#math-pedagogy

1 messages · Page 13 of 1

next relic
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I've always been thinking about giving harder exams then curving students afterwards.

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But idt many students (I'm just across the border, to the north) appreciate that.

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When they're so used to getting 90s on all of their exams.

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Sadly I'm just a TA and I don't get to write exams. :(

long pelican
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It's a very well known problem, you can probably find at least 2 discussions on it in this very channel

next relic
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I'm in ug and not even in math sooo idt I'll be writing math exams soon, but I'm more than happy to suggest questions!

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(I've been writing mock IB exams for a while now :D)

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If anyone's interested: I'm thinking of giving this question to my calculus 2 students. They should've covered ODE by separable variables.

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Thoughts? Too hard? Too easy? Perhaps more scaffolding?

long pelican
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My first impression is that it's a great question
My second impression is that students in a typical calc 2 class don't understand anything about what they are learning, so a question they haven't seen before will need a lot of explaining before they even understand what it is saying

next relic
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In case you couldn't tell I'm very ambitious with this question :p

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I'm afraid the chain rule will beat most students.

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But answers are given in part (a) so hopefully they're able to still proceed with part (b).

long pelican
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My third impression however

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These questions are great for revealing misconceptions you never knew your students had in them

next relic
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Oh?

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Can you elaborate?

long pelican
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Yep, to take a random example totally not from experience, there was a misconception I saw that in the notation f(x), x is inside the set described by f

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This would never have come to light with routine questions about functions (there, they just apply procedures they learned)

next relic
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OH fr??

long pelican
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ya

next relic
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Wow, yeah, have never seen that before.

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The integral in part (c) is also quite tricky.

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So this is definitely a grade A discriminator.

tall bolt
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Yeah I’m in the UK, it’s kinda a blessing and a curse, it actually tests your understanding rather than just memorisation but equally with so much weight on the exam you can sometimes end up with a terrible grade based on a bad day or a hard exam. It’s hard to decide which is better

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In my DE class last semester I just kinda had a bad day on a pretty ok exam so I got a B in that despite really knowing my stuff for it, and my geometry exam that semester was just wildly hard, despite having a 95 in my coursework I scraped an A,

equally though if you’re getting A it does actually show you’ve absorbed the material because to get that grade you will have answered questions which were unseen

next relic
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Uni exams should be treated in a similar way to A-level. @tall bolt

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Either (1) students take the paper, instructor decides the grade boundaries then scale the raw marks, or (2) MS should be changed to make sure the standards stay the same. (This is the way Ireland standardises their LC exams)

tawdry venture
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@next relic update: i let the teacher know i would be grading the tests on an M2+A2 rubric and she said that was brilliant

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cause she was in fact going to suggest M1+A1

next relic
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Thanks for the update, appreciate it!

next relic
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@tawdry venture Don't beat yourself up for formatives. It's not like they're worth a lot in the final grade anyway. :D

tight star
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I feel like “derivative is slope of tangent line” can do a little disservice sometimes
It’s hard to precisely define a tangent, for one
And also the argument usually goes “the secants get closer and closer to the tangent, therefore their slopes get closer too”
But it’s not always true that, if two shapes “visually” get closer together, so do their properties - e.g. consider the usual pi = 4 “proof”

vocal phoenix
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I think it's mostly fine if you treat it as intuition rather than definition.

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I tend to write out the difference quotient and draw the corresponding secant line, and show that as the difference in argument goes to zero, the secant "converges" to the tangent, and the difference quotient converges (without quotation marks) to the derivative

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But I agree that defining the derivative as the slope of the tangent line is fraught with peril due to the imprecision

wise onyx
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You can define the tangent line to the graph of a function at a given point as the line passing through that point with slope equal to the derivative of the function at that point. In this way, "derivative is slope of the tangent line" becomes a tautology.

tight star
tight star
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What troubles do people learning math have with the concept of a function?

winged urchin
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It looks like multiplication but isn't

cloud zealot
long pelican
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By itself, this shouldn't be an issue because what I just mentioned is far less complicated than exceptions of English grammar, but it's compounded by lack of exposure to consistent mathematical language in their environment

winged urchin
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An expression is I suppose, some sort of mathematical notation that somehow is well-defined loosely. An equation is a statement that two expressions are equal (although ambiguous sometimes whether it's meant to be read as stating it is true for some value or many values or perhaps a question of if it is true)

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Sometimes an equation to me can seen more in a true/false kind of dynamic

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Heck even defining equations and expressions in complete generality isn't so clean aha

cloud zealot
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formulas are things where you plug in numerical values for x to get f(x)

winged urchin
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My mind went to things like the pythagorean theorem or trig identities or like... the idea of if we have a quadratic equation in standard form (y=ax^2+bx+c) then we know the x-coordinate of the vertex can be found as -b/2a

winged urchin
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But for the purposes of passing their only math course they need to care about I suppose

gray smelt
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They learn that a function is f(x)=formula. They may get that you plug in values there and get a number.

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But the abstract idea of an input output object isn't automatic.

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Also, all of the functions they see are polynomials and trig functions.

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They won't think of
a|->blue
b|->green
q|->purple
as a function. They won't think of the indicator function of the rationals as a function.

tight star
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One case where this is decidedly unhelpful is something like the pigeonhole principle

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“Forall collections of 3 pigeons with 2 pigeonholes, there exists a pigeonhole with at least 2 pigeons”

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This isn’t really something you’d prove with a function

gray smelt
tight star
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Mhm mhm

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I think one thing I tried to emphasise is that you can be creative when defining a function

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For example i showed them piecewise definitions, and their reaction was “woah i didn’t know you could do that”

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Also showed them how different formulas can give the same function, like sqrt(x^2) and (x^4)^(1/4)

gray smelt
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What are other good elementary examples to drill that home?

The evaluation functions from B × A^B to A?

tight star
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I- elementary?

gray smelt
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Just explain that as functions from B to A and B?

tight star
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I feel like A^B as an object isn’t that elementary if you’re struggling with the concept of a function

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Could be wrong

gray smelt
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Oh, okay, maybe.

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You're probably right

tight star
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So the examples i worked through were |x|, indicator of the odd numbers

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And minimum

gray smelt
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From functions to their value at 0? I feel that this is elementary.

tight star
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They were stuck trying to prove that (0, 1) is open, and i feel like this was partly because they didn’t even know “min(a, b)” was something they could write down

gray smelt
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I think you could get them to understand the function that sends functions to their value at 0

tight star
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Possibly…

gray smelt
tight star
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It is a kind of higher-order abstraction required

gray smelt
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why would the min thing make it harder to prove that it's open?

gray smelt
tight star
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It’s much like having categories whose objects are themselves combinations of objects of other categories

gray smelt
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Perhaps, the function that adds 1 to functions?

tight star
gray smelt
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Do they know what derivatives are?

tight star
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Whereas evaluation takes a function as an input

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The idea that you can take a function as an input is nontrivial, i think

gray smelt
tight star
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Mhm

gray smelt
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Programming helped me a lot with functions.

tight star
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It definitely would

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Though programmer’s functions are a little different to mathematician’s functions

gray smelt
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Not really?

tight star
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A mathematical function is more like a dictionary or hash table

gray smelt
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Any programmer function is a function in the math sense.

tight star
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In programming, you care what algorithm you use to evaluate the function

gray smelt
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Oh!

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I see what you mean.

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Yes

tight star
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Pfft

gray smelt
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some programmer functions are mathematician functions.

tight star
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It’s funny how quickly you turned around

gray smelt
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Unless you use a purely functional language

gray smelt
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But, if they know what derivatives are, that's usually the first function of functions.

tight star
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Yes, but im not quite sure it’s viewed that way explicitly

tight star
gray smelt
gray smelt
tight star
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Mhm, that makes sense

pastel sundial
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Functions in cs are different from functions in math, but if you understand one then you'll be able to learn the other pretty easily

tight star
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Actually i guess on this - why do people have problems when letters get introduced to math?

tawdry venture
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because it's a conceptual jump from numbers

drowsy otter
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I agree with Ann. Introducing letters that represent numbers is a big concept to grasp. This is why some people aren't ready to be taught algebra when it is introduced.

noble hare
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i think its even harder of a jump when you go from "here is an equation with some numbers and one mystery unknown number that we have to solve for" versus full on letter symbol pushing to prove identities or solve equations in terms of variables

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reason being imo that the former is atleast more natural and common for people to think about e.g. "how much more money do i need if I have £27 now and want to have £100 by the end of the month"

long pelican
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I have a new take on this question which is an angle I didn’t approach before. The hot take is that ambiguity explains why students have trouble with variables, as well of lack of proper guidance in navigating it. When letters enter the scene, there’s many ways to interpret a mathematical equation or sentence with letters. Among them are:

  • as a symbol (like x)
  • as a variable (like x)
  • as an existentially quantified number (like in “pi + 2 pi k”)
  • as a universally quantified number (like in “a^2 + b^2 = c^2)
  • as a name for a quantity (I have a apples)
  • as a name for an object (a stands for apples)
  • as a name for a function argument (as the x in “f(x)”)
    Without proper guidance in which interpretation to pick when, there’s eternal fuzziness in the mind when doing any work with variables
cloud zealot
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variables is also kind of a bad name since it connotes something varying somehow

hardy igloo
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Hey. I'm so nostalgic tonight. There was a mod here (at least back in 2021) who did some really amazing youtube videos of really fundamental math stuff, like logic etc? I'm looking for the youtube channel!!

cloud zealot
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aka dmashura

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he has an online presence in several other places too

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is that who you were thinking of?

winged urchin
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That's who comes to mind for me too lol

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He's shared some great videos in this channel

civic peak
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halo

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guys

tawdry venture
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hello @civic peak

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!redir (preemptively)

coral copperBOT
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This channel is only for on-topic discussion. Please take casual conversation to #discussion or #chill.

hardy igloo
hardy igloo
civic peak
tawdry venture
# civic peak wspp

do you have a question to ask that has to do with this channel, or did you just want to chat?

civic peak
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I do have question

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any anyone help me understand trignometry??

west junco
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Педегоги нахуй

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Ушли нахуй из етого канала

lethal leaf
civic peak
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if i could read why whould i ask you

tawdry venture
tawdry venture
hasty parcel
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Hi all!

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I'm an undergraduate student that just started studying modules in an abstract algebra class

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I was wondering if there's a nice way to think about modules similar to the way the 3b1b does matrix/vector space visualization

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my best guess is that there isnt a full equivalent, because of the looser structure, but I was wondering if there's any such way to think of them that comes close, or if this is just a fail

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and if it's just a fail, why?

long pelican
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Not right channel, but from an algebraic geometry perspective you can think of a module over a ring as a coherent family of free modules over points of the spectrum of the ring

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(aka family of vector spaces over residue fields)

midnight scarab
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But for basic things, keeping in mind the very special cases of 1) vector spaces and 2) lattices gives helpful intuition

light mirage
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Hello. Does anyone have general advice for being a teacher's assistant? Im gonna TA for a calc 3 class for engineers

hasty parcel
hasty parcel
lethal leaf
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Grading, running a discussion/review section, office hours, other stuff?

light mirage
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I think mainly grading and office hours, but I'll have to double check

lethal leaf
minor turtle
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and remind them that they exist FREQUENTLY

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otherwise they won’t go opencry

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(seriously reminding them helps)

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(AND they NEED office hours, and it does them good to be reminded that they need office hours, whether it be yours or the instructors)

lethal leaf
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Also with larger classes in my experience, you're going to get questions from people who haven't gone to lecture or learned anything

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And as much as you'd like to spend 30 minutes on teaching them

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You have other people who need help in OH

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So as mean as it feels you need to be firm with these people

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This is something I struggled with alot

light mirage
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wow, I never would've thought I would need to remind them

lethal leaf
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Most people remember they exist the day before exams

light mirage
lethal leaf
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You tell them to look at the lecture videos and the text

minor turtle
lethal leaf
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Now if they looked at that and are still confused that's fine, help them

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But if they haven't even tried

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Move onto another student

lethal leaf
minor turtle
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yeah i didn’t exactly read

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that it was a large class

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how big are your sections?

lethal leaf
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Like if they just don't know a basic derivative rule / definition, it's not your job to recite the lecture or textbook

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I mean I presume it's a large class, it's calc

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If it's small and your office hours are sparse, you can spend more time on each student

light mirage
lethal leaf
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Also a lot of people may have similar questions

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See if you can help them simultaneously

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Rather than giving same explanation 10 times

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Very rarely are 10 people confused on 10 different things

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a simple saying "hey I am going to go over this concept, come to the whiteboard" aloud is enough

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Alot of this boils down to helping the most amount of people possible in the office hour

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If you have frequent fliers at your OH, learning names and faces goes a long way in making students feel more comfortable

light mirage
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Thanks, Ill keep all of these in mind

light mirage
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There should be good ones, and some people will like video lectures more than books

lethal leaf
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Sure or the textbook

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But my point is you should not spend a ton of time teaching a concept that was taught in lecture

minor turtle
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professor leonard YT is one that students like

lethal leaf
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If you have like 10 other students who need help

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If the only people in your OH are like 1-2 students

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That's fine then ofc,spend a lot of time with each one as needed

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If that makes sense

light mirage
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Yeah that makes total sense

long pelican
light mirage
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Also I guess I should do some calc 3 practice problems myself, since I havent done calc in a long time

minor turtle
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not saying that you will

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just saying that over a long semester or a year it might happen

wispy slate
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@wary cedar dm

minor turtle
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regrade requests are so stressful

hollow musk
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hopefully all goes well for you smay

minor turtle
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I just get stressed in general

hollow musk
sacred thicket
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Is this a general form for posts about academic life?

kindred stag
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no this is about math teaching techniques, as the channel description states

pure light
cloud zealot
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niche question for those who teach introductory mathematical logic (say from enderton or mendelson)

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do you have any specific takes on this rant?

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are the concerns raised in the rant real problems, and if they are, how would you address them?

pastel sundial
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As a logic student my takeaway from this rant is mostly "why does anything dealing with formal syntactic proof get lumped in the same boat as model theory"

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I feel like I still have basically no intuition for formal proof system because every time I've learned them it's been in the context of proving Completeness and then stopping. It's difficult to appreciate proof theory when it's presented as just a means to an end.

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I was thinking about this recently because my school teaches basic proof theory at the start of the model theory section of its logic sequence, and this just seems like bad pedagogy. It seems hard to get the the point across that syntax and semantics are meaningfully distinct when you're working in a context when they're literally the same.

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It would be like if an algebra course went into depth on the properties of non-free modules when every example discussed was a vector space.

soft scarab
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hello all!

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does anyone have any tips for explaining calculus to students whose algebra skills aren't very strong?

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i'm a calculus tutor at a community college, and lately, i've been tutoring Calc 1 students. many of them seem to struggle with basic factoring and algebraic manipulation, and I'm not exactly sure how to help them with pretty difficult calculus concepts if they're lacking in algebra!

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thanks for any the help

quasi musk
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Then I'd highlight what they need to put in their mind first, second, third, etc.

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What was super important and what they can hold off on

inland bloom
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I’m in high school, is it a bad idea to teach fellow student calculus if they are in a lesser math class?

boreal stream
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no not at all, the best way to make sure you understand something is if you can correctly explain it. Just as long as you are confident its a good option

quasi musk
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but teaching someone Calculus if you haven't properly mastered calculus can be a bit of a pickle

soft scarab
cedar garden
austere flame
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James Stewart's Calculus has a a "Diagnostic Tests" section at the beginning that I've found useful

tight star
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What difficulties do people have with functions before analysis, and how can you resolve them?

inland bloom
wispy slate
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Is this an appropriate channel to discuss teacher related things such as activities, grading, etc?

halcyon glade
vagrant meadow
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I'd like to collect some linear algebra application problems to show my students what linear algebra can be used for.
does anyone have any good Markov chain problems that have an actually useful application? like the "what proportion of people will live in the city vs the suburbs" is a classic but I'd like to find something more interesting.

sharp tendon
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you can go simpler to solving linear equation systems, like the required mix of gravels from different quarries to get a desired concrete mix granulometry

vagrant meadow
austere delta
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(also, unrelated, but I love that the PageRank algorithm is named so after Larry Page and not because it is used to rank pages)

lethal leaf
tawny slate
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i solved this one using a markov chain

tawdry venture
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or some other topic channel

tawny slate
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who is that directed towards? i dont see anything that warrants going to that channel

inland bloom
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Would it be a good idea to start a self study club?

oblique bobcat
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Yes.

lethal leaf
cloud zealot
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there is a section on markov chains, in addition to a host of other applications mentioned in the text

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there is also a lab manual available

tawny slate
pallid night
sterile scarab
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I haven't read through all of the posts, but I have a very serious question and, unfortunately, there's a wee bit of a story before the question. Right, so, I'm employed at a university as a lecturer, but I always hear the, let me state them to be, "higer-ups" (for example: the dean of the faculty; members of EXCO; the vice principal of teaching and learning; and the vice chancellor), refer to my lecturing activities as "teaching" and my role as a "teacher". I'm very upset about this matter, because I am not a teacher. I have not been trained as a teacher. I do not have any teaching qualifications. I am, simply, a mathematician. The lectures I deliver (apparently, "classes"), are presented to over two hundred and fifty students (in fact, this semester at this new university sees me having, approximately, two hundred and fifty students in my lecture group and this is the smallest group I've ever had). I believe that teaching and lecturing are vastly different and I believe that in order for teaching to take place there needs to be a relatively small number of students/pupils to be taught so that there can be a personal and individualised interaction with the student. This is why, in my opinion, schools have about thirty students per class and not two hundred and fifty. Given that story, what would you recommend as the correct pedagogy to be inline with teaching?

(Also, please forgive me for typing the numbers as words; I happen to be on a rampage at the moment in which I insist that in typed correspondence - not in typed questions and answers for maths problems - that the numbers must be typed as words; I don't know why I've begun this rampage, but I'm doing it for the moment and soon I'll return to the good ol' fashioned 250 instead of "two hundred and fifty").

wispy slate
# sterile scarab I haven't read through all of the posts, but I have a very serious question and,...

I'm not an expert in teaching but I guess if there's so many students maybe there can be small group activities in which the students work together on a small problem set during class. That way they have personalized interaction with each other and work together to solve a problem (use critical thinking). If they can't figure something out, a TA or a professor can go that group a personally help them. Some classes I've attended do this in a form of a recitation class where a portion of students attend so the size is minimized. I don't think a university will downsize their classes because they want to make more profit.

vital prism
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(I'm not sure if it is OK to ask about this here. Is this channel only about teaching professionally?)
What are some interesting topics, theorems or findings that I can use to show a friend that math isn't only about calculating numbers, and possibly make them start appreciating math?
Preferably something I can talk about in length and not just a small piece of information. Assume they have incomplete high school knowledge

knotty sail
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Another answer is Godel's incompleteness theorem (and by extension, formal logic)

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Finally, turing machines and finite automata (computability theory, in general) are also pretty fun

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Depends who you're talking to, and how you explain

vital prism
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Those are all good topics. I think what I'm actually trying to do is show them that math is more than calculation, it is about relations between things, and how we can derive conclusions (theorems) from some foundation using reasoning
For example, how the Riemann zeta function is related to the distribution of prime numbers. It isn't something that we invented, but something we discovered
So I want to give them a taste of mathematical discovery

knotty sail
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OH! Game theory, too, of course

knotty sail
vital prism
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Something like "see? this is what discovering/creating math is like!"

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The hardest part for me is the fact that they lack a lot of the basics, and the fun stuff is usually more advanced

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Like things in analysis for example

knotty sail
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Game theory?

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Discovering a winning strategy could do that

vital prism
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Yeah, I think that works

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Any subject works, actually. I just need ideas for what to talk about specifically

knotty sail
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Heh

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Creating math is usually difficult

vital prism
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It really is

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Sadly they don't know English, otherwise I would just show them some 3blue1brown videos

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@knotty sail what are some theorems that you find really interesting or that surprised you? Maybe I can find a way to derive them in an easy way, and explain the intuition behind them

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Things similar to the Riemann hypothesis

knotty sail
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Hmmm

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I'm not sure I can name anything

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Most cool theorems were cool with a quarter's worth of context

vital prism
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Exactly

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This is the hardest part lol

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They don't know even calculus. Not even that derivatives and integrals exist

knotty sail
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Probably game theory

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Maybe some geometrical proofs?

vital prism
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Oh maybe graph theory too

knotty sail
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Oh, yeah

vital prism
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Graph theory is cool. I was thinking about showing them how a lot of problems in computer science can be reduced to other problems

knotty sail
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Group theory is good for that too...

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But both graphs and groups need prerequisites before you can show any interesting reductions/isomorphisms

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Oh, maybe a countability lesson?

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Cantor's diagonal proof?

vital prism
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I showed them Cantor's diagonal proof, but I'm not sure if they understood what it actually meant lol
Btw, how about combinactorics?

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Is it approachable enough?

knotty sail
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Hm, maybe

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Deriving factorial, nCk, and nPk on your own is pretty fun

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Stars and bars too, with a nudge

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Those are pretty good "discoveries"

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But they do count things 🤔

knotty sail
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Think it's the easiest "discovery" I know

vital prism
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Ahh

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I might just choose a 3Bue1Brown video and explain it to them

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Like watch the video first, take notes, and turn it into an explanation they will understand, and explain it

knotty sail
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Possibly

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Does Grant have a video on combinatorics?

vital prism
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I don't think so

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But there are a few on graph theory

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Like Euler's formula

knotty sail
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a bit esoteric, though

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Euler's theorem would lead to a "why should I care" from most non-math people, I think

vital prism
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Yeah

knotty sail
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Well, good luck!

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My vote is nCk, but that could also be dry

vital prism
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Thanks

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I will try to explain something to them tomorrow

sterile scarab
gray smelt
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whatever the answer is, do that

vital prism
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Most things I find interesting are too advanced

gray smelt
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What about in the past past?

vital prism
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There are some things, but they are all things most people wouldn't care about, unfortunately

gray smelt
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I had assumed that you had someone that would be interested.

vital prism
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No, I am trying to show them that math can be interesting

gray smelt
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Well...for most people, you will fail.

vital prism
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That's true, yes. But I think it is still a good idea to show math from a different point of view from the one people see in school. They may or may not find it interesting, you'll only know once you try

gray smelt
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Oh, I'm all for that, but, one does need to be prepared to give up

vital prism
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From my experience explaining math to my friends, the hardest part is that people don't think much about what they are told or shown. For example I showed Cantor's diagonal argument to a few people are talked about how it shows that in some sense "some infinities are larger than other infinities", but they kind of only remembered this sentence and didn't think about anything I explained lol

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Most people I tried to explain math to only retained 0.01% of the information I gave them and only remembered a few key phrases but didn't think about it any further

gray smelt
gray smelt
vital prism
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Like, they just don't think about it anymore once the explanation ends. This is what I mean

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The person not asking questions is a good sign that they don't care

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When I study math I am left with more questions than I started with

gray smelt
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I remember showing a younger friend of mine in highschool the construction of the field of fractions of any ring

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Watching the lights light up on his face was pretty satisfying

gray smelt
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@vital prism A discussion in another channel made me think of one of my favorite results: that for any conditionally convergent series, for any number you like there is a reordering of the series that converges to that number.

vital prism
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I didn't know about this

gray smelt
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Oh! Want the proof?

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Basically: since it is conditionally convergent, you know that the sum of just the positive terms diverged, and same for just the negative terms

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Thus, if I want to make it converge to your arbitrary number z, then I can try taking just enough positive terms to be above z, then just enough to be below z again, then above, and so on. At each step, I can always do this because the series is not absolutely convergent.

#

Now, my new reordering is an alternating series. Since the original series was convergent, it satisfies the nth term test (that is, limit of terms goes to 0), so my zigzags near z get progressively finer and finer.

#

More formally, if past the Nth term the original terms are smaller than ϵ, then we know that in our reordering whenever we use all of the first N terms of the original series, we will never leave ϵ of z, because we construct our series by taking just enough terms to get to the other side of z and as such won't be past ϵ of z. thus the reordered sequence is Cauchy and so by completeness converges.

But note that the essential argument can be explained to a calc 2 student that doesn't even know what the letter epsilon looks like.

#

The sheer craziness of the result (it certainly made me go "what the actual fuck") combined with the simplicity of the argument (the two things used are that just positive terms diverhes and the nth term test to justify the approximation getting finer)

#

It's also fairly concrete. You can take (-1)^n 1/n as the example.

vital prism
#

Yeah I think this is a good example

gray smelt
# gray smelt It's also fairly concrete. You can take (-1)^n 1/n as the example.

@vital prism the other theorems I love:

Some basic field theory:

  1. That the minimal polynomial of any alpha over a field k must divide any polynomial with alpha as a root (and thus a p in \Q[x] with sqrt(2) as a root also has -sqrt(2) as a root). The proof is just euclidean division.
  2. That for alpha algebraic over k, k[alpha] is actually a field (and thus 1/sqrt(2) is a polynomial in sqrt(2)). The proof is Bezout's lemma.
  3. What is currently my favorite proof, which can be found in Lang, and which Lang says is due to Artin: the proof that an algebraic closures always exists, using the polynomials over a set of symbols with a symbol for each polynomial over k. This one requires knowing that quotienting by a maximal proper ideal gets you a field.

Hilbert's Theorem 90. By this I mean the classical, non cohomological form, and also I'd include the consequence telling you what cyclic extensions look like (which does the heavy lifting in the Abel Ruffini Theorem).

The fact that the lebesgue integral of the derivative of the devil's staircase is only ≤ the net change.

That the modulus of a curve family is a conformal invariant, thus letting you do things like say that two rectangles of different aspect ratios can't be conformally mapped into each other, or two annuli of different radii ratios can't be. (isn't it so crazy that you can map any simply connected domain that isn't all of ℂ conformally to the disk, but you can't even map two different annuli ratios to each other?)

Cantor's Diagonality Argument. But you already mentioned this one.

The Vitali set exists (assuming choice) but is not Lebesgue measurable. (in fact, every null set has a non measurable set).

Note that this list is biased both by what I don't know (as I won't put anything I don't understand, like the Riemann Mapping Theorem or the Central Limit Theorem) and perhaps by whether I've known something long enough for it to the initial shine.

#

I didn't, and don't, find Cauchy's integral theorem particularly beautiful.

tawdry venture
tight star
#

You can make them diverge to plus or minus infinity

#

Or make them not approach anything

#

Actually, by enumerating the rationals, you can probably make the set of accumulation points as big as you want?

cloud zealot
#

@vital prism if you are familiar with galois theory (or at least its history), you could discuss the unsolvability of the quintic

#

you could also discuss its applications to euclidean geometry, such as its use in proving the impossibility of certain constructions

vital prism
#

I watched a video on this a while ago. I could try to convert it into a easier explanation

vital prism
gray smelt
tawdry venture
#

otherwise you won't get the terms in your alternating series strictly decreasing

#

i think

gray smelt
gray smelt
tight star
gray smelt
#

Obviously not.

#

You can't get only the rationals.

#

Ah. Closure of the sequence thought of as a set.

#

So, now, can every closed set be done this way?

#

In ℝ, you could try getting every rational in the closed set.

#

But, then e.g. if you have an isolated irrational you fail.

#

can you have uncountably many isolated points?

#

Oh, wait.

#

second countable is stronger than separable. subspace of 2nd countable is also 2nd countable. so, take your closed set, now you can find a countable dense set. the closure of this set in R will still be the original closed set

tight star
#

riiiight

#

that’s neat!

tight star
gray smelt
#

since second countable iff separable iff lindelof for metric spaces, this works for all metric spaces as long as the original one is separable.

gray smelt
tight star
gray smelt
#

First heard the term from computability theory (which I know ~none of!). So, imagine you have a sequence of programs, where each program is a sequence of steps. I can make a program that goes through all the programs by doing the following:

Step the first.
Step the second, and then step the first.
Step the third, then the second, then the first.
...

#

Likewise with sequences, my new sequence has all the values

noble hare
tight star
lyric fractal
#

Might be a common question, but if anyone here has experience teaching middle school geometry, how do you approach teaching students how to "write down there ideas".

It feels as if im often lead to making students memorize the process rather than understand why we write down each step despite my efforts, even when they get the right idea, once they have to write down there logic, most if not all get stunned at what to even write down, and its a obstacle im not sure how to overcome without making them memorize it.

tawny slate
#

i dont have a "quick" answer for you, but i can explain how i approach this problem

#

geometry is when i think most students are introduced to the idea of proofs, and the reason is because proofs are finally critical for the first time. we can check answers in algebra by just plugging in numbers, but we cant with geometry. we have to rely on proofs to validate our answers

#

so getting students to first understand why they are writing proofs may help if they are resistant to understanding why it is all necessary

#

when i start teaching and going over two-column proofs, the first proof form they learn, i begin by explaining not only how it works, but all of the "valid moves" they can do in the proof

#

for instance, you're allowed to do AB = AB by reflexive property of congruence

#

youre not allowed to do:

  • AB = CD because it looks like it
  • AB = AB by transitive property
    etc.
#

clearly laying out a comprehensive list of rules makes it crystal clear what is or isnt allowed, just like how we have established rules for algebra

#

it also so happens that this is not easy, way harder than it is for algebra, as you need to not only have basic rules and definitions and theorems, but you will also probably need to explain ideas like measure and construction axioms

#

when a student reaches a step they cannot justify using only the explicit rules you have allowed, then it forces them to think about how to break that down further

#

and, if students manage to prove a general theorem, they can add that theorem to the list of "valid moves"

#

treat it like a puzzle or engineering problem in practice to make it less tedious and annoying

#

of course, it always helps to compile a list of examples where students easily make false assumptions by not being careful

gray smelt
gray smelt
#

Oh, I'll add the Shannon Nyquist sampling theorem.

noble hare
lyric fractal
cloud zealot
cloud zealot
#

there's a section on markov chains in here

tawny slate
# lyric fractal ah fair, i see what you mean, ile have to test this approach to see if it makes ...

as a side note, i hope it helps you too
when i tried to compile a list of those explicit rules for my own use, i actually found it quite tricky, because if you use the formal axioms that professionals use, it will be too cumbersome in many circumstances, but if you start somewhere in the middle, it gets convoluted very fast

i learned a lot myself by doing this work and teaching it. sometimes i end up giving my students a problem that is too hard even for myself, and i have to amend the explicit rules list accordingly

#

i still have a lot to learn in this space, and if i had more time, i would do more work in this space
this is the kind of work i think beautifully merges the work that high level professional mathematicians do that is a directly applied to pedagogy for young children, and is the kind of math work that excites me the most

midnight scarab
#

In that connection, I think Givental's translation of Kiselev's textbook might interest you

soft scarab
cedar garden
#

it is yeah

quasi musk
#

is this it?

#

||bad joke had to go for it||

sudden heron
#

Is there a chrome extension for referencing papers that's better than the google scholar one? (it's often incorrect)

stray grail
sudden heron
#

Can you use it just to turn a url into a citation? Since I don't want to use its document-sorting aspect, since I have my own

stray grail
lethal leaf
#

Same

#

Very useful to bookmark

pallid olive
#

find the direction in which the Uf of f(x,y)=ye^-xy at (0,2) has value of 1

rose beacon
#

Is there a term to distinguish between transformations in the domain vs range of a function? For example, to translate all the points of a curve to the right, I'd do (x+1,y). But for a function representing the curve, I'd have to do f(x-1). This feels counter-intuitive yet it makes perfect sense if you think about it. I'm looking for a quicker way to explain this behavior to people

#

I see my peers trip up on this a lot in their code. They do f(x+1) thinking that it will do f(x)+1, and confused why the opposite happens. I guess it's about order of operations?

fickle current
#

It's funny because thinking about it, I would say that explain this fact with an example such as like
(x,x²)->(x,(x-1)²)
or something makes it more confusing. I would say the best way of thinking about this thing is doing so graphically

#

At the end of the day it's a geometric fact about how curves move so yeah

tawdry venture
fading glade
drowsy otter
#

Why does my student struggle with the key of stem and leaf diagrams? The key is worth a mark in GCSE mathematics exams.

I've tried labelling the key as 'key', and I've tried explaining what a key looks like. "1|1 means 11" for example.

The student is okay with making and interpreting stem and leaf diagrams, but struggles to write and interpret keys. How can I explain keys in a way the student will understand?

noble hare
#

Maybe something like presenting them with a steam and leaf diagram without a key and ask them to interpret it. And then however they interpret it you can say its not right bc you have a different key in mind. This might at the very least highlight to them the issue with not including a key.

drowsy otter
#

I'll note that down for next Tuesday (this student's next session).

stray grail
#

when is stem and leaf diagram used? never heard of it

merry jay
#

Hi everyone what I have is a question about studying math more than a math question, so I think it seems to be the best place to ask.

#

I’m taking a major in electrical engineering and I have some doubts about math in general.
Is it normal, or common, to be stuck in a math question not because of the subject you are studying, but because you are missing some knowledge that is a requirement?

Recently I started learning Calculus II, Sequences, and l'Hopital's rule, and often what gets me stuck is algebra, factorial rules, or a simple rule like the one that says we need to make a number n^2 to pull it to the inside of a square root, while calculus and sequences are not that hard.

Is it normal? How do people get over it?

I’m 30-yo and the last time I studied math for real was about 8 years ago, it is hard to remember all those rules.

surreal junco
austere delta
# merry jay I’m taking a major in electrical engineering and I have some doubts about math i...

Before getting to proofs, I think the only issues anyone has with calculus is algebraic manipulations. And it's quite normal to have issues.

I guess the only real way to get better is to practice. Either by working on problems in a calculus setting or going back a bit.

Also, might be easier said than done, but if you spend some time understand why the rules are exist, you'll understand them much better, and also remember them much better, without having to memorize stuff.

merry jay
#

Right now Im practicing factorial rules, for example haha
thank you for answering it.

bright forge
#

is this channel about the hardest maths?

#

what is math-pedgogy lol

vocal phoenix
#

The channel description has some hints

#

And in some ways yes, teaching maths to people is often the hardest part of the job

soft scarab
#

basically this channel for is people seeking tips or talking about how they teach math

bright forge
#

oh ok

wispy slate
#

,rotate

burnt vesselBOT
vocal phoenix
#

I mean, sure, why not

burnt vesselBOT
lethal leaf
pure light
#

i think they used to be more popular because they're very easy to make by typewriter, but computers are just as good at coming up with other representations

placid knot
#

What is the motivation behind teaching Pascal's Triangle in highschool?

pure light
#

easier to find binomial coefficients by drawing out pascal's triangle than from definition in my experience

drowsy otter
#

So, I've done a presentation designed to teach fractions to my 11-year-old student who is behind. I'm going to summarize what I am teaching in the first part (thankfully this slide summarizes everything!):

image.png
#

Am I teaching the right things, or am I missing something?

#

I am using examples, this is just a summary.

turbid zenith
#

The only thing I can think of is to be careful of “part of a whole” because of “improper” fractions

#

Though a slight change in wording could fix that: “Fractions let us count parts of a whole”

#

Which honestly might go better with your analogies, since the denominator is size “pizza” slices you’re counting, and the numerator is how many of them you count

drowsy otter
#

Could you possibly tell me more please?

tawny geode
#

Hey guys, has anyone some knowledge about the relation between the french philosopher GIles Deleuze and Bernhard Riemann's ideas about manifolds? I would like to discuss some ideas arround this topic.

drowsy otter
tawny geode
#

Oh I'm really sorry. I was seeking for a channel concerned about more "philosophical" contents in math

long pelican
drowsy otter
#

My mum (who is a former part-time LSA) suggested this technique to me as a way of teaching fractions.

long pelican
#

I think you misread

turbid zenith
#

@drowsy otter There are some educators that are VEHEMENTLY against the pizza analogy and act like it’s the worst thing ever for a student’s intuition 😛

#

I take those with a grain of salt personally

long pelican
#

The inevitable answer to "What does it mean to take a fraction of a pizza times another fraction of a pizza?"

#

is something like "Don't think about that analogy with multiplication"

#

Yea?

turbid zenith
#

I suppose if you said “any number of equal parts of a whole” you wouldn’t be too far off

turbid zenith
#

Usually involving a grid

#

Every analogy breaks down eventually but it can take you pretty far

long pelican
#

I'm doubtful because the unit of whatever that is needs to be pizza squared

#

A pizza is already a 2-dimensional object

#

Pizza squared will live in 4 dimensions 🤕

turbid zenith
#

3/4 of 2/3

drowsy otter
#

Interesting DM Ashura.

#

From the grid, I can see that the answer is 6/12, or 1/2 equivalently.

long pelican
#

I don't see any pizzas in that picture...

#

and you'll notice the number line analogy featuring here, because the 3/4 is representing 3/4 of a side

#

and 2/3 as 2/3 of the height

#

both of which are 1-dimensional entities

#

length times length equals area, after all

drowsy otter
#

I get what you're trying to say, Icy0. But the pizza analogy CAN still work, especially when teaching young children.

If you have 3/4 of a pizza, and split the entire part into thirds, you can tell just by looking at it that each part is still 1/4 (which is equivalent to 3/12 if done analyytically).

Another example is if you split 3/4 into fifths. Then each part would look TINY (as it would only be 3/20 of a pizza).

#

Do you agree with me, DM Ashura?

turbid zenith
#

It doesn’t literally have to be a circular object. Usually when people are using the pizza analogy they also use rectangles and squares right along with it.

#

And yes, I do agree. It works at a developmentally appropriate level.

long pelican
#

In this case, the 3/4 at the beginning of the example is a literal object (part of a pizza) but the 1/3 (what you're splitting it into) is now an abstract quantity, no longer a pizza or part of a pizza

#

Agreed?

turbid zenith
#

Considering we’re talking about an 11-year-old, I think it’s develpmentally appropriate.

#

It’s 1/3 of the 3/4 of the pizza.

#

You can phrase it to sound more confusing, but it’s plenty convincing to young learners.

long pelican
#

1/3 of the 3/4 of the pizza, is that 1/4 or 1/3

turbid zenith
#

1/4 of the whole pizza.

long pelican
#

The issue here is the child who goes

#

I will pull out 1/4 of a pizza

#

and 1/3 of another pizza

#

and I want to know the meaning of multiplying them in terms of combining them together

#

in some way

turbid zenith
#

If you find me an actual student who asks that I’ll be glad to answer them.

long pelican
#

I would have done that

#

And I probably did, at some point

#

And not as a gotcha question

#

more like real confusion over how to think about multiplication

#

Eventually we settle on: in the expression 1/3 * 1/4

#

the 1/3 on the left is a scale factor, the 1/4 on the right is an amount of pizza

#

I don't think we think of the 1/3 on the left as a physical thing

turbid zenith
#

If so, then I’d probably point out that not all representations make sense in every single context, and in this context yes an area based model would make more sense as it often does with multiplication

#

Just like it doesn’t make sense to say that 5 people x 2 people = 10 squarepeople

long pelican
#

Yes, so fractions become maybe the thing that requires mysterious multiple representations you must be able to tell from context to understand, which is why it's made hard to grasp

#

So what you said there is exactly something I already pointed out

long pelican
long pelican
drowsy otter
#

While I do get that other analogies are good, at the end of the day, we've got to introduce these other analogies slowly, as is developmentally appropriate.

long pelican
#

Maybe what's underlying the criticism of the pizza analogy is the observation that it's made children (temporarily) think fractions have a different role (grammatically or otherwise) in sentences than numbers do

drowsy otter
#

Even though I've been a tutor for two years, I'm still pretty inexperienced with basic level stuff. I personally tend to think of fractions as division (which is true) but it's about being developmentally appropriate with our analogies.

long pelican
#

The phrase "developmentally appropriate" has been overused in this conversation. We're all arguing that something is making something more confusing than necessary for children

#

Being less confusing is developmentally appropriate

#

That's all this was about, at the end of the day

turbid zenith
#

At the end of the day it’s all just saying “how much/how many copies of this do I have”

#

5 x 2 can be 5 copies/groups of size 2 each.
1/3 x 3/4 can be 1/3 of a copy/group of size 3/4 each.

long pelican
#

Excellent! Unifying the concepts of fractions and numbers should be done more

drowsy otter
#

Anyways, while multiplying fractions is something I will get to eventually, I'm currently doing the introductory stages of fractions. You suggested I use a number line from the start, right?

long pelican
#

Yep, either a line you draw on the board or something else that has the shape of a ruler or some other linear representation of numbers

#

And pizzas won't go away but it'll be more like attaching the number to the pizza rather than saying the number is literally parts of a pizza

turbid zenith
#

I am frustrated with the abstract algebra book I'm teaching from.

#

I love its sequence — starting with algebraic structures in general, sort of answering the "groups first or rings first" question with "why not both?" to show the relationships between them, before diving into groups in depth and then rings

#

But I am so tired of definition-theorem-proof, where everything in its full abstract symbol-filled rigor from the beginning before students play with the thing and see "oh THAT'S what's going on"

#

Anyone have any recommendations for a play-first-formalize-later approach to abstract algebra? Especially something along the lines of IBL? What do you notice, what do you wonder, that sort of thing?

#

(Or anyone else have the same frustrations?)

cloud zealot
#

is this silverman

tawny slate
#

i do have the same frustrations in general with most math textbooks

#

its why i started a youtube channel for math to begin with

#

and eventually i want to do more as well, such as publishing a free textbook

turbid zenith
#

And again, I love the sequence, I'm just frustrated with how it's still "all-the-abstraction-up-front-before-you-play"

cloud zealot
#

oh, i loved sibley's intro to proofs book, The Foundations of Mathematics

turbid zenith
#

Anyone here ever read Building Thinking Classrooms by Peter Liljedahl?

gritty tundra
#

i'm writing an article about the real number system for high school algebra students

#

and i'm trying to figure out if i should include 0 as a natural number

#

although it's not inherently a natural number

#

it gets associated with them in some higher level math textbooks

faint cradle
#

At the moment, it's a matter of choice and chosen convention whether zero is counted as a natural number. During my undergraduate degree, the Mathematics Institute at my university did not count zero as a natural number but many other departments and research fields probably do include zero as a natural number. Teaching about this lack of consensus is probably important.

#

I read that the original Peano's axioms started with one as the first natural number but modern presentations start with zero for convenience of later results.

#

The von Neumann and Zermelo ordinals both start at zero and this is also likely for convenience of including it.

gritty tundra
#

i'm also having trouble figuring out how i should identify the difference between a coefficient and a constant

#

because a constant is any known value in an expression

#

and a coefficient is a number that gets multiplied to a variable

fading glade
gritty tundra
#

i just saw a lot of contradicting sources online saying that a coefficient can't be a constant because it gets multiplied by a variable

#

but i just didn't get the logic behind that

fading glade
tepid smelt
# turbid zenith Anyone have any recommendations for a play-first-formalize-later approach to abs...

There seems to be a lot of research in education recently against IBL. I'm not convinced IBL is the right approach. I was told in my teaching programs to do it and most of the modern curriculum I have seen at the secondary level leans into it. Yet in practical terms explicit instruction has just given me better results.

I'm curious what others think though on IBL vs traditional teaching methods.

In undergrad I liked a book of abstract algebra by pinter because I was forced to learn more by doing the exercises and they were quite a bit easier then dumit and foote on average.

I liked a radical approach to real analysis which developed analysis from a historical point of view. I have not read it but a history of abstract algebra by gray looks interesting. I don't know if all students find the history interesting but it was for me.

faint cradle
#

Ironically, the constant term is also the coefficient of $x^0$

burnt vesselBOT
#

mikeliuk

698161944791744573.png
cloud zealot
gritty tundra
#

until we get to the law of exponents unit

tepid smelt
# cloud zealot do you suppose part of the problem might be that more time is spent developing t...

Yeah I'm honestly not sure and this is a hot topic in education research though and more broadly the science of learning. I have just found that scaffolding and gradually building up through explicit instruction has led to better results from my students.

https://educationrickshaw.com/2023/06/29/maximizing-learning-through-explicit-instruction-with-zach-groshell/

I have recently been listening to an education podcast called chalk and talk and a reoccurring theme is that the "old boring" teaching ideas are actually quite effective

Education Rickshaw
educationrickshaw
Maximizing Learning Through Explicit Instruction with Zach Groshell

A few years ago, the Head of Math at my school confided in me that he was trying to “sneak” explicit instruction into his department. The school was full of romantic ideas about learnin…

turbid zenith
#

I use a combination of both.

#

I do find that the direct instruction people tend to misrepresent IBL as being unscaffolded

#

But at the same time the progressive educators act like you should never ever use direct instruction

#

So my hunch is that using both, each at an appropriate time and in an appropriate way, is best 😛

long pelican
#

My two cents with these debates is that most fighters on both sides need to understand that they don't even agree on what the goals of teaching math is

faint cradle
turbid zenith
#

Or from class to class.

tawny slate
#

my two cents is that if there is "research" supporting one teaching method over the other, we need to be very careful about what that research is actually saying

#

we need to go into the nuance of what actually is being defined and measured, how is it correlational or causal, and how it is applied in practice

#

i had a lot of "gut feelings" about teaching that i thought worked, but sometimes found that my intuition disagreed with, for instance, a generally accepted result in neuroscience

#

fixing that ended up improving my instruction

#

not saying that anyone here is or isnt, but personally i just didnt want to be one of those hard headed boomers who think their way is best despite the evidence on the contrary

turbid zenith
#

What gets me about educational research is that it’s so easy to find research that supports essentially whatever teaching method you prefer

#

For every study supporting direct instruction, there’s another supporting inquiry based learning. And people supporting the former trot their studies out against the latter, and people supporting the latter trot their studies out against the former. It becomes a barrage of citation dueling.

tawny slate
#

yeah it's tough, for sure

long pelican
#

Today’s state of affairs in science

solid charm
#

does anyone have any books to reccomend in the style of a mathematician's lament\apology?

#

I really enjoyed lament and it got me thinking a lot about how teaching is done when it comes to math and I was wandering if there any other books of that type

#

also really liked his writing style

tepid smelt
#

I'm a fan of all the pedagogy books by George Polya. Like mathematical discovery. It goes a bit deeper on hueristics then how to solve it and I feel it is more directed to teachers.

I

#

I have not checked it out but mathematical problem solving by schoenfield looks good also.

next relic
# tepid smelt Yeah I'm honestly not sure and this is a hot topic in education research though ...

This is definitely controversial. And Zach Groshell is known for direct instruction, he's a big DI fan. c:

If y'all are interested, I recommend reading this paper from Kirschner, Sweller and Clark: https://www.tandfonline.com/doi/pdf/10.1207/s15326985ep4102_1

In initial stages, DI is definitely better. As students develop proficiency, that's where we start bringing in investigations, modelling or more inquiry-based learning stuff.

#

Unfortunately you don't get to say "oh I just want my class to solve problems and be prepared for the 21st century" without the fundamental knowledge.

#

A deeper treatment of this can be found in Seven Myths about Education by Daisy Christodoulou.

lethal hornet
#

giving my first lecture/talk tomorrow. any tips?

turbid zenith
#

Breathe. 🙂

#

What's the topic/audience?

lethal hornet
#

giving a lecture to some students taking a course called applied modern algebra. i TA for the class and the prof is out of town for the solar eclipse

#

the topic is about quotient groups, and introducing rings and fields. just basic definitions, group work, trying to make it fun and not to make it too boring by showing them some constructions with quotients like circles, cylinders, spheres, the torus, etc

turbid zenith
#

What will the group work look like?

rapid spruce
#

bring candy

vagrant meadow
#

how important is row reduction to linear algebra?

the class I'm TAing for has no exams or quizzes at all. it's just homework and the participation in discussion. and I'm kind of tempted to just ignore row reduction altogether and encourage them to just use a rref calculator (like they probably will if they ever need to use this material someday). they won't have to perform under the pressure of an examination so why waste the time?

part of it is laziness (not wanting to slog through computation and all the cases of the algorithm), and part of it is contempt for the row reduction focus of linear algebra.

isn't focusing on the interpretation of the rref matrix more important than getting there by hand?

cosmic wind
# vagrant meadow how important is row reduction to linear algebra? the class I'm TAing for has ...

This is a rough question, because there are two opposite answers that are opposite.

To linear algebra as a topic, row reduction as a skill is all but useless. Computers can do that. Interpretting it and knowing why the calculation algorithms work is paramount though.

However, depending on your university, the assessment methods will not agree with this. This is largely because concepts that reduce to "can you row reduce and then interpret that" are super easy to test

vagrant meadow
# cosmic wind This is a rough question, because there are two opposite answers that are opposi...

To linear algebra as a topic, row reduction as a skill is all but useless. Computers can do that. Interpretting it and knowing why the calculation algorithms work is paramount though.
yeah this is my thinking. i completely agree.
However, depending on your university, the assessment methods will not agree with this.
the only reason i'm really considering this is that there are no assessments beyond homework and whatever work i decide to give them. so i guess i'm wondering how damaging it is to not go through 100 examples of row reduction in the short time that i have when i could focus more on interpretation of rref matrices.

quasi musk
cosmic wind
quasi musk
#

Which can hurt them down the line if they don't have that background

vagrant meadow
quasi musk
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Exactly my point! I was always given trigonometric formula sheets when I was a student in Calculus, yet when I took physics (and later fourier series) the professors assumed that we knew all the trig identities (addition, subtraction, double angle, half-angle, etc.) like the back of our hand

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I really struggled with that since I didn't put in the time to memorize those basic formulas

cosmic wind
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What major students are you teaching, do you know the breakdown?

quasi musk
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So if you decide to go the calculator route, @vagrant meadow, then give them fair warning that they might need to know this by hand in their subsequent classes

vagrant meadow
cosmic wind
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Hmm. The chem and engineers and econ probably can just use calculators

vagrant meadow
#

not CS too?

cosmic wind
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if the engineers plan on doing anything controls heavy they’ll want a very very strong conceptual background

#

CS too

cosmic wind
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those are the ones who don’t give a crap about the concepts as much

vagrant meadow
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haha

quasi musk
vagrant meadow
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im kind of surprised how many econ there are. i don't actually know that many applications to economics. i'd like to read up on some to give them some applications.

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i'm planning to give them a fun problem to introduce them to polynomial interpolation and least squares. to show them linear has a lot of applications.

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well, i call it fun at least 😆

cosmic wind
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Also! If you’re willing to share, I’d love to see 1-2 activities of yours that you’ve had the most fun with. Imo linear is the best litmus test in a non-mathematics undergrad program of one’s willingness to truly understand what they’re doing versus just pattern matching, which makes it a lot of fun

vagrant meadow
cosmic wind
cosmic wind
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ie $$x’=\begin{bmatrix}0 & 1 \ 0 -k/m\end{bmatrix}$$ for an undamped spring, and have students explain what the eigenvalues tell you about the trajectory, and then from that what can be said about the energy of the system, or whether those eigenvalues vibe with what they know about springs

burnt vesselBOT
#

Space Lizard

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turbid zenith
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It’s not a problem that it’s hand-holdy — it’s good quality scaffolding.

cosmic wind
vagrant meadow
quasi musk
vagrant meadow
# quasi musk Did you make one for DEs as well?

nah, with the quizzes i had to give, there was no time for me to give any problems besides ones provided by the professor. i was able to make my own linear problems because i had/have more freedom for that class.

wet quest
midnight scarab
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Yeah I feel like the best would be to give out a handout with one or two detailed worked examples and then let the students practice

turbid zenith
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Even better if you can have the students work through a handful of well-chosen examples in class to demonstrate particular cases or concepts.

vagrant meadow
tawdry venture
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unpopular opinion: there's very little pedagogical value in having students verify the laundry-list of vector space axioms on a "weird" vector space (i.e. R or some subset thereof with bs operations)

midnight scarab
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Why unpopular?

long pelican
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I like it

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the opinion

vocal phoenix
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I agree

vagrant meadow
quasi musk
vocal phoenix
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It's an amusing example but I don't think it should be presented by anything other than a curious oddity.

#

In most vector spaces a typical student (heck, even a typical mathematician) is going to encounter, addition and scalar multiplication are going to be very strongly rooted in the corresponding operations on R

vagrant meadow
burnt vesselBOT
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eigentaylor

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dawn portal
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one step further: (x + y) := exponentiation, (x * y) := tetration

vagrant meadow
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mostly joking. I agree that this diverts attention from the fact (and potentially confuses the notion) that pretty much every vector space is exactly the same (just F^n in a different form)

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I have literally never seen this vector space used for anything besides exercises proving it's a vector space.

snow pasture
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It's more like a formal definition doesn't it?

#

Aside from analysing the structures and having rigorous definition of what is called vector space I do not see any uses for that in an introductory course.

dawn portal
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from cs perspective a vector field just asks for two methods A, B that satisfy a set of requirements and have a certain type structure (A, B: X^2 -> X); that they're called addition and multiplication archetypically is more to motivate the structure more concretely

snow pasture
#

Concretely, they just want the concrete structure being lay out and explained, I presumed. Since for the current time, I think in computer science, linear algebra is more of applied purpose, so representation and operations on such object is more important than the rigorous meaning and interpretation underlying such.

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The powerful properties (in my opinion) of linear algebra is that it 'scales' the whole problem up. The object it provides us is very helpful for any kind of numerical and complex computational system, vectors, matrices and tensors.

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But that powerful features do not guarantee (almost) that you have to learn a bunch of axioms like that.

dawn portal
#

they're helpful for branching out linear algebra to functional analysis and gutchecking that things still make sense

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in the scope of cs the reals (well, IEEE 32-bit float) act accordingly and nothing weird happens

snow pasture
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Imagine a place where IEEE standard does not exists.

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The teaching would be so confusing

tawny slate
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if you think about it, what is it really saying?

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its saying that positive rational numbers can be uniquely represented in "prime factorization" notation, standardized

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if you take any integer, write out its prime factorization, the exponents of the prime collected forms a vector

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making it "more complicated" by describing this as a vector space might not be super useful, but the idea that a number can be written as either its base-10 or prime fac is critical in math pedagogy im sure people here would agree

#

the vector space contextualization simply makes it more rigorous by giving it some abstract formulation, in addition to giving a connection between something concrete (prime factorization, which students know is useful) and something very abstracted (an infinite dimensional vector space)

#

you can argue at that point whether or not that level of rigor in this case is useful, but you can then go further to say, for instance, that due to the connection, this is how you might apply some esoteric ideas to solve problems related to these basic concepts

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therefore, it is meaningful to actually verify that some more exotic things are in fact vector spaces, as they might have unexpected structure and connections

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if this is enough to get people excited about math, then i think its worth teaching

tawny slate
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pretty mundane if you ask me

long pelican
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Positive rationals under multiplication do not form a vector space over $\bQ$ let alone $\bR$ because $n$th roots of rational numbers aren't rational in general!

burnt vesselBOT
vagrant meadow
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I suppose you could consider it a Z module maybe. 3b1b_pi_think

tawny slate
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so maybe it is a little bit exotic

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still think it is interesting in its own right

austere delta
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It's the free abelian group on countably many generators, so that's something.

turbid zenith
#

So, an update … ended up finishing Building Thinking Classrooms. Most of it passes muster from what I can tell. There are a small few things I don’t see myself being able to do exactly as suggested, but there’s a lot of stuff in it that I do already implement that fits with my experience so far.

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I think my biggest criticism is that I disagree with the premise that “mimicking” is the polar opposite of and antithetical to thinking.

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@snow shoal You were asking about it beforehand so if you want to talk about it let me know.

wintry oyster
#

Is this an appropriate place to ask for academic advice for higher mathematics?

pure light
vagrant meadow
# turbid zenith I think my biggest criticism is that I disagree with the premise that “mimicking...

i agree with this. especially when it comes to things like proofs: often there's really only one real way to do it. there's a lot that can be learned from reading a proof and trying to understand the logic behind it and trying to apply the logic to a similar theorem.
ex. reading a proof for a statement that is true for n=2 and then extending that to a proof for a general n could be a great exercise. you could call that mimicry, but that doesn't mean you aren't thinking.

that actually gives me an idea for a linear algebra problem hmmmmm lol

turbid zenith
#

There are certainly some proofs that can be tackled in multiple ways but it depends on the field

#

I think the book is more talking about things like learning procedures for computations or other processes

tawny slate
# turbid zenith I think my biggest criticism is that I disagree with the premise that “mimicking...

i completely agree, and i think both machine learning and neuroscience has really helped me in understanding what it really means to learn

there's no good comprehensive way to define learning rigorously, so there are actually many different ways to think about or describe it to help us understand it better

one way that i think about learning in the abstract comes from how machine learning works, is that it's somewhere in the gradient between pure memorization and pure generalization. pure memorization would be like if you didn't see the exact same problem as one youve done before then you cant do it, like knowing 2+4=6 but then not knowing how to do 3+4. pure generalization would be extrapolating one answer to everything, like for example, once you know that 2+4=6, every answer to every problem is 6. proper learning requires not only finding the right balance, but also figuring out the meta of how to adjust that balance. go too far in memorization and you learn immensely slowly, go too far in generalization and you accept wildly incorrect statements all the time

a more practical way to understand learning i think came from some neuroscience reading. when you first encounter an idea, it is stored in working memory, which if reinforced, is then committed to episodic memory. when you sleep, your brain tries to take the episodic memory, restructure and shuffle it, and convert it to semantic memory, probably simplifying it in the process. now you have ideas that don't require a lot of active work to retrieve. for example, if you have your multiplication tables memorized to heart, you can use more of your attention on the things that are still unfamiliar, such as integration. the more you practice something, the more the associated neurons are triggered, and will strengthen the synapses between them as they get larger and maybe even duplicate

for these reasons, my opinion actually goes even further, i assert there is no learning or thinking without mimicry

dawn portal
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gradient! ha

tardy ember
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i mean i think i can see what they're gesturing at, but also yeah it is complicated

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like obviously if you make a policy of never repeating back anything that's been said to you, then, ...that doesn't work, because some stuff is just facts you have to know, like definitions

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but there is a failure mode of... being chatgpt, just repeating back words without really having much underlying concept that the words mean anything, or that what they're saying could meaningfully be held to any other standard than whether it gets marked correct

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doing it correctly, you are still to some extent copying and applying thought patterns that you've been shown, but it's at a deeper level than that

#

maybe the distinction is less "are you copying things" and more... what shape is your map of what the subject is

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is it several random procedures involving numbers and weird notation, or is it the study of what's actually true about a certain type of object and then just also some convenient names to make them easier to think and talk about

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...obviously if you just told them it's the second one that wouldn't actually help anything because they'd just go "ah yes here's a new string of words that i'm expected to repeat back", but like, if you actually believe that mathematical objects actually exist and that that is centrally what the point is, and if you've propagated that belief to everywhere it actually makes sense, that does imply a rather different attitude to... the chatgpt approach

vagrant meadow
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briefly talked about eigenvalues with my linear students. one of them saw the lambda and asked if this was related to Lagrange multipliers. on the drive home today i remembered that you do get an eigenvalue problem if you want to, say, minimize norm ||Ax||^2 with the constraint that ||x||^2 is constant.
does anyone have any suggestions for an application that would boil down to that?

and, in general, does anyone have any resources for econ applications of linear algebra? apparently a huge portion of my students are econ majors.

quasi musk
vagrant meadow
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oh this is amazing thank you!

stray grail
cloud zealot
tight star
turbid zenith
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I feel like if you try hard enough you sort of can boil a lot of what we think of as abstract knowledge and understanding as “knowing what words go with what other words”, if only because that’s how you often end up expressing it — in words

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But then the topology of how that knowledge sits in your head ends up being a big part of it as well — how well can you make the connections between the concepts you’re using

turbid zenith
#

Also wow it’s interesting to read two greatly contrasting books one right after the other. Going through Explicit Direct Instruction right after having finished Building Thinking Classrooms.

long pelican
#

What are your thoughts?

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Maybe combine the two and provide explicit instruction on thinking?

tight star
tardy ember
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like when mathematics is written down, often it's in words, but when you read it you would notice if the words didn't map onto anything that makes mathematical sense

tight star
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Mhm, that’s true

tardy ember
#

and i guess you could think about it entirely in words, as long as the words are meaningful

tight star
#

Though - i do have to kind of get used to things that don’t make mathematical sense

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Since I’m much more a physicist lol

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Or at least - i can switch between caring and not caring about that

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E.g. whether I worry about commuting derivatives past integrals or not

tardy ember
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instead of "repeat back random substrings of what you think you remember hearing, chained together into something that vaguely sounds right", which sometimes happens to produce a correct proof by chance but will also happily suggest that to get a table through a doorway you should cut the door in half

vagrant meadow
#

found out the linear algebra class i'm TAing for won't even cover determinants. i don't know what to do

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like... what the hell

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they're just doing one section a day and skipping a whole bunch.

quasi musk
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Done Right

turbid zenith
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Of course each swears that their method is the only correct method and caricatures the other method

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And, naturally, claims that theirs is backed by research, while insisting the other's research has holes in it

long pelican
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Completely unsurprised by that

soft ermine
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im so dumb im in seventh and my teacher in 4th gave us 1 day to understand lol

fallen needle
cloud zealot
#

do you feel that inequalities get rather short shrift in algebra and precalculus classes? it seems beginning real analysis students struggle a lot with coming up with bounds.

hollow musk
#

we completely ignored inequalities and absolute values in my precalc class(es)

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so when it came time for me to do intro analysis going in from basic calc, it was quite the hurdle

midnight scarab
#

Plus, inequalities (and asymptotics) are so important for all of analysis

tawny slate
#

sorry for my remarks being a bit less pragmatic, but while on the topic

i do think it is a missed opportunity too. inequalities would be a student's first taste of math problems that truly require creativity, problems that cannot be solved by rote algorithmic processes, that need intuition and experience

it works the other way around too. whether it is more formal, like trying to prove "if x>y>0, x^2>y^2." or more fun like "prove 2^81>3^49", showing the solutions to these problems i think helps students appreciate just how far the basic principle go when applied

muted hound
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Spivak's Calculus, an intro college calc book, does this well in chapter 1. However inequalities take so much time to teach beyond the algebraic rules

lavish steppe
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Did/does anyone do 1-1 tutoring with kids in ~10th grade (hes 16 years old) and/or has (general) opinions what works?

I'll meet him in a few days for the first time and offered him to send me whatever exercises he is stuck at and questions he has on the topic.

That way I can think a little about his troubles beforehand and we can spend our time addressing his questions during our time together, including him as much as possible.

I dont really have experience with tutoring (except occasionally answering questions on discord or classmates), so im open for anything.

I think theyre talking about the binomial formula for small n at the moment

muted hound
#

get the same textbook he uses for school, get the term syllabus the teacher gave him and make a list of everything he needs before the exam. This way you can predict his questions or estimate where the problem areas are/will be. Grade 10 came from grade 9, which is algebra heavy. He would benefit if you would reviewed the distributive laws, valid algebraic moves and such things with him.

vital prism
vital prism
fading zodiac
#

I need cbest math discord

tepid smelt
#

The cbest is mostly elementary level math. The CSET is high school level and I have heard of teachers struggling to pass components if they didn't study anything math heavy in college. I think you can teach any middle school or freshmen level with just the first CSET which is just algebra 1 material.

On inequalities even going through a basic undergrad I realized how weak I was at them when trying to look at how prevalent they are in highschool contests. I think it's a topic that is not really addressed well at both levels.

light mirage
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How do I explain to someone that if g is defined by g(x) = x^2 then g(p) = p^2

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The problem is " lim[x to p] [ g(x) - g(p)]/[x-p] where g(x) = 1/x^2" is theyre stuck at "what is g(p)"

dusk sentinel
# light mirage How do I explain to someone that if g is defined by g(x) = x^2 then g(p) = p^2

"For a given input, the function g produces some output. If we denote the input with x, then the corresponding output is denoted with g(x). This is just a notation. The formula g(x) = x^2 just means that for any input, the output is equal to the square of the input. So if your input is 1, the output is g(1) = 1^2 = 1. If the input is 7, the output is g(7) = 7^2 = 49 and so on.
There is nothing special about x. We could have used any other "name" to represent out input. So if our input is some p, then the output is g(p) = p^2."

#

this is what I would start with, then expand depending on their response.

white gull
#

If you were to teach a particularly enthusiastic high school kid calculus, would you stick to the ordinary progression of Calculus with a beginning in limits in R^2 or would you take a different path such as beginning with metric spaces and continuity using epsilon-delta def. and teaching a more abstract idea of what calculus means?

cloud zealot
signal estuary
#

Except for teaching a few classmates calculus

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But I think that (even with a very enthusiastic kid) it would be better to start with more intuition based definitions

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And if they’re interested in the more rigorous approach to math then yes, I think teaching them things like epsilon delta would be a good idea

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It sure would’ve solved the problem that I had when first learning calculus: concepts like limits and “getting closer and closer to 0” seemed very… fuzzy

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Things like the epsilon delta definition really show you how you can think about it in a more formal way that doesn’t rely really hard on intuition

marsh compass
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The epsilon delta definition is intuitive

signal estuary
#

weeell

#

now we need to talk about our definition of intuitive

#

When I said intuitive up there I meant that you don't need to imagine it kinda getting closer and closer and then things should work out?

#

Let me rephrase that (well, rather, say something completely different):
The epsilon delta definition gives you a very clear definition for when and how a function converges to something.
Saying "it gets closer and closer to the value" does not. It might seem like it to us, but that's presumably because we also know the epsilon delta definition and thus what that really means

real jolt
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I feel like ordinary progression generally holds them back and bores students that are advanced.

#

Instead of stimulating them.

cloud zealot
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@quasi musk why do you believe a power series first approach is best for an introduction to complex analysis? how do you feel it compares to the more commonly taught approach which introduces the integral theorems and formulae before power series?

real jolt
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It's a more philosophical take on it, but I have taught at a private school for two years when I was tight on money. We used to pull the kids who did very well on tests and were above the others academically and give them more advanced coursework.

#

And just challenge their intellect in general

quasi musk
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For example, a holomorphic function being once differentiable implies its infinitely differentiable is now obvious

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The main disadvantage is proving things like the product rule, chain rule, quotient rule, etc. can be a pain in series notation

#

But the theory works nicely. You can give a quick proof of the open mapping theorem as well

real jolt
#

Also they're already familiar with real analyzis.

#

convergence, differentiation, and integration of power series in the real domain

quasi musk
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Exactly. If you want to see more, please take a look at Marshall's Complex Analysis. It's my favorite complex book, I think the ordering of the topics is correct. Ahlfors is great, but relies too heavily on geometric foundations that no longer exist in students minds

quasi musk
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You need to talk about paths, path independence, partitions, regions, etc.

#

Power series you can do on a suitable open set, and push all the technicality into the computations that tell you something, rather than having this laborious set up to give you beautiful theorems

cloud zealot
#

what are some aspects of ahlfors you consider dated

real jolt
#

The notation and terminology is pretty dated in general, and content wise, I feel like the computational techniques, while still valid, just feel sloppy compared to modern methods

real jolt
cloud zealot
quasi musk
#

I think Conway's Complex Analysis text is much more precise on definitions and theorems than Ahlfors is

#

Also, Conways proofs feel more rigorous as a consequence

#

Ahlfors is great, but it feels like it was written for a different era as Oliver has pointed out

real jolt
# cloud zealot i'm learning complex analysis rn, how does the approach given in your example di...

I'm not an expert by any means but besides the typographical and notation being dated, the actual content is actually from a more classical era of math, as is complex analysis in general honestly, you can see the comparison of a modern lemma and how the standards have changed. The pedagogical approaches have too, with STEM integration and everything.
https://www.math.ucdavis.edu/~romik/data/uploads/notes/complex-analysis.pdf

quasi musk
#

If you try to prove things the way Ahlfors proves things in his text, then many math profs will be able to find holes in your arguments

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Basically Ahlfors is the foundation of the way we think about complex analysis, and many people have tried to rigorize his treatment

#

In different ways

#

So as far as understanding Complex Analysis, it's hard to beat Ahlfors. But it does leave a lot to be desired

real jolt
#

If you already have a good understanding of complex analysis it might be worth moving to more rigorous material though.

#

Which it seems like you do.

quasi musk
#

I've taken grad complex analysis essentially 3 times now, each with a different book. I'd say Ahlfors is probably my favorite classical text

cloud zealot
vagrant meadow
marsh compass
#

yeah, i wouldn't advise learning the definition by heart without understanding the intuition behind it

quasi musk
#

Hard disagree. You won't be able to intuit everything you learn ever, memorization (for better or worse) is a step in the learning process

cloud zealot
#

but some things are easier to remember than others

quasi musk
#

That all depends on your prior background

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And how quickly you pick up things. I think it's totally fine to ask students to know the definition of basic things

tardy ember
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i mean yes, they should know the definition, and also they should know the intuition behind what that definition means

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(or, well, "should" is a loaded concept here, i don't actually agree with the approach of trying to give people knowledge unwillingly
but i don't think having just the definition actually helps that much)

quasi musk
#

Yes, they should know both! But what happens when you have a student that says I know what it means but I can't recite it

quasi musk
#

Giving people knowledge unwillingly

tardy ember
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yep and i think that's bad

#

anyway

tardy ember
#

so in that case i would be a bit skeptical about the extent to which they actually understand it

quasi musk
#

Exactly, so what advice do you give such a student?

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My answer is obvious: first memorize the definition. Then go through the examples, because how can you understand what the examples are if you don't know what the definition is? And then slowly through this process the correct understanding will form (hopefully)

tardy ember
#

...well what i do in practice is ask them "ok well what does it mean then"

quasi musk
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"Oh it's like...you have this graph and this thing here? Or does it go here? Hrmmm I'm not really sure"

#

"The prof did something with like y = sin(x)"

tardy ember
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"...that doesn't sound like you know what it means"

quasi musk
#

So how do you first point them to what it means?

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You first....point out the definition

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Then build on the picture from there

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So even when you're explaining what it means, you inevitably fall back on the definition

#

I try to give my students very concrete advice for improving their study skills

tardy ember
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well yes obviously explaining the meaning will involve giving the definition, that's like the point of what i said earlier

#

the initial point was just that it isn't helpful to present the string of symbols $\forall \varepsilon > 0 \exists \delta > 0 \forall x (0 < |x - x_0| < \delta \to |f(x) - L| < \varepsilon$)'' and say ok the first step is to memorise this entire string so that you can write it out perfectly despite having no idea what it means''

burnt vesselBOT
#

bee [it/its]

695651562542071871.png
tardy ember
#

that isn't a useful skill to have by itself, and by the time it is useful, it will be remembering a concept that makes actual sense instead of memorising an arbitrary sequence of symbols, which is easier

quasi musk
#

The first step is you explain that the need to know that definition. Eventually we do end up memorizing these strings of symbols as what we mean, and these pictures as what we mean, etc.

tawdry venture
#

i think it is best to start with sequences

#

and maybe talk about the concept of eventuality

#

like "this sequence is eventually constant" or "this sequence is eventually positive"

#

building into "this sequence is eventually within 0.1 of L, and it's also eventually within 0.01 of L, and it's also eventually within 0.001 of L" and oh look

vagrant meadow
left swan
#

actually not even sure this is actually from that class since I don't remember anything from that area due to lack of intuition

tawdry venture
#

@green tinsel wrong channel, and possibly wrong server?

green tinsel
#

yeah, mb

tepid smelt
quasi musk
next sinew
next sinew
#

The way I like to teach the definition of continuity is by drawing a diagram with the domain and the codomain and basically the setup is that I imagine the function being a dart thrower that shoots out darts from the domain into the codomain

Generally, of course, darts that tend to be near each other also generally tend to land pretty close to each other but that doesn't have to be the case in general

A function being continuous means that for any dartboard of any size on the codomain, there's always some corresponding pile of darts in the domain such that each dart from that pile lands on the dartboard

The dartboards, of course, only make geometric sense for a R^2 -> R^2 function with the euclidean metric but of course it's just an analogy and the analogy allows you to retrieve the true definition for any situation that you might need

#

but anyway, I mostly came to this channel to ask what you guys think about teaching matrices and matrix multiplication and maybe even figuring out the sign of a permutation in elementary school?

#

Because I feel like university level students struggle a lot with matrices and determinants and all of that, at least in my country, and I think the reason for that is because the first time that the average STEM person encounters matrices is in university

vagrant meadow
# next sinew but anyway, I mostly came to this channel to ask what you guys think about teach...

it's usually covered in high school in the US, but i've never encountered a single student who has said "oh, yes, i remember this from high school". it's always "i don't remember anything about this". probably because they never use it again until they maybe take linear algebra or multivariable calculus. why teach it in elementary school if they're not going to use it for anything? that'd just make it worse. i guess if you teach it again in high school maybe it's better, but idk. either way, there's no room. elementary school curriculum is already getting completely wrecked here in america because of covid and common core/whatever bs.

next sinew
#

hmm thanks for your insight

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In Finland we teach linear systems of equations in middle school however we don't use matrices to represent them

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I think that would be a good opportunity to bring back matrices

vagrant meadow
# next sinew In Finland we teach linear systems of equations in middle school however we don'...

yeah same here. usually its 2x2 systems in middle school and then in algebra ii they explain the more general method for 3x3 systems (but, again, not with matrices).
i'm conflicted on it, because on one hand starting off linear algebra with "hey we can greatly simplify the system of equations process with matrices" is kind of nice (like a good first impression), and familiarity with systems of equations can give some insight to some of the concepts. when i see they're struggling with a linear algebra concept, bringing it back to a system of equations is usually a safe bet to get them back on firm(er) ground.

tawny slate
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i like to introduce matrices after finishing a lesson/unit on systems of linear equations, but only an introduction

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i explain the high level:

  • why bother using matrices (makes notation easier and separates it into its own kind of object)
  • why the properties of matrices are helpful (can immediately see whether a system is independent, or helps visualize the answer better)
  • why it generalizes well and the fact that computers can solve even ugly matrix stuff algorithmically
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maybe as a bonus if im working with more advanced students i will also explain the idea that matrices are transformations

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i dont want to burden them with actual bits they cant really internalize and rememeber, like what a determinant is or how to calculate it, i mostly just want to show them the magic it is capable of after they struggled so much with some of the more nasty problems (systems of 3 or 4 eq and vars)

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the goal here is for them to get excited about quick easy solutions, get them curious, rather than to actually get them to learn about them

quasi musk
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My issues in showing students matrices is the following: they don't use it enough to remember it when it's useful

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also, often times, it doesn't really help them solve things faster

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It's just this tedious thing that they have to do

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I think giving them practice on solving systems of equations with different methods (elimination, substitution) might be more effective

pseudo vine
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I think a lot of students are introduced to them way after they should be relevant

quasi musk
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I also feel that. I took my linear algebra & differential equations course, and didn't have any matrix background; my issue is that most seem to pick up on it fine during the school term

pseudo vine
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haha yeah, it took me until diff eq to even touch on matrices, and when we did, it was pretty much glossed over in five minutes the practicality they even have as a tool

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disappointingly some institutions have their curriculum all over the place lol

quasi musk
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It also varies a lot from instructor to instructor within departments

vagrant meadow
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when you define matrix multiplication as taking linear combinations of the columns (i.e. defined such that the columns are the images of the standard basis vectors), then matrix multiplication is very intuitive by considering the composition of the linear transformations. but when you just teach it as "along the row, down the column" all intuition is lost.

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i wouldnt try to teach that to a high school student, unless they were highly motivated and had a strong grasp of functions (something you can't even expect of calc, DE, or even linear students anymore)

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too many DE students can't take a derivative (and/or don't know the quadratic formula)

hollow musk
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that’s very surprising to hear

vagrant meadow
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@quasi musk can also attest

vagrant meadow
# hollow musk that’s very surprising to hear

imagine my surprise when i spent 30 mins explaining how to interpret the roots of the characteristic polynomial and then when i grade the quiz it turns out they did not know how to solve the quadratic equation that came out of the quiz problem

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3y''+y'+y=0 completely stumped them

long pelican
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some of you might have seen this before

hollow musk
vagrant meadow
hollow musk
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is there anything one can do to fix this, as an instructor/TA?

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I’m (hopefully) going to be a TA within the next two years, and I’ve spent a bit of time thinking about pedagogy recently

vagrant meadow
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it got really tiring to keep having to be like "yeah so it seems some of you with the product rule. that is something i would fix as soon as possible."

hollow musk
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I see

vagrant meadow
# hollow musk is there anything one can do to fix this, as an instructor/TA?

realistically, no. but having a blog and being able to send my students posts on relevant material seemed to help a few of them. i.e. i already had those materials available to me. if you start writing them yourself for them, then that's probably putting too much work into it (though i'm a total hypocrite because i do that. i shouldn't though)

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we only get 50 minute sessions. that's basically no time at all. especially if the professor has things they want you to do.

hollow musk
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50 min is the norm here too

vagrant meadow
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you can help them in office hours, if they come. but you're lucky if they do.

hollow musk
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welp, I’ll cross that bridge when I get there I suppose

hollow musk
vagrant meadow
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i was gonna say, that's the pro strat

hollow musk
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seems like a fun project to do

vagrant meadow
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looks good on resumes and applications too

vagrant meadow
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i was just very proud of this exercise i'm going to give my students tomorrow and i wanted to share it lol (this textbook has the awful convention that row echelon form is called EF "echelon form" and rref is called REF "reduced echelon form")

196803594576592896.png
hollow musk
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it’s to the point and guides you

turbid zenith
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@vagrant meadow seems to be good at that. 😄

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All of her problem sets seem to have that flavor.

vagrant meadow
# hollow musk it’s to the point and guides you

i love problems where you start thinking about the next step and realize you've already done the work for it! it also rewards the students who can see the conceptual connections to shortcut their work.

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i also like part (e) because i dont know if ive ever seen this method taught but it skips over a lot of the busy work sticking to the definition requires.

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and i've seen too many times people in #linear-algebra telling students that the way to find the matrix is to write a generic mxn matrix and write a system of equations in m*n variables and i'm just like nooooooooooo

hollow musk
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you’re a great TA @vagrant meadow

hollow musk
turbid zenith
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I just finished my take-home final exam for abstract algebra and the questions are built like that

hollow musk
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I really liked her problem sets

rough agate
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For the more experienced teachers out there, how did you guys get better at writing questions ?

When I try to be creative and come up with actually good ideas it always takes so long

austere delta
long pelican
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The ideas always come at times other than when you’re sitting down and trying to make problems! The trick then is to write down ideas you get when you get them so you don’t forget

peak zealot
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I am not sure if this is the correct channel to post so let me know.
I had a question on a test which was the following: find a sequence such that an>0 for all n and limsup (a_(n+2024)/a_n)<1 and limsup(a_(n+1)/a_n)=infinity
and while i was preparing for the test I didn't see any type of question that resembles it, or questions of the type " find a sequence such that the following properties hold" and I was wondering if someone know a real analysis book or material that has similar problems

turbid zenith
tawny slate
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also dont be afraid of iteration

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i taught vieta's formulas once and this kid insisted on solving everything using the quadratic formula (which on top of the fact that it ignores the lesson being taught, is a topic we didn't get to yet). while it's ugly and time-consuming, technically works on most all problems, but clearly not what i wanted the student to do

so i composed problems in which vieta's made it very very easy but quadratic formula was immensely difficult. kid still persevered, impressively, so i had to go back to the drawing board and come up with problems for which the difference in difficulty between the two problems was even more insane, sometimes relying even on students using ansatz

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commended the kid on his tenacity but i did also end up getting my point across, and i think it made both of us stronger

rough agate
rough agate
rough agate
rough agate
# tawny slate i taught vieta's formulas once and this kid insisted on solving everything using...

ohhh yeah that's fair. I mean you could also get around students using sub-optimal methods by increasing the difficulty of the tests (that's how we do it here). But yeah I do try to have a few other people (who know content outside the course) help me test the questions to ensure that I have written accurately to the particular course and also, as you said, to make sure everything works as intended

tall bolt
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Big fan of that final question

wispy slate
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wait you guys are math professors?

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dammmmmmmm

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mad respect dude

wispy slate
turbid zenith
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This channel is specifically for discussing how to teach math.

wispy slate
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oh sorry

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bye have fun

soft scarab
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a quick 2 questions for any teachers in here! do you find it better to assign hw sets that you've created yourself or to just assign problems from the textbook? additionally, when making tests, do you like to really challenge your students and make difficult exam questions or do you find that making tests similar to hw is better?

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i've had professors who have done both, and from my personal experience I enjoy it when the hw is curated by the professor and the tests are similar to it, however, I would like to hear your thoughts!

turbid zenith
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I prefer to give problems I’ve created myself, but this semester I often didn’t have the time to do so, so I had to pick problems from the textbook.

soft scarab
turbid zenith
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It depends on the class. Ideally I would for low-stakes stuff though.

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Also, whether you want to give difficult exam questions … I think that greatly depends on how you grade and what you think the purpose of grading is.

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I don’t remember who it was that said it, but something I’ve lived by for assessing students in general is this:

Students are more likely to hit a target if (1) they can see it and (2) it doesn’t move for them.

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Take that as you will, but it’s why I’m largely against curveball-filled exams.

turbid zenith
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Has anyone used Jupyter Notebook as a tool to teach math?

vagrant meadow
# turbid zenith Has anyone used Jupyter Notebook as a tool to teach math?

i once had the same professor for two classes (one an undergrad computing course and a grad numerical linear algebra course) and we used Jupyter notebooks a lot in both. granted, it was for classes where we were using a lot of Python, but I can see a lot of potential in using it even if they aren't expected to know python.

I think it would be cool to give a notebook where they can mess around with some variables and see some cool visualizations/plots.
especially in the age of chatgpt, where formal python training is simply not at all necessary to accomplish rudimentary tasks.
I imagine a curious student could do some really interesting things if you give them a Jupyter notebook to play around with. I think it'd really motivate me, at least.

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I was actually thinking about giving my linear students a project related to SVD based on an... "incident" with a printer. I think a Jupyter notebook would be a fantastic way to do it actually

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long story short, I have sensitive eyes so I do my latex documents white text on a black background. after the incident, I needed a way to convert and organize my PDFs based on if they use black or white pages. so I needed to detect the color of the page. since I was doing this for my linear algebra worksheets I decided to try some linear algebra to solve it, and looking at the SVD did it! I used the ratio of the first two singular values to tell me the color with total accuracy. so I'm thinking about ways to present it to my students (some of them seemed very interested when I went on a tangent about SVD and PCA when they learned about what a projection is, and asked me to talk about it in a survey I gave them).
it's a super basic intro linear class, so it's pretty far out of the scope, but I'm trying to see if I can give them something about it.

ratio_figure.png
sweet pumice
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graduate scheme speaking

vagrant meadow
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basically

\usepackage{xcolor}
\usepackage{pagecolor}
\pagecolor{black}
\color{white}

but i recently coded up a more advanced function system that allows for easier switching

vagrant meadow
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still working on the final version of the code but it wasn't that bad to get a python script to do it

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using an inverted .py file i found online

rough agate
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do you guys have a preferred program on which you write worksheets or tests ?

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i currently use word and latex, it works nicely enough

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ive seen some other people use word and mathtype

austere delta
rough agate
austere delta
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Ah, that's what you mean. Is that actually latex though? Like I guess the syntax is pretty much the same, so doesn't matter

rough agate
austere delta
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I'm pretty sure, that's not latex, but something made by Microsoft. But the result is the same I guess

rough agate
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uhhh

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the syntax is the same

sharp lodge
austere delta
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Sure, but you couldn't import a latex package and use that

rough agate
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but the one thing is that i like times new roman

image.png
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it feels nice

sharp lodge
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I know a good font that you can use, other than Times New Roman.

rough agate
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so thats why i use word

rough agate
sharp lodge
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Try the Blinker font.

rough agate
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also another reason i use times new roman is because the education authority in my state uses it so its like getting people used to it yk ?

rough agate
rough agate
sharp lodge
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You can print in color. And use textured paper.

rough agate
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b&w is cheaper

sharp lodge
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It's not that expensive, when you do it at Office Depot.

rough agate
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im uh

sharp lodge
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I highly recommend you use textured paper. People will recognize it and remember it more.

rough agate
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not familliar with office depot

sharp lodge
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Are you based in the United States?

rough agate
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nope

rough agate
sharp lodge
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Apologies, I made a hasty generalization that you were U.S. based.

rough agate
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nahhh its fine

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im in australia so its not too different

sharp lodge
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Australia is a great place. One of my best friends owns a company in Perth.

rough agate
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ooh thats nice

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nah i wont lie thou textured paper seems just a little unnecessary

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like ive never seen anyone else use it

sharp lodge
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I helped them create a Snap Shows business, and then I created UEFN maps for them. They made $280K in the month of December for their Snap Shows. Quite more, when you consider it in AUD.

rough agate
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280k a month is crazy

sharp lodge
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But yeah, I'll try to stay humble. Apologies if I am humble bragging.

rough agate
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its fine lol

sharp lodge
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@rough agate it's possible if you have the first-mover advantage.

rough agate
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its not the first mover advantage if the official state made exams dont use it

sharp lodge
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I highly recommend the book by the PayPal founder, Peter Thiel. It's called Zero To One: Noted for Startups.

rough agate
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uhhhh

sharp lodge
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Haha, I'm being out of context.

rough agate
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heres the issue

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im not an entrepeneur