then wrote since A = (ai,j) --> |A| = max(sum(ai,j))

lethal leaf
#

then this isn't really the channel for this

still elm
#

i just want to how would you guys ( who were prob TAs or graders ) would mark this up

#

thats it

#

not asking for any math here

still elm
pastel sundial
still elm
pastel sundial
#

Read the channel description lmfao

still elm
#

idk where the fuck i asked for math help

#

but ok

#

ty

pastel sundial
#

In what universe is this not asking for math help

still elm
#

mark = grading

#

i am asking how would you grade my incomplete answer

#

and how much points i woud llose

#

this could be in any subject in existence

#

mark

#

marks

#

score?

pastel sundial
#

That is still asking for help

austere inlet
#

or more generally people with some sort of teaching experience which is this channel's target audience

turbid zenith
#

Anyone have suggestions for review games that work for large classes (100+)?

pastel sundial
#

starting to feel like the difference between mathematicians and K-12 math teachers is that mathematicians don't understand pedagogy and K-12 math teachers don't understand math

long pelican
#

Isn't that an old saying?

pastel sundial
#

is it?

long pelican
#

Hmm I'm sure I read it somewhere

#

Maybe I can find it...

pastel sundial
#

that's very plausibly an old saying I heard somewhere and then forgot but I didn't have a source I was citing in mind just then

long pelican
#

I'll be bold and modify it a bit to: no one understands pedagogy

pastel sundial
#

true

#

I've started volunteering at a local high school to help out in a math class and holy shit the TSM is so real

long pelican
#

I haven't been inside a high school for a while, what's it like now?

#

Maybe it's still exactly the same

#

as 2012

pastel sundial
#

horrible

#

the teacher doesn't really distinguish between correcting people on accuracy, technique, or mathematical grammar

dim monolith
#

high school sucks

pastel sundial
#

like he'll correct one student for getting the wrong answer and the correct another student for writing $4 \times x$ instead of 4x

burnt vesselBOT
#

Gamma is an Algebraic Number

381174875215364109.png
pastel sundial
#

and then when helping a third student tell them what they did wrong was not follow PEMDAS correctly

#

except it's not even PEMDAS it's an even worse acronym he made up

dim monolith
#

PEDMSA

long pelican
#

Order of operations issues in high school

pastel sundial
#

it's like

long pelican
#

very analogous to algebra issues in college

pastel sundial
#

they're solving basic equations right

#

like x+3=a solve for x

#

but it's readily apparent that not a single student in the room understands what an equation is

long pelican
#

Oh my

pastel sundial
#

like I was helping a student and they were like
x - 5 = a
x = 5a

#

a mistake that is perfectly understandable if you view equation solving as formal string manipulation and completely impossible if you understand what things actually represent

long pelican
#

I'm curious if the teacher shared any wisdom or experiences with you

pastel sundial
#

it's like if you wrote an algebra engine in a typed programming language where all of your variables are strings

#

and then transcribed the code to human-readable form

#

and taught that instead of math

pastel sundial
long pelican
#

Darn

pastel sundial
#

besides more or less saying "yeah these students are too far behind for me to teach them the correct way. This is the only way to get them ready for the standardized test"

#

except he didn't say "teach them the correct way" because I don't think he understands just how incorrect the way he's teaching is

#

also when I asked him for a copy of the worksheet the students were using he didn't have an extra copy. And then he said pretty much "it's no big deal these problems are super easy" IN FRONT OF THE WHOLE CLASS

#

imagine not understanding that intro K-12 algebra is one of the hardest topics of math if you evaluate difficulty in terms of the size of the conceptual leap required relative to the average starting point

long pelican
#

Just one teacher though

pastel sundial
#

literally every piece of math I've learned in the past 2 years is trivially easy to teach compared to getting an 9th grader with mathphobia from 9 years of TSM to understand algebra

long pelican
#

Yep, honors linear algebra was way easier to teach than Calc 2

#

Well, not content-wise of course. I remember thinking hard about how to present dual spaces and the proof that determinant of transpose is determinant of the original matrix in the nicest way possible

#

But in terms of meeting students where they're at, the honors students don't need much handholding

#

In contrast, un-TSMing is hard

long pelican
turbid zenith
#

Okay but how do YOU propose to "⭐PROVE⭐" to middle schoolers that a negative times a negative is a positive in a developmentally appropriate way

long pelican
#

I wrote this out in my notebook. Assuming that 2 x -3 = -6 has already been proved. Looks very logical. Something a middle schooler should be able to follow?

image.png
#

The first thing that came to my mind is that I have to use that -2 + 2 =0 because that's how I even know what -2 is in the first place

pastel sundial
#

ooh I like that

turbid zenith
#

That's pretty good! That's doable.

#

Though students have to understand the distributive property

long pelican
#

Pretty sure that is in the common core although obviously expecting it of students in the present time is another story

turbid zenith
#

If there's one thing I've learned about students' understanding the distributive property in years of teaching

#

It's that students think the distributive property is about parentheses

#

Not a relationship between multiplication and addition

long pelican
#

That's pretty interesting

turbid zenith
#

They view it as "when it looks like this, you can do that"

pale wasp
#

What is TSM?

turbid zenith
#

Which I think explains why they want to do things like $\sqrt{a+b}=\sqrt{a}+\sqrt{b}$ and $(x+y)^2=x^2+y^2$

burnt vesselBOT
#

DMAshura

209014240848117760.png
pastel sundial
turbid zenith
long pelican
#

Team SoloMid or Textbook School Mathematics ©

pastel sundial
#

isn't it Textbook School Mathematics?

#

right yeah

turbid zenith
#

Fixed, sorry

pale wasp
#

I see

long pelican
turbid zenith
#

It's like whoever was talking about formal string manipulation

#

Symbol-pushing

#

Okay now THIS I have a problem with.

image.png
#

Because I am absolutely in that "definition last" camp and I will die on that hill.

#

So, I see where this dude is coming from, and I agree in some places, but uh, the more I read it, the more it sounds like it's just written by yet another mathematician who confuses mathematical foundation with pedagogical foundation.

long pelican
#

Are you thinking about the definition of a limit as an example in particular?

turbid zenith
#

Yup

long pelican
#

I had a homework problem where the student was given that 10^-n converges to 0 as n -> infinity and I had them prove, given this, that there exists some n such that 10^-n is less than 0.0001. (This was part of a multi-part problem whose goal was to prove that 0.999... = 1) Was one of the hardest parts of that set by their own admission!

#

It turned it around, instead of proving convergence, it had them use convergence to prove the existence of something else

#

Well I said 0.0001 but actually in the problem it was more like "any number your opponent might give you in the first round"

#

But yeah, that shows the definition of a limit is a very unusual example of a definition in that it's like 1000 tiers beyond what they've seen

#

With the triply nested logical quantifiers

#

This problem was quite nice because it bypasses the horrible algorithmic thing students might have learned from AP calculus

#

and it's like instant / one-step for all of us here but it was the hardest thing ever for them

#

Some definitions are just better not presented in a first calculus class. I think definition of a limit with epsilon and delta might be slightly beyond that line. Riemann sum as some limit of | Delta of partition | of some sum involving x_i is also beyond that line, for sure, because that limit is not even a normal limit

#

I think I decided the definition of limit of a sequence as n-> infinity was behind the line (and taught it) because I think epsilon and N was a bit more intuitive than epsilon and delta

turbid zenith
#

That seems reasonable yeah

pastel sundial
#

isn't Wu's definition of a definition a bit more relaxed in the K-12 context?

turbid zenith
#

I agree limit of a sequence is a bit more attainable at that level

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And by "at that level" that includes college students, at least where I teach

pastel sundial
#

like he makes it clear he's not arguing that we should teach 5th graders that fractions are a "equivalence class of ordered pairs of integers such that..."

turbid zenith
#

My Calculus I class is a college level class and I have students who can't factor or miscancel ALL over the place

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You give them $\dfrac{x^2+4}{x}$, they'll tell you that's $x+4$

burnt vesselBOT
#

DMAshura

209014240848117760.png
pastel sundial
#

smh clearly it's just x

long pelican
#

Isn't it depressing to know they will never get an A in the class unless they receive individual attention to fix all those prerequisite errors?

turbid zenith
#

To be fair a lot of them do get it because I'm at a small school. I teach two Calc I classes of 20 students each.

pastel sundial
#

once you take the asymptotics pill you never go back

turbid zenith
#

So some of them DO get A's because they come to me and go over and revise material

pastel sundial
long pelican
#

yeah but those students who continue to have algebra gaps

#

And probably never participate in class

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Poor them...

turbid zenith
long pelican
turbid zenith
#

"Standards" is often a euphemism for gatekeeping.

long pelican
#

honestly I have lowered my standards too. My homework now mostly consists of 1 or 2 step logical deductions instead of 3+

#

People still say it's hard of course 😆

turbid zenith
#

I'm still figuring out what to do about those students with the algebra gaps, but little by little I make dents

pastel sundial
#

or I suppose perhaps not

turbid zenith
#

I agree it should be a larger scale reform, but it can be implemented in the classroom level.

pastel sundial
#

yeah I was just thinking

#

what would the best way to do that? Like if you get an A on the final you get an A on the course?

turbid zenith
#

Under systematic conditions that allow it at least. I teach at a private school so fortunately I have the freedom to choose my own grading system.

pastel sundial
#

yeah that's nice

turbid zenith
#

If they don't master a Learning Target, they can try again.

long pelican
#

In practice, mathematical skill-based learning targets don't seem to work as well as they should in the long term

turbid zenith
#

How so?

long pelican
#

Let's say you have some list such as: "Be able to take the derivative of various functions." "Be able to calculate various limits" (either specified by type of function or type of limit). Two problems are: 1. where do non-routine problems fit into this? And 2. how do you avoid students merely learning steps to solve problem types?

#

Actually the two problems are very related because non-routine problems are how you test whether students aren't merely learning steps to solve problem types

turbid zenith
#

Well I try to make my Checkups not just be routine problems

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They tend to have explanation and interpretation as part of them whenever possible

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And I judge whether they're understanding based on those, and I assign a score accordingly.

long pelican
#

To be clear I'm only speaking from my experience and what I've seen

turbid zenith
#

To be fair I don't have this locked down perfectly. I'm still working on finding the right balance.

long pelican
#

The main part of my experience I'm working with is all the exam questions I thought were freebies or close to freebies, but get low scores, simply because they haven't seen that type of question before

turbid zenith
#

Yeah that's true absolutely

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I think students need to be deliberately broken of that

#

The expectation that the test or whatever should be just like the homework with a 2 changed to a 3 or a plus changed to a minus needs to be eradicated

long pelican
#

Even the students who weren't expecting it to be that simple nevertheless found themselves struggling to swim when they had to solve a problem from their own thoughts instead of matching with a problem type

#

I like your highlighting of multiple approaches for the same problem in #1. A non-routine problem I'm thinking would be something like: Why is sqrt(f(x)) maximized at the same value as f(x) is?

turbid zenith
#

Ahh I see!

deep kindle
#

That’s what I like about gateway style exams too where you can take the same assessment more than once to show how you’ve improved

next relic
#

In my mind, a nonroutine problem (for non-mathematicians, can't say for maths majors) is one that tests the same concepts in an unexpected way.

#

For example, instead of asking students to find equation of the tangent line you may wanna ask, if this is the equation of the tangent line to this function at x = a what's the value of f(a) and f'(a).

#

Steve Butler's exams are pretty good at this imo.
calc1.org all the way to calc4.

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(Their calc 4 is ODE)

turbid zenith
#

I think at least SOME familiarity isn't necessarily bad

#

Like I'm beginning to worry sometimes that if every time I ask something it's asking it in a new way, the students can't "latch onto" anything

modern mirage
#

What's some good extra credit problems I could give to calc 1 students given they've only seen differential calculus? I'm trying to find some interesting problems that would inspire the students to think.

long pelican
#

You could ask for a parametrization of the set of points that are exactly 1 unit away from the edge of an ellipse

#

You could phrase it as a challenge to prove to themselves (and the reader) whether they think the set of such points is another ellipse or not

modern mirage
#

The students haven't seen parametric equations yet in the class, so I feel like it'll be a rather hard core problem for them to do though.

#

I am thinking for the first problem, it's gotta be some problems they've kind of seen in class before, but not at all trivial. (they've covered only till L'hopital rule recently and curve sketching using derivatives)

long pelican
#

There's a lot of functions where it's impossible to solve for its roots algebraically but where you can nevertheless prove it has exactly one root using derivatives

#

Although if I was giving any advice, I'd make that standard material since it's a pretty big motivation for why to even study derivatives in the first place

modern mirage
long pelican
#

One of my colleagues put such a question on the first exam and he said the performance on that question was absolutely atrocious

modern mirage
turbid zenith
#

Welp, students are struggling with limits still. But I dunno how much of that is difficulty in calculation vs difficulty in concept.

long pelican
#

oooooh

#

Do you remember any evidence suggesting one or the other?

turbid zenith
#

Dunno. Most of what they’ve been doing is computation I guess

#

Their quiz on the concepts is due tomorrow

#

But today we were calculating derivatives using limits and they were struggling with stuff like conjugates

#

Making basically every algebraic mistake in the book

long pelican
#

How are you separating the two? When my students struggle to understand the epsilon definition (which I know you don't do), it's usually a conceptual (logic) failure

turbid zenith
#

So on their quiz one of the questions is asking them to draw a graph meeting certain limit criteria

long pelican
#

Hmm that question looks like one that a student who half-understands or otherwise weakly understands the concept would get right by remembering that your pictures show a continuous graph with a hole

turbid zenith
#

And the other gives them the function (x-k)/(x^2-16) and asks them to find a value of k for which the limit at 4 is defined, and another for which the limit at 4 would be undefined

#

And to explain

long pelican
#

That one requires an observation the start which is to factor x^2 - 16

#

so there's a computation!

turbid zenith
#

yep yep

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The first one also asked a question about continuity actually

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I told them the left hand limit and right hand limit are both 3 as x approaches -2, but the function value is 5

#

Or something like that

long pelican
#

So the way you're imagining concepts and computation, I'd say there's a huge space between concept and computation that most students' troubles probably fall inside

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The little things like making observations (like being able to factor something), doing a step on their own that isn't what they've seen in an example, what steps are valid to make on their own, etc

turbid zenith
#

Yeah. They have basically zero algebra sense

#

And have gotten by in math by following examples exactly

long pelican
#

Sounds about right!

#

I think you might need to do a lesson about algebra and logic and how to do algebra steps on your own and be confident about it

#

maybe more than 1 lesson

turbid zenith
#

How to break students of that when it’s so ingrained?

long pelican
#

My thought during this semester was that they have never experienced doing mathematical steps on their own, and the homework wasn't helping, so I had to spend class time giving them that experience

#

like students at blackboard type of experience

#

You're flipped so you might already do that?

turbid zenith
#

Yeah. Almost all our time in class is spent doing problems in groups

long pelican
#

Hmm my hypothesis is groupwork at the desk only benefits the top student in each group

turbid zenith
#

Though tbh there’s often no group dynamic

long pelican
#

Do you ever have students at the blackboard?

turbid zenith
#

Once in a while

#

Usually to put up solutions they’ve already done

long pelican
#

I have them at the blackboard to generate ideas

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Basically to prove to each and every student that you are supposed to generate ideas for math problems

#

lol

turbid zenith
#

I don’t like pick a random student to work the problem because everyone else checks out when they’re not on the hook

long pelican
#

Totally, which is why I emphasize that the purpose of the blackboard work is to put ideas down and not necessarily to solve the problem

#

although they end up solving the problem with hints from the class anyway

turbid zenith
#

I guess I’d need to see it in action

long pelican
#

Ya it's something I've never tried or have seen tried before but it does seem to help

turbid zenith
#

Because I’m having a hard time visualizing it

long pelican
#

I guess I'm demonstrating by experience the kinds of things you are supposed to do at the beginning of a problem

turbid zenith
#

I’ve never had work at the board go well except from students who are already strong, and I don’t want to demoralize them

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So there’s something I’m not doing right

long pelican
#

Ya 2 of my students were unable to write anything down, not even trivial observations

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I explicitly encourage trivial observations (I don't say the word trivial though), such as expanding a summation

#

everyone else was able to though

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also I put them at the board in groups of 4, not individually

#

to ease the pressure

turbid zenith
#

Ahh interesting idea

long pelican
#

I think one important thing it has taught students is that factoring an expression at any time works and produces an equivalent expression, not just when you are asked to

#

(and also not just when an example of a similar problem is worked out with factoring)

lethal leaf
#

tho TBH on the student end, it ends up one person doing all the writing and talking and the other 3 smiling and waving to the crowd

lethal leaf
#

And I'm not sure how best to deal with it at a higher level when I'm sure many students have similar holes in their algebra skills

#

and poor algebra skills really just compounds when you get to higher level stuff that just assumes you know the algebra

tribal tiger
#

My calc 3 prof tried groups this semester on the board and yeah it was usually one person doing the work and explaining while the rest stood there. We had a fairly small class (the others ended up getting weeded out so to speak) and what worked best was we did in class work in the last 10 minutes or so that would be collected and you'd get bonus points (usually 1 point) and that seemed to work really because no harm if you don't do well and they were able to walk around see who was struggling and where help them at the time with the problem. Then when we got them back we would go over the problem before starting lecture and the professor would cover some of the mistakes people were making.

deep kindle
#

Man I love microsoft whiteboard but you can't make folders to organize whiteboards

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Biggest downside of the app

tepid smelt
#

I observed a discussion based geometry classroom where the instructor had some problem the students had to work on as a class and they wouldn't move on until the class was convinced. He rotated different students to the board and had almost no instruction only to clarify arguments. It was an interesting idea but it was a university high school with about 10 kids in it.

I am encouraged to bring students to the board but a majority of the class zones out. When I encourage them to work at the board in groups you get most hanging out watching one or two kids work. I still find it valuable to wake them up and you get them a bit more energized on the problem. In groups you always get some kids just passively observing and they generally struggle to pass.

I remember in university having to get called to go through a full proof of a HW problem but you were warned ahead of time but it was interesting because you had to answer questions in real time from the professor. I found it very valuable to build confidence in communicating your ideas and think something like this could be beneficial to implement more in if anything to befit the particular student talking.

long pelican
#

I think I should probably clarify what my thinking on this is. It's just to expose everyone (for the first time) what good habits in problem solving are like, so that they can go and practice these good habits when solving the tricky weekly homework problems. Instead of having them practice bad habits that get even more entrenched every week

#

Also I made it very important that by a certain point in the class everyone has picked up the chalk and written an observation of their own on the board at least once

#

At that point I don't need to do it as much, if at all (but it's actually fun, so it became a semi-recurring thing)

cunning raft
deep kindle
#

Oh cool I’ll check it out

turbid zenith
deep kindle
#

My dumbass thought of fundamental theorem of calc before pythag lmao

strange breach
#

Does anyone find it easier to learn math out of their own will (like learning it for fun because you want to) as compared to having to learn it in a place like an academic environment?

long pelican
#

I think so, but you have the issue where the average experience of learning it in an academic environment is just not that good for various reasons

last pecan
cursive elbow
#

I think it's definitely more fun when one is learning out of self-interest, but I also personally like the structure that an academic environment provides (i.e having assignments with a fixed deadline, being able to go talk one on one with someone who knows their stuff (or at least ought to))

south raptor
#

I too find it easier to learn math when I don't have to worry about tests, homework, and paying tuition

south raptor
#

def gonna try this with the kids at my work now
https://www.youtube.com/watch?v=yBtoWmA9SKM

YouTube
Mathematical Visual Proofs
Don't use FOIL! (distributive property area model visualization)

In this short video, we visualize how to extend the classic mnemonic FOIL to more complicated products using the area model. While this model is technically only valid for positive numbers, it provides a better intuition for how to compute products of sums than FOIL, which is limited to the product of two sums. In theory, all of this is just the...

▶ Play video
turbid zenith
#

I love this

#

And hate FOIL

#

Also ... one video left ... and man I am sick of making videos on such a tight time table -w-;

humble raptor
#

this is a rather weird question, but I'm writing a paper for my research class and the topic, the extended delta conjecture (haglund remmel wilson) is a 2 version conjecture, with part proven, and the other part unproven.

the paper has a rationale section, where i'm supposed to explain (to a general audience) why progress made on the valley (unproven) version of the delta conjecture is helpful

#

i'm not exactly sure how to explain the applications of the delta operator, such as galios theory, to a general audience, when i myself have no idea what that is

#

for further context, i am a high school student who is working with one of the authors of the above conjecture to prove the valley version

frosty pewter
#

FOIL meant I didn't actually understand how to expand brackets first time they were taught to me lol

long pelican
#

Throwback to my first problem set of my current Calc 2 class on algebra review/practice
Foiled FOIL on day 1 😎

image.png
turbid zenith
#

I'm beginning to wonder if leaving individualized comments for each student on their quizzes is even worth it, vs just posting my solutions

long pelican
#

What kinds of individual comments do you tend to write? Also I recently read about the phenomenon of students not reading feedback on graded things

turbid zenith
long pelican
#

I see, this student is making something like “single-task” errors

turbid zenith
#

What do you mean?

long pelican
#

Like instead of being stuck on a problem, he’s answering questions incorrectly

#

To be clear, you wouldn’t expect something of the first kind on this problem

#

Unless they’ve completely been missing class

#

And not studying

#

I’d say for errors like these, there’s some incomplete understanding that writing the correct answer down in a comment won’t correct very effectively

#

Nor even providing solutions

#

Maybe he needs to see more examples of these ideas being used mathematically

#

In a logical way

turbid zenith
#

Actually ... it's kind of amazing, a LOT of students drew something that doesn't pass the vertical line test

#

A large number of students submitted something that looks like a rational function with the other part overlaid on top of it

long pelican
#

Dam that’s really useful information for you

turbid zenith
#

These are all take-home open-book/note assignments

#

But they're supposed to be non-collaborative

#

If students are collaborating they're obviously not doing it well 😛

long pelican
#

Ah ha I wasn’t thinking about that

#

The useful information is that they had an unseen deep misconception or incomplete understanding of what a function is

#

That might have impeded any or all things they learned in the semester

turbid zenith
#

Yeah seriously

#

Like ... we're talking about DEEEEEP conceptual holes

#

Like "how did you make it out of algebra" conceptual holes

#

I guess it explains why so many students are doing terribly

#

Plus not turning in work

long pelican
#

Not turning in work is on the other side of the line id say between being an issue that it’s within your “jurisdiction” and an issue that’s out of

turbid zenith
#

Like things have been staggeringly bad this semester overall :/ Like only a handful of the students in each class really "get" calculus and it's as if nothing I've done matters

long pelican
#

Did you get any feedback on the videos from students about them being hard to understand or anything?

turbid zenith
#

Most of the students (who do anything at least) just try to parrot steps

#

Nope, nobody's said anything of the sort

#

Though when I try to dedicate the first five-to-ten minutes of class to answer questions about the video or anything in it, the only thing is "can you go over the two questions at the end?"

#

Nothing else. No conceptual questions, no procedural questions, just "what were the answers"

long pelican
#

I’ve had the thought that seriously trying to reach all students takes many of the same skills as being a doctor

#

Diagnosing gaps that are hard to catch and treating them in a timely manner

turbid zenith
#

Yeah

#

I'm ... lost on what to do in some cases with this class :/ I tried doing exactly that, but like ... the gaps were always like "why on earth would you ever logically think that" or "the words you said make absolutely no sense" kinds of errors

long pelican
#

Haha I’ve ranted about that in the past with another class

turbid zenith
#

Like the students were supposed to leave comments or questions or answers to others' questions on the videos using Perusall

#

And often the questions were like ... nonsensical

long pelican
#

They’re probably not logically thinking and more trying to say things that they think are right or sound right

#

A cure for that is to make logical deduction a part of the class assignments and tests

turbid zenith
#

I try to do that

#

:/

#

I feel like this is how these conversations are going:

Me: (problem)
Response: Try doing (solution)
Me: I'm already doing (solution), I'm still having (problem)

long pelican
#

Hmm maybe I’ll give an example of a logical deduction question: you know derivative > 0 implies increasing. Does it follow that second derivative > 0 implies increasing too?

#

And the answer is not that second derivative > 0 implies something else but they need to provide a counterexample

turbid zenith
#

ah

#

This is gonna sound terrible but I'm pretty sure my students wouldn't get that. They struggle with even finding a basic sign chart.

long pelican
#

I’d bet not many many of them have learned in their life that counterexamples disprove “for all” statements

turbid zenith
#

This batch has been exceptionally weak with like a few exceptions. And I've been desperately trying all semester to bring them up to speed but the kinds of errors I see just make me more and more depressed

long pelican
#

With my students this year I discovered that not many of them have had any problem solving experience in their life which explains why they didn’t solve my freebie problem

#

Which led them to not actually be doing problem solving on the homework

#

So I corrected that in class

#

And the results were dramatically better on the second exam

long pelican
turbid zenith
#

Well you're working better magic than I am if your results were dramatically better

long pelican
#

The other one just checked out and didn’t want to change how they think

#

Well I wouldn’t say I had any special ability. I just lucked out on finding something that worked

#

And also making the right diagnosis

#

I think if I'll be bold, I'll suggest even for your weak students to make the problems more like problems and less like exercises

#

and work with less content too

#

There's something about doing problems (actual problems) that makes things stick a lot better than just exercises

turbid zenith
#

"actual problems" -_-

long pelican
#

Wellllll

turbid zenith
#

Any suggestions on how to come up with """actual problems"""

long pelican
#

You can look at the problems you currently have and modify them

#

Have you seen the matchstick example in Lithner's work?

turbid zenith
#

no

long pelican
#

Got the name wrong

#

It's actually Lithner

turbid zenith
#

still no though

long pelican
#

ok so

#

The example in his paper is as follows

#

Original question was

image.png
#

He made a single change

#

And that single change changed the problem from an "algorithmic reasoning" question into a "creative mathematical reasoning" question

turbid zenith
#

Okay so

#

I have to say there's nothing so disheartening as to be told that the thing that I should be doing is the thing I've attempted to do in many many cases throughout the semester and to have had it not work. I've tried to design my in-class Activities to have that ⭐🌠 OPEN ENDED CREATIVE MATHEMATICAL REASONING 🌠 ⭐ and to have ⭐🌠 DISCUSSION AND PROBLEM SOLVING 🌠 ⭐ be a daily occurrence in class. And to design the Checkups so it would give students a chance to explain their reasoning.

What happened was just ... "What do we do? What's the first step?"

And pretty much nothing I did brought them out of that.

#

My revision:

image.png
long pelican
#

Ooh that’s open ended for sure but the difference between this and the matchstick example that the actual end should be closed, just the middle should be open

#

And I’ll also add that it only works if the beginning and end are easily accessible and understandable

turbid zenith
#

Ahh yes open middle problems

#

Okay so what I'm getting from this is that I'm just shitty at coming up with problems

long pelican
#

Nah not you

#

You shouldn’t need to have to come up with problems

#

Textbook writers should have them

turbid zenith
#

Yes, the textbook I decided to make optional this year because I wanted to go in a drastically different order

#

So perhaps I shouldn't have done that, I should have just stuck with the status quo

long pelican
#

I’ll also add that textbooks don’t have them lol

#

Like this semester, I was lucky that Strang had good problems but I still had to supplement them with my own

#

And for the latter half, the other professor who taught the non-Stewart version provided me with his homework and exams

#

And they were good, but I also made some of my own

#

If you use a book like Stewart the only source of problems is the last 2 questions at the end of each exercise section and the “problems plus” sections

turbid zenith
#

Yup, we have to use Stewart

long pelican
#

ya and most Stewart users probably don't touch those problems

turbid zenith
#

Fought against it in my department but got outvoted

long pelican
#

In principle the textbook makes no difference if you have to come up with your own materials no matter what textbook option they choose, and students barely read the textbook anyway

#

(it's too hard for students to read)

turbid zenith
#

Yeah, Stewart is unnecessarily impenetrable

long pelican
#

I think all textbooks are

turbid zenith
#

It's like trying to drink from a firehose

long pelican
#

Textbook writers have lost sight of just how much mathematical maturity the median student lacks

turbid zenith
#

That's why I tried making my videos, one reason at least

#

The big reason being as I mentioned before, hoping that pushing the abstraction of limits to the end would make things more accessible to students

long pelican
#

Yeah limits are hard

#

I think my last question on the most recent exam is a great question to test their knowledge of limits that can't be fooled by rote memorization and also isn't too hard

turbid zenith
#

Fun. Yeah, it's not too bad if you understand the ε definition and think about what it means.

#

I've been stressing understanding meaning all semester but it hasn't gone well.

long pelican
#

Here's what I believe about understanding: you achieve it from working a hard problem, and nowhere else

#

This is math-specific obviously

turbid zenith
#

So I've been hobbling my students by trying to scaffold from easy to hard?

long pelican
#

Well the scaffolding was necessary

#

I had to scaffold too

#

At the beginning

#

And taper it down

turbid zenith
#

What I try to do is work from easy to hard, and design problems where students are supposed to run into the kinds of difficulties that make weird things happen

#

Sorry, exercises, not """problems""" I guess

#

. . . and then the students can't do the easy ones

long pelican
#

ah yeah, unexpected roadblocks

#

I've learned a lot about what students find hard

turbid zenith
#

Almost every single time

long pelican
#

And "hard" could be something like a 2 step logical deduction for us

#

I feel like your students not understanding what a function is might be a relevant factor to consider when making problems too

turbid zenith
#

By the way, I managed to finish out my in-class activities for the semester, so I threw them all into a single PDF.

📎 PDFsam_merge.pdf
#

Some may have typos we fixed in class, apologies if there are any in here.

long pelican
#

Oh wow you have integration too

#

You did limits, derivatives, AND integrals in a single semester?

turbid zenith
#

Just the beginning of integrals

long pelican
#

And FTC right?

turbid zenith
#

Yes

long pelican
#

Yeah nice

#

A lot of content

turbid zenith
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MAT 131 Calculus I (4 hours)
Calculus I, II, III, and IV form the recommended calculus sequence for students in mathematics and the sciences. The objective of these courses is to introduce the fundamental ideas of the differential and integral calculus as they pertain to functions of both one and several variables. Topics for Calculus I include limits, continuity, rates of change, derivatives, the Mean Value Theorem, applications of the derivative, related rates, optimization problems, introduction to area and integration, and the Fundamental Theorem of Calculus. Offered every fall semester. Prerequisite: Satisfaction of the mathematics placement requirement (Sec. 6.4.1.) or prior completion of MAT 130 with a grade of “C–” or higher.

long pelican
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Honestly it's going to be impossible to have students who don't understand functions and also cover from limits all the way to FTC in a deep fashion

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especially if you follow a standard sequence and standard problems

turbid zenith
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I did technically add in exp/log functions because I don't see a reason in doing them late in Calculus II and pretending students have never learned about them just so we can give a """rigorous""" definition, and also I barely introduced partial derivatives to give a better way to link together implicit differentiation and related rates, a la 3b1b's presentation

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Just at the level of "hey we can hold one thing constant and differentiate the other"

long pelican
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Yeah honestly partial derivatives from last year teaching was one of the super instant things for students to grasp

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I'm guessing this was impossible for them 😛

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turbid zenith
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But that description is from my university's bulletin with official course descriptions
https://bulletin.oglethorpe.edu/13-course-listing/mat-mathematics/

Oglethorpe University Bulletin
Victoria Lindbergh '18
MAT (Mathematics) - Oglethorpe University Bulletin

MAT 111 Statistics (4 hours) This course introduces students to both descriptive and inferential statistics. Topics include describing data graphically and numerically, discrete and continuous distributions, correlation and linear regression, basic probability, sampling distributions, the Central Limit Theorem, confidence intervals and hypothesi...

long pelican
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For part (a), you have to use the product rule AND conclude something from what you got

turbid zenith
long pelican
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That's 2 step logical deduction!

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Ah-ha so you did have some high-flyers?

turbid zenith
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So most students never bothered looking at those

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Yup, a few

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Maybe 2 or 3 in a class of 20

long pelican
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yeah I have some too

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Just another random thought that came to me, understanding can be hard when you are unable to see the forest for the trees

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You can do a guided exercise only looking at one tree at a time

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Which can you make you feel like you understand it "locally" but your understanding isn't flexible and still contains many errors

turbid zenith
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Mhm. I try to address that by making problems bring things back from older sections, and deliberately pointing out how they do that.

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Like the choice to put motion problems halfway through the semester was so they could bring together stuff from both derivatives and integrals, sign charts, etc.

long pelican
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Thinking back to the function vertical line test thing, it was only when they had to draw a function that their misunderstanding of what a function is was finally revealed, right?

turbid zenith
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Honestly I don't know if that's a misunderstanding of what a function is

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Or just an attempt to draw a graph without the word "function" even passing through their heads

long pelican
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Maybe not a total misunderstanding but a big one

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hm let me look at the question again

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Ok so it looks like on the right side of the y-axis he had no problems and on the left side, he drew a line and he said "this line has to go somewhere"

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And he made it wrap around on itself

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Ok I guess it's hard to say by that alone how big his/her misunderstanding of functions is

turbid zenith
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A few others.

long pelican
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Oh my lol

turbid zenith
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That third one was from one of my high flyers who's been nailing everything all semester.

long pelican
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Now I can see a systemic issue

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I see your convention of a limit existing is that a limit of +infinity "exists"

turbid zenith
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Yes. But I've told students that if they were already taught that (\lim\limits_{x\to 0}\dfrac{1}{x^2}) does not exist because it's (+\infty) I'll still accept it but they have to specifically make that argument.

burnt vesselBOT
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DMAshura

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turbid zenith
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My preference is to say that limit is +∞.

long pelican
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I've always said that both of these are true: the limit is +infinity, and the limit doesn't exist

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Existing requires being a real number

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aka an element of the codomain

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If for example the codomain was the projective line, then the infinity limit would exist

turbid zenith
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See that makes no sense to me to say that the limit is +∞ AND the limit doesn't exist

long pelican
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Makes perfect sense to me, because infinity is a concept and not a number 🙃

turbid zenith
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I personally disagree 😛

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Concept and number are not mutually exclusive first of all. Unity is also a concept, but we also say it's a number.

long pelican
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Well here's some formal logic thing I thought of: The limit exists iff there exists an element L of the codomain such that lim_{x\to a} f(x) = L

turbid zenith
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And second of all, just because something isn't a real number doesn't make it not a number.

long pelican
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To amend that to allow infinity in there

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You'd have to say

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The limit exists iff there exists an extension of f to a function from R to R U {-infinity, infinity} such that lim_{x\to a}f(x) = L

turbid zenith
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Sure. But I'm not using language that formal with my students.

long pelican
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But that's what justifies the not existing thing in my mind

turbid zenith
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I'm okay with someone deciding to say that an infinite limit doesn't exist even though I disagree with it — it just seems weird to say that it """is""" infinity while not existing

long pelican
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For series, $\sum_{n=1}^\infty \frac 1n=\infty$ and the series diverges and the limit of the series doesn't exist

burnt vesselBOT
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Icy001

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turbid zenith
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Usually what I see is the limit doesn't exist because the function "grows without bound" or whatever other euphemism the textbook has chosen because infinity gives the author the heebie-jeebies

long pelican
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ok here's a question for you. Do you agree with the following: if limit of f(x) exists and the limit of g(x) exists, then the limit of f(x) - g(x) exists?

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all limits as x->infinity

turbid zenith
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Not if the limits are both ∞.

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For me, any limit laws have the caveat of "if it's an indeterminate form, take more care."

long pelican
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So from the point of view of uncomplicated definitions "infinity => doesn't exist" is more convenient

turbid zenith
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I assume you mean =>, since there are plenty more ways for a limit to not exist than being infinite

long pelican
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Yes I did

turbid zenith
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I don't see it as that much more complicated to say "beware indeterminate forms"

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Since you should be doing that anyway

long pelican
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That would be something for a procedural-based mindset but not for a logical definition

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So like, if the furthest thing you will do with a limit is to compute them, then sure

turbid zenith
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Eh, not really?

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Though again for 90% of my students computing them is the most they'll be doing most likely

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We're talking about a school where you can count the math majors in any given year on one hand

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So when they get to real analysis, they can be more nitpicky about infinity then

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Since that's the point of real analysis

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But I very much disagree with the implication that "procedural" and "logical" are opposites.

long pelican
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Which is sort of annoying from a logical standpoint

turbid zenith
long pelican
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It's just an aesthetic thing which is relevant if we're arguing what the right convention is

turbid zenith
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If I were writing, say, a textbook on this, I'd have no problem with making that caveat once at the very beginning

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Show indeterminate forms, show why they're indeterminate, and then make the caveat that any subsequent limit laws don't apply when they lead to one of these forms

long pelican
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I think we're talking past each other because I forgot to say I was thinking of infinite limits from a mathematical viewpoint and not a pedagogical one

turbid zenith
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Ah. Okay then.

long pelican
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Yeah my mistake. The mathematical debate came to my mind when I saw that you wrote that dividing by 0 is not enough to say that the limit doesn't exist

turbid zenith
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Yeah my mindset here is mostly pedagogical. I tend to disagree with the typical choice to present the fully-sanitized definition as early as possible.

long pelican
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And I wasn't arguing against your convention because if it's not a universal convention, whatever you set in your class goes

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So back to all the non-function function graphs

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It's exactly analogous to when I asked my students last year on an exam to write a probability density function based on some properties of a random variable and they wrote complete nonsense

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It made me think about every single time I said something that I thought was crystal clear but actually wasn't, if you factor in the gap

turbid zenith
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I should ask my students what a graph of a function should look like

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My hunch is that they would mostly know it has to pass the VLT

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But in trying to answer the limit question that just went out the window

long pelican
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Well maybe now they do, but actually the bigger lesson might be you can't predict what they are missing in advance, and rich tasks like drawing a function satisfying certain properties are what you have to give to spot them

turbid zenith
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Because they were thinking of it in isolation

long pelican
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So like, now the VLT hole is plugged but there are still n-1 holes left

turbid zenith
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I think you misunderstand what I’m saying let me try to rephrase because I didn’t say it precisely enough

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I think if I asked students, “did you know that a function graph has to pass the VLT” the vast majority would truthfully say yes

long pelican
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Hmm I agree with that

turbid zenith
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And that the nonsense graphs didn’t come from not knowing the VLT but rather failing to integrate that into the task

long pelican
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But I'd agree with it only insofar as it's semantic knowledge

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but not integrated knowledge

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cuz I know in the algebra I or II curriculum the vertical line test is a huge thing in the introduction of functions

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but not very integrated

turbid zenith
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I also think if I’d given them a circle and asked “is y a function of x” they would have mostly said no

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And if I’d asked why they would have given some equivalent of the VLT

long pelican
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I also agree they'd say no

turbid zenith
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Maybe I don’t know what you mean by semantic knowledge

long pelican
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And that they'd cite VLT

turbid zenith
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But I don’t think this is a misunderstanding of functions

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I think it’s a failure to integrate ANY of their knowledge

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Because they put everything in little boxes and retrieve them when explicitly asked

long pelican
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Yep it's exactly that, but my view is that integrating knowledge is one of the most important tests of understanding

long pelican
turbid zenith
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So I guess this is gonna be a never ending battle

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Because I’m fighting years of training to retrieve things from boxes

long pelican
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This has made me realize something about my own teaching, I had been subconsciously averse to a lot of things without being aware

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Writing problems that encourage compartmentalizing things in boxes was one of them

long pelican
turbid zenith
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No I mean they were

long pelican
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Ohhh

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Yeah, I think this is actually one of the biggest reasons I'm a huge fan of challenging but accessible problems

turbid zenith
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See now I’m thinking of that matchstick problem

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I always see these examples of open middle problems which let student be so creative on their problem solving! But then they’re always like … algebra 1 problems.

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And I’m like okay, I need them to calculate derivatives

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And you occasionally have derivatives and integrals you can write in a few different ways but often when you do that every way but one makes it take like 3 pages

long pelican
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How about some natural optimization problems?

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Like, no forced real life situation

turbid zenith
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So I feel like I rarely see these mythical open ended problems at the level of anything I teach at the college level

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What counts as a “natural” optimization problem

long pelican
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Hmm something you might wonder about if you were thinking about numbers or shapes or stuff

turbid zenith
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Something I might wonder or something they might wonder?

long pelican
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They, I guess

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ciassic one would be maximizing product of 2 numbers given they have to add up to a fixed value, which no doubt you probably already use

turbid zenith
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Because a lot of them have had the wonder kicked out of them

long pelican
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Ah