math-pedagogy

[long pelican]

Nice non-trig proof

[near oriole]

think the circle is irrelevant

yh, nice

[long pelican]

O m gee there’s a much simpler proof

Delete A’

AC = AP + PC = AP + PB > AB

[near oriole]

im just thinking here, does this construction "working" more or less assume this fact or not.

you draw the perpendicular bisector of BC

but like yeah, who's to say its gonna intersect AB or AC

[long pelican]

Don’t need that line either for my proof

[near oriole]

So how r u constructing P

ah ok, using the smaller angle

think im convinced

[long pelican]

The essential ingredient in the simple proof is that the larger angle can be written as the sum of the smaller angle plus another positive angle

[twin shell]

in greenberg this is an exercise using the exterior angle theorem

[tepid smelt]

This is interesting. I tried something similar at a higher performing high school and students were more focused in class but did worse on the first exam than they normally do. This led to parent complaints which led to admin pressing to stick to the basic curriculum provided. When I went back to the basic problems students were less focused and knew they could learn the material on their own.

[long pelican]

Do you write the exams? Guessing not

[tepid smelt]

We have exams from the curriculum provided and they are pretty straightforward.

The idea from admin is everyone in the district should give the same test every year to collect data.

When I tried to go non traditional the exam reflected that and I wrote it. To be clear results were not terrible but at good schools students are used to doing well with minimal effort.

[long pelican]

Did admin base performance on raw exam scores?

For example older exams might have a philosophy of testing individual basic skills without asking for synthesis, while your exam might ask for synthesis and therefore be harder. If that is the case, it is illogical to compare the raw scores

[tepid smelt]

Yes a big problem at the secondary level is the emphasis of testing individual basic skills. I think the issue is this shift needs to happen at the elementary level to have any chance with older kids. We seem to put a lot of effort in trying to fix issues at the secondary level where it should happen more at the elementary level.

I mean it's shocking to me that students don't ever even see a proof until sophomore year of high school in geometry. I think even this concept can be learned at simpler levels much earlier.

[long pelican]

Agree

[halcyon mortar]

What's the best way to introduce the definition of a topology?

[tidal mango]

Probably start in the context of limits. For example, you can write all the different limit definitions in terms of open sets like in this article:

https://www.cantorsparadise.com/what-is-a-limit-really-ee1c256c6544?source=friends_link&sk=3d786eeea26bdfcdf77d09d60aeb0b5e

I would leave the discussion there as a neat fact like Chekov's gun. Later, you can go back and show how all these definitions are related.

Then you would probably want to talk about continuity and what it gets you. For example, applying a continuous function to the elements of a convergent sequence will get you another convergent sequence. Then, you can ask how to define continuity. You can give the Calculus I definition, then rewrite it to include all the limit stuff from earlier.

From there, you can talk about neighborhoods around a point as being "close" to the point, where "close" is whatever you want it to mean or you can focus on open sets because open sets almost always have more than one point like with open balls (though the discrete topology is an exception), so you can use them to talk about the "region" around a point. You want to avoid closed sets in this sense because closed sets can include a single point without its neighborhood, so using closed sets could get you in trouble.

At the end of the discussion, you can talk about the p-adics as an example where you have sequences that don't converge in the normal sense, but do converge in a p-adic sense. If you watch this Veritasium video, for instance, you'll see that he has a golden cylinder model that he uses is the topology of the 3-adic number system.

https://youtu.be/tRaq4aYPzCc?si=VLuj9pJQFD8voJ3P

Likewise, the boxes 3blue1brown uses in his video on the p-adics is the topology of the 2-adics.

https://youtu.be/XFDM1ip5HdU?si=6t_s2m4_U4VkIN-p

This article might also be helpful.

https://www.cantorsparadise.com/an-intro-to-topology-9e0478313b63?source=friends_link&sk=ebf03da0d56777bb635f32907a46fb7e

Regardless of what you do, I would not recommend the random assortment of facts like "Mobius strip has one side, coffee cups and doughnuts, wind speed is zero somewhere on Earth, temperature and pressure are equal at some pair of antipodes, etc."

[cosmic ibex]

I really don't think one should attempt to introduce the concept of "a topology" without some development of metric spaces first. Most things in metric spaces should be familiar from the main examples in real analysis. And then once one does know metric spaces, one can motivate abstract topologies as an attempt to capture what it is two metrics on the same set have in common when they agree about limits and continuity.

[tidal mango]

You can get around that issue by showing people where it breaks down. For example, you can work with finite limits for functions from $\mathbb R \to \mathbb R$ using metric spaces for both the input space and the output space. But then, say you want to take a limit at infinity. In that case, the metric space approach utterly fails since any real number is an infinite distance away from infinity.

[burnt vessel]

TheLandfill

[tidal mango]

Now, your audience or class has to start thinking about how to get closer to infinity without a metric.

[turbid zenith]

I very much agree with this approach

It motivates why anybody would need to introduce the formalism

[tidal mango]

I'm a little iffy about focusing too much on metric spaces at the start because you could get by with the standard $\mathbb{R}^n$, especially just $\mathbb{R}$ and you wouldn't have to cover anything new. You start with:

"Two numbers are close if their difference is small. Mathematically, we say $| x - y |$ is the distance between two numbers $x$ and $y$. Moving on."

Then, when you want to talk about higher dimensions, you say:

"The distance between two points is $\sqrt{ (\Delta x)^2 + (\Delta y)^2 + \dots }$ by the Pythagorean Theorem. Moving on."

[burnt vessel]

TheLandfill

[tidal mango]

Once you've covered the idea of open balls, then you can start extending it to different metrics.

[cosmic ibex]

The role of metric spaces in the story I'm sketching is as a motivator for abstract topologies, namely in the observation that a metric (no matter whether we use the fancy word or call it "distance" or "difference") actually tells us more detail than we really want to care about for many purposes.

[tidal mango]

I don't have a problem with using the fancy word. If you tell students that a metric is a function that measures the distance between two points, they'll get it by the end of the day. My issues are that

\begin{itemize}
\item common metrics don't introduce anything new,
\item it doesn't work for a lot of common limits,
\item and it's unnecessary detour.
\end{itemize}

For the first point, you can't use any $L^p$ metrics in 1D since they're all the same, so you have to work in at least 2D or use unusual metrics. Worse than than having to work in $>$1D, the metrics differ only in the specific shape you draw for open balls, but the proofs for all normed spaces are identical to the standard proofs in Real Analysis with the Euclidean metric. \

For the second point, note that of the six limits you learn in Calculus (finite two-sided, left-sided, right-sided, positive infinity, negative infinity, and sequences), the only one that works perfectly in a metric space is the finite two-sided limit. The three infinite limits don't work with metric spaces since every distance is infinite and the one-sided limits don't work because you have to exclude points near the point of interest. Unifying these limits with open intervals is much easier (I did it without even realizing I was doing Topology.) and more interesting. \

For the last point, you would introduce limits in metric spaces specifically so that you could abandon them. In this context, the motivating problem feels artificial because you could have just used the Euclidean metric to prove the limits for any metric topologically equivalent. It would be better togo for a situation where the metric approach breaks down. That time and mental effort that student spend learning about metric spaces could be spent much better. Students at this level are already familiar with Calculus and maybe even Real Analysis, so they wouldn't need to learn anything else to follow along with the limits approach.

[burnt vessel]

TheLandfill

[tidal mango]

Unless I'm mistaken, the course would look something like.

1. Here's a bunch of different metric spaces that we're covering because they show up in some contexts like king moves on chessboards and travelling across a city with a grid street system.
2. Now, we're going to do what we did in Calculus with limits but in these new metric spaces. It looks almost exactly like it does for Calculus since we can use the triangle inequality.
3. What's similar about these proofs? That's right, we can define an open set as ... and our topology is ...
4. It turns out all those limits we did were the same because the most of the metric spaces we worked with were topologically equivalent to the Euclidean space.

The only way out would be to introduce funky metric spaces like the p-adics, but now you have to explain why we care about the p-adics and all those funky metric spaces and students now have to learn about metric spaces and topologies at the same time.

[bleak skiff]

meeting my first students as a tutor tomorrow

my goal is to not have a nervous breakdown

[twin shell]

pog

private tutor?

for what level?

[bleak skiff]

i dunno how it translates from norwegian, one is in 8th grade in «ungdomskolen» and the second is in the first year of high school

[bleak skiff]

and yes

[twin shell]

hmm
I've only ever tutored thru my uni

[winged urchin]

Ohhh at that level (I don't know norwegian schools) but don't be so worried imo

Be a person. Be friendly. Just chat about what they are doing in school, maybe ask them questions that might highlight common errors. In my opinion at least

[bleak skiff]

[upper aspen]

what do you tutor?

[bleak skiff]

thanku very much

[winged urchin]

Also depends on whether they are better students who are only really getting help to just keep them sharp or students who are struggling

[twin shell]

just basic calc 1 & 2 in the US
although for some reason it usually turns into fractions or algebra

[upper aspen]

oh nice

tbh you could def tutor real analysis

[winged urchin]

Yesss. Tutoring first years involves pre-requisite material all the way back into elementary levels sometimes @.@

[bleak skiff]

i feel i need to read something on tutoring just to feel a bit less lost… i’m confident in my maffs knowledge at the level i’m to help them with, but is there such a thing as a tutoring survival guide?

[tacit adder]

A tutoring survival guide would be pretty cool

I’ve been a TA for roughly 2-3 years but I still struggle maintaining a class occasionally

[bleak skiff]

[winged urchin]

It's one on one yes?

[bleak skiff]

yes, but they’re siblings and the company that arranges everything around the tutoring itself requires we do this start-up session where we go through practical things on how and when the tutoring is to be done and getting to know each other, which i will do with both at the same time

(this is also the part i’m most anxious about, i am socially quite a mess)

but i have a genious plan that if i ever get too anxious i will just start talking about how much i love maths, and that will surely bring my mood back up again

[winged urchin]

Tutoring a class is certainly one thing, but one-on-one is nicer imo.

Again maybe different standards there but a good number of parents I help students of are happy enough that I just get them to engage with math

Ahaha yes!

Sometimes I go on tangents on complex numbers or fractals or Fibonacci, or whatever else fancies you at the time

Not for very long but it is a kind of oo--aaa for students sometimes ahah

[bleak skiff]

the moon is not a rational number, in my case atm

[winged urchin]

You might find yourself also asking like "What do you think?" or "What would you do?" a lot, and these are great questions both for when you're at a loss or otherwise

I like trying to make sure the students articulate why they are doing what they are doing in a problem

Even if they get the right answer, getting them to explain how they reached that point may showcase areas where they are weaker or maybe they are actually getting it coincidentally correct while making an error

[bleak skiff]

haha yes i have seen that before

[bleak skiff]

i think i rather failed at this, and did way too much explaining things in general

[winged urchin]

It's a little tough to comment without much context but it sounds like you feel you did most of the talking? Did it seem like they were listening or able to do problems based on your explanations?

Certainly there are times where I go on more of a lecture style. Mainly when I notice they have a deep gap in knowledge and I basically need to do a little crash course on the topic. Happens more at higher grade levels though. Like explaining trig or exponentials or graphing with transformations

I do think that at younger ages I want the students do be actively participating in the session though. It's far too easy to think "Oh I explained this idea fairly comprehensively, they'll be able to take this and run with it on the tests or homework or whatever"

It's important for them to show you they can do the problems even just from a self confidence perspective

I'd be happy to hear more from your sessions though and give more feedback. ^^

[bleak skiff]

yeah i think this was what happened, the session was just short of 1 hour… next time i will ask more questions and have them work some examples

[bleak skiff]

thank you very much, you will certainly hear from me again

[maiden relic]

I'm going to start tutoring a 6th grader this Friday, any advice? We are going to be going over the 6th grade school curriculum

[pastel horizon]

Try and reinforce what he's done in lessons recently

[winged urchin]

^ yes this. Also try to figure out what kind of student they are.

I tutor younger kids sometimes. Some of them don't really like math or struggle and some of them are already good and the parents just want me to reinforce their learning.

With the first I will generally just ask what they did in school this week. Think of a few problems in that direction to work with them on. Give them some pointers or short cuts and see how they solve them.

The second kind I do ask them what they did but I don't 'drill' them as much on the simpler things. I'll expand on the topic and show them things they'll only see in later years. Sometimes too I like to work through contest problems with them and see how they are thinking through problems and all that

[maiden relic]

Thanks, seems like a good starting point

[near turtle]

https://en.wikipedia.org/wiki/Vandermonde_matrix - This seems to be a common problem to give students early on in their linear algebra education. The "standard" proof (which I've moved to 3rd place instead of 1st) is more advanced than some lecturers I've met have appreciated.

[main horizon]

Good evening, I'm studying the substitution method and I don't understand what an equality would be. Can someone explain it to me? To be an equality would it only need the equal symbol or more of something?

[long pelican]

What are you studying it for? Teaching or homework?

[still rapids]

hello, I wonder about how you are taught division in your country.

I googled a little bit but it seems a little bit inadequate. Do you have some information about your country and how they teach division for children?

Please let me know, thank you for your help.

For starter, I would like to get a little bit about some countries like France, Spain and even Germany.

[tawdry venture]

http://proof.ucalgaryblogs.ca/

y'all

check this out

[viral parcel]

oh this is sick

thanks

[spark coyote]

wow i glanced in here to ask a question and saw this, i use this all the time, i can't believe this actually has a name! holy moley

this discord is gold

anyway

does anyone else here have problems trying to teach / tutor lower level math? I seem to have an easier time with students that struggle with things later in their careers, regardless of age or complexity of the topic, but trig, early algebra, has been really challenging

[tawny slate]

I find that the simpler the concept you're teaching, the more difficult it is because there are only so many ways you can explain why 5 times 5 = 25

[winged urchin]

This looks so cool! Wow. I'll be reading this through for myself and maybe giving it to students that have trouble with proofs. Woof

[tawny slate]

There are, on the other hand, at least 20 different ways to explain trig functions

I find that teaching these lower level concepts is a good test of your math foundations

[winged urchin]

I like the unit circle explanation at least for thinking of the values and solving

But it also bleeds into so many other areas it can just be a cascade of ideas that struggling students barely remember

[twin shell]

https://matheducators.stackexchange.com/ is always worth digging through, in case you're not already doing that

[winged urchin]

Like transformations to sketch them.

Inverse functions and restrictions of the domain.

General solutions where we specify solutions dependent on some n variable we introduce. Abstraction

[long pelican]

I actually find, unlike CosmoVibe, teaching lower level concepts to younger students is easier for a very orthogonal reason, not having anything to do with concepts: younger students are more malleable in learning the good mathematical thinking habits

Of course it's still true there are only so many ways to explain why 5 * 5 = 25, but I also have a different belief about how to approach such an explanation: it is near futile to try to construct an explanation of a mathematical idea in the absence of information about how the student thinks. When you gather enough information about what the student is lacking, the right explanation for this particular student should come to you naturally.

[long pelican]

https://matheducators.stackexchange.com/questions/26959/natural-origins-or-learned-habit-why-do-students-skip-concepts-before-applicati is a really great thread just posted 2 days ago

[twin shell]

i read that thread and when i saw this line i retracted in my seat

But surely there's no issue accepting that not everyone can really understand Topological Quantum Field Theory at its core, right? Why should Calculus be any different?
because i don't accept the first statement in the first place

but i understand the point of the answer

[long pelican]

By "great" I of course don't mean that all the answers are correct 😛 more ilke thought-provoking

[twin shell]

ye

[winged urchin]

What do yall think of this question for first year calculus? (Not even an epsilon-delta kind of course. Designed for general sciences)

"Find the sum of the x- and y-intercept of any tangent line to the curve given by sqrt(x) + sqrt(y) = sqrt(7)"

Also the school implemented a star rating system of 1, 2, 3, or 4 stars difficulty and this is a 2 star difficulty problem apparently to them

I'm helping some students in this course this year and apparently they made a change in the course this year and have leveled up the difficulty in my opinion

I don't know if it's necessarily bad but just curious what yall think of that question and difficulty labelling

[long pelican]

Confusing. Does "any tangent line" mean "all tangent lines" or "pick a tangent line"?

[winged urchin]

At best it's hinting that the answer doesn't depend on which line you consider

It doesn't specific like... consider the tangent line at (a,b) and then ask for the answer in terms of a and b

Just literally what I quoted

[long pelican]

Yes, I mean that as my feedback about that question

[winged urchin]

Oh right yes I see ahah

It is a little fun exercise for us mathies imo but my students are blown out of the water not only by just understanding it but the insight you need in the algebra too

[long pelican]

I am very surprised teachers considered that a 2 star difficulty problem. Must be a school where the teachers teach how to solve problems properly. Just kidding, it's probably similar to other examples they show in class

I do agree it's a nice problem provided it's not similar to examples shown in class or homework, but as with all nice problems, they are too hard to plop in a class where problem solving isn't taught

Although the correct action isn't to not use the problem, it's to teach deductive reasoning and problem solving and then use the problem :^)

[spark coyote]

Maybe that’s part of my difficulty, I’m used to college level students (I tutor there)

Students come from all sorts of backgrounds, I’ve had people almost 30 show up and learn algebra because they’re determined to get a stem degree (and I’ve seen people like this succeed too!)

But regardless of age or of motivation, they struggle hard with the concepts and i find myself struggling hard to find useful parallels regardless of their major nor experience

Maybe it’s a “language” thing?

For example, I tend to find that showing programming oriented students the math concepts as python code (or a close analogue) really gets the gears moving quick

[wintry lichen]

I want to become a math tutor

(Don’t worry, I don’t actually define vector spaces as modules over a field irl)

but I’m not sure how to get into it

[spark coyote]

Honestly it feels like a puzzle

Like teaching the thing is a math problem in of itself, but you gotta find the right approach with the right analogy at the same time as not revealing too much answer

[spark coyote]

Straight up try helping a friend that’s in a class you know well with problems

Don’t do it for them per se, the goal is to get them unstuck

If you find that process comfortable see if your uni has a tutoring center and hires students (that’s what I did at least lol)

[wintry lichen]

I’m not in a uni, that’s the thing

[spark coyote]

That makes things harder lmao

[turbid zenith]

Do these seems like reasonable errors a student would make?

[long pelican]

Variant on (b): ln x - ln y = ln(x-y) and going from there 🤣

[turbid zenith]

I want to avoid mistakes that come from algebra errors and instead focus on errors that come from calculus errors 😛

[long pelican]

I remember a final exam question on logs 2 years ago. The reason people bombed it, aside from the fact it was an unfamiliar type of question, was saying weird things about the log function that have nothing to do with mistakes in integrating or differentiating it

I guess in short, not understanding the ln function at all

[turbid zenith]

Was half thinking of making the first one be "find ∂f/∂x and ∂f/∂y"

And the student still gives 2x + 3y²

Would that be good?

Figure it might make it at least a different "flavor" than part (b)

[long pelican]

I think they're all good as exercises on partial differentiation. Do you know ahead of time this is going to be a problem area?

[turbid zenith]

This is for my equivalent of a take-home test.

And I've seen students make at least mistake c a NUMBER of times

Not being able to tell the difference between constant terms and constant multiples

They want to fall back on "the derivative of a constant is zero"

[long pelican]

Hmm yeah, couple thoughts about that

1. There's a way in which the student can re-derive the constant multiple property by first principles, then go "Oh yeah, that's how it works"

[turbid zenith]

"Re-derive from first principles" is unfortunately not in many of my students' vocabulary 😛

[cosmic ibex]

Might it help to show in some detail in class how it works out if you apply the product rule to c·f(x)?

[long pelican]

2. The difference between + and * is definitely fundamental and I have a peculiar idea that using words "constant term" and "constant multiples" actually goes in the direction of wordiness

[turbid zenith]

The answer to most questions of the form "why don't you just show them <this method>" is "I did, they still make that mistake"

[turbid zenith]

I've probably used like 15 different ways of trying to explain it, those aren't common terms we use or anything

[long pelican]

Sometimes you need to subtract vocabulary rather than adding it !

Reminds me of a skit where someone didn't get a joke and asked "Is there more?" and the joke-teller said "Actually, there's less"

The joke was like
A: [punchline]
B: [didn't get the punchline] "And then what?"
A: "I dunno, they went home I guess"
B: "I don't get it. Is there more to the joke?"
A: "Actually there's less"

[turbid zenith]

XD

Love it.

BTW here's the other question I'm giving on that "test".

[long pelican]

I know the feeling, I derive stuff from first principles and 1/3 of the students get it, the other 1/3 think I'm just showing another example problem to take notes on, and the other 1/3 are completely lost

I had a conversation with the undergrad director where she called this a "language barrier"

[turbid zenith]

Not to mention all the dead puppies

I still have students claiming that √(x² + y²) = x + y

In Calculus III

[long pelican]

See!! Algebra mistakes dominate!

[turbid zenith]

Yes

. . . maybe I should add one more question where they have to find the partial derivative of √(x² + y²) 😛

And they get an answer of 1

[long pelican]

Ok first things first!! Not colorblind-friendly!

[turbid zenith]

THey can do it without the color

On the website you can click a point and it tells you the elevation

[long pelican]

Whew

[long pelican]

For (c), I fear it's not a very mathematical question. Did you define what a function is? And the correct answer to whether h is continuous is no

I mean, the concept of function is mathematical but most of their answers will not be

[turbid zenith]

So we defined a function in terms of inputs and outputs, in the sense that every input (which could be a number or an ordered pair/triple) gives exactly one output (which in this case is a number but in our previous section was a vector)

And like ... I would be happy if they considered things like (for example) cliffs

But ... meh. I'm rather unsure on that question anyway.

I want to ask them something more conceptual about continuity that isn't just doing a calculation.

[long pelican]

I see the intent

[turbid zenith]

And these are graded holistically, where if they get something wrong I comment with feedback and then they can correct it

As opposed to out of some number of "points"

[long pelican]

One thing I can say is that very often, a non-routine math problem tests conceptual understanding better than asking for a written explanation of a concept does

[turbid zenith]

Ahh yes "non-routine problems"

I try to give those often (or at least as close as I can) in class but there's often lots of wheel-spinning

[long pelican]

What's wheel-spinning?

[turbid zenith]

"Spinning your wheels" means you're continuing to try to do something but gettiing nowhere

[long pelican]

Oh I didn't know that!

Yeah, non-routine problems take longer and require a lot more playing around

You can do something guided if you want to reduce the time they spend on it

My take on written explanation is that students treat them like word association exercises

A current calculus instructor told me last week whenever he wrote a true/false + explanation problem about continuity on an exam, students show evidence of not thinking about the definition of continuity at all

[turbid zenith]

Interesting

Like what kind of true false problem?

[long pelican]

Something like "Every continuous function is differentiable"

half students missed that one, probably due to "Oh yeah I remember a theorem that sounds like that, so it's true"

btw if half students missed that one, that's consistent with everyone guessing 🤣

because it's true/false

[turbid zenith]

Yep

Though in my case it’s take home so people could just look it up 😛 I try to give problems that are not just easily able to be looked up

Or where even if they did look it up they’d still have to do it for their problem

[long pelican]

My exams are all open-notes so I also have the same constraints when writing problems

[turbid zenith]

Most of my students do take it seriously and not cheat

All this semester I’m pretty sure

[long pelican]

In my experience, freshmen generally don't cheat while seniors have a cheating rate of like 10%-20%

maybe my batch of seniors was just bad

[turbid zenith]

Anyway with regards to continuity

I would be really happy if students knew that most of our nice functions (polynomials etc) are continuous on their domain, and that for a limit, the limit has to be the same from any direction, and that can be hard to prove, but that if a function is continuous you’re done

[long pelican]

I've asked myself often what I would be happy that students knew

I have to honestly say that I'm happy if students walk out the class with zero misconceptions about the nature of math, and consequently I've made that my number one priority

[turbid zenith]

I care much less about if they can show one of those random quotient functions has a limit at 0,0

[long pelican]

I've found that once they don't have misconceptions about the nature of math, learning math is a lot easier for them

LIke, the content gets understood better as a byproduct

[long pelican]

This is what I would call a "problem type" and students show artificially good performance on these simply because they practice these problem types

And, having practiced that problem type, you can do the right steps without needing to understand much

So yeah, I eschew problem types

[turbid zenith]

Removed that continuity part from the other problem

[long pelican]

I like both! (a) reminded me of a blackpenredpen video I saw recently which posed this problem: you know 0^0 is indeterminate because the value depends on the particular path taken to the origin. Can you find a path, avoiding the y-axis (very important, otherwise it's too easy), so that the value of 0^0 as interpreted as a limit along that path, calculates out to 0?

[turbid zenith]

Interesting!

I don't like to say "0⁰ is indeterminate" though

I say "0⁰ is an indeterminate form" to make sure it's clear we're talking about a limit of the form x^y as (x,y) → (0,0)

If I'm talking about "the value of 0⁰", in my opinion that's squarely 1 😛

[long pelican]

It's tricky, not obvious, and meant more for you or other people in this channel as I would be surprised if your students find such a path

blackpenredpen said himself it took him 6 years to find it

[turbid zenith]

Ahh yeah XD

. . . wow that's actually pretty hard

[long pelican]

😄

[turbid zenith]

My first thought was "okay just make sure the base 'approaches 0 quickly' and the exponent 'approaches 0 slowly'"

But everything I'm doing goes to 1 instead!

[long pelican]

Same thing happened to me!

[wintry lichen]

I think so.

[cosmic ibex]

||t^{1/t} goes to 0 as t approaches 0 from the right, since it is e^{log(t)/t} and the exponent there goes to -infty.
Of course t^{1/t} is not of the form 0^0, but now we know that (t^{1/t})^t = t is of that form, and that obviously goes to 0 too.||

[long pelican]

Nice, log(t) / t blows up faster than 1/t alone but slower than 1/t^2

[austere delta]

What about something like ||exp(-1/t^2) ^ t||, doesn't seem like it be that hard to cook up.

[long pelican]

That works too. Trickiness is relative. Have you thought about how it's funny that just being tangent to the y-axis isn't enough to guarantee the limit is 0?

Somehow this is saying the first order information about the path does not determine the limit of this (continuous!) function

[cosmic ibex]

There's an entire zoo of rational-function counterexamples in multivar real analysis that show how that fails. But perhaps showing it for x^y is more striking, since that's not a function that's made for the purpose of being pathological.

[austere delta]

I mean the function isn't continuous, but yeah sure that is funny

[long pelican]

x^y is continuous in the first quadrant

which is what matters here

[long pelican]

what's the simplest rational function with this property?

[austere delta]

x/y^2 I guess

[long pelican]

I'm asking for a function where you can have two smooth paths with the same derivative at 0 but different limits

let's say restricted to the first quadrant as well

since x/y^2 has discontinuity along the x-axis

[austere delta]

Yes, so the path x=y^2 and x=y^3 for example both aproach the y axis

[long pelican]

Hmm so it is

(t, a sqrt(t)) is a whole family

Now I'm thinking it's more surprising when a function's limit at (0,,0) (a) doesn't exist but (b) is well-defined when only given the derivative of the path at (0,0)

x/y satisfies this, if I'm not wrong. Algebro-geometrically the graph of that is the blow-up of the plane at (0,0)

[cosmic ibex]

I was thinking of something like xy^2/(x^2+y^4). If I recall it correctly, the limit is 0 along every line through the origin, but that's not the limit of the function.

[austere delta]

Yeah, that's a pretty neat example.

[long pelican]

Good that I got this out of the way since I'm assigned to multivarabile calculus next semester

😅

[cobalt finch]

can anyone give a good way of explaining the concept of a Topos to someone who has no background in pure math whatsoever

[viral pike]

It might help to clarify why you'd want to do that. And where you draw the line to pure math.
A bulk of topoi are sheaf categories, so going after sheaves would be one (rather cold and formalistic) way of getting at it. If you want to stay more vague you could view it as a generalization of the set concept, but then it remains to be elaborated how it generalizes it.

[cobalt finch]

my status set them off to ask me 😭

[deep kindle]

So for whatever reason it took me this long to realize that order of operations is just decending order of hyperoperation with grouping as an override

Posting here because I am personally opposed to PEMDAS and BODMAS

[winged urchin]

Sorry I'm poking fun a little but I'm imagining an elementary teacher at the front of class going "Alright class today we will learn order of operations! Or as you will come to understand it, just descending order of hyperoperation ..."

Though I do sometimes describe multiplication as 'super addition' and exponentiation as 'super multiplication' in a tower then say a lot of rules in algebra apply between two levels but generally not when they are not adjacent

Which frankly is probably not terribly elucidating either

[wide ice]

I still find it confusing why one needs PEMDAS or whatever variants there are

I remember that day when the teacher introduced multiplication, she made us do a ton of annoying additions, then presented multiplication as shorthand for repeated addition.

No one questioned why multiplication worked, but apparently we didn't get confused about the order of operations after that. We just fell back to "repeated addition" and translate everything to additions whenever there was a confusion.

parentheses came a bit later, when she formally introduced associativity. It's like, "a magical thing that makes addition get done first"

[cosmic ibex]

"Order of operations" is not really about the operations themselves, but about how to read expressions. In teaching systems that use "PEMDAS" it's generally (mis)represented as being about which computation you must do first, instead of being about the logical structure of expressions.

[wide ice]

quite a bad thing to introduce to some 8yo kids knowing nothing about math imo

i mean, of course you don't introduce Peano's axioms and Cauchy's sequences. But it's easier to explain why things are, instead of just giving some seemingly arbitrary rules.

[marsh thistle]

they are arbitrary rules though, it's just a convention

I'm not sure that I'd be convinced that there's any reason why PEMDAS is better than a different convention, we just agree on one so that we don't have to write too many parentheses

[twin shell]

I mean if you want to be fully honest, the reason we do PEMDAS (as in, this particular order, not the name PEMDAS itself) is because literally everyone in the world writes this way and if you misinterpret it you're fucked

it's just like language

[wide ice]

given how many ppl still misinterpret it, I'd say it failed spectacularly

[twin shell]

if you're talking about those examples like 1/2(3 + 4), I'd say those are irrelevant

because anyone who writes ambiguous formulas without any help from the context to decipher it is probably not someone you want to listen to

[cosmic ibex]

I don't think anyone here disagrees with the convention itself, just with the way it seems to be taught and explained in some places.

[wide ice]

In France there's no such thing as PEMDAS afaik

and tbh idk anywhere else in the world where PEMDAS is taught

not to say it's a bad thing, but... given the world is not in any big troubles, I think there are better ways to teach it without PEMDAS

[long pelican]

Order of operations shouldn't be isolated into its own unit in the curriculum

Rather, teach the rules as the students see expressions that combine different operations for the first time

[cosmic ibex]

Right. So the precedence between +/- and × ought to be taught well before the kids even know about exponentiation, and then P-MDAS won't even make sense.

[wide ice]

well, that's how it was done for me

[winged urchin]

In this direction and towards students having issues with seeing equivalent expressions or just like... what they can do in equations I've sometimes thought of little fun 'creative' projects of getting students to just write some expression or equations in interesting ways

[zenith slate]

We teach it in the UK, albeit with a slightly different acronym that varies from county-to-county...

[cosmic ibex]

Out of actual curiosity, does that mean the precedence between multiplication and addition can't be taught before the kids know what exponents are, or do you have preliminary acronyms for that use that omit the first vowel?

[elfin sleet]

For me (Canada) we were taught the acronym (with the e) before we learned exponents and were told we would learn about exponents later (I remember this clearly because they told us we would be taught exponents the next year but they said the same thing the next year and the year after - turns out it’s actually taught 3 years later - and child me was very disappointed at having been teased with it multiple times lol)

[austere delta]

It's also the convention that allows you to write down polynomials without parenthesis. I'd say that's the main reason, but of course you probably teach the order of operations quite a bit before polynomials

[cosmic ibex]

Just a pity that it doesn't generalize to "allow you to write down rational functions (on a single line) without parentheses".

[austere delta]

Nothing wrong with the fraction line (does it have a name in English?)

[cosmic ibex]

I've usually seen "fraction bar".

[zenith slate]

Kids are taught to simply perform operations left-to-right (as a so-called 'number sentence') before being introduced to the idea of operator precedence. Arithmetic exercises prior to this are designed in such a way that precedence rules aren't required.

For example, one may see a question like calculate [3 + 3 \div 2 - 4]

[burnt vessel]

rat v2.3.0-alpha

[zenith slate]

(BI|BO|BE|PI|PE)(MD|DM)AS is then taught concurrently with the introduction of parentheses

[cosmic ibex]

So that is supposed to yield -1? Until a few years later it magically changes to 0.5?

Small wonder "85% can't solve this".

[zenith slate]

Oh lmao

Well you see what I did here is give an example of where precedence rules are, in fact, required

I don't teach at this level

The youngest I teach is 11yo by which point they are expected to know PEDMAS or whatever

[deep kindle]

Obviously my intention with that idea is not to present that as information to students learning how to do order of operations, but rather understand it more myself in a way that could possibly be presented to students who are developmentally ready to handle something abstract like hyperoperations beyond the third order

I'm finding that it all boils down to simple grouping. Each operation, if expressed as addition, just dictates a certain level of implicit grouping

$2 \cdot 3 - 1= (2 + 2 + 2) - 1$

[burnt vessel]

M. Frost

[deep kindle]

And grouping itself can be used to override this

$2 \cdot (3 - 1) = (3 - 1) + (3 - 1)$

[burnt vessel]

M. Frost

[long pelican]

Fun fact, higher mathematicians have problems with order of operations too. I see G / H x K in many a paper (which is ambiguous), and just today I saw V|X x Spec A where the author assumes the reader knows that | has lower precedence than x

[deep kindle]

In this way, order of operations is just based on grouping no?

Whether presented implicitly or explicitly

[cosmic ibex]

I'm not sure I think "based on" makes a lot of sense here. The purpose of "order of operation" is to define which grouping to use when it isn't indicated explicitly with parentheses. But that doesn't seem to be what you're saying.

[winged urchin]

Sometimes though I think there is a non trivial amount of value to giving something a quick name for students to say

Like oh were doing this with BEDMAS or cross multiplication or common factoring

Though to be fair we don't see students saying oh this is distributive property or or this is the transitive property!

[deep kindle]

It helps students who are learning to evaluate expressions but it's not great for long term recall because it's misleading

[winged urchin]

There's always that trade off between precision and information density though is there not?

We could teach students precise definitions and make them get used to the level of abstraction necessary to grasp it and then maybe they might make less mistakes maybeee

But maybe that would lead to more mistakes in them just not understanding well enough or not remembering some part of the denser theory

I have students who mix up addition and multiplication routinely in their heads but it isn't a misunderstanding there really it's just a misfiring in their brain and them not checking their answer afterwards

Like 3*2 is 5

I'm really just speaking from my own experience and kinda just broadly too without stats so it's not worth that much but I don't think we need a new way to teach multiplication or addition in that case. Or rather I suppose I think with any method of teaching something there will always be the possibility of a student just getting it wrong

[long pelican]

That might be related to the well known phenomenon that a lot (I forget the percentage, possibly most) students forget more than half of what they learned in math over the summer

Which in turn points out the non-robustness of the knowledge that was gained

like the 3*2=5 thing, one side of the math wars believes that the way to turn this into robust knowledge is just a lot of practice, the other side believes ithas to do with having multiplication be part of the rich web of knowledge in your head rather than an isolated island

The default answer to this is "both" but that's such a lazy answer

[winged urchin]

I would probably default to the 'it depends' answer without much thought aha

Like if the person is going to be around that knowledge and use it routinely then it's worth building that dense web

[twin shell]

lazy is not necessarily wrong

[long pelican]

My input on this war is that practice does itself under the right conditions without any willpower required, and that's the ideal situation

[winged urchin]

I still remember a quote from some mathematician in a textbook I had at the time. Something to the effect of when they want to learn something new (in math) they specifically seek as many examples as possible

Which goes in favor of practice in essence

Though we must watch when we start auto-piloting

[long pelican]

Talking about robustness reminds me of Alan Schoenfeld's TRU framework

good read for anyone who isn't familiar with it

https://truframework.org/
Cool, there's a website on it

[zenith slate]

Yeah for sure

Little mnemonics and the like can be extremely useful for recall

Obviously there are flaws with PEMDAS and similar acronyms and the way they're taught

Most notably, in my experience, that it gets written PEMDAS and sometimes teachers forget to teach (or fail to remember) that MD and AS are of equal precedence

The whole PEMDAS thing is interesting to me in general since it's not, like, a real thing

[pastel horizon]

The AS part gets even more nuance than being equal precedence

I would say it's good practice to even have some ambiguous scenarios like 6 - 1 + 2

Try and get them to say what they think the correct order should be

[winged urchin]

definitely 6-3 then 3

;D

[winged urchin]

That's a good point thought gaunter. Honestly sometimes I tell my younger students to write those a 6+(-1)+2 then they can do it in any order and move things around as they like

[winged urchin]

What was 'math' as a school subject like 200, 300 years ago?

I know I could Google this later but also if I could imagine as we develop more math theory the upper ceiling of what might need to be taught in class may raise?

[long pelican]

I know it used to be studying Euclid's Elements for quite some time (this was also a time when fewer people studied math)

[winged urchin]

I guess there was probably always some upper echelon of math education that was pretty well developed

I'm thinking like what did the lowest member in society study in 'math' if they studied it at all I guess

[long pelican]

The move to skill-based math education was a response to the industrial revolution and the need for workers who could do basic math

You should read what some older mathematicians (like Euler) wrote about math education, I know for a fact they all had opinions on it 😛

I soon found an opportunity to be introduced to a famous professor Johann Bernoulli... True, he was very busy and so refused flatly to give me private lessons; but he gave me much more valuable advice to start reading more difficult mathematical books on my own and to study them as diligently as I could; if I came across some obstacle or difficulty, I was given permission to visit him freely every Sunday afternoon and he kindly explained to me everything I could not understand … and this, undoubtedly, is the best method to succeed in mathematical subjects. - Leonard Euler [13, p. 90]

[pastel horizon]

So the inverted model used all those years ago

[zenith slate]

Saying that the best way to learn mathematics is to have one-on-one access to one of the greatest mathematicians of your generation is hardly an insightful take, though

Like I'm not sure how this applies whatsoever to "math education"

[zenith slate]

In the UK at least, very very few children ever received any kind of schooling prior to about 1800

Certainly the expectation of children (in the lower classes) prior to this was that they should be contributing to the household and paying their fair share

Children were used all over the workforce

Besides that yeah, like icy said, you'd likely have been looking at a very "classical" curriculum oriented around reading Euclid and the like

But again, only for children from wealthy families or who happened to live in close proximity to a church school

[wide ice]

Delta-epsilon definition of convergence was introduced by Cauchy 202 years ago, and students at Polytechnique, arguably the best in France at the time, hated it with passion. Now it's the first thing you learn in uni. Many even saw it in high school.

Galois theory was not even a thing back then. It took us roughly 50 years after Galois to have the modern definition of groups. Now it's taught in every Algebra I class.

[wide ice]

I think there's a video on YouTube about old exams to Cambridge or Harvard or something. It was mostly around calculations, because guess what, there was no calculator back then. It was what was most essential to usual jobs.

[long pelican]

I wasn’t making a content related point, just showing an example of an older writing

[zenith slate]

Fair

[pastel horizon]

The other implication was that Euler was visiting Bernoulli with questions rather than recieving lectures directly. I think there's some merit with this approach especially as we move towards more advanced technology

[cosmic ibex]

Especially it must have been much less work for Bernoulli.

[grim spindle]

I have definitely hounded professors with questions before

Most people studying math in an academic setting can gain access to a professor to ask them questions

The fact that it was a highly established mathematician is less important

[deep kindle]

the value a in a given vertex form expression can be identified on the graph by moving one unit to the left or right of the vertex, and then dropping or raising an altitude to meet the curve. That distance, labeled in green in this picture, is equal to the absolute value of a in the equation a(x-h)^2 + k

Is there a name for that feature on the actual graph?

I'm trying to implement this process into a lesson plan and want to try and do so in fewer words than that whole explanation

[near oriole]

So this property is a result of symmetries right --- translate, rotate, stretch a parabola back into y = x^2

[long pelican]

Don't really need the viewpoint of transformations. Algebra works just fine: $a((h\pm 1)-h)^2+k=a+k$ whose value is $k$ displaced by $a$, proving the claim

[burnt vessel]

Icy0

[umbral pewter]

it's a good general tool to know

but yeah, if you're just concerned about parabolas and this particular property, algebra works

[long pelican]

Well you can take the transformation perspective and it's a simple one but only after you've proved the relevant invariants

[umbral pewter]

wydm 👀

[near oriole]

"go mess around with these desmos sliders" is what i wanna tell ppl to do

[umbral pewter]

unironically used to do that in hs all the time

well, just mess around in desmos i mean

[long pelican]

I guess it can be boiled down into one proof: Dilation of the graph of $x^2$ by a factor of $a$ followed by translation by $(b,c)$ gives the graph of $a(x-b)^2+c$

[burnt vessel]

Icy0

[deep kindle]

Is there a more convienent name for this property? I'd like to be able to say that students can use [insert here] property of quadratics to identify the coefficient a in vertex form

As part of finding an equation given a graph

[umbral pewter]

Moreso what you meant by the 'relevant invariants'

[long pelican]

Well like, the "a" term is invariant under translations (obvious to us, but maybe not to first time learners)

[long pelican]

Don't think so. In fact I never saw this in school

I treated it as an exercise to prove that the method works. Would that be a good exercise for the students?

[deep kindle]

Yeah I never saw it in school either, but the mentor teacher with whom I am student teaching has taught it to them as a way to identify a by just looking at the graph

And I want to include that identification/property in a lesson plan that I have for them

[long pelican]

A principle I like to abide by is that I never inadvertently convey to students anything that is "just another piece of information to remember"

I fail in that myself a lot

So, something I think of as a prerequisite to including that trick in the lesson plan is having them prove that it works and be comfortable with the proof

[deep kindle]

I agree completely

[long pelican]

Also, in my experience conveying the nature of math as a connected whole is a lot harder than it sounds. Half the things one might say to that effort ends up not making sense to the majority of students. For example, students who aren't familiar with role of definitions and proofs may (and do) treat the proof solely as another thing to remember

[near oriole]

Oh right yeah, good point. what is the point of this observation in particular?

[near oriole]

if its just this, then although a bit more computationally heavy, I think I would recommend plugging in points and simultaneous equations

[winged urchin]

Do you think some students are more comfortable just remembering more rather than 'rederiving' things they might need?

Like I sometimes like to show students how you can get the tan^2 and sec^2 identity from the sin^2 and cos^2 identity by dividing by cos^2 but I think sometimes that falls on deaf ears or worse aha.

Like I know I liked to simplify how much I had to remember by removing things I could rederive if needed but for some students they unfortunately can't reason to that degree or see the insight or they worry that that will take too long on an exam

Also thanks to all the comments from people before. I read through and have some reading to do from things I bookmarked from that discussion but have been swamped

[wide ice]

Do you think some students are more comfortable just remembering more rather than 'rederiving' things they might need?
Yes, I know some medical students giving up when they see the kind of math I'm doing, even though they have a far heavier workload.

"But it's just memorising!"

In fact, I chose Math (and by extension, CS), because I have quite a bad memory

[near oriole]

yh takes too long on an exam. The worst mentality

[long pelican]

Lack of experience with deriving, compared with tons of experience memorizing. So from that lens it makes sense the skill of deriving needs to be explicitly taught not just left implicit

[gilded imp]

Is this channel for math teachers?

[twin shell]

among others, yes

[gilded imp]

Thank you very much. OMG, thanks a lot. After helping some people in this server, I realize how torturous it must be for you to teach. Again thank you all.

[winged urchin]

Luckily teaching is a pretty great problem solving exercise aha ;D

Thanks for helping people here

[tepid smelt]

I mean memory is incredibly powerful in all fields and you do have to remember a lot in math. Many topics are closely connected and it's hard for students to ease the memory load by seeing those connections. Also it seems that students have an easier time remembering things when they are tied to something visual that leans on their intuition.

[wispy slate]

Nice teacher (Strang)

[long pelican]

I didn't see the edit but I could still tell this was Strang 😎

[cloud zealot]

https://matheducators.stackexchange.com/questions/10021/why-are-induction-proofs-so-challenging-for-students

[wispy slate]

Different ways to represent vector addition

'All' summarises it

[vagrant meadow]

any suggestions for helping my linear students more easily connect the different concepts together?
like they've been solving Ax=0 for like a month but ask them to find a basis for Nul A and they suddenly have absolutely no idea what to do, no matter how many different ways I try to explain "the null space is the solutions to Ax=0 so you just need to solve that"

they seem to be struggling with the earlier vocabulary as well (understandably because there's so many terms now in such a short amount of time). I tried putting a page of theorems and definitions on the last TA discussion worksheet which seemed to help a bit but there's just not enough time in 50 minutes to explain it all as well as I'd like

[long pelican]

Were there many examples of connecting concepts together that they had to participate in (e.g. for homework or in-class) before the Nul A question?

This answer might sound too simple but explicitly teaching them how to connect concepts together might be the solution

[vagrant meadow]

so less time explaining the explicit connections and more just how to make these connections in general?

[long pelican]

For the first part, I guess some tradeoff of content is needed. For the second part, it might have to be more active than you explaining stuff in words

The fear is they hear the words but passively

[vagrant meadow]

not really. if I had more time to make the worksheets I'd like to do more of that. but I only get a days notice to make it and so I just have to transcribe the textbook problems the primary suggests

[long pelican]

So you're a TA!

[vagrant meadow]

right. I tried writing a bit on the whiteboard and connecting concepts like colA, image, span, Ax=b, consistency vs nulA, linear dependence, Ax=0 etc.
but I just don't have enough time to prep or practice :/
I was considering recording a video of me doing some examples and explaining stuff. I recorded a video of me solving a quiz my other class did badly on and that seemed to help some. but it is time consuming.

[vagrant meadow]

haha yeah I shoulda mentioned earlier lol

[long pelican]

Maybe think of what skills the students are lacking. When I say skills, I don't mean "how to factor" or stuff like that. Deductive reasoning? Understanding definitions? Reading a mathematical logical argument? Making their own observations for a math problem? The majority were probably never ever taught those skills in their whole math education

[vagrant meadow]

that's a good point. I should be stressing looking back at definitions more than I have been. I think because this is a non-proof based linear course, I've been emphasizing that to a lesser degree

you don't have to go back and look at the definitions in math until you do lol
and that point does tend to be in linear algebra for a lot of students

[long pelican]

A relevant skill that comes to mind for the Nul A example specifically is fluently switching between thinking in terms of predicates and thinking in terms of sets

another thing very absent from (US) math education

When I think back to when I lectured Calc II, every class where there was a point I re-expressed something in terms of sets I think a good number of students got lost from that point onwards

Includes famously when I wrote that $\int f(x),dx$ is by definition the set of functions whose derivative is $f$

[burnt vessel]

Icy0

[long pelican]

Connecting +C and "a set of functions" is probably really hard with 0 exposure to thinking in terms of sets. Also goes for thinking of a graph, or more generally, a shape, as a subset of the plane

[narrow mist]

is anyone aware of some good resources for writing contest integrals? i’m organizing an integration bee in a few months and want to have (mostly) original problems

[winged urchin]

That's an interesting question. I don't personally know and I imagine that's a fairly niche area so might be hard to come by.

I know there's the general idea of like taking 'easy' integrals and using substitution to make harder integrals

Throwing in some even/odd integrals maybe not centered on 0 could be a good question I imagine

Sounds like a cool thing to run aha

[vagrant meadow]

this is incredible advice. i feel like that concern was on the periphery but i hadn't really articulated it. since the class is full of CS majors i've been thinking about maybe defining Nul A like a Boolean.
(Ax=0) = (x in Nul A)
or an if statement or something.

[near oriole]

teach kids quotients

[wispy slate]

Wouldn't it be equivalent to ask, just to add some help to them, to solve for b find x for that b, when b is the zero vector ?

I like this paragraph, from Strang

[cosmic ibex]

"Solve for" is always followed by the name of the thing you don't yet know before you start solving.

[wispy slate]

I find the columns' transpose notation confusing as well, in some contexts

[wispy slate]

Recording is great since they can re-watch. Or just link someone's video about that topic.

[silk ridge]

has anyone here tried playing around with manim when trying to explain things to others in a more formal setting?

[vagrant meadow]

so something I've been struggling with when doing my TA sessions for diff eqs is that I feel like I can't adequately explain how to check if something is a solution to the diff eq. like it just seems so obvious: "plug it in and see if you get equality".
I don't think writing it symbolically would help, with the whole
phi(t) a solution iff phi'(t)=f(t,phi(t)), for example. that would just obfuscate it for a lot of them.
so how do you explain that a solution to an equation satisfies the equation? there's clearly some disconnect where that isn't immediately obvious to them

[cosmic ibex]

Hmm, that sounds like the disconnect must really be in understanding what a diffeq is in the first place?

[supple quiver]

as you mention plug it in. A diff eq is an equation involving a derivative or order words rate of change. To solve the equation you need find the appropiate function or relation. If upon substitution you get the LHS (left hand side) equal to right hand side (RHS). If LHS != RHS you will get a contradiction. e.g. 2x+3 = 6. The equation holds only for x=1.5 but no other value of x.

[cosmic ibex]

(Note that this is math-pedagogy so it can be assumed Taylor herself definitely knows how it works. The question is how to make students grasp it too).

[austere delta]

What I would do would just be to show an example maybe. So you just say

phi satisfies the equation if when we plug it in the equation holds, for example [simple phi] satisfies [simple equation] because when we plug it in [after simple calculation] the RHS equals LHS

While [other simple function] doesn't satisfy [same equation] because RHS and LHS are different at [input where they are different]

[long pelican]

Ah yes this issue is everywhere. As with issues like this you should think about what got brushed over in K-12 math education. Function equality, being comfortable functions as objects come to mind. If I was a student and “function” was “equation” to me, differential equations would be rightly mind-twisting: equations of equations?

Also, equals sign itself in a differential equation is confusing and even more so without a proper treatment of function equality, because all throughout their life they were told that the equals sign can mean substitute, simplify, or solve, which is an unmathematical way to think about things.

[winged urchin]

Kinda pretty neat arbitrary picture aha

I like it

[wary ether]

What's a good way to say "does anyone else know the answer?" without making someone feel guilty for being an active student?

[wispy slate]

Use Socrates and just ask through there

You will also spot students that won't answer, but know the answer.

[austere delta]

Maybe something like "cmon I know more of you have an answer". Then it seems more like you're expecting more hands in the air, not that you don't want the answer the active student is giving.

Assuming you actually expect other people to know the answer at least

[wary ether]

That’s a good idea thanks jagr

[oblique bobcat]

Does anyone know how to create a graph on excel in order to view the distribution of grades for an exam? I have a big table with everyone's name in one column and their score in another, and I can't seem to figure it out.

(well okay not excel actually libreoffice sheets because I don't have windows but you get the point)

[merry gyro]

You'll start to notice the students that know the answer but weren't sure to raise their hand or were slow in doing it.

Instead of asking an open ended question for participation, start calling these students out by name.

[oblique bobcat]

Right now when I do this I just get this, which gives one value on the graph ffor each student. Which is not helpful at all. I'd like the grades on the bottom, and the number of students with that grade on the side.

[wary ether]

I’m reluctant to pressure people. Do you try to judge which students respond well to being asked by name?

[oblique bobcat]

What I tend to do is never let someone answer twice in a row. "Ah, let's see if we can see any new hands." If that doesn't work after a minute or two, "Alright then most-talkative-student, what's your answer?"

(I don't actually call them most-talkative-student, but I don't want to dox their name)

[lethal leaf]

In my experience that doesn’t solve the issue

Like it works but I don’t get new hands

[merry gyro]

Nope, the smart and confident students in the class will answer appropriately and we move on. The smart and non-confident students will be unsure of the answer and the focus is to help build their confidence through answering loud and confidently. I'll have them re-state their answer out loud with more purpose in their voice. The ones who are neither smart nor confident, are also the ones who feel stupid and left behind and rather tune out and stare at the wall. I have these students work out the answer out loud with me and then praise them afterwards, sometimes indirectly such as "Okay so does everyone know how Student got that answer?" This helps build up their internal confidence that this is something they are capable of doing and they're not stupid.

[lethal leaf]

Sometimes this works after I’ve built a rapport and they know I’m not gonna be mean or something

[merry gyro]

Some students will start to be more proactive in the class, the others just have to be called on still forever

[lethal leaf]

Praise is super important I agree

[wide ice]

There's also a category of students really smart and so bored that they can't bring themselves to answer the questions because it's trivial. I'd say rare, but idk if anyone else has experience with this category.

[proven cape]

That was me in a lot of UG

I was in classes way below what was interesting to me and I didn't pay much attention

[deep kindle]

does anyone here do interdisciplinary curriculum with math and physics?

[pallid night]

Do you know R ?
Cos that's what I'd use honestly.

It'll generate a nice histogram for you which you can modify by altering the number of break points in the command as well as a bunch of other stuff without much effort.

[oblique bobcat]

I do not, unfortunately.

[patent stream]

I'm curious if anyone has further answers to this

[lethal leaf]

I mean all students are important

But like

These students clearly aren't the target of the original question

What's far more common in my experience is the people not speaking up don't know the answer

And are afraid of speaking up

And I mean me saying "it's ok if you're wrong, we will all learn from it" doesn't seem to help much

[long pelican]

It's for this reason I don't ask any equivalent of "Who here knows what 5 times 9 is" anymore. Class participation is usually exploring a question by a student (these are really good when you can think on your feet about such things), asking something non-factual-based like "Who has heard of Leonhard Euler before?" or doing problem solving sessions

[patent stream]

That makes sense, though I think asking for understanding of a specific question is still useful. Do you tend to break down problems and ask for next steps when solving? I think that's a good way to ensure that one student doesn't dominate

[long pelican]

I used to do that but it became an unwitting reinforcer of the notation that problems come with specific steps to remember, so I changed the entire problem solving education approach

also "knowing what's the next step" can be, from an unenlightened student's point of view, pure knowledge the same way "what's 5 times 9" is

[patent stream]

That's fair

[deep kindle]

Not directly related but a colleague of mine likes to have students volunteer to come up to the board to work out problems, and to balance it out that student then gets to pick who goes up next

Maybe a similar concept could be applied to answering questions verbally idk

[dark haven]

I’ve encountered this technique too. Sometimes it has the negative effect of impeding students from attending class for fear of being called upon

[violet timber]

I think this can be decently common

There can also be the motivation that someone else should answer it and then use it as a learning experience

[maiden relic]

How to introduce arithmetic with negative integers to a 6th grader? I've tried introducing it via a few real world situations like loaning money and visually through the number line but I feel like both of these didn't work

I'd also appreciate more simple examples... basically anything that can help the student get a feel for the rules instead of having to remember them mechanically

Also a more broader question, how does an amateur tutor figure out how to explain something on the go when they realize the student has some missing background... like is the best idea a tactical retreat, cover something that doesn't require the background and come prepared with explanations for the missing background in the next session?

[cosmic ibex]

Debts are very abstract; I think I would try to lean heavily on the idea of possibly-negative numbers as a difference between things that already make sense.

For multiplication (once addition and subtraction are familiar) one idea could be to start with showing that multiplication by a positive number corresponds to the difference between non-neighboring terms in an arithmetic sequence. By then it should make sense to the student to have arithmetic sequences with negative common difference, so that explains the "positive times negative is negative".

For "negative times something", one somehow has to motivate the idea of looking at diferences when we take jumps to the left in the arithmetic sequence.

[long pelican]

I prioritize identifying the missing background first, especially how deep the hole is

[maiden relic]

I'm a bit worried my student will get more confused if I introduce the concept of an arithmetic sequence.. it would be nice to have a shorthand method of recovering the rules (not a mnemonic, something conceptual but also really straightforward)

[cosmic ibex]

I wasn't suggesting that you introduce the words "arithmetic sequence". :-)

[noble hare]

Temperature is possibly a better analogy than money atleast for addition, I imagine it breaks down for students in the same way as other analogies though, especially for multiplication

[long pelican]

For new tricky concepts such as negative numbers, the question may not be so much what analogies are used or how many, but rather how you interweave the analogies and the deductive reasoning

(Notice I said deductive reasoning instead of rules; usually school focuses on alternating between analogies and rules in its instruction, leaving deductive reasoning by the wayside. This means that one's math knowledge consists of memorized facts, vague analogies, and practiced rituals, without many connections between them)

[maiden relic]

Thank you for the suggestions everyone!

[frail summit]

Hi, everyone. How would you introduce an 8yo to the concept of raising to power? I did it by using the example of cell division, but I'm not sure it was very effective.

[frail summit]

Could you give more details about what didn't work when using the number line?

[winged urchin]

Honestly I find powers are usually not so much a problem for students to understand at least at its simplest level.

I do usually remind them of how multiplication is just addition with a given number of copies of one number

I'm not sure if it would occur to me to even bring in the number line necessarily but I'm curious if there is something nice there

Sure they sometimes mistakenly do multiplication with exponents like say 2^3 is 6 but they usually recognize the error in my experience

Cell division is typical to introduce when talking about exponentials though. Exponential growth and all that

[cosmic folio]

guys if Un= the sum of (minus 1 )*k / K! from k =0 to n then U2n is ?

[wary ether]

This channel is intended for discussion for teachers, tutors, TAs, and professors about math teaching techniques. This is not a channel to ask for math help.

[cosmic folio]

my bad

[frail summit]

The question about the number line was related to @maiden relic 's question about how to present negative numbers to a 6th grader.

[frail summit]

So, do you introduce rising to power just by giving the definition "repeated multiplication" ? Without any real life examples?

I feel like it would be best to start from a real life need, at least at a young age as age 8.

[long pelican]

Starting from real life doesn't work for everyone

In fact a bold hypothesis I'm currently entertaining is that the success of real life motivations in K-12, by the data, has more to do with far more teachers being more comfortable talking with those things, thus being more successful as a result, compared to logic and deductive reasoning which far fewer teachers are trained in teaching well

[frail summit]

Logic and deductive reasoning sounds great. How do you apply it to this topic specifically?

[long pelican]

To teach something (here, exponents) based on logic and deductive reasoning the first place to start is to ask yourself the question "how is exponentiation defined, really?"

As well as: "Let's take all the properties of exponents I know. How did I come to know that? How did I derive them?"

[turbid zenith]

Ugh I need to vent a bit. Nothing like learning in my pedagogy classes about what makes effective pedagogy, and then seeing absolutely NONE of that in practice in my grad level classes.

My Applied Mathematics class is giving me trouble right now because all we ever do in class is watch the professor put his notes up as a PDF and read them word-for-word. It sure would help if we, ya know, DID math in class.

[long pelican]

Dang I have a friend in med school and his math classes in college are exactly like that (reading powerpoints word for word), and the exam averages were like 40's despite all the exam questions being straightforward (he showed me the exam questions)

[turbid zenith]

Yeah, for one thing it becomes impossible to sift through like 45 minutes of background information to get the 5 minutes of how to do the damn problem

[long pelican]

are the notes like pdf lecture notes similar in format to this?

[turbid zenith]

Pretty much yup

They're almost word for word what's in the book as well

So it's an utter waste of class time to just rehash it instead of having us actually WORK the problems

Not to mention there's zero distinction in these recent chapters between "nice to know" and "need to know"

[long pelican]

Even the first page is pretty hard to read for me

Don't have enough domain-specific knowledge to figure out what this equation is saying -- I doubt the RHS is actually a constant

[tawdry venture]

next week i'll be doing a test lesson for 7th graders and the topic i was given is systems of (linear) equations. i have the written materials i need, but one thing i could use assistance with is time management

the lesson is 40 minutes, how long should i devote to what part of it?

ping me if replying please

[frail summit]

I'd say that no matter what you teach, what is important in order for your teaching to be effective is to connect with your students and to teach in a way that makes them feel competent. (i'm not a student, but that's what I noticed by observing my daughter's learning and development)

[frail summit]

I thought the start point would be "why we need to define exponentiation". This would be more logic to me. At least with young ages.

[wide seal]

5m intro
10m operations you can do
20m strategy to solve
5m extra Qs

ime the hardest thing any k-12 student is building intuition for problems where it isnt always straightforward (although they often don't know how to classify that they lack intuition)
so i feel that you want to spend most of your time going through examples and strategies, what is the path to solve this problem

hopefully the "tools" (adding/subtracting equations, multiplying by a scalar, etc.) aren't something students will get tripped up on for very long, although it'll take practice either way.
but if you give them the tools, but dont give them strategies, they are banging hammers against brick walls.

[long pelican]

That happens before the logic part

[wide seal]

if its just exponentiation it might be helpful to revisit that addition is repeated counting, multiplication is repeated adding, and then exponentiation is the notation for repeated multiplication

to put some physicality to it you can talk about grouping of some items (do you have cubes/etc?)

there's a kneejerk to put something to real life, and a kneejerk to try and talk about it formally as best as you can

what an 8yo really needs is something physical to look at

the irl scenarios are best left to those gross common core textbook questions :p

[frail summit]

This is what I was looking for. Examples that show the need to define this new operation/concept.

[frail summit]

Thanks for your answer. For the suggestion about using cubes - do you mean exploring how can we find out the number of cubes inside a cube?

[wide seal]

no i mean literally counting cubes

but that too

you can set up like that

but like place them in a grid

you can represent multiplication like that grid

now how do we do grid of grids

(or one big cube)

[frail summit]

Ok. Thanks for the idea. I'll keep it in mind.

[wide seal]

kids cant ask many meaningful questions in english, there's very little metacognition you can work with
give them something physical and they can use that to ask questions

[frail summit]

I totally agree with using something physical.

I have one more question. We know multiplication is repeated addition. How can we apply this explanation when we multiply fractions? e.g. 1/2 * 1/3 - Again, I am looking for an explanation suitable to age 8.

[wide seal]

imo dont be clever and lie a bit
separate out the top and bottom and track both

[long pelican]

For the last question, you may enjoy this book: https://bookstore.ams.org/view?ProductCode=MBK/79

for the question of motivating the need to learn exponents, you might find it yourself if you imagine conversing with someone who adamantly claims that this 8 year old does *not * need to learn exponents, ever

[frail summit]

Wouldn't this would be like rote memorization?

[frail summit]

Thank you for the recommendation. It looks great!

[wide seal]

i mean the way you handle multiplying fractions (from a childs perspective) is mutliply all the top numbers and multiply all the bottom numbers, then say top over bottom

[winged urchin]

Yeah and the notion of multiplication that kids are familiar with isn't so digestible with things that aren't whole

[wide seal]

how you add is rote memorization, we just do it so much it becomes second (third, fifteenth) nature
certain things will be rote until they feel natural

[winged urchin]

2x3 is 3 2's added together

But 2/5x1/3 being what.. 1/3 2/5's added together?

[wide seal]

yeah you end up in a bit of a circular definition if you try the repeated addition idea verbatium

1/3 * 1/2 im adding 1/2 a third of the time... 1/2/3... oh

[frail summit]

I'll ask my child to see what her intuition tells her to do.

[wide seal]

abstraction doesnt really happen until age 12, its probably best to assume anything you think your child or other 8yos do is less abstraction and more memorization

[frail summit]

I was thinking to use the idea of repeated addition just because the child is already familiar with it. No mather what, one should be able to prove that when multiplying fractions you multiply top x top / bottom x bottom

I have observed my child over the years and with both addition and multiplying it wasn't memorization. Perhaps when you say memorization, you actually think of automaticity?

[long pelican]

Age 12 does not apply to everyone

[frail summit]

Also, once a child works with numbers (an abstract concept) mentally, can't we say they are already on the abstraction road?

[wide seal]

i deleted the second line

whoops

the teacher parent enrichment loop can definately speed things up

[wide seal]

its hard to say

[tawdry venture]

in 40 minutes?

[long pelican]

Also, what's your working definition of a fraction such as "1/3"

[wide seal]

that reply was to smth else

[wide seal]

here

[frail summit]

What's your opinion on constructivism in mathematics?

[wide seal]

let me rewrite this

[wide seal]

Traditional implementation of constructivism has been studied to be bad across the board. Students retain less with minimal guidance. Some students will also learn a less structured framework, and make mistakes along the way, since it wasn't corrected in a meaningful way.

The philosophy that people learn based of their original framework, i believe to be correct and a strong argument for active learning. But its a cycle. If you don't try to present the strong addition to a framework, then a student can get lost. You strike a balance with modern active learning: Let students build a framework naturally, but you still give strong guidance when needed. This allows a student to have some foreign framework to ask questions off of, then work on it on their own to adapt it to their own knowledge base.

In math: You cant follow traditional constructivism, students will fail. You cant just lecture at them, students will lose interest. You have to make a balance of lecturing a framework, which may be some white lies, leave some room for students to figure out small aspects on their own, but make sure everyone is on the same page. That second step, can be done through a multitude of ways, and is best done with other students + activities.

[frail summit]

Thanks a lot for sharing your opinion on this. May I ask when was decided that the traditional implementation of constructivism is bad?

[wide seal]

the progenitor paper: https://www.tandfonline.com/doi/abs/10.1207/s15326985ep4102_1

[frail summit]

Thanks!

I see it's from 2010.

I first heard about constructivism in 2019-2020 by reading some articles and a book written by Constance Kamii. It made me a good impression and I have tried to use its principles with my child. Based on my personal experience, it worked fine - according to me, my child seemed to have a good understanding of mathematical concepts (e.g. place value) before they were studied at school, but I was wondering how far can one go with just being offered lots of opportunities for exploration and practice.

[wide seal]

1:1 is a bit different

Just cause you can be more attentive to their needs

If your strategy is working, keep at it.

[wide ice]

You can have prodigies. There are many cases throughout history like that

[frail summit]

Thanks for the encouragement!

[turbid zenith]

Thanks for sharing this, @wide seal . I've been looking into Inquiry-Based Learning in my liberal arts math class next semester, and this gives me some things I should consider as I'm redesigning it.

[turbid zenith]

I do think that some blend of direct instruction and constructivist learning is probably what's most effective, but it depends on what's being taught and what the goals are

[long pelican]

I'm quite confused about the differences in meanings of the terms "constructivism" and "inquiry learning" and why "inquiry learning" is considered bad now. I don't use or think in these terms myself, so I'm open to explanations

[turbid zenith]

Also, here's a direct response to this article.

[long pelican]

oh tack on a third term: "active learning"

[wide seal]

Pedagogy buzz words

The way people thought it was good to learn was to make kids ask all the questions and use problems to jumpstart it (this is inquirybased/constructivism)

And if you can give specialized attention to every student, it's fine

But in a class of 15 or above (aka everywhere) it's hard to see when students are failing

[turbid zenith]

So here's how I see it:

• Constructivism: The idea that students learn best when they "connect the dots" themselves, because having done so makes it stick in your brain much better.
• Inquiry-based learning: A particular implementation based on a constructivist philosophy. The idea is that students are given tasks that are supposed to lead them to be able to "connect the dots." The implementations I've seen (and the ones I want to implement) include a lot of scaffolding, guiding the students to "notice" the right things
• Active learning: The idea that students should be doing something in class rather than just listening to the teacher lecture. Not quite the same thing as the above but not entirely orthogonal either.

[wide seal]

^

[turbid zenith]

So like "active learning" for me in my Calculus classes means that I want my students actually working on problems (preferably at tables where they can work together or ask each other questions) for a good chunk of class.

But yeah even if you do adhere to a constructivist philosophy, there has to be some guidance, it can't just be "here you go, have fun, we'll see what you've learned in an hour"

Which is the point of effective scaffolding

[wide seal]

It's all weird philosophy and psychology that people can argue about for hours

I think after covid some real effective strategies and implementations are starting to emerge

[long pelican]

I see

I have been focusing more on answering the question "how do you effectively measure whether someone has understood the mathematics" because a bad answer to that question does not pave a good path to answer the follow-up question "which teaching strategy works best"

In what I've seen, if you measure by not-too-well-designed tests, the teaching strategy that works best tends to be teaching to the test

Training on examples that are extremely similar to questions that will appear on the exam...

I mean, these philosophical "active learning" approaches are definitely better than teaching to the test but how are we currently measuring that?

Currently I'm measuring by having exams where at least half the questions are problems the students have not seen before, and grading based on how much progress was made in solving the problem rather than on a strict rubric (and the bar for an A is 75% rather than 90%)

[long pelican]

Based on these descriptions, I think they all will work well but there is a strict precondition: that the students are not behind in their prerequisites. If they are (spoiler alert: they are), you can still use these but the main problem will not be which strategy you use but whether you can identify the missing prerequisites quickly and whether you can design problems in an adaptive way to respond to what you discover.

[turbid zenith]

Oh yeah, the prerequisites thing is definitely an issue.

My Calculus III students are STILL sometimes trying to say that (a + b)² = a² + b².

And the thing that I'm dealing with now is students having forgotten integration techniques

I had a student write that $\int \dfrac{x}{1+x^2},\mathrm dx=\dfrac{\tfrac12x^2}{x+\tfrac13x^3}+C$

[burnt vessel]

DM Ashura

[turbid zenith]

Which I guess is the same "everything is linear" problem

[long pelican]

Just had a thought about the "everything is linear" problem

It might be these students have a history of relying on shortcuts and never really understood why the actually linear things are linear

[turbid zenith]

Oh yeah absolutely

[long pelican]

Question is who is going to help them address that...

[turbid zenith]

What if I've tried to help address it and it just doesn't seem to stick? :/

[long pelican]

They probably need their foundations rebuilt

including understanding of logic and how logic is used to prove that actually linear things are linear

One possibility is if enough people in the class have the same issue you could spend some time rebuilding their foundations

Doing that would go outside the syllabus for sure

[turbid zenith]

It's rarely enough people all at the same time

But one day it's one student, the next day it's another one or two students

What I've tried to do is couch it in talking about the Distributive Property — the idea that students think it's something "about parentheses" when really it's about the relationship between multiplication and addition

And in Calc I, I can't count how many times I had a student suggest doing something like √(a² + b²) = a + b and going "you know what? Let's see if that's true, let's try it with a = 3 and b = 4, nope that didn't work"

[long pelican]

Oh, maybe you can have them work it out on a problem set. The benefit being that it takes time to understand linearity, and a couple minutes of class time is nowhere near enough, but homework time may be enough

[turbid zenith]

But it goes in one ear out the other :/

[turbid zenith]

Yeah, maybe I just need to design a problem set about linearity heh

[long pelican]

Yeah, something like:
"Read this online thing about linearity"
"Prove distributivity of integrals over addition"
"Disprove distributivity of integrals over division"
[... more examples like the above]

[turbid zenith]

Asking students to prove it might be a bit much but I see what you're getting at

Demonstrate with examples, perhaps

[long pelican]

There's a professor where I work who adamantly believes that college freshmen non-math majors are capable of proving things if taught

The only thing I can say is that when I taught a semester of calculus based on his curriculum, it seemed true

[turbid zenith]

Well I'd love to see his curriculum then

And see how that applies to my population

[long pelican]

sure, I can share his lecture notes

well I haven't exactly gotten permission so I will share screenshots instead

Oh wow I have something better

This is his 9th homework

I didn't use this one (I thought it was too hard for them) but I did have some (slightly less) hard questions in my own problem sets

[turbid zenith]

Yeah wow this is way above what I ask <_<

[long pelican]

yeah, this professor's curriculum is quite out there

[turbid zenith]

But if you've got a class that can go there, then run with it!

[long pelican]

Oh yeah the point of my anecdote was to suggest that it's worth it to consider teaching what a theorem, definition, and proof is

If my semester was any indication, they are capable of learning it, and it helps a lot with understanding of concepts too

[turbid zenith]

Sure, it makes sense ... I haven't had as much luck :/

Although maybe I"m just not seeing what they have learned. I just have been kinda disheartened by how things have gone the past couple years

[long pelican]

I have heard state schools (not sure if your school is one) have a larger population of students who don't care than non-state schools

but you can detect who doesn't care by, say, who doesn't come to class and rarely submits homework

[turbid zenith]

That's its own issue ... I'm talking about a private school though. It's not that my students don't care, they definitely do. It's that there are such gaps in their knowledge and confidence

[long pelican]

One semester I gave the class a pre-semester survey with questions (not problems) such as "Do you know what it means for two functions to be equal?"

If you have questions like that ranging over the curriculum you might be able to develop a picture of what background needs addressing before day 1

Got any ideas for possible questions?

[turbid zenith]

Mostly algebra things, to be honest. Like do you know how to raise a number to a negative or a fractional power.

[long pelican]

I'll show you my questions

(this was for linear algebra)

[turbid zenith]

Which I've had to help a LOT of students with in Calc I

[long pelican]

I guess I mainly targeted skills that are less likely to be talked about explicitly in the curriculum

[turbid zenith]

That might be a good idea to do a survey like that. I'll have to give it a shot.

[surreal warren]

Any recommendations on teaching lay people masters level maths? Or on self-teaching the same? Trying to understand to what extent being a great mathematician can be taught, vs being innate

[lethal leaf]

What's the goal here and what's the background of the lay people

Some expository article or presentation will be different to trying to write a longer form curriculum/monograph

[tawny slate]

I like problems like this. Conceptual tests that arent super difficult

[long pelican]

Thanks!

[simple zenith]

Hi there,im gonna start tutoring soon and would like to better educate myself on best practices and in general on math pedagogy,any sources worthy of recommendation?

[frail summit]

What age ranges? If it's for elementary, maybe you can have a look at Jump Math'sTeacher Guides.

[simple zenith]

im more inclined towards hs tbh

[surreal warren]

Fascinated by the idea of the US state department language courses/ the Martians and how consistently their background led to excellent mathematicians. Started wondering about why firms don't have 'quant bootcamps' for example. If that makes sense?

[lethal leaf]

the Martians

What

[surreal warren]

Basically what Szilard called Oppenheimer and friends

Something about the Hungarian teaching system produced a cohort of incredible mathematicians. Gyorgy Marx wrote a biography of the group (including himself) called 'the Martians', Von Neumann used to joke about it and so did Polya

(Obscure Oppenheimer trivia alert ahaha)

[winged urchin]

Ohh I love reading a bunch of discussion here with icy and ashura ahah

[winged urchin]

You know I have a feeling you'll never get rid of the linear problems or other nonsense like that.

For two reasons really:

• carelessness. Heck even we can make silly mistakes at times. Multiplying something wrong or just missing something. To correct this, students need to be mindful enough to check their work and consider whether their answers 'make sense'. But this is a hard skill to teach really.

And

• the idea that it's better to have something rather than nothing. I believe for some students they spend hours trying to do a problem, get nowhere. Then they are in a time crunch and find a thing they are not 100% sure is true but accept it anyways since it solves their problem. I know I myself did this sometimes in school. Albeit in higher years but still I would get something and not be entirely convinced it's correct but still submit it since I had spent so much time on it already

I think as far as my tutoring is concerned in teaching styles I'm definitely an inquiry-based tutor with a dash of Socratic method thrown in.

I tend to get students to work on problems and throw them hints or explain topics if they have a large enough gap in their knowledge. Then I try to ask them questions about their work or thought process. Try to get them to explain what they are doing

But of course I am just a tutor and I get to work with my students 100% so this style doesn't work so well perhaps in a class

Though oral exams kind of hit on a portion of this though even that is infeasible in large classes.

I do love getting students to explain their work though. I really try to get them to be specific. What exactly are you doing in the equation. Not just canceling or moving terms but what did you add to both sides or subtract from both sides or multiply or divide from both sides?

[turbid zenith]

Idea I had for a question for my final exam in Calculus III. What do y'all think? Anything I can do to make it more clear?

(These would be worth maybe 1 point each of a 100-point final)

[long pelican]

(c) is ambiguous 😆

[turbid zenith]

How so?

We've stressed distance is a scalar, displacement is a vector

[long pelican]

ok maybe not

how about some mathematical formulas at the end?

[turbid zenith]

$d=\int_a^b\Vert v(t)\Vert,\mathrm dt$

[burnt vessel]

DM Ashura

[turbid zenith]

What do you mean?

[long pelican]

Given $v\in\bR^3$, the value of $(v\times v)\cdot v$

[burnt vessel]

Icy0

[turbid zenith]

Hmm that could be interesting

[long pelican]

I've made a similar question on homework where some formulas didn't make sense and it was an exercise (for some, a first time actually reading notation in detail) to understand how it doesn't make sense

[turbid zenith]

(And by the way, "type checking" is something we've talked about in class explicitly)

[long pelican]

dang you're ahead of me there

or wait

type checking meaning vector vs scalar

[turbid zenith]

Yes

[long pelican]

I remember one of the nonsense expressions was $\frac\partial{\partial x}f(g(x,y))$ where $f,g\colon\bR^2\to\bR$

[burnt vessel]

Icy0

[turbid zenith]

Oh fun

[long pelican]

My thinking with those questions was along the lines of "You should have this skill [parsing notation slowly] if you reached here"

Of course, they didn't

I think it's a good lesson to every future teacher that the median student never picks up mathematical skills that aren't exactly those assessed on tests in school

Nice python screenshot!

wonder if python is more intuitive than calculus to most 18 year olds nowadays

I checked python's floor sends floats to ints, and it does, whew. Lean does too. I know C++ doesn't. So C++ is the outlier in this aspect.

[turbid zenith]

Huh interesting...

[modern trench]

What is floor(3.14E38)

[cosmic ibex]

Python's ints seamlessly change representation between machine words and bignums.

However,

>>> import math
>>> math.floor(3.14e38)
3.14e+38
>>> type(math.floor(3.14e38))
<type 'float'>
>>> 

[long pelican]

Must be a python 3.11 or 3.12 change!!!!!!!

[cosmic ibex]

Wait, the Python I get by default is 2.7.17.

Python3 agrees that math.floor(3.14e38) is 313999999999999997086945350577234640896.

[long pelican]

Guess it's a python 3.0 change

according to Bing AI

Hello can I have 313999999999999997086945350577234640896 apples please

They have taken us for absolute fools

[turbid zenith]

Man ... making tests can be difficult

[long pelican]

Sure is, I once spent an hour trying to come up with a real-life-situation differential equations problem that uses integrating factors (I think...) and it turned out to be so creative the students couldn't handle it

I still don't regret making it

[winged urchin]

Man it is beyond disappointing how little some students are able to even attempt algebra or just manipulating equations.

I know it falls into the more tedious kind of mechanical learning. Factoring these things, expanding these, power rules, log laws, all that kind of mundane unsatisfying math we sometimes need to get through problems

[long pelican]

I've thought about the analogue where instead of manipulating equations for me it was using the excess intersection formula in intersection theory. In this case my reluctance to apply it came down to not understanding enough about what it is and what the proof was, and also more generally why one should think it is true and why one should think to use it. There's also the aspect that I haven't seen enough examples of it being used. I do believe the same phenomenon is playing out with this student and algebra

[winged urchin]

It just makes me want to drill students on solving random equations essentially so they can see different varieties of manipulations but it is so boring for students that they just turn off during those questions I think

I haven't heard of excess intersection formula aha

[long pelican]

You can find it in Fulton's intersection theory book 😄

There's normal bundles involved, if that helps

[winged urchin]

I remember in my early years I'd spend so much time just trying things in equations. What if I log both sides now? What if I use this log law? What if I plug the equation into itself? Aha

But some students I don't feel 'try' different things like that. Which is fair I guess. Im a mathematician so of course I have some interest in doing these things

[long pelican]

probably uncertain about what the legal moves even are

[winged urchin]

Yeah I sympathize with them and understand that's likely the issue I just am at a loss sometimes

I would say they don't try hard enough sometimes or don't focus but that's maybe the reason in some cases but definitely not all

And I don't like to assume the worst of my students anyways

[turbid zenith]

@long pelican May I PM you? I'd like to show my first draft of my final exam to another pair of eyes but I don't want to post in public

[long pelican]

Sure

[pastel sundial]

it occurs to me that I don't actually understand related rates

ok "don't understand" is an exaggeration. I know the steps to solve related rates problems and I can explain them to my students, but I don't feel comfortable with what's going on at a deeper level

ok "I don't feel comfortable with what's going on at a deeper level" is also an exaggeration. In any specific problem, I can "see an isomorphism" between the steps in the textbook related rates approach, and the steps in a technically different approach I'm more comfortable. But I'm not comfortable with that isomorphism in general on a deeper level.

It's similar to the whole separating dy/dx thing

[long pelican]

This language suggests you are enjoying your category theory rabbit hole

🤣

[pastel sundial]

not sure what channel to ask about this on since it's about my own lack of understanding, but it came up through teaching

[pastel sundial]

I can see the isomorphism, but I don't see why it's natural

[long pelican]

Maybe one way to approach what the symbols "dy" and "dx" mean are as cotangent vectors on the relevant curve cut out by the relevant equation

If the curve is 1 dimensional (and smooth) the space of cotangent vectors at any point is 1 dimensional, hence dy/dx has a meaning, being the unique scalar which when multiplied with dx gives dy

[pastel sundial]

is this about related rates?

[long pelican]

I thought so

dV/dr * dr/dt = dV/dt stuff

right?

[pastel sundial]

not sure

[long pelican]

With the cotangent vector interpretation the cancellation literally works

[pastel sundial]

wait what are cotangent vectors again?

[long pelican]

Linear functionals on tangent vectors 😏

[pastel sundial]

right

I'll need to think about it. It seems promising (though maybe less so as a way to explain related rates to calc 1 students)

[long pelican]

Fun fact, I have mentioned dividing vectors (in a 1-dimensional vector space) before in this server and it gave me unofficial crank status for a bit of time 😁

[pastel sundial]

lmao

[long pelican]

I enjoyed a similar look on my graduate students' faces when I discussed dividing elements of a G-torsor

this semester

[pastel sundial]

woah you're teaching a grad class?

[long pelican]

yep

A whirlwind tour through algebraic schemes, stacks, and derived stacks

[pastel sundial]

are you tenure track then?

[long pelican]

Nah not yet

Still gotta learn enough to get into the state where I can get obsessed with a specific research area in the langlands program

[pastel sundial]

I see

[lethal leaf]

Division of complex numbers 👀

a 2-dimensional vector space

also quaternions

[long pelican]

But both of those examples require extra structure!

[lethal leaf]

do they?

Do you mean extra structure of defining multiplicative inverses?

[long pelican]

Multiplication itself

[lethal leaf]

Oh you mean that with just with any field, you get "divison" from scaling

so no extra structure

[long pelican]

Yes but also the slightly stronger claim that division on 1D vector spaces does not require a choice of isomorphism with the base field

[lethal leaf]

ah

[cosmic ibex]

field : 1D vector space over it : scalar multiplication == vector space : affine space : addition

[frail summit]

If in the decimal system 10 units become a ten, 10 tens become a hundred and so on, how does the naming work in the binary system? 2 units become a what? Are there names for the positions of a binary number? How do we call the positions of 2^1, 2^2 and so on?

[lethal leaf]

For binary people usually talk about the "i-th bit"

Half the people zero index it half the people one index it

[austere delta]

I mean, the names two, four, eight, sixteen, etc work perfectly well.

[vocal phoenix]

Especially sixteen, being six plus ten

Very reasonable in binary

[frail summit]

So the binary number 111 would be 1 four, 1 two and 1 unit? I don't know what to say. I would say it sounds confusing for someone who is just getting started with a new base number system.

[frail summit]

Sorry, is this sarcasm?

[austere delta]

Sure it's four plus two plus one. Why is that confusing? Wouldn't it be more confusing to make up completely new words for four, two and one?

[winged urchin]

I'm equally confused by this tbh. I don't know why 16 being 10 plus 6 is very relevant to binary

I think most times a new abstraction is introduced it sounds 'confusing' to a student. Being a good math student is about not just reading a definition and saying ok that's that! No it's playing with the definition. Working with different examples until it starts to sound natural to you

[austere delta]

Understating binary and positional number systems in general can be confusing sure. But I'm not sure the fact that we have words for the numbers two, four and eight is the confusing part.

[cosmic ibex]

I think the complaint was that the word "sixteen" etymologically suggests the speaker is thinking in base ten.
Which I don't think is a particularly valid complaint -- it's not like bases are something you have to swear eternal allegiance to, and just because you have a practical reason to speak about base two doesn't mean there's any expectation to eradicate the way you already think about numbers from your mind ... on the contrary, if there's a pedagogical point that ought to be made, it is that numbers are the same no matter how we notate them for any given purpose.

[surreal warren]

You would just say ' one one one'

It's analagous to reading out a phone number or morse code

You won't know the position until the number is done being read

Morse code is the analogy used in code by Charles Petzold, he spends a few chapters explaining different number base systems. I found it a really accessible explanation

[tepid smelt]

Are you using a book for the course or just notes you come up with?

[long pelican]

Neither

[long pelican]

I'm lecturing off already-existing lecture notes but I am kinda drawing from a lot of sources

non-exhaustive list

[austere inlet]

waaaa

teaching grad class must be a challenge in itself

[long pelican]

It is in many ways that are different from the challenges of teaching an undergrad class!

[vagrant meadow]

yeah i've been curious about what kind of challenges that grad classes pose in contrast to UG. i imagine chief among them being that knowing the answer to any question off the top of your head is a much much much bigger ask. like atm my goal is to teach at a community college, but teaching a grad level class is also something i'm interested in as well.

[austere delta]

I'm teaching a grad class for the first time next semester. Exciting, though I'm a little anxious about the exam.

[long pelican]

No exams here, as is standard for pure math topics classes

[austere delta]

Wait, so no grade?

[long pelican]

Every grad student gets an A, undergraduates have to do some project

No undergrads though

[austere delta]

So you have mandatory attendance or something? Or you just give out free As

[long pelican]

Yeah, free A's. The grad students here have to know what they're doing to even be in the program and pass quals etc... anyway

[austere delta]

I guess

[vagrant meadow]

what subject?

[austere delta]

Representation theory of Artin algebras

[long pelican]

Wow that is pure math too. You're expected to have exams in that?

[austere delta]

Yeah, all courses have exams

There are no qualls though, so the whole system is a bit different I guess

[winged urchin]

I had exams in my grad courses personally

[austere delta]

PhD students don't get graded though, just pass/not pass

[winged urchin]

Both masters and PhD level at two different universities there were still exams

Though a couple were just oral. The professor just asking questions and you explaining in a sit down setting

Definitely had some written ones though

Though even as a PhD student my grad courses weren't like specifically for PhD level. They were with masters level students and even some undergrad

[austere delta]

Yeah, there no restriction on who can sign up for which courses at my university

[quasi musk]

Damn all classes have to have exams? Even topics courses?

I feel like everyone would fail my NT theory class if there was an exam

[austere delta]

I mean the exam doesn't have to be difficult. Surely you learned something that could be demonstrated in an exam.

[cosmic ibex]

What's the difference about "topics courses" and other courses?

[quasi musk]

Topics or elective classes in grad school are, to my knowledge, classes that aren't required "qual" classes. Usually they're geared towards that professors research or research interests

e.g. one topic being offered at my school this year is ergodic theory

or another topic I took was knots & 3-manifold topology

[cosmic ibex]

I think that must be distinctions that didn't exist in the time and place I was a grad student.

[quasi musk]

My requirements are 4 qualifying course sequences, 2 qual passes, and 4 electives (quarter system)

[lethal leaf]

what about qual classes?

[long pelican]

quals don't have associated classes here

[lethal leaf]

oh huh

strange

[long pelican]

Well there are classes with the same titles

but they don't match quals questions with what happens in those classes

[small scaffold]

I am leaving my CC after 15 years, to follow my spouse overseas.
She got an offer we’d be fools to refuse.

[turbid zenith]

Ahhhhh quals

I have those at the end of this semester

And I am terrified

Gotta love officially sanctioned academic hazing 😛

[wise onyx]

[void lynx]

woah wait are you a grad student teaching grad classes 👀

[long pelican]

Nah already graduated

[quasi musk]

To bring it slightly back to pedagogy, I've been thinking about how a well run "qual prep" summer class goes

[turbid zenith]

I think mentioning that quals are academic hazing is still talking about pedagogy. I don't think they have pedagogical value. 😛

[quiet jackal]

why not?

perhaps not

but it is not like they are exams in a course or something

you need to force students to build a solid foundation somehow

[quasi musk]

Interesting. Even if you think that the quals themselves have no pedagogical value, you also think that if there were a qual prep class, it would also have no pedagogical value?

e.g. at my institution, over the summer more senior grad students help prepare the youngins for their quals in a classroom environment that meets twice a week for an hour or so

[cloud zealot]

quals vary by institution. some institutions give written exams, while others give oral exams. are you against quals altogether in general, or are you against certain qual formats?

[turbid zenith]

I guess I only know my own institution's format where I'm doing my PhD

But essentially in my school it's just like "Hey you know you took a final? Guess what? Now you get to take another one but harder!"

In one case literally two days away from each other

Oh there's also Georgia Tech who like ... brags about how few students pass their quals sometimes 😛

[quiet jackal]

yes both of those are idiotic

but that is not universal

[quasi musk]

I wouldn't say idiotic, they have their pluses and minuses; it's nice that there's an option to take the qual right after you finish the class

Rather than having to wait months & spend the summer studying lest you forget everything

Also if it's well known that this particular school has hard quals, and you do well on the qual, then you could have some room to get a particular advisor, transfer schools, etc.

[quiet jackal]

but then why not remove the final exam

maybe instead offer a practice qual or something

[vagrant meadow]

well some people take the course but not the qual for one thing.

and the midterm in one of my classes was mostly just questions from previous quals. so it was kinda of just qual prep 11 months before i will be taking it.

[quiet jackal]

11 months lol

[quasi musk]

But the final exam is preparing you for the qual. Overall I suppose that I have a more positive outlook on quals than Ashura. Just to make sure you know the basics before you move onto research

[quiet jackal]

i agree that quals have some value

maybe a lot of value

probably oral quals are much better than written

[turbid zenith]

Making sure you know the basics doesn't mean making an artificially hard test that you have to do under tight time constraints.

[quasi musk]

That also varies a lot from institution to institution. Mine seem like very reasonable questions with reasonable time given

Hopefully yours will be more similar to that

[vocal cypress]

$\DoubleSin$

[burnt vessel]

Blaved
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[south moat]

hey, got a question for you all. How many homework assignments do you give for your courses?

my school is asking for 20 assignments a week per class per subject per week

curious how other institutions manage their teachers

[deep kindle]

20 a week for each class?

4 assignments a day?

[south moat]

sorry, mistype. We have 4 subjects. I teach Alg 2/ pre calculus/Ap Physics/...Drama...

so that means, 4 fully graded assignments per week for each subject.

Before it was only 2 graded assignments (homework) and the rest was just participation/general stuff.

[deep kindle]

Fully graded as in summative?

[small scaffold]

I believe “fully graded” as in “I need to sit down and really look at this” as opposed to “complete? Looks about right? Check!”

[deep kindle]

That seems a little insane

[wispy slate]

(a) Show that the objective function can be rewritten in a much simpler form, to wit,
F(s,x;β) = w−css−cxx−α[1− p(s)]D(x).
(b) What additional assumption would you make, if any, to ensure that (s,x) = (s0,x0) is the unique local solution to the optimization problem? Explain.
(c) Derive the FONCs and SOSCs of the optimization problem. Explain carefully how, in principle, the solution, say s=s∗(α,cs,cx,w) and x=x∗(α,cs,cx,w), is derived.
(d) Prove that p′(s)> 0 and αD′(x)< 0 at the optimal solution. Do these conclusions make sense? Explain.
(e) Prove that p′′(s)< 0 and αD′′(x)> 0 at the optimal solution. Provide an economic interpretation of these results.
. (f) Prove that s=s∗(α,cs,cx,w) and x=x∗(α,cs,cx,w) are positively homogeneous of degree zero in (α,cs,cx).
(g) Prove that Fsx(s,x;β) < 0 at the optimal solution. What does this imply about the economic relationship between self-protection and self-insurance

can any of you help me with this?

[wispy slate]

Absolutely anyone

[long pelican]

This is not a help channel, so you are asking in the wrong place

[pallid night]

We can help you with discussing the pedagogical value of setting such a question to students as a teacher but we can't help you solve the question (at least not in this channel)

[tepid smelt]

That's a great way to burn out a teacher. I have only ever graded hw at the secondary level for completion. I don't think having four hw assignments a week is insane assuming it means hw everyday other than Friday.

[wispy slate]

Cool

[frail summit]

Did you check the latest PISA results for mathematics?

[long pelican]

https://news.ycombinator.com/item?id=38530357
This also came up on hackernews

pretext

The pandemic's toll on schooling emerges in terrible exam results

Hacker News · Dec 5, 2023, 1:21 PM

[lethal leaf]

I have a friend who is doing HS teaching now (not math but the policies she's talking about are school wide)

It seems to be really really bad

Students just don't seem to care.

She has a policy where any student can turn in work until the end of the semester and she will take it for full credit (so even if it's 10 weeks overdue) and she still has students not submitting and failing

I imagine that similar lack of caring is extending to all these tests and stuff

[frail summit]

Still, there are countries that saw growth even with the pandemic.

There must be more than this.

Wait, so are you saying the result are worse because students don't care, not necessarily because their knowledge is decreasing? (although, I'd say these 2 can very well also go hand in hand)

By the way, how about the nordic countries (Finland, Sweden) that were very much praised for their schooling system. They now seem to go downward (although they are still above average).

[winged urchin]

I think Ive noticed a general increase in misconceptions in first year university. But it's just anecdotal. I'm in Canada too

Cancelling terms in fractions when there is addition/subtraction in the way.

Ideas that you can't cross asymptotes.

Difficulty or inability to solve even linear equations sometimes

[lethal leaf]

I don’t think it’s correct to say students are fundamentally losing knowledge

Rather they just don’t care enough to obtain new knowledge

[winged urchin]

We're always losing knowledge through our meat sieve brains @.@

[wide ice]

Education is that which remains behind when all we have learned at school is forgotten.

[frail summit]

This doesn't sound good. Do you notice a pattern? students educated in the local education system versus international students?

[frail summit]

You are right. That's what I meant to say. Thanks for putting it into words better than I did.

[vagrant meadow]

so i feel this example is representative of the thing i struggled the most with my first quarter as a TA. in my DE section, we were covering constant coefficient 2nd order homogeneous diff eqs a few weeks ago, which i feel like is a relatively simple topic (in terms of the actual execution of the problem)

1. solve a quadratic
2. interpret the result and write it in the right form based on the type of roots
   i thought, "these are DE students. they've passed calculus 2. clearly the trouble is going to be in part 2". but... no. many of them didn't even know the quadratic formula. and it's like i don't know what to take away from that experience. that i can never underestimate the gaps of knowledge my students have? i'm just not sure if it's a good thing to go in expecting that DE students don't know the quadratic formula. it feels like if i actually thought that, it would mean my expectations for my students would be so absurdly low that it's almost insulting. but... i guess if it's accurate, then it's not that bad? 😬

[winged urchin]

To be fair most of the time I think problems are designed specifically so you don't need quadratic formula in highschool and even first year calculus courses

But then again I think that goes to the lack of initiative to learn what they need perhaps. They should know that the quadratic formula exists and so if they need it they can search it themselves

[winged urchin]

I dont actually ask if my students are international or not. I suspect local though but I think it's fairly widespread.

Another big problem area I think now that Im mulling over it is making diagrams.

I have some difficulty getting students to realize that two parallel line segments of length a and b can be made to have a triangle between them with a height of a-b (or b-a)

We need to teach geometry again perhaps aha

[long pelican]

Why did you say the line segments are parallel? If you are going to move them to make a triangle anyway

or I'm lost

Yeah I'm lost

I did what you said though

[winged urchin]

Sorry I wasn't clear

Think like a rectangle on a right-angled triangle

Related rates type speed problem

[long pelican]

My thoughts about this: simulating a student:
I see on the worksheet: Solve x^2 + 8x + 11 = 0
Thoughts: ok this is an equation. I don't exactly remember how I solved this equation though, it's been a while. Actually I forgot almost everything about equations. Factor? I forgot factoring too, I probably won't even try. Guess and check? Math teachers hate that. I'm just lost about everything and don't know how to even begin!

If lucky, and if the keyword "quadratic" appeared somewhere, they might remember there is something called the quadratic formula but don't think that I'm recommending that you put that keyword on there, because being lost like that is a problem and we shouldn't try to mask it with mnemonics or memory recall devices

[winged urchin]

I think it is valuable and a good exercise as a teacher to sit in your students shoes like that though it feels like there aren't a lot of good solutions in the moment here

But it is a problem more throughout grade levels

As an aside it's funny that yes usually guess and check is frowned upon since you're not showing your ability to solve. Like how a student can see that 2 solves 2x-4=0 but might not be confident doing the steps

But then in calculus when you first see differential equations or really simple integrals sometimes all they have to 'solve' is a guess and check type thing

parallel line triangle thing i was describing before

[long pelican]

Ooo

I was confused because I drew them horizontally

lol

[winged urchin]

trying to explain diagrams in words is a fools errand really

aha

[vagrant meadow]

it wouldn't be a problem if I didn't have to administer a quiz. otherwise I wouldn't care at all. because yeah they can just make a note "oh over the next week I gotta review solving quadratics if I don't want to fall behind" and no harm done. but with a weekly quiz, these holes in knowledge have real consequences for many students. I want to set them up for success, and I feel like not anticipating that DE students would struggle with solving 3r^2+r+1=0 is a failure on my part to adequately prepare them.

I feel like this class has given me a real distaste for quizzes. I can't tell if the overall effect is actually positive, but it feels negative. I wouldn't be surprised if it was actually instrumental in keeping many students from failing, though.

[wide seal]

this is what my sister does often, including the college she works at adjunct 💀

she managed to get really good retention on avg, but students she loses, are just lost

¯_(ツ)_/¯

[north wraith]

Hi 🙂
I'm not sure if it's an appropriate channel to ask such a question (since I'm a student myself) but mods, feel free to remove 👉🏻👈🏻🥹

what non-fiction books do u think any 'successful' student / undergraduate should read?

something along the lines of Ultralearning (Young), Deep Work (Newport) and Learning How to Learn (Oakley)

sure, not every single piece of advice from these was useful (to me at least), but I definitely feel like learned 1 or 2 new things at the very least from each of these

[frail summit]

Thanks for asking this. I'm curious, too. Always eager to hear some good book recommendations.

[wide ice]

I recommend The Defining Decade by Meg Jay and Late Bloomers by Rich Karlgaard

Mostly to combat imposter syndrome