math-pedagogy

[winged urchin]

What I do definitely like about the cups is the use of an empty space instead of x

As a tutor I find empty squares help some students understand substitution more than using variables

I guess it looks like fill in the blank questions or something

[long pelican]

I've definitely used empty squares instead of variables in more than one occasion in my lectures

Gosh, I'm liking my analogy between reading math for yourself and reading sheet music for yourself more and more

[pastel horizon]

I kind of disagree on that, my opinion is that you're the expert in the room you should be the one teaching and instructing them rather than leaving them on their own

With discovery learning it has the risk of them 'discovering' new 'theorems'

[real mauve]

they probably don't mean just tossing them into cold water. the lecturer should in any case propose a reading schedule with topics and book sections/chapters to cover by a certain date

[twin lichen]

yeah i didn't mean totally exploratory, i just meant that they should understand that the teacher cannot shoulder the burden of learning for you

it's not a passive thing, you have to be doing it and learning it yourself

[real mauve]

i honestly don't think this is a large issue at HS level

but it does carry over as an attitude into under and postgrad, and there it turns into a big problem

[quasi musk]

Yeah, unless you're teaching calculus to young children

Like under 13 or so, I'd avoid using manipulatives like that

It can be fun for like an off beat kinda day to just get through the symbol pushing

In general I've never been a fan of these types of things, since I think students in calculus should be expected to be able to read examples and figure things out on their own

[stark pine]

A course I took in the first semester of my first year in university had an incredibly large discovery based portion. Assignment questions were phrased as "do this specific computation. Generate more examples and do those computations. Do you notice any patterns? do you have any conjectures? prove as many of your conjectures as you can". An early assignment had us build our own axioms for the integers in Coq. A majority of the class got a huge amount out of it.

There's not that much difference between a grade 11 or 12 and someone in their first year of university. I think that, with a sufficiently competent teacher, discovery based learning can be incredible

[pastel horizon]

I think it's a fair point to be made for further or higher education it's not something I would do for secondary level

Arguably you could kind of discover the pattern for them and guide them to it as well

[stark pine]

this was what the prof did, he just expected us to do more on our own. Like, the given exercises were deliberately made as illuminating as possible

[zinc wigeon]

What do you think about Paul Lockhart’s critique of modern mathematical education?

[pastel horizon]

I haven't read it, source?

[strange bronze]

its very pretentious

that isnt to say its criticisms arent valid

but i am not a fan of its suggestions nor its writing style

[strange bronze]

https://www.maa.org/external_archive/devlin/LockhartsLament.pdf

i guess in hindsight its suggestions arent that bad

but it still rubs me the wrong way with the whole "mathematics is the most sublime of all arts" shtick

:/

id understand it more if it was written for a general audience, hed understandably feel the need to justify a further focus on actual math

but it isnt

it was written for the MAA

and i think it ignores plenty of valid criticisms of the pedagogy of higher pure mathematics as well

like, it never explicitly says this, but one could read it as an argument in favour of bourbakism-plus-pictures

which is... controversial to say the least

again though, i think a lot of his criticisms are legitimate

grade school mathematics sucks

(especially geometry, yuck)

hm, on reread its a bit less bourbakist than i recall:

[...] the simple and profound ideas of Newton and Leibniz will be
discarded in favor of the more sophisticated function-based approach developed as response to various analytic crises which do not really apply in this setting, and which will of course not be mentioned.

but still seems very focused on def-theorem-proof-example and def-claim-counterexample

which is pedagogically dubious even if its the norm in higher math

[lyric fractal]

how would you help middle schoolers build intuition for geometric problems my brother excels at algebra but sucks with that
not sure how to approach teaching him
maybe more practice to help him visualize it better?

[pastel horizon]

@strange bronze He comes across as really pretentious in my opinion and it's very bold to assume that secondary education is just rote learning formulas

[real mauve]

that snippet reeks of the copypasta in this server

[tepid smelt]

He makes valid criticisms but the issues with k-12 education are so massive I don't see any way he addresses a solution. I love discovery based lessons but they take a lot of effort to plan good ones and its a much slower process where most teachers already are not getting through all the required content.

[pastel horizon]

Don't think I've ever seen the proof written this formally in secondary either

[long pelican]

I read it in high school and I wholeheartedly agreed with it then when relating to my experience talking with fellow students about math and how they learn it

People who went through math education in K-12 hate math iff they did not explore it outside school

The only thing I'd change is that we don't necessarily have to do the discovery learning route to make students appreciate it. Do music students have to discover how to read notes to enjoy music? I think a lot of frustration they have is that they don't even understand the rules of how to read and write it. (This is like a music student never even learning how to read sheet music, or what a chord like C major means) This is because teachers and early grade textbooks don't teach how to read it and even use incorrect math syntax themselves. Hsung-Hsi Wu calls this "Textbook School Mathematics" (TSM)

So students end up unable to make sense of it at all except by going the "remember what to do next" route

In fact, discovery learning where the stage is incorrect mathematics / not even mathematics may even be detrimental

[vague parrot]

I wish you could tell this to the students at my school. I'm working at my old highschool as a maths tutor/support teacher. Some of the students here expect you to either solve the work for them or are only willing to listen to you so far as it's the answer that you're giving them

Like I prepared a little demonstration on how mean and variance will change when you change the dataset (multiply or shift by a number) but noone really cared and just wanted to rote memorize it 😦

[blazing sandal]

Did you had ask why they didn't cared about it?

[pastel horizon]

Just intellectual laziness I'd guess

Idk if it's the higher years maybe you can say you expect them to start working like a college/uni student like they will be doing after they leave

[blazing sandal]

Sometimes this comes to my mind:

https://www.youtube.com/watch?v=UIKGV2cTgqA

[stark pine]

I feel like he spends so much time talking about the importance of playing and examples that he's not in favour of this style

[tawny slate]

I had to write proofs like this in 8th grade

I also require my students, in the process of learning geometry, do some form of proofs like this

Either the two column proof shown here, or a paragraph proof (freeform)

[pastel horizon]

Ahh fair enough. Idk I think at such a young age it's a bit unfair to make them state the proof so formally, but they should give reasons

If you don't force them to give reasons they'll just say any old angle facts

[wispy slate]

i think the two column format can often mislead people

lots of times i had students complain

that it’s obvious that it’s true but why do you have to write it with all these weird rules

and they don’t get that you are doing by writing a proof is giving the reasons why something is true

[covert hearth]

I definitely had to write 2 column proofs in 9th

[fierce charm]

What's the source of this screenshot?

Oh it's from Lockhart's lament, disregard that

[long pelican]

It's become pretty clear at least to students that the 2 column proof is of negative value to introducing the actual idea of mathematically proving things. It's pretty strange that educators and high school teachers remain ignorant

[winged urchin]

I'm curious, I know it's commonly shit on but it's also a common tactic if you believe an equation is actually an identity is it not?

Like if I am trying to see if it really is an identity I do work on both ends at the same time

Since sometimes trying to go from one side to the other can be very un-intuitive

So in that way, is it not a useful tool in that capacity?

[long pelican]

Ah that’s different from two column proofs

[winged urchin]

Ohhh you're right I had the wrong thing in my head

Something like that as an example I guess

Just curious what you think (or others think), what is wrong or bad about these? It can't be bad to know the reasons why you can do certain steps in a proof right?

Is it just that like... students should 'get it' intuitively rather than... uhh... pushing the symbols?

[long pelican]

Half right

There is not so much wrong about this if taught ONCE and used once as a gentle introduction to proof

But this is their only proof experience in all of middle and high school and that’s bad

And in one year only

Besides trig identities (that’s a very limited type of proof) and maybe proof by induction

So I’d say the real bad reason is the lack of authentic proof experiences in math

[wispy slate]

the format is far too structured

i think there is this thing where whenever someone sees structure

there must be some reason for this

and and they think things like

a proof has to have these properties

a proof is just an explanation why something is true

look how intimidating those first three lines are

when all it says is

Since the triangle is isosceles we know they the left side and right sides are equal, also the left and right angles are equal

[wispy slate]

“it can’t be bad to know the reason why you can do certain steps in a proof”

this i think is exemplary of why this two column proof stuff is bad

you are a describing a proof as some kind of operational object where certain steps are allowed

this does not make sense

you shouldn’t be thinking what am i allowed to do

you should be thinking why is this true

you are allowed to say anything, just certain might not be true

moreover the two column proof makes it seem like there is one right answer

[wispy slate]

so like if i was writing this proof i would write it like

We want to show that the two top angles are the same. It is enough to show that the left and right triangles are the same. So it’s enough to show that two sides and one angle or the same, (or that one side and two angles are). Since the big triangle is isocles the two long sides are equal, and so are the bottom angles. But the triangles also share a common side. So we have two equal sides and two equal angles. The triangles are the same and hence so are the top angles.

[pastel horizon]

It makes it seem like proofs have to be a boring formal procedure when they can be really interesting that's the problem I think

[pastel horizon]

I think as well you should cut them some slack like the first two lines are really obvious, should they be expected to write facts and say "by definition"? I'd say if they had drawn the correct diagram that's enough in my opinion, at that stage in their learning

[pastel horizon]

For an 8th grade level/KS3 and 4 UK I would say line 3-5 are enough and maybe a quick statement that AE = EC and the angles are equal so BE must be a perpendicular bisector

[wispy slate]

I really don't get your point

[pastel horizon]

Like if they're expected to write all definitions, even if they seem obvious, it's easy for them to forget something and then you're not really assessing the depth of their understanding

[wispy slate]

Sorry i still don't understand what your point is

[pastel horizon]

Like if you want to check they understand an isoceles has 2 equal sides, you could more efficiently check that based on their diagram instead of them writing in a line of proof

I suppose as well you could technically write the exact same steps for an equilateral triangle and still reach the same conclusion

[wispy slate]

i'll try one last time

i still don't get your point

more broadly what are you trying to say

[pastel horizon]

They've had to include basic definitions as part of a formal proof. This isn't something I agree with since at that level they might forget basic things even though they know what an isoceles is.

I would rather assess this by getting them to draw a diagram and then apply their reasoning to the diagram

Like would you really deduct a point because someone forgot to write AB = CB by definition even though they've shown it on the diagram?

That's what it seems like in the model answer

[willow scaffold]

depends how many marks the question is out of

and whether you're the only person marking the work

[covert hearth]

Ok 2 column proof has been argued against here.. what do you guys think of tableau type proofs?

[long pelican]

Honestly never used them, seems on first glance like it'd be useful for extremely complicated logical statements of the kind you don't usually see in math at any level

[tepid smelt]

I mean I show the two column proof because its in our curriculum and it might show up on state testing. I don't like them for all the reasons stated. I sadly don't spend much time on proofs because it is so different then how they have been trained in math their whole life. I really have struggled with proofs for kids at the HS level and sadly have not pushed it much due to lack of time and emphasis going forward for 99% of students. I think students in calc ahould see them more though as those students care about math and are capable of learning proofs even though it wont show up on AP testing.

That class is also likely where you will push a future student into studying math at college if you do teach more proofs. It is kinda sad that so many colleges don't even touch proofs in that class let alone at the HS level where you have more time and smaller classes

[quasi musk]

In my experience, even the calculus teachers don't know epsilon delta that well

[long pelican]

Hey guys, a question that occurred to me: By what age would you expect the average math student to read the statements "$f(n+1)=nf(n)$ for all positive integers $n>0$" and "$f(n)=(n-1)f(n-1)$ for all positive integers $n>1$" and be able to readily confidently say (after some reading and parsing) that they say the same thing, under:
a) the current education system
b) an ideal education system

[burnt vessel]

Icy001

[long pelican]

I think the answer to a) is never (based on my experience) and the answer to b) could be as early as like 10 years old (fourth grade)

I think that a) is never because the current education system does not train students to read math with quantifiers in any way, shape, or form, and when faced with such a collection of sentences they are trained to think in terms of algebraic manipulations and "what to do"

[pastel horizon]

10 years old might be pushing it tbh. Maybe 14

In the UK system I would say currently no as well, the average maths student doesn't need to know function notation to pass foundation tier

Higher tier students at 15/16 would be able to

I know in China they train children to learn algebra alongside number theory but is their system really ideal? It also results in a lot of burnout and pressure

[tepid smelt]

Here in san Francisco proof school probably has the ideal curriculum in the world I have seen. I would say for sure by 8th grade when they are taking an intro to proofs class but likely even by 6th grade where they are seeing proofs for the first time.

Look over their curriculum they have the right idea behind math education imo. Those students are taking graduate level math classes by the end of high school.

My 9th graders do learn fuction notation with sequences but the average student wouldn't pick that up sadly even after being shown it because again they are so conditioned to think a certain way for years. I do agree that more emphasis on reading math should happen earlier its just hard to make up the ground when we have so much material to cover.

I think tying in more programming can help with this actually even at an earlier age. I think programming and math should go together in the k-12 system especially considering what skills are valuable now.

[quasi musk]

$50k/year

Nice

The curriculum is very good, it's just not cost effective

In my opinion things like Russian School of Math or Johns Hopkins CTY are much more cost effective at achieving the same goal

[pastel horizon]

Yeah programming and maths definitely go hand in hand

[wispy slate]

what counts as graduate level maths?

[tepid smelt]

That is hard to define but they offer things like point set topology and algebraic geometry which could be considered graduate level classes. I think of undergrad as calc/linear/algebra/complex and real analysis as the core. Then electives like number/graph theory stats/prob/diff/pde etc first courses. So i guess graduate level would be a 2nd+ course on a topic or requiring core classes like analysis/algebra. Though this is my subjective bias.

[wispy slate]

wow

thats impressive

[quasi musk]

It's unlikely to be at a graduate level sophistication

Probably very surface level, but still formidable

[wispy slate]

yeah definitely

honestly i find it impressive even just to think think that kids are exposed to

there is this kind of maths where we only care about shapes up to bending and strechting them

[covert hearth]

I know you guys were discussing 2 column proofs in the past.. If you want to hear something horrible, one of my professors does 2 column proofs for really complicated stuff.

[twin lichen]

There's nothing wrong with 2 column proofs

they are a perfectly cromulent way to present certain kinds of arguments, or rather a certain style of argument is easiest to write as a 2-column proof

like there are some arguments where i find it difficult to imagine that the argument would be easier to read or understand recast in a different form

here's an argument from categories for the working mathematician

if a computation just involves like

recasting terms a bunch of times according to a chain of equivalences

idk

what are you going to do

it's like

some proofs in analysis involve just chaining a bunch of inequalities together

if that's the proof, that's the proof!

there's not much you can do about it, unless the idea is that we should pretend that the essential idea of the proof is something other than "Chain all these inequalities together"

[winged urchin]

Certainly when one knows an intuitive or less symbol pushing explanation to a proof then that is more ideal but yeah, sometimes all you have to rest your argument on is a bunch of algebra and use of already established results

[long pelican]

We're in #math-pedagogy, and there's plenty wrong with 2 column proofs as they're currently implemented in education

Students leave high school with completely wrong ideas of what proof means.
Fun example: Someone says "I hated proofs, I liked solving for x better". Do they realize that the method in solving for x is a proof that there are no other solutions?

10th grade algebra teacher in the high school I used to, on the topic of integrating proofs into their class: "No, I would never ask them to prove something, because proofs are scary and students lock up. I would rather ask them to explain or something like that" [moral of the story: proof and explanation are two different things]

[winged urchin]

I think there are some implicit assumptions in what kind of problems and what methods you're talking about in your statement about the solution method being the proof of no other solutions. Or at least, I'm sure a student could 'solve' a problem and not get all solutions. But I think that's mainly a distraction to the conversation

I will admit that I think proofs are misunderstood for sure. Even in myself I think I noticed I used to be less confident in what is and isnt a proof, for sure

[molten urchin]

Yeah, and what students probably mean when they say they can 'solve' but not 'prove', they mean algorithmic plug-chug by the former and having to think a bit on their own in the latter.

[winged urchin]

I am a little hesitant but I think I would say that a student's ability to explain something in math is probably of higher value than their ability to prove something. But on the flip side I do notice sometimes... too high of trust placed in explanations. A student will explain their perception of a problem and believe based on their understanding that whatever statement we're looking at 'must' be true (or false, I know I dont have an immediate concrete example in my head for this)

Proofs are also kind of like writing proper, faultless sentences in a particular language whereas explaining something is more like just being able to write something that gets across your point more or less, so it'd make sense that students might get turned off when (sorta like) forgetting a comma gets them a mark deducted (at best) or ruins the whole proof (at worst)

Also for proofs I find there is kind of a 'messy toolbox' problem. Sometimes mathematicians talk about our 'toolbox' of theorems or whatever. And students are expected to know a certain amount of pre-requisite material whether they remember it or not.

Once you want to tackle a proof you need to be aware of (or look through) the toolbox you have. This is where students really struggle I find. Their 'toolboxes' are messy and they may not even have a very good idea of what's in their toolbox

I find as a tutor, even just writing the identities that may apply to our immediate area can help their not quagmire as much (writing the log laws if we're working on those; writing elementary derivatives or integrals; or writing some of the key logical relationships)

Two column proofs seem to be, at their heart, just explaining your steps. I don't think we want to lose that, no? Or do we think that a student just explaining more loosely to be ideal? If we want students to explain their steps, what would an alternative to two column proofs look like?

Is the stigma of two column proofs coming from the method or is it just proofs as a whole (to highschool level students) that make it seem like a bad method?

[long pelican]

I meant solving an equation, for example:
2x + 3 = 0 implies 2x = -3 implies x = -3/2
That method gives a fully logical proof that if there is a real number x such that 2x + 3 = 0, then x must be -3/2. It actually rigorously eliminates other solutions. Hence being a proof that the only possible solution is -3/2

Obviously in the US I wouldn't expect a single high school graduate (except the ones who learned math outside school) and maybe 30% of high school teachers to realize that connection on their own

[molten urchin]

Right, I agree the distinction is not understood, but I feel students know how to move symbols around for solving equations, but tend to have a tough time when the arguments are abstract.

At the very least I didn't have a fun time in middle school geometry, fortunately in HS my teacher did some proofs in basic set theory.

[long pelican]

The thing is they are not tested in abstract arguments, and instruction in abstract arguments is practically zero. So for argument's sake we can't rule out the hypothesis that they will actually be successful in abstract arguments given proper instruction and practice in such

All we know is that they are bad at it after going through the current system

... which is more or less expected

[wispy slate]

The largest problem is that students are taught backwards; often they are presented an equation or an algorithm that solves some problem without understanding the derivation or reasoning behind it

The "solving equations for x plug and chug" is just an algorithm they learn

[long pelican]

I find that this image summarizes the weaknesses of incoming college math students in the US very well

These issues have existed for a long time, make no mistake about it; it's also obvious teachers are (if unintentionally) encouraging such practices due to many reasons (one being standardized testing pressure); people in charge have tried a lot of reforms but they all don't solve these weaknesses. Hmm...

[wispy slate]

the first bullet is very accurate

[winged urchin]

Ahahah

This reminds me

So I tutor, and often I'll try to poker face their guesses

And sometimes I purposely act contrary to whether they have the right answer or not, though only with a few I've really gotten to know

Like, highschool, late elementary level

And while ultimately I am aiming for them to tell me why they're right, it does get one of them frustrated in the moment. Not in a very serious way but ya ahah

[wispy slate]

means the gears are turning in there

[molten urchin]

I think there was some discourse before about assigning students things to read and discover on their own. I feel doing more of that could be productive, because most of these problems seem to stem from spoonfeeding each and everything.

[astral laurel]

Do you mind explaining more? Also I assume 2-column proofs refer to aligned equations with equality of expressions at the left and either some reasoning/text, or even referencing earlier numbered equations at the right.

If I were writing solutions at an undergraduate level I thought 2-column proofs are a good way to tell undergraduates they can and should support their work directly with direct reference to supporting axioms

[pastel horizon]

This is the two column proof I think

[astral laurel]

Yeah essentially the same

[pastel horizon]

Undergraduates, sure. I wasn't a fan of using it at secondary level though

[astral laurel]

I'm at a more applied-math side rather than pure-math person, so I think my context slightly differs. At a secondary level locally, I'd think 2-column proof are still good for geometry as shown in the example. A danger of this is memorisation of geometric rules but I'm not sure how damaging this would be

[pastel horizon]

I didn't think it was a bad tool to structure arguments, I thought it was bad as a first introduction to proofs for someone in secondary since it makes them seem like a formal process and dull when proofs can be exciting and interesting. Once they've developed sufficient maturity then it's good to reintroduce it

[astral laurel]

But triangle geometry is dull and boring, and it's taught endlessly in secondary school education

I'd say many questions at a secondary level are also not complex enough for much explanation. You can algebra/symbol-soup it and it won't look that unreadable

It's possible to replace statements like "f is convex, so since f'(x*) is 0, x* takes a minimal value" into raw symbols, even if this is rare and would look like a highly mechanistic answer to a typical secondary-school find minimum cost of container material question

[pastel horizon]

Yeah agree

[pastel horizon]

Agree to disagree here. But one thing's for sure, adding in pro formas makes it worse

In fact I'd even go a step further and say you should replace as much as possible with symbols. Later on, they'll be expected to do so anyway

[astral laurel]

I'd say it's mostly symbols for formal writing along with some accompanying exposition as is standard in academic writing

[wispy slate]

What exactly is mathematical rigor

[long pelican]

Yeah so to clarify, if some textbook or paper presents a direct proof in two column form, it’s still a valid proof and might even be a good presentation

The problem just lies in how teachers implement it to “teach” mathematical reasoning and logic and proof writing

[pastel horizon]

Yeah, it's like they're saying all proofs need to be a pro forma

[wispy slate]

Have you read "A Mathematician's Lament?" It's exceptional

https://www.maa.org/external_archive/devlin/LockhartsLament.pdf "What is happening is the systematic undermining of the student’s intuition. A proof, that is,
a mathematical argument, is a work of fiction, a poem. Its goal is to satisfy. A beautiful proof
should explain, and it should explain clearly, deeply, and elegantly. A well-written, well-crafted
argument should feel like a splash of cool water, and be a beacon of light— it should refresh the
spirit and illuminate the mind. And it should be charming.
There is nothing charming about what passes for proof in geometry class. Students are
presented a rigid and dogmatic format in which their so-called “proofs” are to be conducted— a
format as unnecessary and inappropriate as insisting that children who wish to plant a garden
refer to their flowers by genus and species"

[long pelican]

Yep, exactly that!

[ancient elk]

does somebody have a good book recomendation in math teaching for deaf people?

[long pelican]

I’m deaf and I just used the same books others use 🤷‍♂️

[long pelican]

I did have a proper support system in school that ensured I could understand the teacher though

[deep kindle]

How did that work, if you don’t mind me asking?

Was there someone using sign-language or etc.

[long pelican]

I was given a FM receiver and gave the teacher the microphone at the start of every class

To help me hear the teacher

[deep kindle]

Ooh that’s kinda cool

[wispy slate]

Did you go to/are you in university? If you're deaf the likelihood of us sharing an alma mater goes up like... A lot 😂

[long pelican]

I went to MIT, graduated in 2016!

[ancient elk]

That's actually really conforting to read.

I'm in the end of my graduation and this student enrolled in math last year. How ever, most university teachers have no idea on how to teach or behave whenever he is in class

This boy also doesn't speak/read portuguese (i'm Brazilian), he uses sign language.

By july, one of my professors invited me to a study group in math teaching for deaf people, we hope to come up with some resources for his aid.

[long pelican]

I would suggest to try to interact with him a lot even if he isn’t saying much… especially if he’s not saying much… for deaf people like me it actually takes a lot of courage to talk when you have a decent chance of not understanding what people are saying

Takes even more courage to talk when you are with a group and can’t hear what’s happening even though it looks Iike you’re right there and listening

[wispy slate]

nvm then 😭 anyway I coded up the thing that I wanted in sagemath, it computes e^(kt) where k = x + (p(x)) in R[x]/(p(x)) but only for p(x) = x^n +/- 1 unfortunately lol
I will have to play around with more stuff to generalize it if possible
And to get the coefficients for each term in R(k) it uses the DFT idea you gave me fam ty

[long pelican]

Good luck!

[wispy slate]

It doesn't print things very nicely nor does it replace expressions with trig/hyperbolic trig functions but it works!

anyway wrong channel LOL

[slim thistle]

2nd bullet point I imagine is students cant pick up on genral themes and patterns or modify examples and play around with them

[flat cargo]

this is really good

I can vouch, HS geometry killed geometry for me

[long pelican]

Actually got a good example of this today; my student was asking me about this question on this week's homework, and asked me "So does it want me to find the derivative of 1/x^2 or the antiderivative of 1/x^2?"

re the 2nd bullet point, "while they can usually read the template examples, they cannot read the text to extract conceptual information"

[winged urchin]

Ah yes. Seeing the d/dx and just assuming they have to differentiate

The part of me that doesn't want to give the student the benefit of doubt asks how long they spent thinking of their own question

Did they just look at the question get this immediate thought in their head then just thought to ask you instead of figuring it out for themselves

[long pelican]

It was on homework so she probably tried to do it herself but couldn't understand what it was saying

The most likely thing that's happening here is that she [a typical student] is used to knowing what to do because the problem's instruction looks like one in worked out examples in lectures or easier homework questions

so she never practiced reading math to get what it's saying

Whereas for us, it's transparent that "solutions to dF/dx = 1/x^2" can only mean that F is an antiderivative of 1/x^2

[winged urchin]

There is some context there right. Like... If we had F then that exact same statement would actually be asking for values of x that satisfy that

Unless you used a triple equal sign. Then it absolutely couldn't be that

[long pelican]

The theme of the week is differential equations fortunately

[winged urchin]

Ya for sure. We don't necessarily want to get too nitpicky with notation since context should remove ambiguity

I wonder if even just a simple "what are all solutions, F, to"

[long pelican]

That would've been better in hindsight

[winged urchin]

But I have to imagine that there were problems asking to show something solves a differential equation...

Which should've queued them up for this at least

[long pelican]

Don't think that's the main confusion though

it's that she didn't know how to parse this

"does it want us to find the derivative of 1/x^2 or antiderivative of 1/x^2?"

The famous "does it want us to..." question

-> This isn't like any example I'm familiar with so I can't tell what it wants me to do

[winged urchin]

I'll just say that if I had to bet on what really is the true reason she asked that question, I feel like I would be most confident saying that it's because of a lack of effort (or no effort)

But that doesn't move the conversation forward very well. And although some students don't put effort into their homework, I can believe some genuinely would be confused

Like if I were in person tutoring with them I'd ask like.. what do you think "dF/dx" means?

Hopefully they identify it as the derivative of F

Then I'd try to get them to read out the statement then

"The derivative of F is equal to 1/x^2"

If we have realized we want F then we see that whatever the answer is, it's derivative should be 1/x^2

Then perhaps we could take random functions for F and do their derivatives (although that now makes it seem like it does want us to differentiate but that's just one solution method)

But hopefully after trying a couple functions they'd realize they want to kinda go backwards in this differentiation process rather than forward so we need the opposite of a derivative

I might've also used the language of inverses

Our F is kinda stuck in this derivative

Think of that like a function, generally if we want to reverse a function we use its inverse and integration is kind of the inverse of derivatives in that way

[long pelican]

I took the route of asking her what the definition of an antiderivative is (the first half of the class was on integrals and techniques of integration...)

hoping she'd say { F | F' = f }

and then realize the definition matches the equation exactly

She couldn't state the definition though

She did say area under the curve but I pointed out that's a definite integral

I wasn't surprised though

I knew it's very commonplace that a student in a class spends weeks on techniques of integration, aces the test, but turns out to be unable to state the definition of what he's even calculating

[winged urchin]

Though isn't that sometimes a criticism of math? Or at least something bemoaned to a degree? What seems like rote memorization.

You sometimes get students complaining about that sort of thing for instance with IVT. State the definition of the intermediate value theorem

[long pelican]

Not quite, I wasn't asking as if she was supposed to memorize it

[winged urchin]

And even if they remember the idea of IVT. Like they can draw a graph to show the idea or whatever. They might still mix up the definition cause they aren't careful enough

[long pelican]

I was more asking for her to quickly reconstruct the mathematical definition based on her understanding

in her own words

or whatnot

[winged urchin]

Right

I wonder if she could've answered a more specific question

Like what is the antiderivative of x

Or perhaps when she said it was an area I'd ask if the antiderivative is always a number then?

[long pelican]

That tutoring session looks like it'll take a while 😛

[winged urchin]

Yeah I take a fairly Socratic approach in my tutoring usually ahah

[long pelican]

We haven't even gotten to the part about why the domain being R - {0} is relevant in the problem!

[winged urchin]

Just asking questions over and over until they realize what's right

[long pelican]

It's an extremely complicated problem if you have to strain to remember the definition of antiderivative tbh

in the previous part she computed (basically) that ker(d/dx) on Diff(R - {0}, R) is 2-dimensional instead of 1-dimensional

But in this part later on she claimed the solutions are -1/x + C

despite having gotten a 2-dimensional kernel in part b

eeeeeee

[winged urchin]

It does have a kind of implicitness to it that I notice in a lot of questions that gives students problems.

They can calculate a derivative but if you give them the result and ask what it could have come from they just get confused

Oh lordie ahah. When their answers are contradictory and they either don't notice or just stick with it cause they don't have time / run out of ideas

[long pelican]

Yah, that and the pervasiveness of +C being automatic

without using brain

[winged urchin]

I tell my students to at leastttt mention the problem with your answers if you notice one

Maybe you get a pity point or half point lol

[wispy slate]

Hello teachers! Do you have any experience in explaining implication to students aged 13-15? What worked/didn't work well?:)

[winged urchin]

I've tutored a few students in logic courses. Although that is a fairly young age so it might take more time.

Id personally try to use real life examples that are as clear as I can. Like

If it's raining then it's wet outside

Vs

If it's wet outside then it's raining

You could ask how these statements are different or what extra conditions might one need to make them always true. Also showing how the truth of an implication changes when you reverse the arguments

I'm assuming they've gotten into like the and/or operators and negation possibly?

I'm actually curious what a student at that age is doing learning logic, might help give advice since it seems like a sort of an extraordinary case

[long pelican]

To add to it, the most important ingredient is time. Allow them lots of time to digest and think about new logical ways of thinking

First time I ever read the epsilon delta definition of limit, it took me 3 hours of going back and forth between reading the definition and going through the examples until I got it, but nowadays it takes me the same amount of seconds to understand it as to read it

[winged urchin]

Yeah. Examples are really important. Reminds me of a quote in a textbook I vaguely remember. Something about when [the person saying the quote] wants to learn something new they start by collecting many many varied examples of that thing. Very good advice

[wispy slate]

Nonexamples too are really good for more abstract definitions

[wispy slate]

Those examples don't really explain why 0->1 is true for example (which is often the hardest to understand). Best example I've found was: "If you pass the exam you will pass the course" - the 0->1 case can be interpreted as passing the exam wasn't the only condition for you to pass the course, but it was sufficient.

[wispy slate]

let's say I learn some math stuff right?

and

I learned it from a book which wasn't rigorous at all

how do I know what stuff to learn so that I can like get the mathematical rigor behind it like the theorems and proofs and stuff?

pedagogy is defined as "the method and practice of teaching" and I'm teaching myself mathematics

[austere inlet]

no, read the channel description.

[wispy slate]

fair enough

[patent beacon]

What levels/ages do people usually teach here? University, high school, primary?

[vagrant meadow]

i can explain and tutor diff eqs and linear algebra really well, but today i was completely floundering trying to explain that sqrt(x^5)=x^2sqrt(x) not x^2*x. does anyone have any advice on explaining the simple stuff? im just horrendous trying to explain basic algebra and arithmetic

[vagrant meadow]

idk about other people but i tutor college/university students

[winged urchin]

A couple ideas off the top of my head:

• If they agree that sqrt(something) = something^1/2 and are okay with their exponent laws then perhaps I could lead them down this route to seeing the exponent ends up as 2.5, not 3

• In highschool at least they are perhaps familiar with mixed and entire radicals (2sqrt(3) vs sqrt(12)). Perhaps showing how sqrt(12) breaks into sqrt(4 3) then sqrt(4)sqrt(3) then 2sqrt(3). Something they can also check with their calculator

• I could perhaps talk about how square root is an operator, how it's doing things to stuff in the expression. And how it acts on things in a product is it applies itself to each term in that product. Thus if we see x^5 as a product of x^4 and x we see we want to act on x^4 and x. Now we are probably comfortable seeing that sqrt(x^4) gives the x^2. If they believe the sqrt(x) gives them x challenge them on that. "So you're saying the action of sqrt on x leaves it as itself? So the square root of 4 is 4, you claim?"

What did they seem to be having trouble with? Were they not believing you or something?

[patent beacon]

Nice, I'm a high school teacher just wanted to know where people are coming from in terms of who they're teaching, because I know my pedagogy changes even across the year groups I teach

I imagine most people here are College/University level

[winged urchin]

Oh right, sorry. I tutor pretty much at all levels up to the end of undergraduate uni

I've had students in elementary, high school, etc..

[long pelican]

I'm one of the 2 main instructors for a freshman calc 2 class this semester

[vagrant meadow]

i actually kind of tried all three of those avenues you described (didn't spend as much time on the 3rd), but nothing seemed to stick for the student. I'd explain why that was wrong and they'd say "okay", and then a minute or two later when I ask them the same question again (related to a different part of the problem such as "what is sqrt(y^5)") they would tell me that sqrt(y^5)=y^2*y even though I have the correct result written out multiple times on the screen...
I've been in situations where I know my explanation is good but the student just doesnt get it because of something on their end, but I don't think that was the case this time; I think my approach just sucked.
Perhaps I was skipping through these different perspectives too much because I couldn't tell which one I should really focus on? I think I imagine a better tutor feeling these different approaches out and catching on to which one seemed to work best for the student, but I just didn't and it was painful for everyone involved.

[long pelican]

Did you ask him to show his derivation for sqrt(y^5) = y^2*y ?

[sinful thistle]

I'd try something like:

• Square root is unique (as long as you take it positive). To see that, just intuitively: if you pick something even just a little bit smaller than the square root, then squaring it will give you something a little bit smaller than squaring the square root, and same thing if you pick something bigger, you get something too big.
• Check that sqrt(ab) = sqrt(a)sqrt(b) by simply checking (sqrt(a)sqrt(b))² = ab and using uniqueness.
• Let a = x^4 and b = x, then you get: sqrt(ab) = sqrt(x^5) = sqrt(a)sqrt(b) = sqrt(x^4)sqrt(x) and then to simplify sqrt(x^4), you use uniqueness again

But this is assuming the person you're teaching is familiar enough with basic logic to follow such a reasoning (in particular, the uniqueness argument is probably not obvious for the average highschooler).

[long pelican]

The hardest students to tutor are also going to happen to be the ones who have not done serious logical reasoning with math, and have accumulated lots of misconceptions

so I would not be surprised if he could not follow that

[sinful thistle]

Yeah, that's fair 🤷

[pastel horizon]

Secondary, 11-16

And yeah you're right, there's a lot of ideas I don't necessarily agree with but then they'll say it's for undergrad level and then I think "fair enough, sounds ok"

[pastel horizon]

I've only tutored once but I had a challenging kid to tutor who wouldn't really engage. At least I managed to get his school to give him intervention

If they're engaged at least that's something you can work with

[tepid smelt]

High school here also. I have tutored from first grade to college also. It is an unusually difficult year for me with the learning loss and built up trauma from last year. It is also depressing how my students who are struggling the most and have had the most trauma are almost all black/brown students. Its like the secondary level is a clear picture of whats wrong with this country.

[tepid smelt]

These are also the most rewarding to work with as you start to see growth. Having tutored the opposite also where you sit back and they just crank out there hw with no issues and only are in tutoring because parents in that income bracket have all there kids in tutoring.

[long pelican]

I wonder if not engaging can also be a symptom of being totally lost and unable to comprehend your explanations, which suggests serious gaps in understanding. Could also be that he has fully learned the attitude to math his teacher indirectly taught him which is that problem solutions are always provided by the teacher and he never has to think of one on his own

[tepid smelt]

Now imagine how challenging that would be with half a class full of kids just like that but you can't spend quality time with them all. Thats just a small part of the difficulty of teaching Sometimes with these kids they need time to get comfortable with you. Often parents force them to go and its a reminder of something they struggle with and want to avoid that feeling. Start with really easy problems and build up their confidence and get to know them a little. Tutoring is a personal interaction you should try to learn a little bit about what they like outside of just school.

[vagrant meadow]

I'd ask "How are you getting that" and they wouldn't really have an answer.

[vagrant meadow]

Yeah I think I'd have completely lost her if I went in talking about uniqueness. I tried the sqrt(x^5) = sqrt(x^4)sqrt(x) but its like she couldn't follow any sort of explanation i gave. I write things like that on the screen but it's like it was going in one ear and out the other. I don't think I was able to engage her.

[astral laurel]

I don't understand, isn't x^2 * x just x^3?

[vagrant meadow]

For context I work at my previous college's tutoring center. So these are supposed to be short-ish interactions to answer quick questions. I think I would have taken a different approach if this was a 1:1 hour long tutoring center

[astral laurel]

Unless you mean x^{2x} but that's a whole other thing you can just show to be different by counterexample

[vagrant meadow]

Yes, it is. How she thought that was a sensible result for the square root of x^5 is something I'm still trying to figure out.

I had it written in like 2-3 different places on the screen and yet every time id ask she'd give the same incorrect answer

[astral laurel]

Maybe don't 'explain' it

Just say that since the coefficients don't match

They are not the same

You can try instead of square rooting, write explicitly as x^{1/2}

And index-rule from there

[vagrant meadow]

On reflection I think she may just be a student who would benefit far more from a 1:1 private tutor than going to a tutoring center. The misconceptions and missing foundational understanding was just too much to fix in one session.

[vagrant meadow]

Right, I tried that. But she, unsurprisingly, lacked understanding of basic exponent rules.

[astral laurel]

The first thing I would focus on is correctness before understanding - or at least maybe that's more effective locally since people are results-driven here

[vagrant meadow]

It felt like this was the first time she'd ever seen the square root written as raising something to the 1/2 power

[astral laurel]

Even if they don't 'understand' ^(1/2) I don't see why they would persist after being told it's wrong, on a comparison of coefficient basis

In a sense I'm on the same boat as you I guess

But I would try to drill the correct result I suppose

They don't have to 'understand' the rules as much as follow correct algorithms with respect to indexing I think

Understanding would be more like transformation from standard space to logarithmic space, or something like that, or understanding some definition of e^x

But that can come later

[vagrant meadow]

I don't know about the word "persist". I'd explain why its wrong and show her the correct way and she'd say that she got it, but asking again just a little bit later would result in her repeating the same mistake.
I think I also disagree with the notion that doing things correctly is more important than understanding. But I suppose that's just a philosophical difference. The student probably couldn't give two shits whether or not she actually understands it if she can get it right and pass the class 🙄

[astral laurel]

It's not that I think it is more important on a philosophical level, but I do think it is important to fulfil verifiable educational policies

To put it this way: I am obligated to teach a syllabus to students, but I welcome students' questions of all levels and will admit if it surpasses my knowledge, and I enjoy sharing things even if they are out of syllabus, but it tends to cause panic

[vagrant meadow]

True, but even without a higher level explanation far beyond them that makes this lower level stuff intuitive, I still think it's possible for algebra students to understand the things that they're learning. The rules of square roots are very simple and quite straightforward to show, especially with examples. I think if a professor explains it well (and the student is actually paying attention) then the student can understand all of those rules at an intuitive level. That being something I believe would serve them far better than just being able to reliably do the rote computations correctly because they've done it enough times. :/
But I acknowledge that it's just a difference of opinion.

[long pelican]

Before the start of this semester I would have agreed that professor explaining well is the most important thing. But mid-way into the semester I now believe well-designed homework problems is the most important thing! The reason for the change is I noticed that I would lose students no matter how clear and beautiful my explanations were, simply because even simple logical arguments and simple impromptu unpracticed calculations took too much time to do and process for them. Meaning that the only way I could lecture and not lose them was to never prove anything

To fix that, I put hard problems on the homework that forced them to spend time thinking about logic and math for themselves

#calculus message like this one!

Check back in a few months to find out the results of my "experiment"

[winged urchin]

I like including an example in the question

Have you had any complaints about these problems yet? =p

[long pelican]

I asked them how they found homework 5 (which is when I really ramped up the difficulty) and they all said it was HARD, but not in a complaining sort of way. When I asked last week if the class was taking a lot of time, they actually said about the same as other classes

Homework 5 featured the following question

aimed at calc 2 students 😛
I was very impressed by the submissions and the amount of correct work (at a glance; I'm not the grader)

[winged urchin]

A fair few hints

Probably a good thing, one maybe could argue that you'd want students to be able to navigate themselves but that's probably too harsh for calc 2 students

[long pelican]

Yep, yep

[winged urchin]

Did anyone get the second part of 4e wrong? That seems exceptionally simple

[long pelican]

hasn't been graded yet but 4e was actually the worst-performing question from my quick glance

[winged urchin]

Dang, that's unfortunate

It doesn't have hints, I suppose

[long pelican]

Yep

[winged urchin]

But that's gotta be the mathematical illiteracy coming in more than anything

[long pelican]

I agree

[winged urchin]

You know, I wonderrr if some students had a variable confusion issue there

I wouldn't be entirely too surprised if someone said we can't use substitution there because both statements have 'x'

And if they did substitution they'd have written some other variable (likely u)

But again, that's just understanding the language

If that was indeed a problem

[long pelican]

That's a good demonstration of why this type of problem on the homework is so important and how it affects how they receive my explanations

If homework was all easy: when I explain that you can use different variable names as the variable of integration, they'll copy it down in the notes as a rule to remember

After doing the gamma function question, hopefully they understand it's logically obvious why

I wonder...

Maybe math education in K-12 is lacking not necessarily because teachers aren't good at explaining
Maybe it's because 99% of students don't do a single real problem in their homework

K-12 homework is usually all rote exercises

solve this equation x20, factor this quadratic x20

So their mathematical mental models of everything are extremely limited and weak

This now leads to the question: From where are we supposed to expect teachers to find good homework problems?

I created these out of my own head

but that requires a high level of content knowledge

[long pelican]

And for high level classes, homework involves word problems and such but it's still completely rote: just follow the textbook examples and solve them in the same way

[vagrant meadow]

Very good point

[pastel sundial]

is this for high level classes at the hs level or are you also talking early undergrad?

[long pelican]

High school level

[pastel sundial]

specifically, are you also referring to proof questions that are just definition pushing

[long pelican]

Hmm those are rote but useful too to start with

I don’t really have any complaints about college proof based classes from what I’ve seen

The students I have are fresh out of high school and they need serious catching up as to what math is even about

[pastel sundial]

is this for a first year undergrad calc 2 class?

[long pelican]

Yeah

[vagrant meadow]

I think I just have a problem in general with assigning math homework, at least at the college level/in upper level classes. By the time the student is taking calculus, I think they should be responsible enough to decide how much practice they need in the class, and assigning homework can end up pointlessly taking up the time of students who don't need the amount of practice that is required of them (experiencing this in the DE class I have to retake because the original was applied diff eqs).
But looking at it from the perspective of using provided worksheets as a tool to teach the concepts rather than just reinforcing them, I'm starting to change my mind. Especially if the textbook doesn't supply questions of this quality.
If I am to become a professor, your worksheets provide a model that I really admire.

[vagrant meadow]

That gamma function question is just... mwah

[vagrant meadow]

I very much look forward to hearing how the class does overall

[long pelican]

I’ll be sure to let you know!

[tepid smelt]

I absolutely agree students should be presented difficult/interesting problems. I get inspiration from contest math at the k-12 level. Though the problems at the k-12 level are so much more then just presenting good problems. I like how you provided appropriate hints also. For difficult problems you need to find the right balance of struggle to keep the students engaged but not shut down.

[long pelican]

One of the the teachers at my high school 10 years ago used AMC problems as challenge problems for any students who finished the class work early. All well and good, but I read a paper yesterday that cited another paper which actually found that having creative problem solving homework benefitted the poorly performing students the most! Perhaps the idea of “challenge problems for the bright students only” is completely wrong…

The type of task that gave the most benefit had the following constraints: the solution path needs to be non-obvious, the solution needs to be constructed by the student himself, and there should be an appropriate means by the student to justify their answer and solution as correct

The last one means that open ended “collect data in real life and do some calculations” type projects were no good

Harder contest math problems fit the criteria perfectly

[patent beacon]

With my teaching, I approach it as being a balance of fluency and understanding (teaching 12-18 y/o's). So in terms of homework, sometimes it does go down the path of repetition just because it's a piece of content that you must be fluent in because you'll encounter it so much that you want it to be second nature almost. At the same time, setting something that's a bit more problem-solving (which here has been talked about as being the difficult/interesting problems) mixes it up and gets them to stretch their understanding.

At the same time, I'm aware of the motivation of a teenager because there will be student who aren't motivated to do work outside of class so there's that consideration also. But things like, if I'm teaching indices for example to 13 y/o, I might set the question of Find the only solution to a^b = b^a as homework, something along those lines. Won't take too long, but they need to do some investigating. Even younger kids can get "Find me a square whose Perimeter = Area, then do the same thing for a rectangle."

[long pelican]

Those 2 examples are pretty good for their age. Do you have a repository of such problems or do you try to make them up as you go?

Also, another purpose of stretching their understanding in homework for me is a selfish one: so that they won't get lost in my explanations anymore, because they have practiced the logical thinking and math literacy necessary to understand a proof that A implies B

For example, if such a proof uses an algebraic step or a u-substitution as an intermediate step, it's sure to lose them if you don't give them exercises to practice thinking in that way

namely using algebra or calculus to actually facilitate a proof of a theorem or other similar goal, rather than being an end in itself (like on a typical homework or test question)

[patent beacon]

I usually just pick them up as I got and tuck them away for the right moment. I don't ask them to prove anything, but getting them to discover those interesting math facts but experimenting is my purpose. If you want to steal them and then extend them once more, get them to find the only right triangle with and area = perimeter 👌

Next year, I want to bring in some more justifying, reasoning and deductions by way of Venn diagrams, so I'll see how that goes.

For older students, Fermi Problems have been a nice homework assignment. Gets them working with standard notation, they can ask and estimate solutions to some interesting questions. How much does the Pyramid of Giza way? How many litres of water will we need to fill up every room in the school?

The "Highest level" of homework I've ever given is an Olympiad Question tk my Extention Mathematics class. They had a few weeks to do it and they could collaborate, it was nice that they felt the satisfaction of answering that type of question.

So hopefully a bit more logic. I've been thinking of just setting them up with
All A are B
C is B
Therefore, C is A

Get them to fill it in with whatever they want

[long pelican]

The unique thing about olympiad problems is that they package all of the parts of a good problem into one, whereas Fermi problems are open-ended but leave you unsatisfied because you can't really prove your result

Of course, olympiad problems are a tad too hard but maybe not as hard as you think, given that you're able to provide hints as necessary and they get a lot more time than the students actually taking the olympiad exams get

[patent beacon]

That's why I only give them to Extension, because they're more into spending time thinking about maths and are keen to learning and understanding rather than applying a method 👍

[long pelican]

If my ongoing experiment works out, the conclusion from that is that giving them to regular students might benefit them dramatically too

There's already literature supporting that, surprisingly

You just have to find the authors that actually know what problem solving is

and are able to distinguish problem solving homework from homework that "looks" like problem solving but is actually rote

[mint lark]

Can you link this study?

This confirms my suspicions but I want to throw it at people

[long pelican]

https://link.springer.com/content/pdf/10.1007/s11858-017-0867-3.pdf

In addition, it was found that this performance
difference was largest for the students with the lowest
cognitive proficiency (measured by standard psychology
Principles for designing mathematical tasks that enhance imitative and creative reasoning 943
1 3
tests; Operation span and Ravens APM). In other words,
it was the students with the lowest cognitive proficiency
that had most to gain by CMR practice compared to AR
practice. This finding contradicts the common belief that
tasks requiring creative reasoning are more suitable for
high-performing students

[mint lark]

Thankie

[lethal sapphire]

Hey, I'm a high school student and I'd like to start a tutoring service in my area. I know there are a lot of deficits in the public school system in terms of math education, and what I've picked up on most is the inability for students to solve problems they have never seen. Both from tutoring friends, or from what I've noticed in my classes, people often have no idea how to approach a problem, and often simply give up. When someone says "yeah I have no idea how to do that", everyone usually agrees and the teacher goes "yeah that's ok, we're going to learn". Which isn't inherently bad, but that general attitude just seems to hinder problem solving and be more focused on 'just do what I tell you'.

The other is a lack of pandering to the needs of higher performing students. I assume this is a problem in more than just my district, as it's a pretty wealthy area with good schools. From my own experience, I was always bored in math. I can remember back to like the second grade, finishing the classwork in like 5-10 minutes and literally just creating problems for myself out of boredom. I knew the concept of a variable and would play around with it for fun and solve simple equations. Looking back, I wonder how many similar students there are who could benefit from a more personalized education. For me this got better as I learned algebra, and I was content with my classes. Soon, however, (and this is primarily what I'd like to help people with) I found my 8th and 9th grade courses easy, and I started to learn more. After a year of self-teaching, I began to get quite good at it, and rapidly progressed from taking geometry in the summer going into 9th to moving to precalc the second half of my freshman year. However, I had gaps and overall self-teaching wasn't the best.

This moves into what i would like to do. I want to give kids who were like me, very interested in learning math and progressing the opportunity to do so. I want to focus on problem solving, possibly proof writing (though im not too great with them myself, so I've still got some to learn in that area), and being able to learn higher level math, or the math they wish to learn. If a kid comes in and he wants to learn calculus: great, I'll teach him calculus. I want to help kids get interested in math, and be able to enjoy it. I want to give them a proper, personalized education in a way that school, or even places like RSM or Kumon fall short.

So, where do I start? I'm sure there's a lot of things I've gotten wrong, misinterpreted, there may be many flaws, etc., so where can I go to learn about how the current system falls short, where it needs improvement, and what can be done about it? Is this really something that would be of use to people, or is it such a rare case that it may be too difficult? Really, any information you have would greatly appreciated :)

[long pelican]

You should look at Art of Problem Solving if you haven't, I think they are solving the same problem you are

[tepid smelt]

As a teacher your right my high performing students are not a concern for me. They are such a small portion of any class though and with 30+ kids many of which are years behind and dealing with all sorts of tramua they are my main concern as I need to help them pass as we are evaluated on pass rates. The art of problem solving is great. In particular they have a app alcumus which offers good problems taken from math contests. I encourage my high performers to set up an account and do extra problems with alcums on there phones. I also have old amc tests they can work on. Its not perfect but it helps them with boredom and gain better problem solving skill.

Brilliant also offers free accounts for teachers and has some really good problems created by various users I can assign students through google classroom.

I have near 200 kids total and maybe 5 to 10 max who would be high performers so logistically I do cater to the majority who hate math and are not even close to grade level due to our system of passing everyone and little accountability before HS

[pastel horizon]

If you're tutoring you'd usually only have one student anyway. It's so much easier to cater to their needs

[long pelican]

Do you teach proof school? If not, I must've misread a comment from earlier

[tepid smelt]

No I teach at a large city school. I just mentioned that school as one that has a good math curriculum and what could be possible with the right size classes and time.

[long pelican]

Ooooh I see

Does being evaluated on pass rates pressure you to drill the weak students on practicing problem types to build muscle memory on doing similar problems?

[tiny dove]

This may have only been my personal (regional) experience, but why is the concept of function not really introduced at the highschool level?

Students may have been told what functions are, but I think there is an extreme confusion present between the concept of "function" and "formula".

Why do we wait until Real Analysis to show students that a function from R to R can be completely crazy and doesnt have to be defined piecewise by formulas

Is it just too abstract?

[analog token]

I think that's because you need set theory

In order to "formalize" it

[tiny dove]

Well what if you just don't really make it formal. I don't know how hard basic set theory is for students if you do it very informally. Is the union or product of sets a difficult concept?

[long pelican]

Algebra 1 students are taught the funny definition of a function being a relation with a unique element having first coordinate x for each x in the domain. But the students are nowhere near fluent in this definition and none of the practice problems even use or mention it in a non-superficial way

[mint lark]

imo teaching students this definition is a bad idea (tm)

reject modernity (formalism) return to tradition (intuition)

[long pelican]

The definition I would find most appropriate for first-timers is "A function is a domain A, codomain B, and a rule which assigns to each element of A an element of B."

One can feel free to adjust the presentation of domain and codomain in that definition to be givens or whatnot

It might sound intimidating but it's the minimum level of rigor needed to ensure that they don't get a wishy-washy notion of function that bites them in the future

[tiny dove]

Yes this is totally fine, no formality needed. But saying "rule" can already be confusing. I think it is an improtant thing to realize that functions don't need to follow a formula or rule or anything. They can be literally any assignment. I think any basic programming experience teachers a teenager a better understanding of what the concept "function" refers to than the math curriculums I've seen

[tiny dove]

Because they don't really "get" the definition, exactly

[long pelican]

yeah, an improvement would be "A function is a domain A, a codomain B, and an assignment of one element of B to each element of A"

By the way, has anyone else had the shock of half their class writing utter mathematical nonsense on a problem in an exam when you thought they were doing fine in the class up to that point?

[tiny dove]

I never taught myself but I remember my math teacher from high school getting really angry at the class after every exam and telling us how stupid our mistakes were... I wonder what it felt like for him.

[long pelican]

So one of the questions was, essentially: Given that $f(x)=\Pr[X<x]$, what is the PDF of $X$ in terms of $f$?

[burnt vessel]

Icy001

[long pelican]

About half the students wrote $p(x)=\int_{-\infty}^\infty f(x),dx$ which is nonsense for more than one reason >.<

[burnt vessel]

Icy001

[long pelican]

It's like they don't even read their own math

[tiny dove]

ouch

see, the problem here is not one of algebra or deep concepts

[long pelican]

Is this shocking experience common? You think your students are doing wonderful and learning just great, and then they completely shatter your illusions on an exam

[tiny dove]

it's a problem of fundamental understanding of functional relationships between objects

I think but dunno Im not a pedagogue

[winged urchin]

I get utter nonsense in some material from professors lol

I just helped someone with a logic course and a problem on their practice midterm was to consider the statement

If x^2 > y^2 then x > y and x < -y

Then in another part it looked like the practice midterm was really using and properly (as an intersection)

Sometimes I smh hard even at stuff from teachers/profs lol

Sorry for any derailment. I just got back from that and it was on my mind

[long pelican]

I'm not sure what's nonsense about that sentence tbh

Is it that the midterm grouped it like (If x^2 > y^2 then x > y) and (x < -y)?

Ohhh

I bet students read it quickly and said "yep" that's true

but what's really true is x > y or x < -y

not x > y and x < -y

[real mauve]

i would say it's important to be willing to meet in the middle for some things

[long pelican]

x>y and x<-y can never be true (at least if y is positive)

[real mauve]

simply thinking "oh my students suck" and doubling down on what you're doing or becoming more strict will only make it worse

since it's not working, whatever the reason

[winged urchin]

It's actually really annoying, I asked the student to ask the prof to clarify because although it should be clear from context (hopefully) it really isn't good

[light pelican]

I feel like this experience is common among a lot of teachers. It can be really shocking learning how far students can get into a math program without understanding a lot of the basics. And their answers are just a realization of that. And I’m not sure how you can overcome that as a teacher as you can’t really go through the fundamentals with them in detail because you have your own material to teach. And on goes the cycle

[long pelican]

Fighting 12 years of indoctrinated nonsense basically

[winged urchin]

The best thing is when they try to answer a question entirely in words

99% of the time they're going off the rails

And just trying to use words that sound right

[long pelican]

Yeah

Probably putting the blame 50% on their 12 years of indoctrinated nonsense and 50% on softball homework that allowed them to complete it without fixing their nonsense

I started off the semester taking previous homework questions (and textbook questions) mostly

That might probably have been a big mistake

[tepid smelt]

Yes sadly not to mention state testing results typically inprove with this style also and we are evaluated based on those scores. I would prefer to go slower and deeper but we have to cover so much and the system encourages the drilling

[long pelican]

Hm hypothetical question, if state tests just gave the AMC 12 test every year (somehow adapted to each grade level), how would teaching styles change? For the better? For worse? Is it even possible to fake-teach students to do well on the AMC 12? If it isn't, then the only way left is to teach real math

It sounds nice but I'm not sure if there aren't unexpected consequences

[astral laurel]

I'm not sure if using deceptive notation is grounds for slamming students

[long pelican]

I do not see where notation is deceptive. The notion that capital X is not the same as lowercase x should be standard knowledge, and besides, capital X is a standard notation for a random variable anyway. Are you arguing that f usually stands for the pdf, hence students who did not read the problem assumed f was the pdf? I don't see how that's deceptive at all. We shouldn't be (and we aren't) teaching students a letter always means a certain thing

Besides, the nonsense they wrote down like the equation I wrote is independent of anything in the problem

[astral laurel]

I don't see how that's deceptive at all

Complains about people writing $\int_{R}f\dd{x}$

...

[burnt vessel]

ShatteredSunlight

[astral laurel]

You would unironically be believing in this mess of a 'proof' as well

I literally do not need to tell you the context, I can just assume you understand what each letter needs to be by its relations with others

[astral laurel]

The alternative to this is symbol soup, which is definitely not how I recommend questions be answered.

Anyway somehow, this channel is now used for flaming students when student-blaming is the worst mark of a 'teacher' - Focus on teaching material in a proper way. In no way will people improve by people needlessly badmouthing them or their technique. If you have so much time to write about them negatively, spend time with them if they are receptive and if they are not receptive (doesn't take a genius to know why), find out how better to make them more receptive.

[pastel horizon]

It's like complaining about someone from a foreign country making simple grammatical errors in English

Nobody will have perfect mathematical fluency when they're still learning, it's your job as a teacher to instill that in them. One of the things I've quickly learned is to never be surprised at misconceptions, it's almost a puzzle in a way to figure out why they think that way and how to guide them to the truth

[pastel horizon]

It also seems to be mainly used to bash the US education system when there's also other systems around the world with positives and negatives worth discussing

[wispy slate]

You're absolutely right but I believe the point of that particular discussion was to point out the failures of the system as they present in students rather than roast the students

like if you were talking about how awful the effects of radiation on survivors of Nagasaki were to demonstrate the evil of nuclear warfare, that's actually quite the opposite of victim blaming

you right tho and I cringe when people use this channel to snark at students

[real mauve]

#math-demagogy moment

[long pelican]

Ok I think you thought I literally just wrote that without context as the question. The real question was longer and talked about a random number generator

[long pelican]

Preaching to the choir here, I’m talking today about how I’m addressing these issues in my lectures and homework design.

@astral laurel somehow you single handedly turned this environment into a hostile one… everything you said was not necessary. Are you a teacher? I assume so. Let’s be civil

[long pelican]

Symbol soup and word salad are kind of the same problem

[long pelican]

And yeah, this. I am surprised at the extent of their weaknesses because I didn’t know what to expect, having assumed K-12 education would improve since I was in it. It is known that they aren’t quite where they should be, but not clear to what extent, and I’m never thinking to fail students or blame them because of that. On the other hand it is a monumental challenge to get them to understand the language of math, and I am searching and experimenting to know the best way in the short time I have if there is one . I think you (ShatteredSunlight) put some bad words in my mouth?

[astral laurel]

Are you civil when saying people do not read what they write?

[long pelican]

Lastly, the complaint was not what you wrote. In their expression, they defined the PDF to be a constant because the integral from -Infty to infty of a function is a number. Secondly, they just saw in part a that f was increasing, so this integral diverges

[long pelican]

I’m not gonna be able to get you to stop being hostile, so let’s take this conversation in a different direction. Do you have any insights into why that particular wrong answer might make (at least, the most) sense to a student?

[astral laurel]

You can just correct their mistakes that the integral in R being a number does not mean the PDF is uniform. If you are saying they are willingly saying something contradictory, then I think it's a case of "I wrote 2 contradictory statements but I sure hope one of them is right"

[long pelican]

Hmmm, I wasn’t expecting them to think of a uniform distribution. Anyway lecture’s coming up in 30 minutes so I’ll hopefully have an answer then!

[astral laurel]

For example, if given a question and I write both x=1, x=2 (does not happen for normal real numbers), I don't think the correct interpretation is that I don't know real numbers to not know a real number cannot take two values, but I hope that either =1 or =2 could be correct

[long pelican]

Ehh would your answer change if I said part a was to prove that f is increasing?

So they know that for a fact

[astral laurel]

Ok fine, if there is a monotonic function, and they are claiming the integral on R converges (or is represented by) to a real number, it's very illogical

[long pelican]

Yes indeed

[astral laurel]

You should point that out instead

Literally draw a non-decreasing function, and show its Riemann-Stieljes integral does not make sense if assigned any number if taken to infinity

Without more context I can't say more than directly show why wrong

[long pelican]

I will, but I would also like insight into their minds about why that was the most common response

Yeah that’s fair

[astral laurel]

I'd find that very weird considering it means that they think some integral of infinities converges without good reason, since I would think integral involving infinities seem less likely to converge (and I don't think they are assigning special numbers to the non-converging integrals)

Maybe you can say something to that effect? After all, I'm pretty sure there is a rigourous analysis statement on the tails of a function if the integral converges

@long pelican In any case, I apologise for my language. I only know too well my own mathematics is subpar. I personally wouldn't like it if my teachers/professors joked at my subpar mathematics at my expense. It's just that I felt the direction of the conversation was not going in a good direction.

[long pelican]

No problem 👍

[pastel horizon]

They are most likely confused by the definition of a pdf and just integrate over R and hope for the best

[analog token]

(what's a pdf ? For me it's a file format haha)

[pastel horizon]

Probability density function

[analog token]

Oh right, thanks

[pastel horizon]

The question says P(X<x) = f(x), find the pdf in terms of f

It seems like here f(x) is actually the cumulative distribution function

That's probably something they didn't appreciate

[long pelican]

Just got back from lectures, the students said some things!

Most of them did understand what f meant here (so this didn’t trip them up) but they didn’t realize the wording “find the pdf in terms of f” was a nontrivial problem instead of a memory recall question

So they didn’t find any recalled equations matching the prompt and wrote down something that they hoped makes a little sense

Thinking back, this kind of makes sense because a lot of homework problems were step by step and each step was more or less an instruction on what to do

So we had to take away that structure if we wanted them to practice outside-the-box thinking

[pastel horizon]

Yeah, and that question is more throwing them into the deep end compared with that

[long pelican]

Good learning experience for me!

[pastel horizon]

Guided problems are great scaffolding though to introduce a concept

Maybe next exam you could instead put "show that the pdf is ... "

[long pelican]

Yeah so a more balanced mix of the two

[pastel horizon]

So you're not telling them what to do but you give something to aim for and you can assess their understanding

[long pelican]

Tbh I actually had your wording to begin with and thought they’d immediately recognize f as the integral of a pdf from -infinity to x, making it too easy

That fits with it being outside the box now because recognizing something like that when you’re not explicitly prompted to proved to be something they didn’t practice

[pastel horizon]

You could also penalise them if they gave weak arguments for why it had to be that integral

You would expect the <x to be a big clue anyway that it's an integral up to x

[long pelican]

<x being a big clue was exactly what I thought lol

[astral laurel]

I don't think they'd exactly think in terms of $f(x)=\int_{x\in\left{t\in R: t < x\right}}...\dd{x}$

[burnt vessel]

ShatteredSunlight

[pastel horizon]

That comes from the definition of a cumulative probability function though

Integral of a pdf from -inf to x

[astral laurel]

When given an inequality you just 'shade' a whole region, like this

[long pelican]

The … is a good point too, the pdf wasn’t given and they had to do the step of creating a notation for it

[astral laurel]

I'm not disagreeing, it's just like, I don't think they are that symbol-savvy yet

[pastel horizon]

Yeah I guess creating a function could be confusing

[astral laurel]

Not say symbol-savvy, even just understanding sets perhaps

[pastel horizon]

It makes it seem like you got the wrong answer since you haven't stuck to only using symbols in the exam question

That's my opinion anyway I don't know if you expect students explicitly to define their own notation

[long pelican]

I can find the exact wording of the problem

[astral laurel]

maybe really confident ones?

[long pelican]

https://media.discordapp.net/attachments/694677515645747201/902056605808472125/unknown.png

[pastel horizon]

Oh then yeah that's a fair question

[astral laurel]

I still find $f$ a bit iffy...

[burnt vessel]

ShatteredSunlight

[astral laurel]

Would $F$ be too obvious a signal

[burnt vessel]

ShatteredSunlight

[pastel horizon]

And if you get b wrong you're screwed in c as well

[long pelican]

You could do c by directly saying the median is the value of m such that Pr(X <m) = 1/2

I did use lower case f to make them have to use a different letter for the pdf, which reinforces that letter names aren’t tied to anything

[astral laurel]

PDF doesn't make median that special

I'd try something like define a map $\gamma$ that assigns to each real number the probability $X<x$

[burnt vessel]

ShatteredSunlight

[astral laurel]

Like, something out of the way

[long pelican]

I wonder if that’d be even more confusing due to it looking more like Greek,.. pardon the pun

[astral laurel]

In this case I feel like you're asking them to recognise $f$ as some arbitrary thing that does $R\to R$, which is hmmmmm

[burnt vessel]

ShatteredSunlight

[long pelican]

It also catches students who memorize definitions symbolically

[astral laurel]

How do you expect answers out of a by the way

[long pelican]

Because they’ll write integral of f

Anything that made sense got full credit

Like being less than y automatically implies being less than x

Average points on a was 53% which isn’t too shabby given the grand scheme of things

[astral laurel]

Then I'm interested in the case a correct but b wrong

As you said

They proved monotonicity, or at least recognised it

[pastel horizon]

I'm surprised they saw a 6 marker as a recall question

[long pelican]

Yeah that’s exactly what puzzled/shocked me

[astral laurel]

Did you teach them derivative of CDF = PDF

[long pelican]

Not explicitly

[astral laurel]

I mean I guess you did, but maybe they didn't take it to heart

Oh wow

[long pelican]

I did teach them the other way around

[astral laurel]

Oh the integral

These students recognise the reversability of these two things?

[long pelican]

They’d get 5/6 points for doing the integral relation

[pastel horizon]

Tbf that makes more sense

[long pelican]

Yeah it’s FTC1 which they had their first exam on

They’ve also explicitly practiced interpreting

Probability of wait time more than 5 hours is the integral of the pdf from 5 to infinity

[astral laurel]

For c did you accept like $f^{-1}(0.5)$

[burnt vessel]

ShatteredSunlight

[long pelican]

That was the intended answer for c!

[astral laurel]

That's kinda the "I recognise this so this is the direct answer"

[long pelican]

But also writing 1/2 = f(m) got 8/9 as well

[astral laurel]

I feel like c rewards smart students

Rather than....mathy students?

I don't know, I don't have a better way to describe it

[pastel horizon]

Well it says express the median explicitly

[astral laurel]

I would give full marks for this, but up to you

oh hmm

[pastel horizon]

That implies an m = equation

[astral laurel]

I say this because I think a smart student can somehow reorganise their thoughts more quickly, and recognise the direct answer

[long pelican]

Hmmm possibly

There’s no clear standout smart students in this class interestingly enough

Those who did well on the first exam only did average on this one

[astral laurel]

I don't like rewarding smart students

It...penalises diligent students

[long pelican]

And those who got this problem right didn’t do that well on the first exam

I think that’s evidence I’m not rewarding smartness necessarily

[pastel horizon]

That's what you'd expect in undergrad

They're all the top students from their high school/sixth form

[long pelican]

You mean you’d expect a few clearly smart students or no?

[pastel horizon]

They're all equally capable so you wouldn't expect someone to stand out unless they are genius level

Which is not a word you'd use lightly by the way

What separates them usually is how committed they are to studying d

[long pelican]

Hmm

[astral laurel]

Usually the coursework filters the lazy geniuses with just passing marks

Actually I take that back, geniuses do well on exams, and not particularly well on projects or homework

Like, any project, requiring effort >2 weeks

[pastel horizon]

I haven't met anyone I would consider a genius, I have met very smart people at university though

[astral laurel]

Anyway, back to this, I'd celebrate the ones who did well on this at least

I'd say those who do well have the correct basic intuition

[pastel horizon]

They all clearly understood cumulative and probability functions

If they scored high

I think 54% is a fair average

[long pelican]

Stats were 53% part a, 22% part b, 27% part c

[astral laurel]

54% is kinda low TBH

One thing I'd look out for is twin-peak phenomenon

[pastel horizon]

Not in the UK, it's like a middle 2:2

B and C aren't so good though that's not even a pass

[astral laurel]

Where you get a concentration of marks around top 30% and another concentration around bottom 30%

[pastel horizon]

But you have to differentiate 2:1 and first class students somewhere I guess

[astral laurel]

Education is for alll though?

[pastel horizon]

Well it's a degree either you do the work and pass or you don't

[astral laurel]

You can say you want to reward the ones who do well, but it is possible education is failing those who failed the course

Well this gets a bit more subjective

it depends on their effort levels

[pastel horizon]

The reward for people going above and beyond is getting exceptional or strong passes

[astral laurel]

And although one professor has told me, if someone is putting in effort and it does not show in grades, it does suggest the person is not suited for uni. I have mixed feelings on this

[pastel horizon]

That's probably true sadly

Like I said before, everyone should be equally capable of passing at uni

It's not something everyone even gets the opportunity to do

It's kind of like how in the NFL, they would be a mix of athletes that dominated in high school

[astral laurel]

Context matters I suppose. If this is literally a math major core course, then it's really important that one passes. There are cases where universities do compulsory-common classes, and that's like hmm....

[pastel horizon]

I can't really make an opinion on the US system but the UK system is much more reasonable, you only study towards your major

[astral laurel]

Basically the question above is not hard (it's not Putnam, no logic magic jumps, etc.), so non-passing grade is not a good sign

[pastel horizon]

In that case, there's no reason why you shouldn't be able to at least achieve a pass on all your units

[astral laurel]

Can I ask you about the UK

[pastel horizon]

Sure

[astral laurel]

If Cambridge math is on a different level

[pastel horizon]

It's not really

[astral laurel]

And I suppose Oxford too

Since Cambridge Y3/Y4 stuff are public

Or at least I can find them

And then I see Martingales and I'm like what

UNiform convergence what

[pastel horizon]

Oxbridge isn't really that much better than another Russel group

I went to Manchester

[astral laurel]

So would you see martingales in 4th year?

[pastel horizon]

Tbh I didn't study maths I studied chem eng

[astral laurel]

Specifically, martingale theorems and proofs

o RIP

Ah well I don't have a pure math background so

[pastel horizon]

I know about pdfs from A level stats

[astral laurel]

I can't remember my A-levels

[pastel horizon]

But all my units for chem eng were compulsory it's not like I got to study an optional unit in French to make up credits

[astral laurel]

Like if I even had CDFs/PDFs during that time

[pastel horizon]

Although you can take some French classes if you choose a year abroad

That's a flaw with the US system in my opinion you'd have to study for a minor degree

Like what's the point? Just seems like a filler to make up credits

[long pelican]

Fortunately I still think most of the students are of a passing standard when compared to past semesters’ exams

They did quite well on familiar problems

Maybe I’m too generous idk? Lol

Past years’ exams had more ratio of familiar problems to unfamiliar problems

[long pelican]

Any thoughts on this definition of limit I came up with which is perhaps more kid-friendly than the epsilon-delta definition but just as rigorous?

We say $\lim_{x\to a}f(x)$ exists if the intersection of the ranges of $f$ restricted to a basis of open neighborhoods of $a$ is equal to a singleton and we call the limit the element of the singleton

[burnt vessel]

Icy001

[long pelican]

And one for infinity:

We say $\lim_{x\to\infty}f(x)$ exists if $\bigcap_{a\in\bR}f((a,\infty))={L}$ for some $L\in\bR$, and we call $L$ the limit

[burnt vessel]

Icy001

[long pelican]

I haven't thought this through completely so it's possible that one could prove that these aren't even equivalent to the rigorous definition of limit, but I'm somewhat confident they are

Okay, this breaks if the range doesn't include the limit in question. Drat, but that could be fixed maybe

Another problem is $\begin{cases}\frac 1x & x\neq 0\0 &x=0\end{cases}$ where the limit at 0 according to that "definition", bummer

ok I can salvage this by requiring the ranges to form (or be contained in) a basis of open neighborhoods or punctured open neighborhoods of the limit

yeah now this sounds more like epsilon delta but without the epsilons and deltas. In fact, this is getting at the lim sup - lim inf = 0 definition of limit. So I didn't actually think up any new presentation. But do people think lim sup - lim inf = 0 (as in, supremum and infimum of the ranges) is a more kid-friendly presentation for calculus class than epsilon-delta?

[burnt vessel]

Icy001

[quasi musk]

what do you mean by singleton?

What's a range?

How do you restrict it

wait how do you know that it'll just be one point

etc.

That's how teaching that will go

[long pelican]

We don't know it'll be one point o_O

In nice cases we can express the condition as "the size of the range (max minus min) shrinks to 0"

It's at least something in between "Just take it on faith sin(x) has no limit as x approaches infinity" and "let epsilon > 0...."

[tepid smelt]

I don't see any reason not to just show the normal definition.

[long pelican]

For a first calculus class, about 5% of students will get the normal definition usually

[light pelican]

This seems very inaccessible for those who don't know the epsilon-delta defition of limit. It requires understanding ranges, intersections, and worst of all, bases of open neighborhoods. For a beginner to limits, the epsilon delta definition is probably the most understandable rigorous definition. It does take time and thorough explanation and interaction, but its the most accessible.

It's also a nice step into formalization, which is almost a prereq for understanding bases of neighborhoods

[long pelican]

I should've clarified that the formulation in terms of intersection and basis of neighborhoods was for you guys, not for the students, just to get the idea across. The idea being how to explain why does sin(x) not approach a limit, but e^(-x^2) sin(x) does, if you don't have the time to introduce the epsilon-delta definition to them and they've never seen that formalization before and they aren't going to understand it? (Think, like, non-math majors or high school/middle school people)

That's what got me thinking about how all this picking epsilon and delta business can be encapsulated by taking the range of the function under increasing (a,\infty) subsets

On the other hand, it is a good idea to show even non math majors that formal math isn't scary and can be understood... in that case I'm all for epsilon delta. But the execution of it is pretty tricky though and I haven't seen it done well

e.g. in calculus textbooks, epsilon-delta just reduces to the recipe of picking delta as an algebra function of epsilon, which is all that students typically get out of it (all of the recipe and none of the logic)

I live in the United States if that provides some context

[light pelican]

Ah, I see. Well I agree, i think walking them through this without the terminology is a good way to help them obtain intuition for a limit. It's the route that I try to take when explaining it to people

(this is also just my response to a light skim of what you've written, not a particularly deep analysis of the approach)

[long pelican]

Yeah my communication could be better, because I just thought of all this like a couple hours ago. The main impetus was that this morning I had an impromptu office hour and part of that had me using sin(x) as an example of a function without a limit, and I realized that although I could intuitively explain it, if the student challenged me to convince him rigorously but using only what he knows, I'd have totally been at a loss.

like if I brought out epsilon delta, he would be lost, and if I stuck with the intuitive explanation he could be unsatisfied or not totally sure whether I'm being rigorous enough

[topaz scarab]

I got a hot take. I think middle school kids should learn PDE
so many of problems in the world can be formulated as PDE, so it is kinda a big deal.
The way to teach it without learning calculus 1st, is through game of life, Conway's
PDE is just like conway's game of life, except that:

1. the value is R, instead of just binary life or dead
2. You might have more than 1 value in a grid
3. Your transition function is different.

That's it right? We can teach everything in terms of PDE

[pastel horizon]

Do they even learn ODEs?

[quasi musk]

So what you can do is teach recursion equations

And that can be a model for simple ODEs, or linear PDEs (The ones where the factoring trick works)

[topaz scarab]

nope

[topaz scarab]

YES! recursion equations on a grid!

That's an extermely powerful model for the world
And then teach how eveyrthing, from weather forecasting, climate model, modelling bridges and sky scrapper, to car engine, and quantum field theory, can be reduced to recursion equations

[quasi musk]

I think you might be over-stating it haha

but they are pretty neat

[astral laurel]

The jump from recursion to climate is really big

[topaz scarab]

Why linear only?
If you expand the grid to 5x5 or 7x7, instead of 3x3, I think you can get ALL pde?

[quasi musk]

No

[topaz scarab]

how so?

[topaz scarab]

thanks ^^

[quasi musk]

You can't get all PDEs from recursion equations

Well

Maybe you can

But that doesn't mean it's always a useful way to go about it

I haven't taken a computational methods in DEs class

Hopefully I'll be in one next year

[topaz scarab]

isn't the numerical approach is basically reducing it to a recursion equations?

[pastel horizon]

Yeah you end up with a kind of lattice

Bessel function, finite difference and Fourier series are the limit of my knowledge when it comes to PDE

But they should definitely learn ODE before tackling PDEs. They should even learn vector calculus first. That's too many pre requisites to cover in middle school alone

[frosty flame]

Well, you would start in preschool ofc

[light pelican]

We should probably teach them category theory so they have a formal foundation of mathematics before they start using things like addition, subtraction, and multiplication.

[topaz scarab]

My whole point is, skip all of that.
Just learn Conway's game of life, change the 3 stuff I mentioned before, and you get a "PDE" (or the discrete numerical approximation of it)

[long pelican]

I think about 10 middle school teachers in the entire country know enough about this to teach it

And the path to learning it for themselves isn't very clear to them; there isn't a standard textbook with a big publisher that their school can point them to

So... draft a textbook? 😛

[topaz scarab]

That's my plan

[long pelican]

I wait with bated breath

[topaz scarab]

I just wish I can get pre-paid for this...

[light pelican]

Said every entrepreneur

[topaz scarab]

I think you are underestimating them. In my experience nearly most of math teachers I have ever worked with (at least high school math teachers), know their math quite well.

[long pelican]

Elaborate! There's a huge range of what people consider "know their math quite well"

[topaz scarab]

calculus 3 at least?

[long pelican]

Not content wise, more like...

Mathematical thinking wise

[topaz scarab]

I see... Fair enough, there is a broad range for me to say anything about it hahaha

[real mauve]

heyo, anyone wanna rate my shitty attempt at a self-contained explanation of gradient descent on linear least squares problems? it's a jupyter notebook, so i'd have to share the file. edit: aimed at engineering students that probably won't have much knowledge beyond basic linalg and multivar calc

[long pelican]

Oooo how are you sharing the jupyter notebook with students? Do they each have their own jupyter server?

or schoolwide server?

or something else

[real mauve]

and i'll also give a short talk on teams or something like that sharing my screen

[long pelican]

I see

[real mauve]

anywho, i'll share the file here. i'm open to feedback, esp cuz my notation might be crap.

get your free nitro today!

(as the title and intro imply, it's the first part of a series that will lead into l1-regularized optimization for sparse recovery and compressed sensing. that's why i take a weird route into gradient descent using proximals)

[astral laurel]

I don't have much feedback, :\ but I notice you don't have Jensen's inequality on convex inequalities, I suppose it's not that relevant
Additionally I see the part about grad descent, I feel like the analysis is very close to rate of convergence of gradient descent so you might want to go a bit further there

[pastel horizon]

Yeah I have no idea about this I just know the more 'traditional' route

It still doesn't seem very efficient like why not just teach vector calculus? I'm not convinced the other way is better

[strange bronze]

there is a huge gap between conways game of life and an actually useful pde

if you think you can demonstrate how to bridge that gap pedagogically, go for it

if you cant, the point is moot

what are you hoping kids... do with their pdes?

[topaz scarab]

Any reason not to use google colab? So I don't have to install stuff on my computer?

[strange bronze]

its a jupyter file lmao

[pastel horizon]

You can run python in docker

[real mauve]

granted it has no code atm, but it will in the near future. it's supposed to have some latex and some code together

[pastel horizon]

That's my go to, sure you have to download docker itself but after that you don't need to download anything else to your computer

[topaz scarab]

Because:

1. this is a kinda important concept that is applicable everywhere, you want to teach it to more people earlier, like middle school
2. Knowing this will give a much better motivation to learn calculus and linear algebra. for most STEM student, PDE is kinda the end game for a lot of applications. Why not show the end game early so students are not blind.

[strange bronze]

do you really think youll be able to teach kids to solve a single useful pde

[pastel horizon]

You could just as easily do that with something like the heat equation

[topaz scarab]

This is the gap:

PDE is just like conway's game of life, except that:

1. the value is R, instead of just binary life or dead
2. You might have more than 1 value in a grid
3. Your transition function is different.

What do you think?

[strange bronze]

okay explain that to middle schoolers

[pastel horizon]

And say "here's a useful problem you can't solve yet but you will be able to by the end of the year"

[topaz scarab]

Yes, heat equation won't be very hard.

[strange bronze]

bear in mind that most of them dont know what ℝ is

or what continuous means

[pastel horizon]

The heat equation is easy conceptually but hard to solve

When you're learning it at least

[strange bronze]

how tf are you presenting the solutions of the heat equation

you have an idea but i havent seen any substance

[real mauve]

here's a pdf version. the pages and formatting were decided arbitrarily by whatever procedure my browser decided it liked

[topaz scarab]

They know what contious means, I have taught it.
Not in the proper mathematiacal sense, but in simple sense as in descriptive statistics: continous variable vs categorical variable, that's all there is to it, no need to go as far as R

[strange bronze]

???

im so confused

how would you even describe to them a solution to the heat equation

how would you explain what "solution" means

[topaz scarab]

Give them a grid of number. to calculate the next time step, just do weighted average of the neighbours (and themselves).

[strange bronze]

students struggle with learning what a "function" is

much less the idea of what a function solving a system means

so you want to work with discrete time steps and have them do each step manually???

[pastel horizon]

That's not even a solution that's just a numerical approach

[strange bronze]

for some really weird, botched numerical solution that no one would ever use in practice since we have far better numerical pde methods?

[pastel horizon]

I think it can be useful to leave it as a long term problem for motivation but don't just jump straight in to solving itt

[strange bronze]

i dont understand why you think thisd be more useful than what theyre currently learning

this is, necessarily, a replacement after all

there are parts of the curriculum that could certainly be cut

[topaz scarab]

I guess in an exam, just let them calculate few cells to make sure they know how the whole things work.

[strange bronze]

but id rather not cut 1 useless thing for another

[pastel horizon]

I still think there's too much material that needs to be covered before you can actually solve an ODE let alone a PDE

[topaz scarab]

That's all I'm teaching to middle schooler

[strange bronze]

but... why

[pastel horizon]

Then you're just cherry picking parts of maths you think are useful

[strange bronze]

its not a good numerical method

no professional would use it

[topaz scarab]

It is not

[strange bronze]

and i cant see a situation where itd be handy for a layman

[topaz scarab]

Because:

1. this is a kinda important concept that is applicable everywhere, you want to teach it to more people earlier, like middle school
2. Knowing this will give a much better motivation to learn calculus and linear algebra. for most STEM student, PDE is kinda the end game for a lot of applications. Why not show the end game early so students are not blind.

[pastel horizon]

What about non stem students

They are gonna get screwed over imo

[light pelican]

The heat equation might be a bit too abstract to give the kids any sort of motivation. They won’t be super excited to learn more about the heat equation in 10 years

[strange bronze]

i feel like the "end game" could be shown more efficiently with a 15 minute youtube video on heating rods and hitting drums

[topaz scarab]

Because:

1. this is a kinda important concept that is applicable everywhere, you want to teach it to more people earlier, like middle school
2. For STEM students: Knowing this will give a much better motivation to learn calculus and linear algebra. for most STEM student, PDE is kinda the end game for a lot of applications. Why not show the end game early so students are not blind.
3. For non-STEM students: they would have a better approximations of how STEM actually works, rather than "solving equations for x"

[strange bronze]

and i wouldnt necessarily be against that, though it still feels kinda silly to show it so far in advance

[real mauve]

i wouldn't say no one uses it, it's your run-of-the-mill finite differences scheme

but explaining why the cells need to be a certain size is another issue

[pastel horizon]

I mean, solving equations for x comes up in STEM more than you'd think

[strange bronze]

but edd its discrete time steps

[pastel horizon]

It's just that you assume STEM students know how to do that

[real mauve]

yeah

[topaz scarab]

At the end, you will give them a bunch of different examples like: weather and climate modelling, structural engineering, fluid dynamics, (combine those two and call it literal rocket science), quantum field theory.

[real mauve]

it's called FDTD

[strange bronze]

like are you planning on teaching recurrences and crank-nicolson?

[real mauve]

either that or separation of variables

[pastel horizon]

I wouldn't necessarily say PDEs are the endgame either. The endgame is mathematical fluency so engineers and scientists can focus more on the physical applications rather than wrestling with maths

[strange bronze]

like im not sure showing something so far in the future is pedagogically valuable

[topaz scarab]

no need to go that far

[pastel horizon]

Engineering graduates are very good at solving problems and getting answers. Where they really earn their money is the actual design

[strange bronze]

imagine a 12 year old being told "yeah the math youre doing will eventually be useful if you go to university in mechanical engineering or theoretical physics for 4 years"

[topaz scarab]

In a certain sense, I agree.
But I think this is still much better than what middle school (and most adults) think of what actual engineers are doing right now.

[strange bronze]

that seems... very counterproductive if the 12 year old isnt interested in those things

"oh i dont wanna be a fancy scientist so i dont need to try in math class"\

[pastel horizon]

The objective in middle school is just guide them towards fluency, reasoning and problem solving

[astral laurel]

Oh, I remembered one more thing, You could try to show p-norm balls for various p. Could be pedagogically useful

[topaz scarab]

How is that different from a 12 year old being told to solve quadratics?

[pastel horizon]

If they grasp all three of those areas they'll be successful in future mathematical education

[strange bronze]

12 year olds are being taught quadratics?

[pastel horizon]

They shouldn't be

[strange bronze]

when i was in 7th grade we didnt even know what a polynomial was lmao

[pastel horizon]

Even year 10s struggle with it

[strange bronze]

we did like

[pastel horizon]

And they're like 14-15

[strange bronze]

fractions, surface areas

rationals and reals

and linear equations i think

[astral laurel]

As a final note I think nbconvert does fine, but you might want to do equation-referencing eventually. Not sure if notebooks can do that. I use rmarkdown which can, but basically still working with crippled LaTeX as long as you're not working with raw LaTeX

[pastel horizon]

So if a 12 year old can understand how to solve a quadratic, well hats off to them they're a lot smarter than most of the kids I know

[topaz scarab]

Well. I did learned it as a student, and I did taught it as a teacher... But maybe it may varies from places to places

[strange bronze]

in any case, there exist applications where you need to know quadratics but not pdes

but i cant think of any example of the converse

since knowing the basics of quadratics is kind of a prerequisite for actually understanding pdes (formally or numerically)

[pastel horizon]

All they need to know is they are skilled problem solvers in my opinion

[topaz scarab]

Ok, so you guys are saying that, being able to see the world in some kind of extremely rudimentary finite element method, is not useful for most people?

Even though so many problems can be formulated that way?

[pastel horizon]

It's not that it's not useful, it's that you're sacrificing a big portion of the curriculum that might be better focused on

[strange bronze]

if you want to put together a draft of what this would actually look like

what the lesson would look like, what test questions would look like

how much curriculum time itd take

where it fits in the ordering of things

how you'd teach teachers to teach it

go ahead

but as-is i cant see it being useful nor feasible

[pastel horizon]

Like yeah the biggest problem is you're flying in the face of a tried and tested curriculum

[strange bronze]

getting teachers to do it in particular seems... impossible

most math teachers in my country havent taken a math course beyond calc 2

[topaz scarab]

Okay, let's put the feasibilty aside.
No point of even thinking about it, if it is not useful right?

[pastel horizon]

Saying it's not useful is like saying teaching triangle geometry isn't useful

[strange bronze]

i think those two points are inherently related

[topaz scarab]

Okay... That's is very different from my experience, but if that is the case, I can see why this might be an issue...

[pastel horizon]

You can use it for some problems, the question is, is it more valuable than something that exists in the current curriculum

[strange bronze]

since the more difficult itd be to implement, the more usefulness youd need to justify it

if it takes 15 lecture hours, it better be worth those 15 lecture hours

thats like 3-4 weeks of classes!

[pastel horizon]

Like I said before, currently it seems like you're just cherry picking parts of maths you like or don't like

[astral laurel]

Tried and tested? Really? More like huge amounts of inertia and insufficient will to change

[strange bronze]

there are a lot of flaws in the current curriculum

[astral laurel]

I'd say there is room to improve curricula, but I took the PDE one as a joke because I also find it difficult to believe

[strange bronze]

but im unconvinced this is a positive addition

even if it were possible

[astral laurel]

Then again, disregarding all the PDE/whatever, approximation as an area could be useful, but it is more engineering than anything. Mathematical approximation is hard and I would not recommend teaching before undergrad

[strange bronze]

if we "put feasibility aside" then lets just teach 'em a proper PDE course with multivar calc and everything

i think we could afford to have more numerical approximation stuff in grade school

more useful than 2-column proofs at least

this seems like a bizarre place to start though

its a more sophisticated technique for a more technical and involved setting

[astral laurel]

TBH I do hope Taylor series gets compared more to actual approximation-methods, to show how the 'best'-ness of it is very mathematical than practical

[strange bronze]

to justify it, youd have to introduce basic numerics well before IMO

[pastel horizon]

The risk with approximations are some students will think that's the actual answer

[astral laurel]

But pi =3.141

[strange bronze]

grade school spends like 8 years working with problems only in the literal simplest possible setting

[light pelican]

How would one go about teaching numerical approximations without assuming some calculus knowledge

[strange bronze]

with no extra conceptual weight

[astral laurel]

You need increasing levels of justification to use more digits of pi

[real mauve]

the problem i see with this is explaining convincingly that approximations are needed in the first place

[strange bronze]

fair point, idk

[real mauve]

all of a sudden you toss at them problems for which, for some magical reason, there is no direct way to find an answer

[pastel horizon]

I personally think the problem is more the implementation

[astral laurel]

Quick compute maybe?

I'm not sure how Elliptical integrals are numerically done these days

But approximations are nice than actually solving integral equations

[pastel horizon]

Yeah, it's quite a difficult concept explaining only a tiny percentage of problems are actually solvable analytically

There'll probably be some students that see it as a challenge

Pi might actually not be a bad example to use for motivation to approximate answer, as long as you can explain it has an infinite amount of digits and you can't write it as a fraction.

And don't teach the fractional approximations obviously or you'll end up confusing them

[astral laurel]

but they are so good

[pastel horizon]

I don't understand why

[long pelican]

Honestly pick a random math topic to replace literally anything in the current curriculum, assume every single teacher can teach it well, and it's an improvement over what we currently have

[topaz scarab]

I propose cat theory

[long pelican]

Playing around with discrete grids is pretty concrete and hands-on so can be pretty fun for kids

One does not even need to mention the connection to PDEs

[pastel horizon]

I love manupulatives

We used counters for teaching negative numbers and it really helped them grasp it

Even changes the way you see them

[long pelican]

Category theory being taught well to 12 year olds everywhere hmmmmmmm

Very interesting proposition

[real mauve]

you could just do linalg instead

or basic stats

[long pelican]

to a 12 year old category theory is just graph theory with composition laws

I guess

[pastel horizon]

They don't teach stats in the US?

[strange bronze]

AP stats is typically optional

some local curricula might cover likee

whats a normal distribution

[topaz scarab]

we do teach basic stats right? By basics I mean descrptive stats

[strange bronze]

whats a z table

how do you use em

and thats about it

maybe random topics like p value

but this isnt a federal standard

[real mauve]

you can go beyond that. all i remember from descriptive stats is that making histograms by hand is a sin

[pastel horizon]

Yeah getting the scale wrong screws you over

[real mauve]

go into probability distributions more in detail or something

explain pds, cdfs, see different distributions and not only the normal

[pastel horizon]

Well first they need a solid understanding of what probability actually means

[topaz scarab]

Whoever said I was cherry picking, maybe you're right.
But I think my choice is excellent.
If anything, the alternative topic would be statistical inference, but I don't really see anyway to teach that to middle / high schoolers.

[real mauve]

you can explain that too

[astral laurel]

People would complain about the breadth of it, I guarantee you

[real mauve]

i would say any pde is more advanced than basic statistical inference

[pastel horizon]

We have a GCSE stats course in the UK as well

[long pelican]

pdfs are kinda tricky for 12 year olds

[pastel horizon]

GCSE is 15-16

[long pelican]

p(0) = 0.4 has nothing to do with any probability being 0.4 unless you talk about integrals

[astral laurel]

They usually learn integrals then, so that's not an issue

[topaz scarab]

Maybe I shot myself in the foot by calling it PDE.
What I meant was simply: extremely rudimentary finite element method.
just grids, and numbers on grids, and rules on how to change the numbers

[pastel horizon]

We save that for later. But tbh, it's not that hard to just teach them how to use the table

[real mauve]

you can introduce that with simplified riemann sums along with integration and differentiation

since anyway they're approximations to integrals and derivatives, precisely what you wanna do

secant methods and discrete approximations to integrals, sure

[astral laurel]

I'd think Riemann as a concept is taught in integration

It's a standard lecture/teaching

[real mauve]

iterative methods for pdes,

[light pelican]

If you’re gonna go that route, focus on giving them the intuition for the 1D case just so they have it. Probably no need to go into specifics like higher dimensions and stuff

[long pelican]

https://i.stack.imgur.com/sUIbf.png

Good final project @topaz scarab ?

This is from Artin's Algebra

[astral laurel]

Like, the GCSE kids (age equivalent) won't be asked formally about Riemann but I think Riemann is the way to teach it

[pastel horizon]

Yeah it pretty much is

I think it's also good teaching integration as the opposite of differentiation

[topaz scarab]

Yes, I think I would definitely start with 1D heat equations. Maybe for structural engineering too?

[light pelican]

Just very basic awareness and understanding of dynamical systems might not be a bad addition. Just to expose them to it

[topaz scarab]

final project for?

[long pelican]

The unit on discrete differential equations that you're proposing

[topaz scarab]

Well... my idea is to teach this before calculus so...

[long pelican]

Oh, you can ignore the calculus stuff

Only the first paragraph is about calculus

[pastel sundial]

Icy how would you reform the K-12 math curriculum if it was up to you

[long pelican]

I don't have a dream curriculum, but since curricula are built off of standardized tests, I'd like to see what happens if we make standardized tests have more outside-the-box problems, which change from year to year. Or get rid of problems altogether and make it about reading comprehension of mathematical proofs