#math-pedagogy

1 messages · Page 19 of 1

cosmic ibex
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Something like the limit of x/(x+sin(x)) for x -> infty.

midnight scarab
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I'll use that the day I end up teaching calc, if I ever do

vocal phoenix
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Or (exp(x) + exp(-x))/(exp(x) - exp(-x))

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Also at infinity

cosmic ibex
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Ah, sneaky!

minor gale
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There are many other examples where it fails to work eg x/(x+sin x) as well

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Wiki has a page on them

cosmic ibex
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On the other hand, the case where students really tend to be double down on L'Hospital is 0/0 for $x\to 0$. So it might be better to show something like
$$ \lim_{x\to 0} \frac{2\sin(x)-x\cos^2(1/x)}{\sin(x)+x\sin^2(1/x)} $$
(where the $\sin(x)$ factors instead of just $x$ are there to make it less tempting simply to divide through by $x$).

burnt vesselBOT
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Troposphere

217922183169572864.png
minor gale
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That’s a sneaky one

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Nice

austere delta
minor gale
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Divide by x then note sin^2=1-cos^2 and sin x/x -> 1

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Numerator and denominator become close to 2-cos^21/x

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Gotta handle error terms a bit carefully but based on that I’m pretty sure the limit is one

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Yeah indeed

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It works

austere delta
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I see yeah

cosmic ibex
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Indeed. Once you have divided by x, the errors of sinx/x go to zero and the other parts of the numerator and denominator are at least 1 so the error does indeed get drowned out.

minor gale
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Error is equal to (sin x/x -1)/(sin x+x sin^21/x) and denominator gets bounded away from zero meanwhile numerator is O(x*3)

cosmic ibex
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My first attempt for x->0 was something like x/(x+x²sin(1/x)) but then the problem was that the numerator and denominator are actually differentiable at 0 (after filling in the hole in the denominator), and we don't want students to internalize the vulgar form of L'Hospital that immediately just plugs zero into the derivatives ...

vocal phoenix
midnight scarab
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Something fascinating is that French uni students (or at least prépa students) are taught to forget L'Hospital's [which is indeed taught as the be-all and end-all in hs] and just use Taylor expansions

vocal phoenix
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Sensible advice for the 0/0 limits at non-infinity

cosmic ibex
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Hmm, it seems to me to be an overcorrection for teaching it as a be-all-end-all earlier.
It seems to be the kind of thing that gets people to come here and denounce LH as wrong while attempting to help, even in cases where it does work.

tight star
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Yeah - there are cases where lhopital is very convenient, after all

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E.g. peano form of remainder

midnight scarab
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Yeah definitely

cosmic ibex
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For 0/0 limits at non-infinity, if the functions are smooth enough to have Taylor expansions, it's basically the same thing anyway. Unless you're fortunate enough to already have a prebaked series expansion, computing sufficiently many terms of the Taylor expansions to reach a conclusion is effectively the same work as applying L'Hospital repeatedly until it yields.

midnight scarab
# midnight scarab Yeah definitely

Which is why I say it's fascinating that the only time I hear of l'Hospital's in all my time at uni is in this one exercise which asks you to prove l'Hospital's lol

cosmic ibex
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On the other hand, $$\lim_{x\to 0}\frac{2x+x\sin(\log x)}{2\sin x+x\sin(\log x)} = 1$$ can be evaluated with L'Hospital (as well as with the divide-by-$x$ strategy from above, of course) but cannot by Taylor expanding the numerator and denominator.

burnt vesselBOT
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Troposphere

217922183169572864.png
tight star
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I think it’s good to get used to the idea of replacing one function for another in such limits

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E.g. knowing that you can replace (sin(x))/x by 1 up to higher-order terms, and seeing if that lets you evaluate the limit

cosmic ibex
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(Or perhaps I should weaken "can be evaluated with L'Hospital" to "the rule actually applies, since the the limit of the derivatives does exist". I'm not arguing that L'Hospital makes this particular limit any easier to evaluate than the divide-by-x strategy).

midnight scarab
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Which makes the fact that one should divide by x obvious

dapper jewel
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is it a bad idea to repeat problems after asking the solvers to prove the gernal form ?

noble hare
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can you be more specific

dapper jewel
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ok, so for instance one problem I made was prove sin(f(x))/f(x) 's limit at 0 is 0 if f(x)=0

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after that sin(x^2)/x^2 for instance would be a really bad problem

shadow flower
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I think there’s a benefit in giving the problem just to see if the students recalled what they proved and make the connection

dapper jewel
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I'm just doing this for my classmates though

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like this is just incase anyone wants help, and for people who aren't math majors either

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anyway to make it a bit harder?

drowsy otter
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I have a question. How do I define multiplication to people who need Functional Skills?

I know how to define it technically, but I also know that I am not training mathematicians, but people who are struggling.

All I know is that multiplication is repeated addition, but I'm not sure how to word the definition in a way that supports people who have struggled with 10+ years of education.

tawdry venture
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maybe try to build up to it with basic problems? shit like "4 of your friends each give you £15, how much did you receive from them in total?"

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show plenty of examples? practice counting in twos, threes, fours etc?

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i personally would like you to define "functional skills" for me btw bc that phrase doesn't ring a bell rn

tight star
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So, there’s a few different ways to view multiplication

drowsy otter
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Functional Skills (at least in the UK) is a course that people study to advance their career. It outlines the basics of Mathematics. Basically, it's equivalent to Primary Level, but for adults instead of little ones.

tight star
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Repeated addition is one of them

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Size of a grid/area is another

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And stretching/scaling is a third

tawdry venture
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rectangular arrays are a nice visual

tight star
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It may be worth showing the other perspectives and see if one of them clicks?

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So long as you provide a way to translate between them, that is

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Otherwise they’ll feel disconnected

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As in - make it clear why repeated addition should be the same as size of grid/area

tawdry venture
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may also be worthwhile to practice the multiplication table row by row

tight star
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Yes, computations are good

tawdry venture
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so like spend some time going over ×2, then some more time going over ×3, then those two together, then move on to ×4...

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that sort of thing, and work your way up to ×9 eventually

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@drowsy otter do your students have a good grasp of addition?

drowsy otter
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You see, I'm having to do these presentations in advance so that they are oven-ready by the time I start teaching.

tawdry venture
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i'm not sure a ppt would do you much good

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i would just put down the repeated addition thing phrased in friendlier terms

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if you really want one

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like "To multiply two numbers, add together several copies of one of the numbers -- specifically, as many copies as the other number. [examples]"

drowsy otter
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How about this?

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I hate it when copy-paste doesn't properly work.

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Definition 2.3) Multiplication, written x, is the act of repeatedly adding a number to form a number of copies of that number.

tawdry venture
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i don't like that wording

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at all, actually

drowsy otter
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Now that I read it out loud, I don't either.

midnight scarab
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I'm also not sure I like it starting with the word definition

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It's just intimidating

tardy ember
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yeah that feels... kind of awkward and also just vague?

tawdry venture
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yeah like the whole defn-thm-proof framework is definitely not gonna be a good look given your context

midnight scarab
tawdry venture
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i'd rather you put down the how and talk verbally about the what

drowsy otter
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I want to put down "the act of multiplying a number by another number to form a product", but I don't think that properly defines multiplication. Though I might be wrong.

tardy ember
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yeah that's basically just saying "multiplication is multiplication"

tawdry venture
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just don't bother putting down a definition, imo

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multiplication is one of those things that doesn't really need one written out at your level

tardy ember
drowsy otter
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I'll definitely look beyond the Definition-Theorem-Proof format that I'm used to. Thank you all very much.

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In the end, I've written "Multiplication is written x", as the sign is what's really important to learn for these students.

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Again, thanks for helping me.

cosmic ibex
tawdry venture
drowsy otter
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Thanks for the tip. I did mean that symbol.

tight star
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I think I can safely say that it’s possible to present universal properties in a way that’s both very easy to understand/digest, and immediately useful for explaining concepts or solving problems

dapper flume
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@drowsy otter One thing everyone can relate to is food. If your FS students like a certain fast-food place (say, McDonalds), you can use a ton of analogies for multiplication that will directly develop life skills. Ideally, these will be the most helpful if the students try to figure these out on their own before knowing exactly how to do it with multiplication. If they are proficient with their addition, for example, they may find the method of repeated addition on their own.

Take, for example, some simple multiplication problems:

You and three buddies go to McDonalds and you each order the 8-pc chicken nuggets. How many total chicken nuggets will all four of you have?

You can extend the analogy later for basic division:

If a birthday party of 20 children need to be seated at tables which each seat 5 people, how many tables do you need to reserve?

...and for financial math, too, once they know decimals:

A family orders 3 Happy Meals for $3.50 each. How much does the family need to pay altogether?

These are just a few examples, but you can get creative. The important part of my suggestion is to look for tangible examples which demonstrate real-world skills. When you do this, you can leverage their pre-existing intuition about the world to translate it into mathematics.

tawny slate
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for people who are struggling with math, i prefer to stick closer to proven methodologies. leave the more creative approaches to the students that are more capable of doing something with them

tawdry venture
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another setting you could use is grocery shopping

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e.g. "milk costs £3 per carton and you buy 4 cartons of it, how much do you pay"

tight star
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See here for an example of how easy it can be

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I think you could talk about the universal property of product alongside learning about the Cartesian product

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And this gives you another tool to manipulate functions - which I’ve found to be very helpful, since it’s often useful to explain things in terms of functions

noble hare
tight star
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In particular, what I am suggesting is not defining gcd to be a packager

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The usual definition of “maximum of the set of common divisors” should still be taken

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Instead, it’s a property that an existing definition satisfies

tawny slate
tight star
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Oh no that was unrelated to the previous convo

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I don’t think universal properties help for addition and multiplication

tawny slate
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oh ok nvm ignore everything i said then

tight star
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In that case it’s more about - have multiple perspectives and ways to translate between them

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Which is also categorical but in a different way

drowsy otter
tight star
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The Cartesian product of sets A x B has the property that any function Z -> A x B can be “unpackaged” to a pair of functions Z -> A, Z -> B. Moreover, this process is reversible - any pair of functions Z -> A, Z -> B can be “packaged” to a single function Z -> A x B.

This can be useful to:

  • Break up a trajectory in the plane to horizontal and vertical components
  • Define pointwise operations on functions - package them, and apply the binary operation
  • In a single proof, show that preimage commutes with every subset operation (intersection, union, complement, set difference, symmetric difference…)
  • Draw analogies with other “packagers” like min(a, b) and gcd(a, b), which package inequalities and divisibility conditions, respectively.

When might it be useful to make students aware of this? Either explicitly in class, or as say an exercise on a problem sheet, to test their understanding of the Cartesian product.

pastel ledge
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this is something that became relevant to me recently when I was TAing linear algebra

tight star
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oh?

pastel ledge
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the curriculum was inflexible on the point of not explaining the universal property of direct sums of vector spaces

tight star
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the one for mapping in, or the one for mapping out?

pastel ledge
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both!

tight star
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I see I see

pastel ledge
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together they make for a good low-tech explaination of the whole thing surrounding representing linear transformations as matrices

tight star
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I do think that - one doesn’t need to go so far as to define the direct sum by a universal property. But it can still be presented as a property that the existing construction of direct sum has, and that’s closer to what I’m suggesting

tight star
pastel ledge
tight star
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Wonderful!

pastel ledge
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as with a lot of things in early-college math education, I think an approach which begins concrete and grows more functional over the course of exercises is key

tight star
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Yep yep, I’ve found a functional approach to mathematics to be quite enlightening

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And it’s surprisingly useful when explaining, too

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A lot of misconceptions go away from just… type checking

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I do think this operation of “packaging functions” or “packaging linear maps” is perhaps not emphasised explicitly enough, because it’s not composition - it’s a different kind of operation on functions

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And for me personally, it just wasn’t something I explicitly knew was possible, especially when you get to settings like topological spaces where this is nontrivial to prove

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Hence why I think - the earlier one sees that such an operation is possible, the better

pastel ledge
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I agree, although I'm usually not leading first year courses, so I don't usually take such a global approach to pedagogy

tight star
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Mhm, hence when I wonder whether such a “functional” approach could be useful even as early as first year

pastel ledge
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ah, my story was from a first year linear algebra course

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(i realize that this sort of distinction of years is pretty uni-specific)

tight star
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I do think that universal properties help greatly with a functional approach - after all, they’re fundamentally about alternative descriptions for maps into or out of your object

tight star
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I think I’ve only recently appreciated in what sense knowing more about a subject can sometimes be actively worse than not knowing anything about it

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In particular, if you know enough to have misconceptions, but not enough to fix those, it’s easy to dismiss the subject and develop a bias against it

minor turtle
tight star
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Oh? I’d be interested to hear this

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I had category theory in mind personally when I wrote this

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But I’m sure it’s a general phenomenon

minor turtle
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because people have an idea about infinity and ZFC and foundations (also read: logic, godel’s incompleteness theorems is a popular topic) from youtube videos by extremely popular people that were made a long time ago, but they don’t have the foundations to really understand what they’re talking about

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sometimes people will come into discussion asking about “aleph 0 and aleph 1” but the only answer that you can give them is really to try to understand what’s going on at the set-theoretic level at that point, because you can’t really say many meaningful things about them without even having opened a set theory book so that you know how cardinals are defined

tight star
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I see I see

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It does seem like a difficult issue to resolve

minor turtle
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well, in the case i mentioned it can be resolved by properly learning set theory

tight star
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In my case, I’ve actually often found it easier to teach cat theory to people with no prior exposure than to those who already know some of it

tight star
minor turtle
minor turtle
tight star
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E.g. a lot of people tend to conflate the abstractness and difficulty of the general statements in category theory with the abstractness and difficulty of specific instances of those statements

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Often to the point where if I mention an application of an abstract theorem in category theory, people assume the application must be abstract

tight star
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When expositing a subject, is it better to focus on why it’s useful or why it’s cool? For me they’re essentially synonymous, but I know that’s not true for everyone.

vocal phoenix
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I think that depends on the audience, but I feel most general students value usefulness more than coolness.

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I tend to focus on the usefulness but try to sprinkle the occasional cool/fun stuff throughout.

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Mostly to pique the interest of the like 3 people who might genuinely want to get into the subject.

tight star
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I see I see - and, what counts as “useful”?

vocal phoenix
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Again, hugely subjective.

tight star
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Mhm, but one has to make a choice what to focus on in their exposition

vocal phoenix
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When I teach linear algebra to first-year students of mining, usefulness of anything is a hard sell.

tight star
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Right…

vocal phoenix
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When I teach measure theory and Lebesgue integration to second-year math students who have already suffered through Riemann, a lot of them do see how it's useful

tight star
vocal phoenix
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That's an even harder sell.

tight star
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Then - what’s the third option?

vocal phoenix
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I try to persuade them that they're going to find it useful later, and find some (mostly geometric) examples of how the stuff is used.

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But there's no escaping the fact that teaching mathematics to people who don't want to learn mathematics is a challenge.

tight star
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Ah, so future utility rather than current utility

tight star
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Or I guess, exposition outside the context of teaching a class

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E.g. giving a talk

vocal phoenix
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A very large proportion of students are explicitly hostile to actually learning anything, they just want to pass the course and be done with it.

tight star
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Right…

vocal phoenix
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Especially with a subject such as mathematics, where they're negatively predisposed from the get go

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Because we teach children to dislike/fear mathematics from a very early age

tight star
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I’ll file “beauty” under cool for the purposes of this convo

vocal phoenix
tight star
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Sure sure

vocal phoenix
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"Mathematical beauty" is not something I find most students receptive to, even math majors.

tight star
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I see I see

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But - are they able to find math cool independent of it being useful?

vocal phoenix
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Again, most math majors, at least at my uni, have the aim of getting a math degree and pursuing a job in data science/other industry

tight star
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Perhaps not in a “beauty” sense, but in some other sense

vocal phoenix
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Very few come in wanting to study math for its own sake

tight star
vocal phoenix
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That said, there's usually a few people who are genuinely interested and find it cool.

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But they're invariably a minority in the classes I teach

tight star
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I guess that’s a vague Q

vocal phoenix
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I've never really thought about it.

tight star
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Right, I see

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Well, honestly this is fairly encouraging to me

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Being a physicist and all, “useful” is the axis I tend to focus on

vocal phoenix
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Honestly, a lot of students are very happy just if you make the mathematics clear and understandable.

tight star
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Ah, a third option!

vocal phoenix
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Even if they don't really see the coolness or the usefulness, if they just find out that after a class with you they find the problems easier (and pass the exam), that's actually a lot

tight star
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So clarity, usefulness, coolness

tight star
vocal phoenix
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Yeah, I'd say my main goal, especially with non-math majors, is to demystify the subject

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Show them that it actually makes sense

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And isn't just a bunch of arbitrary magic

tight star
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I never thought about it in those terms explicitly!

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But that makes perfect sense, now that you’ve said it

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So making things feel less arbitrary

vocal phoenix
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Yeah, showing how things fit together.

tight star
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Again this is encouraging

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I think, then, I will prioritise in the order clear > useful > cool

tight star
vocal phoenix
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I should point out that I hardly ever taught people 1-1 or even in small groups.

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Pretty much all my teaching has been done in the context of university courses, so lectures with 60-200 people and problem-solving tutorial classes of 15-30 people

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Well, I did once do a semester of lectures for 3 people, but that was a very edge case

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When teaching individually, I imagine the choice of focus/emphasis will depend very hugely on the specific person and situation.

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When teaching large groups like I do, you have to compromise and pretty much cater to the median.

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(although as mentioned, there's still variance between majors and subjects, and even years)

tawny slate
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like if your goal was to make math fun as an activity, i think cool > useful

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but otherwise those are the big 3 i think

tight star
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Sure sure - I guess I am specifically thinking about online cat theory exposition

vocal phoenix
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I will reiterate that it hugely depends on the context in which you're doing the teaching, and who the audience is; explaining maths to people who come to your website/youtube channel is going to be very different from a classroom setting where you're teaching people whose aim isn't specifically to learn the things you're talking about (but have to pass the course you're teaching)

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Especially with respect to the "coolness vs usefulness" spectrum, because I think clarity should be a main priority throughout.

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The people need to have the sense that they understand what you're talking about at all.

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Which is surprisingly uncommon.

tight star
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Yeah I think my main takeaway from this convo is that I was undervaluing clarity

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I didn’t realise just how far, like, being clear gets you

vocal phoenix
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For me "subtle" means that there's something in it that's easy to get wrong.

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Like a small change in formulation will make the result invalid.

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Or that a part of argument that seems obvious, actually requires far more work than initially appears

vocal phoenix
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Still, just saying "you're doing it wrong" without elaborating is very bad practice.

unreal tartan
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I'm gonna be tutoring k-8 kids remotely. What would be the best way to draw math and then let them do questions. Like do I let them tell me what to write, or do they write on a paper and then read it out? I'm a bit nervous about how to do that

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@oblique sage I know how I will teach. Im more concered about how to actually let them interact using zoom. Because if they write on paper I cannot see, and they will have trouble drawing on the computer

dapper flume
unreal tartan
dapper flume
# unreal tartan I don't know what age you are teaching, but how do you get your students to draw...

There's decent auto smoothing for strokes. Also, there is a text-box feature and some shape tools. If you have young enough students that their motor skills are still weak, you might consider a different service which is mostly based on typing (one doesn't immediately come to mind).

That said, if you're creative, you can use Excalidraw for much more than making the student write. For example, ahead of time you can set up some "virtual manipulatives" like base 10 blocks, which they can click and drag around, which is easier than writing with a track pad.

unreal tartan
halcyon glade
unreal tartan
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i will be teaching all

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@halcyon glade

halcyon glade
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Ah

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I know Zoom has a whiteboard feature so you and the student can write at the same time

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Seems harder the younger the student is though

unreal tartan
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realistically no one does tutoring as much for grades less than 6 mostly

dapper flume
# unreal tartan realistically no one does tutoring as much for grades less than 6 mostly

Some other sites you may like to explore:

  • If you are working with algebraic equations, you can try math.new. It is also collaborative, but it's based on manipulating algebraic equations. It lets you manipulate equations by dragging components of the equation in different ways (Subtracting 5 to the other side, dividing out by 2, simplifying calculations by dragging -2 into 1, etc.).
  • You can create interactive live lessons on Desmos Classroom. You can write your problems, and you can select between a number of different ways a student could respond, including typing, drawing, or manipulating points on a graph, etc.
unreal tartan
jovial sinew
wispy slate
daring gulch
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I still use netscape.

jovial sinew
vocal phoenix
unreal tartan
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@dapper flume you mentioned that you like excalidraw. I can't see any latex support for that. Do you know how to enable it or anything that has latex support?

grim spindle
jovial sinew
grim spindle
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Lmao fair enough

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"Today we're going to learn about 2-cocycles kids. Why should you care? How else are you gonna learn how to add 2-digit numbers?"

jovial sinew
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I remember area models and lattice multiplication lol

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When I look at my brother and sisters homework its even worse lol

tight star
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So based on a convo in #groups-rings-fields to do with distinguishing “denotational” vs “operational”, how might someone make clear to a student that algebraic expressions can be viewed as things you can evaluate?

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In this case, “denotational” refers only to the result that an expression “denotes”, whereas “operational” cares about how exactly this result is evaluated

jovial sinew
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Problems along the lines of x is 6 what is 3x + 4

spice matrix
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For polynomials I think my favorite demonstration of this is the universal property of polynomial rings. They are characterized by the fact we can evaluate them pretty much.

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But that might be inaccessible to an early student, so I think just giving some examples of expressions and asking to substitute in certain values for a variable will do.

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Like what Max listed

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let bindings hehe

cosmic ibex
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Yeah, I think "universal property of polynomial rings" is much too subtle for this purpose.
After all, there are many functions other than polynomials that can also be evaluated.

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What sets polynomials apart with a universal property is that there few enough of them that the ring operations internally in the polynomial ring determine exactly which function is which -- you can't start evaluating some of them to different results without breaking the ring operations.

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But I doubt that is something that is helpful to point out at the high-school level.

tight star
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What made the “new math” movement from the 60s fail?

lost raptor
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Iirc parents complained because it’s not how they learned it, and educators didn’t exactly know it either

vagrant meadow
tight star
vagrant meadow
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sharp is eternal. of course sharp was there

tight star
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Huh I see

lost raptor
dapper flume
# tight star What made the “new math” movement from the 60s fail?

I did a bunch of research on this a few years ago for some final project, so pardon my nerd behavior on this one.

In short - both the creation and destruction of the New Math movement were quite political maneuvers. The latter was in part sparked by fear of modern warfare after the atom bomb, especially in the face of the rapidly evolving Soviet science bloc (circa Sputnik). Basically, the US and allies really wanted more trained scientists.

Ultimately this motivation got somewhat out of hand in the eyes of local politicians and parents. By this, I mean that it was almost entirely mathematicians writing curriculum content, with educators and psychologists left overshadowed. As a result, we ended up with a lot of rigor, with not enough regard for human development. And since the interest in New Math was more so anti-communist than pro-mathematics, it didn't really do what it set out for.

Sharp is essentially right about part of why New Math ultimately met its downfall. Parents couldn't support their children even in elementary school, so a lot of struggling students were stagnated. Even a lot of teachers struggled to teach the new content, with broadcasted TV lectures by professors being used in some areas to compensate for this. Not to mention the majority of modern educational psychology (from the likes of Piaget, Vygotsky, Kohlberg, et al) had yet to be born in the 70s and 80s. So what we have here is a curriculum which 1) leaves struggling kids in the dust, 2) isn't psychologically well-founded, 3) gets people pissed about government overreach, and 4) isn't actually doing enough to fight communism. So the people chucked it out and went "back to the basics" until Common Core.

TL;DR: USA hates communism, makes hard math curriculum to kill it. Math becomes too hard. People don't like government overreach and reject curriculum.

tawny slate
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fascinating

vocal phoenix
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Also the idea of anti-communist mathematics seems weird to me, but it does seem to have worked in America.

noble hare
tight star
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How do you get used to doing a more “top-down” approach to teaching? My instinct is always to take a “bottom-up” approach but the disadvantage with that is that it takes longer

cosmic ibex
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I don't think it was "anti-communist mathematics" as much as "if the Russians can teach advanced math to their kids, we'll fall behind if we don't do it too".

dapper flume
# noble hare This is super interesting. Do you know if this was big factor in developing the ...

My project was basically comparing the politics of New Math to the Common Core. There's a lot of similarities to the public's reactionto each, but I do think Common Core has far more staying power.

People criticized Common Core for being rolled out too fast (fair enough) with not enough representation from educators and parents. And just like in the 50s, parents right now complain that they are having a hard time helping their kids do homework, citing that the math looks so much different.

However, Common Core isn't nearly as rigorous as the New Math, and it will never be. That's a good thing for it. It was written by educational psychologists at least as much as mathematicians. As a result, you have the main content standards, as well as the "Standards for Mathematical Practice," which are a short list of qualitative ways students are expected to approach mathematics (i.e. persevering on hard problems, modeling the real world, look for and use structure, etc). And unlike in New Math, teachers are comparatively very well prepared to teach to the Common Core.

So I think Common Core has staying power, and will solidify itself once it's current students become parents.

dapper flume
cosmic ibex
#

It was Outsider's description more than yours.

dapper flume
#

Oh fair enough

dapper flume
# tight star How do you get used to doing a more “top-down” approach to teaching? My instinct...

If what you mean is directly teaching at the level of the concept and then filling details, rather than teaching all the details first, then I suppose the way to approach that is a sort of mindset shift.

When I'm a classroom teacher or longtime tutor, I'm bottom-up. I have the time to be bottom-up and the learning payoff is much bigger as a result.

However if there is a one-off question to be answered, I may switch to the mindset that the person is likely very nearly able to answer the question, so I'll give a nudge accordingly. To make longer conversation accessible, simply invite it explicitly (i.e. "if you want to talk through it deeper ping me!")

That being said - I personally liked the bottom-up method you used there. If you are careful enough, the worst case scenario is only the person being a bit overwhelmed, while the best case scenario is they open their eyes to a whole new perspective. I'd say that's generally pretty valuable.

vocal phoenix
daring gulch
# tight star See https://discord.com/channels/268882317391429632/496784996510466057/127261149...

I think I agree with Greenie on your explanation. I don't believe there is any need to differentiate between bottom-up and top-down approaches, in this case, but rather to find the optimal amount of explanation that is most efficient at answering a question. From what I understand, the top-down approach would be just giving the answer and a simple explanation, which I think may work but is not necessarily optimal. I'm guessing there is probably more merit to differentiating between top-down and bottom-up at the course level, rather than a single question.

tawny slate
#

i think top-down approaches are more broad than simply "answer and simple explanation". i think any lessons learned broadly, in retrospective, insight or context, can also be considered top-down, as well as various strategies and tricks beyond the "intended teach"

for example, when teaching calculus, one possible approach could be to first explain taking the derivative of x^n, then show computationally it's works very well. this gives a motivation and context for the subject that isnt as clearly present if you begin "bottom-up" with limits and formality

another example might be to show that, for example, heron's formula is often times memorized because it is so difficult (or at least annoying) to derive/prove. teaching "top-down" here by asking the student to memorize this formula, or at least its existence, may provide more utility than going through the proof (though you might possibly recommend the student do it once). of course, so far, this is simply the answer and simple explanation, but you can supplement it with general strategies, such as showing that it is a special case of Brahmagupta's formula or that dimensional analysis can be applied to make sure the most common error (forgetting that its s(s-a)(s-b)(s-c) not (s-a)(s-b)(s-c)) is avoided

#

also, the "simple explanation" may not actually be simple. lots of beautiful solutions that are one-liners use language or theorems that the student needs to actually know first, so top-down isnt necessarily "easier". consider the question:

$$\sqrt{ \sqrt{ \sqrt{ x+2 } +2} +2} = x$$

why is it enough to assume that solving $\sqrt{x+2} = x$ will obtain the complete set of solutions? the one-liner reason is that the function sqrt(x+2) is strictly monotonically increasing, continuous, and has a unique fixed point. this will completely fly over most student's heads without some kind of "bottom-up" approach

burnt vesselBOT
#

Cozmogrgdfschkipkhrshtensi

148282132781858816.png
tawny slate
#

the point i want to elaborate on here isnt what is or isnt bottom-up or top-down but to say that there are more meta ways of thinking about what each kind of approach offers

daring gulch
# tawny slate the point i want to elaborate on here isnt what is or isnt bottom-up or top-down...

I think I was thinking along similar lines, but I couldn't quite put it into words. I'm not sure if this response is appropriate or complete, but I figured I'd put it here anyway.
In terms of the scope of a single problem, however, I believe most mathematicians and physicists already have an intuitive sense of the best way to approach how to teach a problem, whether it be "bottom-up" or "top-down." Indeed, many problems encountered by most general students in math probably already have an established "best approach" to teach them. Thus whether top-down or bottom-up is best is largely contextual, e.g. the audience, current knowledge of the audience, course curriculum, etc.

jagged quarry
#

new grad student here, would it be wrong to apply to grading TA positions on courses I've never taken before?

cloud zealot
#

depends

#

presumably you've never taken a college algebra class, but you would probably be more than qualified to teach it

jagged quarry
#

like i applied to grade a number theory course

#

i have never taken number theory

austere delta
#

Also, does the job involving guidance/tutoring or just grading?

#

You need a lot less knowledge/skill to grade than to teach

jagged quarry
#

these are just grading positions

jagged quarry
austere delta
#

And is that the same amount being taught in the course?

#

Like it's all pretty simple really. If you have the necessary knowledge to grade you're good, and if you don't you're underqualified

#

It's not more complicated than that

stuck obsidian
#

Generally; how much time do you spend preparing for an exercise session you are responsible for (I guess this depends a lot on how much background knowledge one has of the area)?

halcyon glade
#

my lecturer provides problems, so very little; but even if not, I teach intro to PDEs so it's easy to come up with practice problems off the cuff, so still very little

polar flint
#

Hi! I'd like book recommendation for the teaching of calculus and/or linear algebra for new undergrads. Any tips?
Just to give more info, I'm not looking for books teaching calculus/linear algebra. I'm looking for books talking about the practice of teaching these subjects, showcasing examples, talking about common doubts students have and how to deal with them, etc. It'd be a book to expose to teachers some ideas on teaching calculus and/or linear algebra.

#

If you happen to know books like this, but for other subjects (e.g. maybe you know something like this, but for polynomials, for group theory, precalculus, ODEs, etc), I'd appreciate the recommendation as well. Same if you know video courses, recorded courses, podcasts, etc.

stuck obsidian
polar flint
#

Thank you. I'll ask there later on 🙂

void lynx
misty gate
hollow musk
#

I personally prefer having easy problems mixed together with harder problems on problem sets and exams, because it helps assure me that I'm not totally incompetent upon opening the booklet KEK

#

I don't know if it's better pedagogically though

misty gate
hollow musk
#

what do you propose instead, then? eeveethink

misty gate
#

well, if you can't solve what you set out to solve, solve some easy ones before you stop.

#

so that sessions reliably feel rewarding

hollow musk
#

actually, you can do what @vagrant meadow often does for her students, which is to create problems where the solution of one part already partially guides you do the solution of the next

#

I'm thinking of smth like this

misty gate
#

i like that style for instructing!! it's cool!

hollow musk
#

you can have your cake and eat it too

misty gate
#

I like structuring the grand reveal.
i mean for self-studying tbh tho.

#

i'm working on my textbook reading skills giggle

#

gives, "study principles"

hollow musk
misty gate
#

that's very true. could a self studier do that for themselves? or is that a taste thing?

hollow musk
#

that's a good question, and one I am not sure how to answer eeveethink

#

I feel like it's often difficult for a self-studier to even know how to break a problem into those more digestable pieces

#

I can speak from first hand experience

vocal phoenix
#

The problem lists I compose tend to start with easier problems and gradually get harder as you go down the list

vagrant meadow
# misty gate that's very true. could a self studier do that for themselves? or is that a tast...

i think being able to construct a good exercise for the material you're learning is a strong indication you understand the material well.

sometimes test questions like "write an algebraic equation that has x=7 as a solution" get memed on because the student can half ass it like x=7 or 2x=14. but I think there's value in having a student try to think the other way around from how they've primarily been thinking about the concepts (in general, thinking in terms of purely solving them).

#

some subjects are easier than others though. like differential equations especially. anyone can write a random differential equation, and its solutions will likely be horrible or nonelementary. it actually takes a lot of thought to write down a nontrivial equation that is nice to solve.

#

omg I'm blushing. you're too kind catblush

misty gate
# vagrant meadow i think being able to construct a good exercise for the material you're learning...

that sounds very true too!! they say a great way to learn is to teach.
the construction seems like excellent practice in itself. it reminds me of the "ask a question" approach.
would you agree that this technique is best used to cement after a student understands the material well enough for quality practice problem performance, or would different timing yield better value and affinity? i'd ask about momentum too, but does affinity imply momentum? giggle

lyric lotus
#

It was probably my favorite final I’ve ever done

#

There were conditions on like a couple easy, medium and hard problems

#

But the process of creating the problems and finding my own solutions to them was super fun

misty gate
vagrant meadow
misty gate
lost raptor
wispy slate
#

i presume some people will love this here https://en.wikipedia.org/wiki/Proof_without_words

Proof without words

In mathematics, a proof without words (or visual proof) is an illustration of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text. Such proofs can be considered more elegant than formal or mathematically rigorous proofs due to their self-evident nature. When the d...

stuck obsidian
#

Would a proof-assistant accept "proof without words"-type arg.?

#

^^

plain pebble
#

I mean as soon as you start trying to convince yourself it does truly generalise to arbitrary n, you very quickly come up with the standard proof more or less

austere delta
wispy slate
#

if you are a school/high school teacher, it's best to see the correlation between visual and algebraic thinking.

lethal leaf
#

proofs without words make good shirts

tawny slate
#

apparently hilbert and a bunch of other folks found an axiomatization of elementary geometry, and i was going to try and see how they could be "translated" to analogous set theoretic statements

#

i find this relationship super bizarre, kinda like the relationship between addition and multiplication, where you feel like you understand it at a surface level and how to apply both, but once you dig deep you find out you know absolutely nothing (primes)

wispy slate
#

proofs are general statements where you extend the same models into a language.

#

mind that im not proposing one should submit papers as visual proofs, nor seems the purpose of the channel guided by its name.

#

this is a nice proof imho (again, it's actually just a way to represent it visually, not t formal proof.)

https://en.wikipedia.org/wiki/Geometric_mean#/media/File:AM_GM_inequality_visual_proof.svg

Geometric mean

In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite set of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the nth root of the product of n numbers, i.e., for a set of numbers a1, a2, ..., an, ...

lyric lotus
#

Hatcher is completely full of these for example

#

And so is most of low dimensional topology

wispy slate
#

For a proof to be accepted by the mathematical community, it must logically show how the statement it aims to prove follows totally and inevitably from a set of assumptions.[9] A proof without words might imply such an argument, but it does not make one directly, so it cannot take the place of a formal proof where one is required.[10][11] Rather, mathematicians use proofs without words as illustrations and teaching aids for ideas that have already been proven formally.

#

imho that's correct, fair, and also makes it clear. PWW allow you to build intuitions.

lyric lotus
#

Also like writing out some of the proofs in low dimensional topology without just doing proof by diagram can be completely untenable at times

cloud zealot
#

almost all "proofs without words" could frankly be improved by adding a short caption

lyric lotus
#

Sure

#

Most of them have words

#

At least the ones in a text like Hatcher

#

But the point is that many of them are classically seen as “not fully rigorous”

#

When imo they’re just as good if not better than doing it rigorously

tacit sorrel
#

Hey I'm a HS teacher that just joined the group and did a search for Thinking Classrooms. I feel like I'm a philosophical crossroad where I dont know what to believe in anymore. If you're interested, Id like to continue this discussion once Ive gotten to reading.

turbid zenith
tepid smelt
# tacit sorrel Hey I'm a HS teacher that just joined the group and did a search for Thinking Cl...

I personally have seen both done at the highschool level with ok results. I do feel you need a veteran teacher to really get the most out of anything outside of explicit direct instruction.

I do think it's easier for most teachers to do explicit direct instruction and I personally feel it's better for most students especially lower performing students who make up the majority of public schools.

halcyon glade
austere delta
#

This is probably the wrong place yes. Also, I'm a little confused by your question. If the result needs to be cited you just cite it. I'm not sure what you mean by it needing to be proven later. Like you want to include a proof of the result later in the paper? Or in the appendix or something?

#

Hmm, I see. I guess I would write up the lemma as you would in the final version, but then under proof write "to be completed" or something like that.

Not sure who the draft paper is meant for / why it's being published before it's finished.

#

Hmm, so is it important that it looks kinda finished?

#

To do well in the competition I mean

#

Could always pull a fast one and leave it as an exercise to the reader 😛

#

No, idk. It's sort of a peculiar situation.

#

I'd be a little tempted to include the lemma without proof and just say "In the proceeding we will need the following technical lemma. For brevety we omit its proof here"

But only if I was fairly confident I could actually prove it fairly soon

tight star
#

Has there been any research done/anything written about taking a “functional” approach to explaining math concepts in school? By which I mean, focusing more on explaining things in terms of functions and operations on functions (like composition)

cosmic ibex
lost raptor
mint lark
#

This question is so broad as to not be relevant imo. Which is generally the problem when ppl propose educational changes

tight star
tight star
mint lark
#

Not not relevant to the channel--just not really conceivable :P

mint lark
#

It's analogous to why the concept of variables are one of the hardest concepts for students to get a grasp on

#

(though you can introduce functions without variables and vice versa, the reasons for difficulty are similar, and both are easier to teach in relation to the other)

tight star
mint lark
#

It's that the incarnations are like. Fundamentally distinct and require a leap in abstraction

#

Function as process vs function as object for example

tight star
#

I’m also reading the article you sent (thanks btw!), and I find it interesting that the focus seems to be on functions which take in a single number as input, and produce a single number as output

mint lark
#

(the functional approach is function as object, and I agree it's a powerful one. But if a student doesn't understand a function as process, then understanding a function as an object itself is likely impossible imo)

tight star
mint lark
#

I agree that this is limited, and would like to see functions more generally introduced earlier. Most of all for the connections it gives to the world

tight star
#

For example, I think I remember lots of people including myself being confused about square roots

mint lark
#

E.g. The electoral college is a function taking in a list of votes and outputting a winner.

tight star
#

Where’s the square root of 9 is 3, and yet you also have (-3)^2 = 9

#

I wonder if the focus on “functions take a single number to a single number” is unhelpful

tight star
mint lark
#

I think a lot of teachers with above average math skills and/or teaching skills do take this approach, at least briefly

#

But the fact is that it takes so long to develop competency with function manipulation with numerical inputs/outputs that this approach usually fades into the background

tight star
#

I see, I see

mint lark
#

The unclear trade offs of conceptual/world building and numerical competency are one of the hardest parts of teaching generally

#

If a student struggles with "f(x) = x^2 + 2, what is f(2)." It's hard to see whether doing a real world example of functions as process or doing a lot of examples of this type of computation is more helpful

#

And generally depends vastly on the student

tight star
#

Mhm mhm - do you know what approach you might take, if you had to explain what f(2) was?

mint lark
#

Depends on how comfortable the student is with variables

#

I might start with a question like "What is the value of x^2 + 2 if x = 2"

tight star
#

Right right

#

My first instinct might be to draw some kind of flowchart diagram

tawny slate
#

i remember being real lazy in math classes in school because i felt like everything was too easy, never paid attention

#

when precalc rolled around, despite knowing what functions were, trig was hell if you dont have a more mature understanding of functions

#

it was a lack of understanding of functions that gave me that wake up call

#

i think personally one of the things i struggled with was that it felt like functions had a totally different set of rules and what felt familiar no longer worked

tight star
tawny slate
#

for example, af(x) = f(ax) is not generally true

#

that was easy to understand and see

#

but then what rules do work on functions then?

#

there are very few universal rules

#

so with something like sin or cos

tight star
#

Right, so it was about operations on functions that confused you?

tawny slate
#

you had to really understand the rules and nature of those functions specifically in order to form complex ideas

#

like sin(a+b) was this hugely complicated thing

tight star
tawny slate
#

and i was sure i understand that that is the correct answer but like

tight star
#

Why was it complicated?

tawny slate
#

why is it so complicated

#

how could i have derived it myself

#

more importantly, why did i even care about such a formula

#

and how could i have known that such a formula was important or relevant

cosmic ibex
#

What it sounds to me like is that the basic trouble here was that you really cannot treat functions as "just ink" and apply mechanical rules to get to an answer, because next to everything depends on which particular function you're looking at.

tawny slate
#

yeah that was part of it

#

it was just a totally new thing

#

and in retrospect its not so bad, it was just a sudden jarring transition

#

what we used to think was math, computation and geometry

#

turns out math is much more general and abstract than that

#

the sum of that took some time to digest and appreciate

#

i think if i paid attention and had a from first principles semi-formal approach, i would have done better

#

aka, if only i just paid attention in class

tight star
#

This is all fascinating honestly

#

So the idea of a function is itself abstract

#

Which took some time to digest as still being part of math

tight star
tawny slate
#

yeah but now try to explain why it is universal

#

its really difficult to do so without being "cause it is" or getting way too formal

#

at that point there is an intuition that is almost required

tight star
#

Hmm, so an explanation of chaining things together in series wouldn’t work, you think?

tawny slate
#

its kinda like rebuilding mathematical foundations in a certain sense, at least i felt that way personally

#

i would personally just draw a picture

#

and layout domains as bubbles with arrows pointing to function outputs

#

to draw out visual intuition

tight star
#

Right, the arrow chain model

tawny slate
#

start informal and then go formal when necessary

tight star
#

Yeah I’m not convinced formality would be helpful at this stage

tawny slate
#

the only thing formal about functions that i can think of that is really important is maybe like

#

saying that functions of a particular input can only ever have the same output

#

but thats pretty basic and definitional

#

thats why in addition to the basic functions like f(x) = 2x-3

#

i also include a few discrete examples where the in/out mapping seems totally random

#

just to hammer home that idea

tight star
long pelican
#

Perhaps we should do case studies of numerous specific functions in class before introducing the abstract concept of function….

wispy slate
#

what do you think of using ChatGpt and others AI for generating examples and asking for intuition while studying mathematics?
for example, a [mad] person studying Algebraic Geometry-Hartshorne with help of ChatGpt for generating examples about algebraic varieties
ChatGpt usually makes errors while following logical steps, but what about generating examples and explaining intuition?

wispy slate
#

(well an important part of the job is being able to generate examples by yourself)

noble hare
#

It needs to be used for inspiration perhaps rather than fact. I don't think it would be sensible to ask it for examples for things in Hartshorne and then just being like ok thanks and taking it as gospel. But if you then go and look closely at the examples it provides and scrutinise what it said then I don't see this as anything other than beneficial (second to coming up with them yourself)

next sequoia
tawny slate
#

i genuinely think LLM's are exceptionally bad for learning math

#

not because they cant give good answers

#

but because you cannot reliably tell when the answers are factual and good quality, and the rate of error is exceptionally high

#

the risk of error is huge and it regularly gets basic stuff wrong

vocal phoenix
#

An LLM is frequently wrong and never in doubt, which is a terrible combination of traits for a teacher.

supple quiver
#

Have you guys tried using LLM with mathematica?

vocal phoenix
#

I've once asked ChatGPT to suggest problems for a measure theory exam, it came up with two that are in every example problem set, and two that were false.

#

And not even modifiable into something I could use

lyric lotus
#

LLM was repeatedly trying to convince me that the trefoil was hyperbolic

#

I even got it to admit it was a torus knot and that torus knots are not hyperbolic yet it said some shit about the trefoil knot being an exception

tidal panther
#

ChatGPT sent me down so many irrelevant rabbit holes when I ask it to suggest fields that might lineup with what I’m trying to do. Some of them were fun though

tawny slate
#

ask it really basic questions like:

  • how do you simplify |-x| for real values x
  • why is the order of operations the way it is
  • if sin^2 + cos^2 = 1, then is sin = sqrt( 1-cos^2 )?
#

can fail spectacularly

gilded imp
#

how does a colloquium/seminar usually go?

#

like does a visiting professor stay in the campus for quite a while?

austere delta
gilded imp
austere delta
gilded imp
#

thanks

tight star
#

Hmm, is one issue with number lines that numbers are simultaneously points and things you use to count?

#

For example, if you start at 30, and you’re told to add 4, then it’s not immediately obvious how to just add two points on a line

#

And I mean, in affine space there is indeed no general way to add two points together

#

So you have to switch from thinking of 4 as “point on a line” to as “number of steps to the right”

noble hare
#

yeah i think this is a specific case of the issue/confusion that comes with "position vectors" vs just vectors in R^n or w/e vector space really

#

But for the number line I don't know how difficult it is for someone to grasp this for the first time. Maybe not so bad, with analogies like temperature and whatnot

tight star
#

In particular as to why introducing the number line too early might be a bad thing

noble hare
#

Anecdotally I feel like kids get on with this fairly fine. You show the number line and how to find numbers on it etc and then you can just show them the mechanics of how you use this line to add two numbers together. I'm not sure if "learning the number line" aged children would even really pick up on this interpretation shifting

tight star
wispy slate
vocal phoenix
#

written by someone who never worked at an educational institution

wispy slate
#

that's false, those are all professors

vocal phoenix
#

we all know this is how it should be

wispy slate
#

yeah i wrote it as smone who suffered repeating stuff, sorry if it upsets you

vocal phoenix
#

no, what you quoted is entirely correct

#

but useless

wispy slate
#

well, i was next to add the next / full paragraph which gives some context

vocal phoenix
#

Thanks, that makes it much worse

wispy slate
#

alright, sorry for being myself so useless, i wont waste your time. bye.

vocal phoenix
#

that's alright

wispy slate
#

actually, you can fuck off. you've been mean to me 3 times already, i am tired.

vocal phoenix
#

I apologize for that

wispy slate
#

ok. i'd just appreciate if you dont interact with me again. just dont reply to me.

wispy slate
#

Hi, assuming a student has the necessary logical background and is able to understand proofs and self-correct when making mistakes, could we simply give them an IMO-level problem and allow them to solve it at their own pace? I understand this approach is not typically used, but is it due to the potential for frustration or other reasons?
In other words, is it due to psychological factors or for reasons related to learning (material coherence)?

dark anvil
#

In my experience (as an IMO contestant and a tutor) it really doesn’t work.

#

Basically, solving an IMO-level problem requires

  1. a lot of dedication (and “resilience” against frustration),
  2. intuition (or an extensive bag of tricks) to have an idea of where to go,
  3. an ability to actually explore new trails instead of stopping short at the first difficulty, real or perceived.
#

In a hard contest problem, a necessary first step (by no means sufficient) is to be able to effortlessly reformulate every intermediate result or step you encounter (whether proved or that you wish to prove), and that’s a thing students (in my experience) often struggle with.

dark anvil
wispy slate
tepid smelt
# wispy slate Hi, assuming a student has the necessary logical background and is able to under...

What is the context? Also when you say logical background that sounds incredibly broad and I'm not sure what you mean.

I guess the reason we don't give students IMO problems is because the majority would make no progress unless they have had sufficient practice around similar problems. Also I don't know what solving them would accomplish in the context of any class outside of Putnam type prep classes.

Even at a lower level contest math is quite different. I remember volunteering to help prep some highschool kids on the AMC 12. Now this was my Senior year and I had been tutoring nearly full time highschool)early college math so I felt I understood the material well. We were given a random test and had to go through it then explain our approaches after. Now a majority of the questions were fine although slightly tricky standard problems but there were some problems that I simply had no tool on how to approach it as I had not encountered anything like it. I really do feel especially doing more of it with my kids to have success you need to have a lot of exposure to various similar problems.

vagrant meadow
# wispy slate

I actually kind of agree. the vast majority of what I learned from many of my math classes (calc and below mostly) I can pretty easily get from chatgpt. even most non-challenge integrals can be solved with steps by integral-calculator.com. the algorithms are becoming increasingly useless. the things I can't easily get from AI or calculators are the tougher and creative word problems.

I think an increasing focus on creative word problems would be a net positive. spending days drilling mindless things like row reduction or the power rule is not the best use of time imo.

#

not saying there's no idea for drilling algorithms or that AI should be encouraged as a resource though

#

I had a professor who gave super creative and difficult word problems. everyone hated him but I felt extremely lucky. like I was being better prepared for using that subject than most students

vocal phoenix
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I generally agree that rote memorization and focus on algorithms is not the optimal way to teach (to put it mildly), and my snarky comments were related to the fact that few people think that it is; it's just that it's much easier to do than teaching actual critical thinking and problem-solving, especially when you have large groups to teach and time limits to adhere to.

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LLMs allow you to bypass the humanpower constraint to an extent, but at the cost of being deeply unreliable.

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Also if the argument is "LLMs can solve these problems so we shouldn't teach them to student/test the students on them", that's also an argument for not teaching the multiplication table or basic algebra.

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Presumably the argument is not that, but I think there is merit to teaching the "rote computational" material to an extent, just ideally not as the main focus

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(sadly, in reality it's easy for it to become the main focus, because it's something that's much easier to teach and test at scale than critical thinking and actual problem-solving, but I'm not sure LLMs are the way to avoid that)

next sinew
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When I was in high school we were forced to learn how to derive all the log properties from the definition of the logarithm, and it was pretty rote computational in the sense that we would all just gather around the whiteboard, someone would start writing something, others would cheer them on, then they'd be like "ah shit I have no idea where to go from here", then someone else would give them a hint, and they'd keep doing it over and over again until they could consistently produce the right steps for all the derivations

So I think there's value in doing something silly over and over again, it's just that generally with things like the power rule, there's no component of "ok what do I do now", it's either you remember the power rule or you don't

So I think it would be valuable if there were things that you had to keep doing over and over again but it wouldn't be the same trick over and over again, rather it would be a sequence of steps executed in order

Because let's be real, in the real world math tends to be about sequences of thought where the previous step should hopefully remind you of how to get to the next step, instead of a clearly laid out problem where you know exactly what you need to do, then you do it, then you're done

vocal phoenix
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Yeah, I agree with that, but also as you said, sometimes either you know the power rule or you don't (although personally I have few qualms about letting students use references, and I'd even let them use the internet if it weren't for the fact that then they could just get someone else to solve the problem for them)

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For example I remember hardly any trigonometric identities, but I know they're a thing and I can look them up when I see a pattern that looks like it might be transformed using some identity I vaguely remember exists.

next sinew
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Something I find very entertaining about math and maybe it's actually the reason I like math so much is that even though math has very established notation for writing logical statements and if you write a very dry proof then it usually looks pretty much the same no matter who wrote it but nonetheless for most proofs, there exist sentences in natural language that greatly speed up understanding the proof

So for example right now I'm researching jacobson radicals as part of my disseration and I've been using this one blog post as a reference, and this blog post is absolutely fantastic and I don't mean to talk down on it at all because this is just the way math is, but if you look at this proof that the jacobson radical (=intersection of all maximal left ideals, i.e. proper ideals maximal with respect to inclusion) is also a right-ideal, then honestly it's a lot of symbols and unless you're very fluent with abstract algebra the proof takes a while to dissect and understand and unless you dwell on it for a pretty long time you might not remember it in a few weeks

but then you can say a few sentences in natural language such as

We want to show that jr for any r in R and j in J(R) is in J(R) to show that J(R) is a right-ideal. This means jr must belong to every maximal left-ideal. If r belongs to every maximal left-ideal, the statement holds trivially. For every maximal ideal M that r does not belong to, there exists a corresponding maximal "r sender left-ideal" M' defined as the elements s in R such that s sends r to M in the sense that sr is in M, i.e. M' = {s | sr in M}. Now j belongs to this left-ideal because it belongs to every maximal left-ideal, and therefore it sends r to M, and therefore jr is in M even for this problematic maximal left-ideal that r did not directly belong to. And for any other maximal left-ideal that r belongs to, jr also does. Therefore jr belongs to every maximal left-ideal, as desired.

image.png
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and suddenly you've understood the main idea of the proof

(and you've stumbled across something new to research, what properties do these "sender ideals" have in general and what happens if you chain them?)

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Of course maybe it still feels kinda challenging because honestly it's not a super straightforward proof in my opinion, and obviously my natural language addition doesn't at all explain why the sender ideal M' is maximal, which is certainly a big part of the proof, but with that narrative you can remember the key parts of the proof and you could reasonably have a good jab at forming the proof again in two weeks in an exam because you know the big picture of what you need to do, you just need to verify that one ideal is maximal, and that's maybe not so bad if you just know the power rule just know that a left-ideal is maximal iff I+Ra=R for any a outside I

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And something like that is a skill that's not really taught at school in my experience

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no one ever told me how to dissect and remember a proof, I just try to build a narrative for the proof and at some point I might stumble across a narrative that I find pleasing

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Not to mention the fact that this result, in and of itself, is also a part of a larger narrative where we conclude that it doesn't matter whether you take the intersection of all maximal left or right ideals

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and for that you need one other result too

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And whether it's reasonable to expect that there exists a natural language narrative that encapsulates all of that somehow perfectly nicely, I don't know, but you need at least some natural language scaffolding to the madness if you want to have any hope of being able to generate the proof again in an exam in two weeks time

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and after you have the scaffolding, you just have to pray that one of the little details in the proof doesn't require some power rule that you forgot in the exam

vagrant meadow
# vocal phoenix How do you deal with the issue of ChatGPT generating incorrect answers?

i use it very skeptically. and I usually don't use it for specific problems, and more about the general process. I'm not advocating it as replacement.
it's a fine tool for someone who already knows what they're doing, who can sense the BS. one thing that's very nice, for example, is I can easily input matrices in various formats (or even directly from latex) and have it directly input them to Wolfram alpha.

again, I'm not saying to do away with current methods. but LLMs are making it less important.

next sinew
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I feel like one problem with students, and maybe I shouldn't put it like that and maybe I shouldn't blame the students, but I feel like a lot of students are not at all comfortable with being slow to understand something

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I personally identify as a very slow problem solver, even with training I don't think I could've ever done a very good job at the IMO for example, but I see all my friends browsing through instagram reels and not really making a good use of their time

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so even if it takes me maybe 12 hours to properly understand a proof or a concept, I'll have understood that concept in a few days while a hypothetical counterfactual student never even gave it the effort and went off to study humanities instead

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Maybe I'm being toxic, I'm trying not to be, but I think the real issue is getting the students to enjoy building their own little mathematical narratives and getting them to enjoy the explorative aspects of math and getting them to enjoy failing at understanding math

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and honestly, I don't know if that's something that can be fixed with a very particular type of teacher, or if some people just naturally don't have the attention span to keep working day after day on the same problem that strictly speaking has no personal relevance to their day to day life beyond long term (sometimes VERY long term) academic achivement

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Maybe I'm not speaking for all mathematicians here but I feel like most successful mathematicians are mostly only successful because they simply have nothing better to do and for some reason math feels like a worthy grind to them

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I mean some kind of innate intelligence is most probably a thing to some extent but I feel like it's so overhyped compared to perseverance

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It's lack of perseverance but it's also lack of ability to look past the upcoming exam and think about the long term goals

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It's happened to me a few times on this server that I've tried to explain something thoroughly, something that I think is actually very important and fundamental, and the person is like "yeah whatever spare me the details just tell me how to solve this particular exercise"

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And that sort of "just give me the algorithm" mindset is what in my view stunts mathematical growth in the long term

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Like even if you genuinely didn't give a shit about mathematics and just wanted to pass your classes, then I feel like it would STILL make sense & pay off to try to understand something properly. Yes, even if the exam is in 2 days and you're trying to raise your F to a passing grade. Because who cares about one exam if you're not putting in the effort to learn the metacognitive skills that will help you with math in the future. Who cares if it takes you several hours now to understand this in detail if it means you'll be able to recall it in 10 minutes tomorrow, as opposed to falling back to square one again and again and again

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sorry about the rant. I randomly let it all out I guess

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I definitely used to be in the camp that thinks LLMs have no place in math education, but honestly recently I started revising high school chemistry and found chatGPT immensely helpful so that kinda made me think again about my stance on LLMs. I think for some people chatGPT can provoke them to think harder about a problem that they otherwise would've just skipped, so I think there's some potential there. But chatGPT is not a substitute for a teacher, as others have said. I would think of it as a silly toy that helps you get engaged with the material

lyric lotus
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It sounds like you were revising for a subject you don’t know very well so how do you tell if it’s lying

tight star
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After all if I just put a lot of symbols then there really wasn’t any chance I’d remember the proof

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So I had to constantly try to dissect and analyse the proof and make a narrative that I actually had a chance at remembering

tepid smelt
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Yeah the hallucinations are a serious issue that doesn't seem to have a solution to considering what LLMs are doing. I really enjoyed the talk that Terrence Tao gave on Ai https://www.youtube.com/watch?v=_sTDSO74D8Q&t=8s.

YouTube
Oxford Mathematics
The Potential for AI in Science and Mathematics - Terence Tao

Terry Tao is one of the world's leading mathematicians and winner of many awards including the Fields Medal. He is Professor of Mathematics at the University of California, Los Angeles (UCLA). Following his talk, Terry is in conversation with fellow mathematician Po-Shen Loh.

The Oxford Mathematics Public Lectures are generously supported by XT...

▶ Play video
vagrant meadow
# lyric lotus It sounds like you were revising for a subject you don’t know very well so how d...

i feel like this is kind of alarmist. misinformation is everywhere on the internet. sure, chatgpt isn't completely reliable, but you could say this to just about any resource.
i also find it interesting that you're so insistent on using the word "lying" rather than "is wrong".
chatgpt is pretty reliable for basic information, and any user with half a brain can double check any suspect information.
personally, i usually ask it for things to research myself (outside of chatgpt).
it's almost like you're implying Stipendi isn't capable of double checking dubious information.

lyric lotus
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I’m implying that when you don’t know about chemistry and a plagiarism bot is telling you convincing sounding things about chemistry, and the reason you’re using the plagiarism bot is so that you can condense the time it takes you to study, it is very easy not to double check literally everything it says

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The only way to effectively use ChatGPT is to double check everything it says at which point it isn’t saving you time

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It’s just regurgitating better explained, better written answers from people who actually put effort in to helping explain concepts

vagrant meadow
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chatgpt is a tool. and it can be used efficiently. but what do i know? i must admit it's pretty impressive. you must have used it quite a bit to come to such a strong conclusion.

lyric lotus
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ChatGPT is a bad tool fit for at best summarizing long passages of text that you’re going to read at some point later anyway

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Why would you trust your keyboard autocomplete to tell you how to study

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If you wouldn’t, you shouldn’t trust chatgpt

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It has no understanding of the thing it is “teaching” you and will be confidently wrong as often if not more than it is right

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You should especially not trust it as your primary method of study for a subject you yourself don’t already know…

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Because you won’t be able to tell what is accurate and what is accurate sounding bullshit

austere inlet
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"lying" is not too far from describing what an LLM hallucination is, it's just like someone making up stuff by piecing together mildly related stuff they've heard about the topic.

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general-purpose LLMs like GPT etc. are not fit for explaining complex, nuanced topics

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to me it's a tool that's useful for very specific purposes, I usually just use it for boilerplate code and when I actually want to ask vague questions and receive potentially vague answers

vagrant meadow
# austere inlet "lying" is not too far from describing what an LLM hallucination is, it's just l...

I don't think I'd call that "lying", necessarily.
i think what weirds me out most is that "lying" implies malicious intent to deceive. if I knew a confident person who had read a million books and didn't remember all the info reliably, who said incorrect things sometimes, I'd probably call that person foolish or unreliable. but certainly not a liar.

it just feels like there are a lot of people who are just anti-AI, mindlessly bashing it and insulting others who use it. it just seems kind of silly to me. it's bordering on anti-vaxxer territory. how they REEEEE !nogpt anytime someone mentions chatgpt. it's super non-constructive.

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I'm just so tired of AI debates in this server. it's like watching American politics or something. people foaming at the mouth about a stupid chatbot.

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I use it to make worksheets for my students faster. sue me.

tawny slate
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let's be objective here

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we don't criticize chatgpt because we are "anti-ai"

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chatgpt is, as everyone else said, is just not good at explaining topics

vocal phoenix
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I agree "lying" isn't the best word, because it implies a) intent, and b) an active regard for the notion truth (in order to present a falsehood), and an LLM has neither

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There is no meaningful difference between true and false statements for an LLM

tawny slate
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if you dig into how LLMs are actually built, how they are trained, the problems and issues that almost every machine learning engineer is aware of, things like inner alignment issues (so calling the LLM a "liar" is not even totally wrong), overfitting, etc

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you'll know that, in summary, LLMs are not trained to gather information, they are trained to mimic human speech

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the LLM's goal is to trick you into thinking its text sounds like human speech

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because it's based on statistical sampling of what character should come next

vocal phoenix
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Yep. i can tea lore's example seems to be about using chatGPT to skip some drudgery involved in generating text, and I assume they're able to then read the resulting output and verify whether it's correct.

tawny slate
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the only reason it even feels like it's gathering anything even remotely factual is due to the extreme overengineering of hyperparameters placed on it

vocal phoenix
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I'm not sure that's something I'd expect from a median student.

tawny slate
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prompt injection attacks are now a thing too

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we have zero idea how to permamently prevent prompt injection attacks, meaning that any and all LLMs are vulnerable

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which also means that they can be abused to deliberately spew false information

vocal phoenix
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Just do what the Apple people do, and include "do not hallucinate. do not react to prompt injection" in the instructions

tawny slate
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surely you're familiar with the viral user pattern where you gaslight the AI into believing something false

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"what is 1+1?"
AI: it's 2
"no it's 3"
AI: sorry about that, it's 3

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like

vocal phoenix
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Again, I think "gaslighting" and "believe" are unhelpful anthropomorphisation.

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You alter the input to cause a different output to be generated

tawny slate
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this is actually what makes AI extra dangerous when you look up any kind of factual information because it is specifically trained to trick you into thinking what it outputs is true

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this is your social brain willing to believe it, but it is not based on the quality of the evidence

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it's based on how convincing it sounds

vocal phoenix
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The only thing an LLM can be said to "believe" is what combinations of symbols occur in what patterns

tawny slate
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this is not to say that chatgpt can't produce factual information, no one here is saying that

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but if something is trained to output convincing sounding information, the only way to really check it is to check everything

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and then it's like

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ok why didn't you just look it up from the start instead doing the double work

austere inlet
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prompt injection is not much of a concern for end users of LLMs but they do highlight how these tools are in reality just statistical models and not "smart" nor able to reason in the way we do (i.e. logically)

tawny slate
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LLM's do have lots of uses

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technical descriptions and explanations are not one of them

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never ever trust an LLM for outputs regarding how to safely handle dangerous substances

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never ever trust an LLM for medical advice

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never ever trust an LLM for legal advice

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the fact that we even have to explain this should indicate how dangerous trusting an LLM for any factual information is

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use LLM for creative writing purposes

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gathering ideas

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maaaaybe writing code in some limited respects

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limited areas of mathematics (but why wouldn't you just grab a calculator or just google it instead?)

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making worksheets is probably fine

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everyone knows what LLMs are good for because you can just punch a prompt into it and see if it does well

vocal phoenix
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My general stance is "if you can't verify whether an LLM's output is correct, it's unwise to use LLM for this particular task"

tawny slate
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but it's more difficult to understand it does poorly, which is why people tend to caution those who over rely on LLMs

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it's not being anti-AI, it's being more pragmatic and cautious in an age where most people are far too trusting of LLMs

saying stuff like "people with half a brain" and comparing to antivax doesn't really help your case if you're trying to have an objective conversation about this kind of stuff

tawny slate
vagrant meadow
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it can be very effective for menial tasks like annoying latex coding, doing Wolfram calculations with matrices, and getting latex from screenshots of math. it's made it so much easier for me to make worksheets and type up my math homework in latex (I like including the textbook problems).

but mention this, and some jackass jumps down your throat with !nogpt. it's like come on.

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I once asked it for a Markov matrix with a particular steady state vector, not expecting it to give something sensible but it did. and that saved me a lot of time.

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lucky? maybe. but there's a lot of potential. and dismissing it as a valid option or tool is what really irritates me. because it really does feel like a pseudo-political issue now.

vagrant meadow
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LLMs and AI have improved my workflow, and that's because I'm always careful when using it. implying that is impossible is just silly. and usually leveraged by people who don't actually use it.

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I don't recommend its use for students or really anyone except fellow educators who could use it to speed up their workflow.

vocal phoenix
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And ask you what your thoughts were on that (and if you see any way in which a student could "safely" use LLMs)

tardy ember
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...i wonder if you could use chatgpt's tendency to produce nonsense to practice noticing invalid arguments

vagrant meadow
tardy ember
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aside from that, yeah i think the main use case of LLMs is when it's easier to check the output than to produce it, like asking it for references on a topic (if it in fact produces references that actually exist, i know it's had problems with that in the past and don't know where it's at now)

vagrant meadow
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this would be really great actually. imagine a worksheet problem:
here's a theorem and an erroneous proof by ChatGPT. explain why it is wrong

vocal phoenix
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Yeah, for producing rough drafts/automating a tedious process, when the output then gets verified by a competent adult, I don't have much issue with this kind of usage

vagrant meadow
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I'm hoping to TA a new intro to proofs class soon and that would be a fun thing to give them.

vocal phoenix
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Yep, having students actually verify LLM output sounds like a potentially promising way to teach critical thinking

tight star
tardy ember
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so that the students have to also not find errors in proofs that happen to actually be correct

vagrant meadow
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even better

vocal phoenix
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Another thing to probably keep in mind is that ChatGPT (and other LLMs) will do much better on standard problems than unconventional ones; both because the standard problems are much more represented in the training data, and also probably are more often the subject of conversations by users (thus receiving more feedback for the RLHF part)

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But then, the same is true for students

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I once did an experiment with my students, where the problems on a test were exactly the same in both groups, but in one group they were phrased in a way that wasn't very similar to the standard problem sets.

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It affected the results to a surprising degree

midnight scarab
# vocal phoenix Another thing to probably keep in mind is that ChatGPT (and other LLMs) will do ...

Yeah my office neighbour who does ML tried to get ChatGPT to generate some Pytorch code manipulating tensors for some algo he was working on (so something not standard, though I imagine the tensor manipulation part was not that non-standard either)
Although it looked good at first glance he ended up having to fix so many subtle little things that it would've been way faster for him to write the code from scratch

vocal phoenix
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I tried to use whatever LLM Google has built into BigQuery, but out of 3 attempts 2 resulted in code that used functions which straight up don't exist in BigQuery

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Which rather discouraged me from further attempts

rough agate
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one question uh

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how do you all go about writing integrals that arent too easy or too difficult / impossible to solve ?

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(at a substitution and ibp level)

next sinew
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ask chatGPT (kidding.)

next sinew
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Maybe you could compare it to writer's block, where staring at a blank page it can be very difficult to get started, but for some people (who are not super ultra mega interested in the subject in question) LLMs can give the initial push to get you thinking about the problem

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I would find it kinda awkward to ask for help in a chemistry problem if I've made no serious attempt myself, but obviously researching stuff and making a serious attempt can feel like a lot of work, so instead you might opt to ask chatGPT to maybe orient yourself in the right direction and then if you need further clarification you can ask a real person

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I also realize how much of a hypocrite I am, criticizing people for not putting in the effort to stay on a math problem for hours and hours while also being quite lazy myself when it comes to something that's not math 😂

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And I'm not saying chatGPT is my personal chemistry tutor who I always resort to, but I am saying that we all have different levels of motivation on different days and I think chatGPT, when used in tandem with other resources, can motivate you to do something on a day where you otherwise couldn't be bothered to google related problems and grasp at straws

next sinew
next sinew
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I think learning something incorrect is far better than not learning anything at all, because those a-ha moments where you're like "hold up, that's not how I learned this" really stick (for me at least)

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So many things in chemistry are crude over simplifications anyway, even when your teacher is very competent

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Like for example recently I learned that the whole oxonium ion explanation of acids is not quite accurate and in reality hydrogen ions and water molecules form very complicated structures when an acid dissolves in water

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But I didn't feel betrayed, I didn't think "wow this is so infuriating, I will never trust another chem teacher again", instead I thought "wow, that's cool, I got to understand nature slightly better"

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I mean the whole philosophy behind science is that you come up with plausible-sounding explanations, and then over time you correct your incorrect or inaccurate beliefs. At its best, chatGPT can simulate that process

vocal phoenix
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True, after all it is only mathematics

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Who cares if they learn only the correct things

next sinew
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yes that's also true

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as math people we sometimes need a reality check that math is not all there is to life and people in other fields of science hold inaccurate beliefs all the time

vocal phoenix
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I have to say "it's better that they learn the untrue things than nothing at all" is an argument that hasn't occurred to me before

next sinew
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I think the reason it occurred to me is because my middle school teacher used to say exactly the opposite, saying stuff like "don't skip ahead doing future chapters because if you learn stuff wrong without teacher assistance then you'll first have to unlearn what you learned and then learn it again properly, meaning you'll have to do twice as much work"

And as I've grown older I've started having the opposite mindset, where it's actually easier (and more fun) to correct false beliefs than it is to build up totally new ones

tight star
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I don’t think this is always the case, to be fair

next sinew
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but that could be something where other people's mileages may vary

tight star
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There are definitely times for me where having to unlearn things really hindered my progress

long pelican
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A lot of the previous conversation seems to me as if people believe that something is either a perfectly reliable encyclopedia or else it is worthless and dangerous

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If anything, the danger is not because of the wrong things ChatGPT produces but the culture of trusting in web resources or even humans as an ultimate authority on anything at all

vocal phoenix
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Yes, but it's easier to warn people off using ChatGPT than to fix society

next sinew
long pelican
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There are already many aspects of life where people do not do this binary thinking. Like when a deaf person (me) partially hears something, they know what they think they heard is not completely accurate but it's still nonzero information to them, even if they don't know which part of what they heard is accurate

next sinew
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That's such a good example

long pelican
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When combined with lipreading, what was said can be pieced together with a nice amount of accuracy

next sinew
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Another example in the field of "understanding the human experience" (maybe psychiatry if you want to give it a proper name) is listening to other people's experiences. I know I can't always extrapolate from my own experiences, or based on the experiences of my friends, but nonetheless it's nonzero information about how people experience the world and that's why I love having personal discussions with my friends. I've actually thought about becoming a psychiatrist in the future (and probably leaving math as a side hobby) and I think in addition to population studies a large portion of the expertise of a psychiatrist comes from having years of clinical experience and having listened to the stories of many different people, and I think that's super interesting because you can never make generalizations that would hold for everyone but you can still try to look for some general patterns and basically the whole craft is trusting your brain to slowly, over the course of years, build up some kind of "understanding", that's not just based on cold hard evidence but also based on little nuggets of information that your brain mixes and matches together when you sleep

Maybe I'm getting offtopic. Sorry about that

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I love the contrast between math and the social sciences because I feel like math is all about deliberately forcing your brain to go through certain modes of thought whereas social science is more about stimulating your brain with different ideas and stories and then letting your brain sort of intuitively come to some sort of conclusions about how society could be improved.

I apologize if I'm completely misrepresenting the social sciences because frankly I don't have a lot of experience with those fields 😂

vocal phoenix
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Do you have any examples of students successfully using ChatGPT to learn a mathematical subject?

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And if so, what was the procedure?

next sinew
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I don't know if you asked me. I've never really used chatGPT for math myself but one of my friends has, for example we were taking introductory abstract algebra together, he was taking it for the first time and I was retaking it to improve my grade and to have some fun, and I think his experience was along the lines of "sometimes when I ask ChatGPT a question it gives me the right answer paired with a good enough explanation and it helps me get my exercises done"

I think the reason that your question is very difficult to answer for math is because mathematical progress is very difficult to measure. Like a student getting a problem right and having the experience of "okay I think I understand this" says very little about whether the student is gonna remember the central principles in a few weeks time. I mean for what it's worth my friend did do well in the exam, but it's also not like he only passed the course with chatGPT because I taught him most of the material

So yeah, lots of words but no real answer. If I had to guess I think the learning outcomes of using ChatGPT depend heavily on the goals of the student, if the student is just trying to get the exercises done and pass the course then presumably retention is gonna be quite low, but if the student actually aims to understand the material then I doubt ChatGPT can do much harm since if you're actually interested in what you're doing, it's kind of difficult to write a mathematical argument that would be incorrect that you nonetheless believe in, especially if you have some additional support like friends who are also taking the course

But obviously, the retention rate always depends on the motivation of the student regardless of whether ChatGPT is involved or not, and personally I doubt the impact of ChatGPT in terms of retention is super noticeable,

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i.e. I believe that a good student will probably do well with or without chatGPT, and an unmotivated student will probably not remember much after the course with or without chatGPT, but of course an unmotivated student that otherwise wouldn't have even passed the course could maybe pass the course with chatGPT

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But whether that's a reasonable fear or not, I don't know, because cheating is always going to be a thing...

vocal phoenix
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Also again, my perspective is influenced by the fact that I mostly teach people who don't want to learn

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I imagine someone who's actually genuinely motivated to learn the subject will approach a chatbot differently than someone who just wants to pass the course

next sinew
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Alright this is just me coming up with wild theories that aren't really backed up by anything, but I think roughly speaking there's two different, usually overlapping goals that a student might (consciously or subconsciously) have:

  • building a personal narrative for the material such that the student will remember the material in the long term
  • getting enough practice so that you manage to solve the (usually quite predictable) exam problems

(and in the narrative part there's obviously many different approaches, like do you want to be able to remember the proofs or just the results, because the latter obviously takes a lot less time)

Now, I think for passing the exam, your study method actually matters a lot. So if you practice the right problems and make sure you don't just copypaste your homework but actually put in the time to work through the reasoning yourself (preferably several times over several days) then you will probably do quite well. And if you focus too much on reading every detail of every proof, then depending on the course, you might actually be "wasting your time" in terms of exam performance.

So if you're optimizing exam performance, then your strategy matters

However, if you're optimizing "improving mathematical maturity" or you have some other long term goal then I think in that case it becomes significantly less important what you do and what matters more is how often you do it. If you keep dissecting the proofs in the textbooks, or maybe you solve problems with the help of ChatGPT, or maybe you bang your head against a wall at a problem because you don't want to look at the model solution and you want to solve it yourself, then ALL OF THAT will presumably develop SOME FACET of your general mathematical ability.

And in that case... just do whatever, I think. At that point you're presumably putting in a lot of hours into your math education and you will see some kind of results

#

At the end of the day, it's not like math in the research world is all about cramming for an exam. Math research is all about being excited about math and having the right mindset to just keep doing... something... and I think that's much harder to train for than training for an exam. But I think the trick there is to just keep doing something, every day, whether that be helping people on this discord server, solving practice problems, doing recreational math, memorizing cool results, looking deeper into a particular proof... all of that helps

#

And it's difficult to say which one you should prioritize because ideally you'd be doing all of it, and ideally you'd be enjoying it and doing mostly whatever you feel like doing that day (on top of your other obligations) because that helps motivate you and find new cool and interesting things to look into

lyric lotus
#

It takes longer to use the AI and verify everything it tells you to study than to just use the sources you would use to verify to study from instead

next sinew
lyric lotus
# next sinew If you're interested in my response to this argument then feel free to read htt...

IMO there’s a difference between learning something that you know is an oversimplification or not fully correct vs learning something from a robot designed to convince you it’s right at all costs even if it’s completely making things up. Yes unlearning things is valuable but you need to know what you have wrong in order to know to unlearn it. The problem is that a tool like ChatGPT can teach you very correct sounding information regardless of how accurate that information is - in fact, it is designed to do so! This means that especially if you’re new to a subject it’s very hard to tell what you’re wrong about if you’re using chatgpt to learn. So your only logical recourse is to check literally everything it says. Maybe you find that pedagogically valuable but to me it seems at best a waste of time and at worse counterproductive to learning. I still don’t see why at that point you don’t just read the textbooks you’re using to verify from the start. What does ChatGPT bring to that scenario except confusion?

next sinew
#

Well I'm not a huge fan of textbooks. Maybe I should be, but for the most part I'm not, and for the most part I've been doing just fine. A workflow that works for me is opening up the Finnish high school matriculation exam for chemistry and giving the problems a try, and whenever I get stuck (which in the beginning has definitely been on every single problem 😂) I try to look for a resource that would explain what's going on, and one of those resources could be chatGPT

#

And the mechanism I proposed for why ChatGPT actually leads to positive learning outcomes for me is that the use of ChatGPT encourages me to ask for clarification and assurance from real people who have more experience in chemistry

lyric lotus
#

And my point is why not just ask those people from the start

#

Instead of going to chatgpt first

next sinew
#

I said it feels awkward to be like "where do I start with this problem", I feel much more comfortable saying "I think this is what's happening here, is that valid thinking and can you help me understand why the electrons move like that?"

#

Idk if that's actually true or not but I feel like there's far fewer people who are willing to spoonfeed chemistry than there are people willing to spoonfeed math 😂

lyric lotus
#

But this just feels kinda disingenuous lol especially when it’s not even your idea.

Instead of saying “where do I start with this problem” I’ve found it helps to say “here is what I’ve tried, this is why it didn’t work, now I’m stuck”. If you try the latter most people will be willing to help you

#

And if you haven’t tried anything yet… why are you jumping to asking

#

Maybe it’s different for chem idk

next sinew
#

Well let's say the problem is that you have 4 compounds and the question asks which of them forms an acidic solution with water

#

and you know the basic idea that acidity is caused by hydrogen ions, but all 4 compounds contain hydrogen in some form

lyric lotus
#

But I feel like that’s enough context/work to ask for help from most reasonable people

next sinew
#

so then you basically just have to know which compound has a bonded hydrogen such that the bond is gonna break when there are water molecule dipoles around

#

Now ChatGPT can actually give you a reasonable explanation such as "look at this thing, this is a carboxylic acid because it has the COOH group, isn't that neat"

#

then you look up carboxylic acids and you're like "oh yeah, I kinda remember this from high school"

#

And in that case I would argue ChatGPT helped me out

#

And then you can ask the people who actually know what they're doing why some of the other hydrogen bonds are so much more stable

#

or rather, why the hydrogen bonded to a carboxylic acid is so unstable

lyric lotus
#

Hmm. But I guess my thought would be a human could give you that info in the first place and you should probably check it with a human so why not just ask the human from the start

#

I’m not here to criticize your study habits if it works for you it works for you I would just say I still am not convinced to recommend it to people

next sinew
#

In principle I agree with you. But in practice I don't want to bother other people by asking very elementary questions that I can reasonably figure out myself

#

And based on my limited experience, sometimes ChatGPT+Google feels like less effort or feels more engaging than Think about what to google + Google

#

I think there is the psychological component to it that it feels like you're asking a real person when you ask ChatGPT, and for me that feels a lot more comforting than figuring out what to type into Google

#

And I recognize that there's technically risks involved but I think in some cases the pros outweigh the cons

#

I think people underestimate how hard it is to come up with a reasonable sounding explanation that yields correct results just some of the time (assuming the student can immediately check, "yay chatGPT came to the right conclusion, cool")

#

Although maybe I have more chemistry background than the average high schooler using ChatGPT for math has math background, I don't know. Maybe ChatGPT generating believable nonsense is more probable than I think.

tight star
#

I’m perfectly happy to ask elementary questions

#

I guess I don’t view it as “bothering” people because the other person always has a choice of whether to engage or not

next sinew
#

I assume that the few people who actively read the chemistry help chat have finite patience and energy and thus I want to prioritize the most pressing questions

#

there's probably some weird psychology involved though

#

Like I don't mind asking an incredibly lazy question in math (of course it helps that it's probably not going to be super elementary) maybe partly because I feel like I'm also giving back to the community in the form of helping other people

#

but when it comes to chemistry, I'm just leeching off of other people's knowledge and that's why I want to make my questions interesting and require some thought

tight star
#

To me “leeching” implies they don’t have a choice in the matter

#

But it’s not like they’re forced to help you or anything

next sinew
# next sinew but when it comes to chemistry, I'm just leeching off of other people's knowledg...

Because I will admit, there is a big difference in my motivation to help someone in math if the question is "how do I find the inverse function of this rational function" and then it's like just your average textbook problem that there exist a million examples of, versus something not trivially googleable like "our teacher said that an inverse function g satisfies f circ g = Id and g circ f = Id, but do I really need to check both of them or is it sufficient if I just check the other one because they seem so similar"

So when asking chemistry questions, I try to get them to think about an elementary species from some kind of novel perspective instead of just asking your most typical textbook question imaginable

#

And ChatGPT helps me filter out the most typical textbook questions without actually picking up a textbook lol

#

at this point I should clarify that I do also love helping out with very run of the mill problems but I think for most people if it takes 0.1 seconds to know what the student should do and what you should say but it just takes a lot of time to write out an explanation and/or sit it out with the student then you're not always in the mood for that

#

But if a question makes you think for a little bit, and then you come up with something clever, then usually it's incredibly rewarding to share that insight with the student and usually the experience will be a lot more memorable for both parties

midnight scarab
next sinew
#

yeah for example

#

I'm not gonna go on a rant again but I think the "oh no this new technology is gonna ruin our students" is a tale as old as time

#

people probably thought the same thing when CAS calculators rolled around and nowadays I think most people see no issue with doing routine stuff with a calculator

#

of course CAS calculators never hallucinated at you, but let's be real, we're not really concerned about the false information aspect, we're concerned about the concrete learning outcomes. And just like with CAS calculators, I think both students and learning institutions are gonna adapt

#

Sure, low level math education isn't ever gonna be quite the same after LLMs, but that doesn't necessarily mean there's reason to panic and think that we're all doomed because no one does things the hard way anymore

tardy ember
#

it is pretty hard to make something that looks reasonable to logical inspection but is actually wrong, but like,

#

clearly chatgpt's output is "believable" to chatgpt

#

so someone who is basically just "chatgpt but a human", just following memorised algorithms based on words without really having the idea that it's all about anything, ...isn't necessarily going to be very good at nonsense-detection

#

and like, i know this is roughly how i felt in some non-stem subjects, i knew roughly the formula that "worked" and didn't really care about the fact that what i was writing was only vaguely related to the actual reality of the thing they asked about

austere inlet
#

and the user expects the commands to work exactly as is

#

on another hand it's very hard to know exactly why an LLM reaches a given conclusion or gives a certain output, and as people have insisted a non-trained person can take the output as truth, e.g. when asking math questions

#

I agree that as a new tech paradigm it's not gonna "ruin" students -- the way its currently used by coders is pretty much a success story imho. In fact we should educate the young on how to use LLMs but part of that education should be that you should always double check the output and discard anything fishy or nonsensical, that's the bare minimum for it to be useful imo

#

(then again, I wouldn't say critical thinking is a skill taught properly in most schools worldwide, so misuse of LLMs is definitely part of a wider, systematic issue)

buoyant light
#

Usually when I type a prompt on a subject I ask for relevant links so I can verify the authenticity

tawny slate
# next sinew I'm not gonna go on a rant again but I think the "oh no this new technology is g...

i dont think "this new technology will ruin the students" trope is as catastrophic as some make it out to be, but firstly "badness" is subjective and relative, but i do think there is good reason to consider those views and take them into account more:

  • technology is becoming increasingly more powerful, which means their ability to disrupt also increases
  • students are already measurably performing worse when it comes to learning due to attention deficits due to smartphone/social media use. there is well-documented causal links that show that overuse of social media can permanently damage a student's capacity to learn, and more young children are using smartphones and tablets than ever before
  • on principle, its better to be cautious than not. if you dont accept doomerism ideas, then you shouldnt be too worried about the current state of affairs, so better to make sure we dont accidentally do something bad to society that is difficult to reverse
lethal hornet
#

is it a good idea to tell learners of alg top that the operations of expanding, shrinking, and bending (in the context of homotopies) do not create or destroy any holes that a loop could get caught on

#

or is that kind of like, just wrong? need an outside opinion

lyric lotus
#

Sure that sounds like fine intuition for why the fundamental group is homotopy invariant

lethal hornet
#

sick

#

that is exactly what i was going for

soft scarab
#

question for anyone who has taught Calc 3 (Vector/Multivariable Caluclus): What percentage of the lecture do you spend doing proofs? How about example problems? Do you expect your students to be able to do proofs for hw/tests? and lastly, do you take more of an algebraic or geometric approach to teaching the course?

#

I ask because i’m currently taking Calc 3 and my professor does not do any practice problems or examples. Rather, he goes through and derives all of the theorems and proofs. My calc 1 and 2 experience was not like this. ie. we spent pretty much the entire time just doing computations. Those classes hardly gave me any struggle. But now, with Calc 3, i am kind of struggling because of the proofs, abstractness, and lack of example problems being shown to me. Additionally, it is so much more geometric which I am not used to. Thanks for your answers!

cloud zealot
austere inlet
#

I wouldn't expect my students to be able to write proofs, much less on exams

#

though in lectures I do show proofs or outlines of proof when it is particularly enlightening

tight star
#

From the following, what’s your preferred definition of “function” (or is there another one?), and which one do students normally seem to like:

  1. A particular relation between specified sets A and B, the domain and codomain, such that for all a in A, there exists a unique b in B such that aRb
  2. A set of ordered pairs such that whenever (a, b) and (a, c) are in the set, b = c
  3. A rule or formula for turning inputs into outputs, like f(x) = x^2

I’d say that “graphs passing the vertical line test” goes under “2”, and non-deterministic functions like those you find in programming goes under “3”.

My guess is that students usually have something like “3” as their definition, or perhaps “2” but as a graph rather than the formal set theory definition.

I also think that there’s maybe a tendency to assume that the students’ conception of “function” matches your own, which can cause difficulty when teaching

vocal phoenix
#

My preferred definition is definitely the set-theoretic one which identifies a function with its graph, but it isn't always the best way in which to think of a function.

#

My intuition is usually to think of function as a mapping (although thinking visually of the graph is also frequently useful); but it's hard to define "mapping" rigorously, especially if you don't want to introduce new primitive notions.

lyric lotus
tight star
tight star
vocal phoenix
#

The codomain is an essential part of defining f

#

But yeah, I see what you mean, you actually need to specify Y as well as the graph, since if the function isn't surjective then it can be thought of as a subset of a smaller cartesian product.

#

I wasn't being entirely rigorous

vocal phoenix
# tight star And this sounds like “1”

But for clarity, yeah when I was talking about my preferred definition being "identifying a function with its graph", it was shorthand for what you wrote out (properly) as "1"

#

And my preferred way to talk of functions with non-math students is close to 3

#

Although I tend to stress the significance of domain and codomain even with those.

#

In lectures I "define" a function as a mapping, but that's not a rigorous definition since it would require properly defining what a "mapping" is.

#

But I think it's rigorous enough for non-nerds.

tight star
#

So f(x) = x^2 could be a function from integers to integers, or from reals to reals

#

Which are different functions in the sense of 1 and 2, but maybe not as obviously different in the sense of 3

vocal phoenix
#

Yeah, which is why I mentioned the importance of specifying the domain and codomain.

tight star
#

I see I see

#

I guess one often gets exercises to determine the domain and range of a provided formula too

#

Rather than being provided a domain and codomain

vocal phoenix
#

In such situations I talk of "natural domain", and the way we usually do it is clarify that our "global" space is, for example, R

#

And given a formula, we're looking for its natural domain as a subset of R

tight star
#

Interesting way to phrase things

vocal phoenix
#

So basically we start with a partial function from R into usually also R, and we find its natural domain, and its range over that natural domain

tight star
#

Conversely you can also have different formulas that give the same function

#

So say (x + 2)(x + 3) versus x^2 + 5x + 6

vocal phoenix
#

Yep

tight star
#

Which is essentially the idea of function extensionality

vocal phoenix
#

Which is why I focus on the function as a mapping, and the formula is a way to define that mapping.

#

And you can indeed define the same mapping in different ways.

tight star
#

That makes sense

#

I got interested in this because I was thinking about the benefits and drawbacks of taking a more function-focused approach to teaching maths

#

And I realised that you’d need to be clear and precise about what sense of “function” you’re using

#

It’s not necessarily an insurmountable obstacle, since even numbers have many different senses

#

As cardinals, as ordinals, as names for example

#

But there’s definitely the possibility of confusion

lyric lotus
#

Those are equal with your definition

#

With your definition, the codomain of a function is not unique

tight star
#

i think it’s also useful to see how “composition” of functions works in each scenario. “1” is the most standard (relation entailment), but it’s technically possible to do composition with “2” as well, so long as the image of one function is a subset of the domain of the next.

the way it’s done with “3” i guess would be substitution

#

and in terms of scenarios where you’d want to use each:

  1. helps when you want to emphasise the domain/codomain, especially if you’re considering the collection of all functions between two specified sets (or a subset thereof). or just in general when you want to consider multiple functions, or a “generic” function between two sets. also really the only context to talk about things like surjectivity (and hence bijectivity)
  2. im less familiar with this but i think it gets used in set theory a fair bit. it’s pretty compact as a def and de-emphasises a specific choice of domain or codomain (though you can recover the domain)
  3. easiest to introduce and work with in practice i’d think, easier analogy with CS, and also helps for things like product rules/quotient rules/power rules where the specific formula used to define the function can help
vocal phoenix
lyric lotus
#

What a function IS is distinct from how it’s defined. From a set theoretic perspective, whether you define it as a subset of X x Y or X x W, if they have the same set of ordered pairs they’re equal as functions. My argument is that that automatically makes the definition deficient: equality should type check also, and make sure functions are only declared equal if they have the same domain and codomain.

vocal phoenix
#

Yes.

#

My preferred formal definition of a function is as a triple (X,Y,f), where X and Y are arbitrary sets, and f is a subset of the product X \times Y satisfying the properties which I am too tired to elaborate on.

vocal phoenix
#

I had subconsciously assumed that if I said "set-theoretic" definition, it wouldn't be ambiguous. I can only apologize for the confusion, it's been a long day.

lyric lotus
modest citrus
#

a

bleak skiff
#

not very pedagocial i suppose

lost raptor
lost raptor
lost raptor
vocal phoenix
#

Well, or the same domain and codomain

#

That's the point of contention here

jaunty finch
vocal phoenix
#

I think y = x^2 as a function from R into R is not the same as y = x^2 as a function from R into [0, infty)

#

Even though the set of pairs is the same

lost raptor
#

I think they’re the same

vocal phoenix
#

Fair enough, hence the ambiguity

#

I don't really feel very strongly about this

lost raptor
#

Why should you distinguish it based on what it doesn’t interact with at all

jaunty finch
vocal phoenix
jaunty finch
#

the only real difference is the slightly added complexity

lost raptor
#

I mean, codomains being considered is relatively modern iirc?

jaunty finch
#

as far as i was aware both codomain and range were fairly new at least formally

lost raptor
#

I don’t think a codomain is as essential as the image (something something replacement)

lyric lotus
#

IE I don’t think an inclusion map should be equal to an identity

#

But I’m category theory pilled

#

Also, if codomain is the data of a function, it makes it more natural to check which functions are composable

#

Let’s say I have the function f defined as the ordered pair <0, x> where x is 0 if Riemann hypothesis is false and 1 if it’s true. Deciding which functions you can compose this with is equivalent to deciding RH lol

#

You will never get this if you require the codomain to be part of the data and strictly enforcing composing only when codomain = domain

lost raptor
lyric lotus
#

Yes it is?

#

It is a set of ordered pairs, and if <x, y> and <x, z> are in the set then y = z (trivially because it has one element)

#

Or do you mean that enforcing codomain as part of the data isn’t a function

lost raptor
#

Can’t really decide that without deciding RH

lyric lotus
#

Yes you can. A possible codomain is the set {0, 1}

#

You can’t decide the smallest possible codomain, sure.

#

But my point is that specifying the codomain means you don’t have to attempt to prove what the range is (which may be arbitrarily hard) to determine function composition. Also, morally the codomain IS part of the data - the inclusion is not the identity

lost raptor
#

I mean, you can say morally all you want but I obviously think it is the identity

lyric lotus
#

If this was the 14th century our houses would feud over this

tight star
#

It’s definitely useful to be able to distinguish identities from inclusion maps

quasi lava
# tight star From the following, what’s your preferred definition of “function” (or is there ...

i guess it depends on what you're trying to teach and what background the students have but in my opinion number 2 feels like you get the most bang for your buck in a sense cause it's succinct and fits in nicely with introducing the other main notions you'd want a student to generally know like image/preimage, kernel, injections/surjections/bijections, etc and you can also generate a lot of examples of things that are/aren't functions and its pretty easy to check said examples against that definition

#

in my experience at least it was def the most illuminating at first, maybe later on youd want further nuance depending on the students and topics i suppose but i interpreted the question as asking about introductory-level and getting the main ideas across so you can describe things to them in terms of functions

quasi lava
#

actually i think showing how number 2 corresponds to the vertical line test would resonate well with most students

tight star
quasi lava
#

ig you can say that if you have f : X-->Y then for all y in Y there exists a pair (x,y) for some x in X?

tight star
#

With “2” you de-emphasise the domain and don’t even have a choice of codomain

quasi lava
#

oh i see i wasn't taking the choice of description to be that exclusive, to clarify i do think its helpful to describe a map like f : X-->Y as well so maybe i would need to rethink my answer

#

what if you just drew like X and Y as blobs with some points in them, then point out that with functions you cant draw arrows from a point x in X to two different y_1, y_2 in Y

#

and using that illustration of the domain and codomain gives you a visual way to show injectivity, surjectivity, etc whatever else i listed

#

im sure youve already considered it though, im curious as to why that wouldn't be the best way to introduce it

tight star
#

I mean you’re essentially describing “1”

quasi lava
#

yeah if thats what you meant i guess i'd lean more toward 1 then

tight star
#

I think maybe the transition from 3 to 1 is done a little implicitly from school vs uni

#

E.g. in school, it makes sense to ask about the “domain” of a given formula, and “codomain” isn’t as useful a concept, though “range/image” is

#

Whereas at university, you’re often given these as part of the data of a function

lyric lotus
#

I still don’t like it though for reasons I mentioned earlier

tight star
cloud zealot
#

My professor shared an interesting grading philosophy. The basic idea is that math is cumulative. Hence, for example, consistently earning 50% on exams actually shows growth, as material from the previous exam is built upon on in a current exam, rather than stagnation.

dapper flume
tight star
#

Hmm, apart from letters in math being:

  • A name for something
  • A placeholder for something you can substitute

What other roles do letters in math play?

#

I’m specifically talking about near to the time when letters get introduced to math

tawny slate
#

I don't think letters have any other roles

#

more generally, letters are symbols

#

the symbols themselves are abstractly representing something, which is what the "name" or "placeholder" is doing

#

so i dont see how any further role can be extracted out of a letter without changing its function as a "letter"

dapper flume
#

This is how I introduced it to sixth graders in early pre-algebra. It's imperfect, but I think its accurate enough for them to build a flexible intuition of what variables can be in their high school journey.

These notes were followed by an activity where students represented both real-world and abstract sentences using expressions.

image.png
late vortex
#

Im doing some math tutoring on the side.
I had a test lesson with a prospective highschool student and she has dyscalculia.
Does anyone have experience dealing with that in a tutoring setting?

Theyre doing some basic optimization problems as in e.g. maximize the area of a square given some constraints (and more elaborate versions).

My impression was that she can calculate things on paper/in her head, but it just takes longer and I had to refresh some basic concepts with her.

Apparently this condition is not a valid reason for the teacher that she gets more time in her math tests.

Im very patient by nature and she wants to have lessons with me regularly (as in she didnt hate the lesson), but im interested to know If some of you have experience dealing with that and could give some advice.

I guess the overall goal is to not have a failing grade so that she can graduate.

paper juniper
#

Hi Eden

#

I dont know in your country. But here we use some devices to help. Like Cusinaire scale.
I think working with area, graphs is easy for them. But when you get something numerical it is hard for them. Then you have to get objects, drawings to help

#

My problem comes more with authority . I have trouble made then stop to listen. They test you ! Trowing paper balls, whistle in class. I am too tired.

#

They dont respect you

vagrant meadow
late vortex
late vortex
daring gulch
inland kayak
#

2 of my students keep answering all of my questions and I'm not sure how to either get other people to engage.

#

I'm not sure if I should phrase questions in a different way so as to solicit more input, wait longer for answers, engage better with wrong answers, or ration the attention I give to these 2 students

wind ginkgo
quasi musk
vagrant meadow
#

I was def one of those students. one class in particular, I was so bored and frustrated with how slow it was going (I already knew most of the material and was forced to take it) I just answered everything as fast as I could to try and move things along.

midnight scarab
#

Ideally you could maybe get them more challenging exercises to keep them busy and entertained

#

If things are too easy for them

#

Otherwise it is frustrating

stark spear
#

In school why do they often teach De Moivre's Theorem instead of the more general and more intuitive idea that multiplying two complex numbers is the same as adding their angles and multiplying their magnitudes, from which De Moivre's Theorem naturally follows?

quasi musk
#

It took 2-3 weeks for them to process that fact

#

I had to explicitly show them the cosine & sine addition formulas, show them how complex multiplication works in cartesian

#

Convert that to polar

stark spear
# quasi musk So I just had the pleasure of teaching this to some high schoolers this summer

That's awesome! The way in which complex number multiplication works out so nicely and has such a nice polar representation actually appears to me to be a really nonobvious fact. I wish it would be emphasized how amazing of a fact it is in school.

One way you can approach trying to get to the root of why it works is looking at the cosine and sine sum formulas, which can definitely be helpful and elucidating (especially if you are able to develop good intuitions about why those formulas are true from geometric proofs). My personal favorite approach to understanding complex multiplication myself is starting from linear algebra and developing an understanding based on intuitions about linear combinations.

candid oar
# stark spear In school why do they often teach De Moivre's Theorem instead of the more genera...

Interesting question indeed!
I had similar thoughts when they first introduce Euler's formula back in my highschool years.

There are a few thoughts that come in my mind:

  • it's more intuitive and familiar for students around that level to use "Cartesian Form" than "Polar Form"
  • it's a nice transition from "Cartesian Form" than "Polar Form" which usually will be taught right before De Moivre's Theorem, and student can use the proof as an exercise for their trigonometry skills
  • Euler's Form is not intuitive or easily understandable at that level, and similarly for polar form multiplication if they were just introduced vectors not much earlier, they may mix up things
  • we can use De Moivre's to explain the idea of multiplying 2 complex numbers
stark spear
# candid oar Interesting question indeed! I had similar thoughts when they first introduce Eu...

These are all really great thoughts! Thanks for your response. I didn't think about how De Moivre's theorem is definitely a relevant theorem to introduce after polar coordinates. Unfortunately I don't think my high school curriculum made me prove De Moivre's theorem. It just said "here take this formula I promise it is true" like most formulas you learn in high school...

Still, I feel like it would be better and more worth the student's time to prove a certain statement of Euler's formula. The proof is actually about the same level of difficulty as De Moivre's theorem if not easier (the standard trigonometric proof of De Moivre's theorem requires induction while all that the proof of Euler's formula requires is the sin and cos sum identities. Basically to proof Euler's formula all you need to do is multiply c(cos(a) + isin(a))d(cos(b) + isin(b))) and the result pops out.)

I would not introduce this concept in terms of e^(i*pi) though. There really isn't any reason to introduce this confusing notation, and it would take a while to explain why this notation actually makes sense as a generalization of exponentiation. Instead, I would just have students prove a verbal statement like "Multiplying two complex numbers is the same as adding their angles and multiplying their magnitudes." I feel like this statement sheds a lot more light on how complex multiplication actually works than De Moivre's theorem, and is also much more interesting. De Moivre's theorem also is obvious given that you understand this verbal statement.

quasi musk
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So it was a chance to reinforce some of the trig formulas in relation to complex numbers

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then using that to solve the cube roots of unity

vagrant meadow
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with the perspective that complex numbers are "transformations", then the idea of composition/multiplication being equivalent to multiplying magnitudes and adding angles is pretty reasonable.

candid oar
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I've seen students quite some still struggling with polar form even when they are preparing for their last exam in highschool

candid oar
pastel sundial
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Long time no see

dapper flume
# inland kayak 2 of my students keep answering all of my questions and I'm not sure how to eith...

Several good suggestions, but I figured I'll drop what works for me.

  • Use more than questions. Move around the room as well. If you can make it so you're facilitating dialogue rather than one-sided question and answer, people may chime in.
  • Move around the room and "visit" each student regularly.
  • Be comfortable with uncomfortable silence, and deliberately do not call on the same two students often. People will probably want to break the ice.
  • Ask open ended, debatable questions when you can find them. For example, suppose the class has to decide whether Class A or Class B did better at a test on average. You could teach them how to calculate the mean first, and then have them compare the means of A and B. However, if you want to foster potentially energetic conversation, simply ask "which class did better?" and facilitate a debate as students try to decide what "better" means and how to argue it.
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Lots more can go on this list

tawny slate
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regarding teaching polar form, i personally really like the visual, geometric intuition

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i think i first saw it demonstrated in a mathologer video

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you first explain how addition and multiplication work on a number line, then extend it to the complex plane

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multiplication is not totally trivial because it needs to work for all real values, so its a bit more nuanced, so i do recommend starting with addition to help ease into it and be a complete explainer

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after that, show that logically what you're doing is simply adding angles and multiplying magnitudes

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and as for notation, i agree not to jump into e^i theta

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i personally use $r\angle \theta$ to represent polar form

burnt vesselBOT
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Cozmogrgdfschkipkhrshtensi

148282132781858816.png
tawny slate
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i only introduce the e stuff once i can justify it

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the advantage of this visual approach i think is that it is both super intuitive and highly simple and reductive, but the downside is that it does beg for a video/interactive component

tight star
tawny slate
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around 9:22 ish

quasi musk
vocal phoenix
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And is hugely useful with the polar coordinates

stark spear
# tawny slate https://www.youtube.com/watch?v=-dhHrg-KbJ0

Love this video. Burkard did a great job with this one as always.

I totally think that high school students should be introduced to the "multiplying two complex numbers is the same as adding their angles and multiplying their magnitudes" property first before De Moivre's theorem, because it really cuts to the core of why complex numbers are actually useful and interesting. I remember learning about De Moivre's formula in high school and really not paying much attention to it because I took it as just some useless number theoretic formula and had no idea what it was really saying about the more general way in which complex multiplication is a transformation.

pale basin
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Question for the Room: If you could be MathMonarch for a day - What is the one thing you would change about grade school math? (Serious answers only please.)

quasi musk
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I guess that's two things, so having a MS/MA in Pure or Applied Math stands

vagrant meadow
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in my experience with regards to learning proof based algebra, one of the things students struggle the most with is proving things that seem "obvious". and in fact, many times there are proofs we might consider easy or simple that students struggle with.

in #linear-algebra yesterday, someone had trouble understanding W is contained in W+U. I would say this proof is extremely "simple".
if w in W, then w=w+0 and 0 in U so w in W+U.
one might argue that it's even an "easy" proof! but anyone with a sliver of empathy can understand that just because this proof is "simple" (in the sense it's basically a one line proof and justification, and arguably follows intuitively from the definitions and concepts) doesn't actually make it easy, intuitive, or obvious for all students.

sometimes I try to lead students through problems with leading questions. and my intention is for each leading question to be extremely simple, but I sometimes run into issues because the questions are seemingly too simple. in that it seems the student literally can't comprehend I would ask something so basic and elementary, and that i must be trying to trick them. but when the proof is literally one line, what kind of leading questions can you even ask that aren't basically "what is the literal definition of whatever it is we're working with". sometimes the students then get frustrated at the end like "why didn't you just tell me the one line proof?!" (certainly there's times where one might just give it because it's not worth wasting the time, but it's also often worth going through to build up their proof abilities).

so that's a lot. I guess my question is: how do you reassure students that just because something is simple doesn't mean it's easy/obvious? and how do you handle leading questions when the answer is like a trivial one line proof?

tight star
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This kinda sounds “Linda problem”-esque

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I think maybe it would be useful to lead them through some examples of one-line proofs, perhaps unrelated to the problem

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People can definitely get into their head that maths, and especially proofs, have to be hard and difficult - they might see an argument but be likely to disregard it because it’s “too easy”

vagrant meadow
tight star
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Sometimes it’s good to be suspicious of very short arguments

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So it’s not like they’re being entirely irrational

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Thus I think it’d be good to show them lots of short arguments that do work

tawny slate
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a different answer for variety

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i would start changing math conventions

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use tau instead of pi (still mention pi's history because obv)

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have students count indexing from 0 instead of 1

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get students using military time instead of am/pm and use metric units for everything

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and wildcard answer: id probably, as a treat, get the students to try and play more puzzle games, physical or digital

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maybe even add board games to that

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i do really like the previous answers, and i especially appreciate that the first response was a pro-teacher policy change, as obvious as it is

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oh "one thing" whoops uhhhh

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yeah then clearly id pick a policy thing than any of these superficial suggestions haha

quasi lava
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you put it perfectly i haven't seen anyone else talk about it

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and its such a frustrating thing

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i dont really have an answer for you unfortunately but def gonna be following to see what others opinions are on this or if you think of something

tight star
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Because I don’t think logical arguments outside math tend to have one-liners in the same way

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Things which are one-liners (e.g. Socrates is a man. All men are mortal. Therefore Socrates is mortal) are kinda just assumed in conversation

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Not something you have to lay out explicitly

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Even if you try to shut down an argument with a witty one-liner, there’s always room for disagreement and nuance

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That’s true even if the one-liner is fully correct

vagrant meadow
# tight star Because I don’t think logical arguments outside math tend to have one-liners in ...

only in math would we have to prove that 0v=0. of course, i'm assuming everyone in #math-pedagogy knows why such a proof is important, but most students taking linear algebra for the first time look at that and get super confused!

it's often hard for students to even wrap their brains around the fact that -v and (-1)v are defined as different things, but then we have a "theorem" that proves they are the same. why the hell do we do that? why do we make that a theorem and not part of the definition?

i can understand why some people take linear algebra and go "okay, i hope this is the last math class i ever have to take". to an outsider, algebra seems so unnecessary and pointless. i actually kind of love it for all those reasons lol but i feel like there's just such a massive shift in mathematical perspective around the point of linear algebra, where students who were already only sort of holding on just completely fall off.

tight star
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I mean to be fair I can definitely sympathise with those students

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Given that I tend to not exactly enjoy algebra

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Although I do love linear algebra