math-pedagogy

[turbid zenith]

Pointed out a lot of the properties we’d like open sets to have in a more general setting

And then in the next section defined a topological space, noting that the definition of open sets is built to take those things we like and hold onto them

[south vigil]

Oh hey that's a good way of doing it

[turbid zenith]

I agree with you, it’s a MUCH more effective way to do things.

Another example is I have a book on abstract algebra that has you play with “algebraic systems” in general first before it defines groups and rings — You play around with examples of sets with various operations to see what properties you get, and then say “Hey when we have this particularly useful collection of properties, we call it a group/ring”

The traditional way seems to be:

1. Set out the completely sanitized definition right at the beginning, with all the caveats that keep the weird edge cases at bay specified up front
2. Maybe give some motivation for why that’s the definition, but not always
3. Examples
4. if you’re lucky, applications as an afterthought (this applies mostly to things like calculus etc)

I would argue that a lot of students would be better off if things were done pretty much in the opposite order. Actually just gave a talk to that effect.

[cosmic ibex]

I agree that giving motivation is crucial, but I'm not sure I agree the reader should be burdened with trying to absorb the motivation without yet knowing what it's the motivation for (unless they cheat and read ahead so they can understand the motivation).

My experience when I read sources that present a long sprawling discussion before they deign to tell me what the concept they're aiming for is, is that I get extremely annoyed at it. I end up shouting impatiently, scanning ahead until I find the darn definition, and then jump back to the start to see if all the sprawl actually helps me understand it, now that I have a mental framework for reading that discussion and know what to look for.

[south vigil]

Hmm, although that's valid, I will point out that what you're saying is that it's annoying more than that it's ineffective, right?

Well, maybe not. You're also saying that being informed on what the explanation is for helps you figure out what to build a framework of

Okay, I can see that

[south vigil]

What I hate is that number two is the most important one

Well, numbers one and two are

[winter phoenix]

can anyone help with logical math?

[south vigil]

I think with abstract math, the correct approach is to build up an intuitive explanation to motivate the reasoning you will do, and then use your experience with rigor to make that reasoning work

But I also worry that because that second part can be applied without the first, i.e. deriving things rigorously without a motivated approach, it's a skill that gets overemphasized

[turbid zenith]

I guess it depends on what you’re trying to do

Some people wanna just cut to the chase

[south vigil]

But that only works for solving problems, not understanding math

The chase isn't answering textbook questions, it's knowing what's going on

[turbid zenith]

I think if you’re already “drinking the Kool-Aid” as it were you can handle having the motivation come second

But for students who are still getting a good feel for why math matters, I think leading up to the definition often seems to be what works

[south vigil]

Of course you can handle it. You just shouldn't

You can just define random bullshit as axioms, and as long as they're consistent hey you got math

[cosmic ibex]

I find it very inefficient to deal with motivation without knowing what it is the text is supposed to motivate.

[south vigil]

That's valid

But my problem is that the motivation is so often underemphasized in the extreme

Despite being the whole point

You can get VERY far in math just by being good at manipulating rules

[cosmic ibex]

That's a different problem from whether there's a definition in front of the motivation you're lacking.

[south vigil]

Sure, but that's my actual problem

[cosmic ibex]

I can totally get on board with wanting motivation.

[south vigil]

The reason I said something different is because I had to think out loud to realize what it actually was

[cosmic ibex]

The worst is when there's a long, ambling, chatty exposition of something where it never gets quite clear whether they're describing a motivating special case, or a typical example, or actually general truths about the mystery concepts -- and then suddenly, with nary a paragraph break, the text says "... and this is then known as a left-handed semi-simple antimorphism" and jumps directly into discussing applications and further generalizations. Without ever deigning to state a self-contained definition.

[south vigil]

Yeah that can be bad

[fallow siren]

I had a few grade 9 students who would stop mid sentence to complain about not knowing what a word meant despite the fact that the sentence they were reading was defining that word. It was baffling to me. There were other issues though.

[turbid zenith]

This is useful. I’m definitely gonna have to make sure I’m not doing that when I do things.

[elder venture]

I'd not like a full page of motivation only like a single paragraph.

[tardy ember]

imo it... depends somewhat on what the thing being defined actually is

if it's "curvature", the reader probably already knows at least somewhat what that is, you can just point out that that's what's being defined, then define it, then maybe say something about why that has anything to do with curvature if it's not already obvious

if it's "a category", that probably isn't a concept the reader is already familiar with (unless they already know the definition), so i'd say more like, give the definition, talk about it a bit, give some of the easy examples (Set, abstract algebra), that kind of thing - it makes sense imo to give the formalism first here because they don't already have intuition for what a category is supposed to be (or at least not explicitly enough to easily point at)

if it's "a monad", then uh, i have no idea, i know what those are but i have no idea how to explain it

[south vigil]

A monad is an entirely fictional concept that exists purely to stroke the egos of those who think they understand it

[long pelican]

It's interesting to note how guides to monads aimed toward programmers are much more understandable than ones from a mathematical viewpoint

[cosmic ibex]

In the former case you have the luxury of being able to assume that objects have elements without needing to apologize. :-D

(And "parameterized type" feels a lot more concrete to the programming side of my brain than "endofunctor" does to my mathematician side ... never mind that the former doesn't in itself imply anything about functoriality; it does provide a nice mental scaffolding for organizing one's understanding).

[vagrant meadow]

i know it's hard to say in general, as i imagine the answer is different for different concepts, but how much definition and discussion would you say should precede the motivation?

for example, i'm toying with the idea of introducing linear transformations to students who only really know calculus using the examples of derivatives and integrals to motivate linearity (for an intro to linear algebra).

[cosmic ibex]

From a personal and subjective point of view, I believe I would prefer learning from a text that always starts by showing me a technical definition. It might then very well need to immediately admit something like "the first time around, this definition looks impenetrable and/or unmotivated" and dive into some motivation, discussion, and examples. Or even "half of the words in this definition will be unknown to you yet, so we'll need to spend 10 pages of defining auxiliary concepts before it even makes sense". I still feel it will be easier to internalize those auxiliary concepts when I've seen a context they're going to fit into.
Or, hmm, perhaps I should moderate that -- I can live with some text coming before the definition saying, e.g.

Our next task is to define the concept of blablabla, which tries to make precise the vague notation of such-and-such. For example how do we distinguish between things like [...] and [...] where one seems to obey a principle of [...]? It turns out that it works well to define: [DEFINITION]
But I'd like to at least know the name and a high-level "type" of what we're working towards, rather than just see a cavalcade of random examples of something unknown, and only afterwards be told what I ought to have noticed about them.

[vagrant meadow]

that's fair. personally, i agree. but i've found that as someone who best understands through abstraction, there's a big divide between the explanations that make sense to me vs most other students. that divide just seems to grow larger and larger the more i tutor, as well.

i think my worry is that by showing the "scary" technical definition too soon, their eyes will glaze over and they'll just give up trying to understand. however... putting the goal up-front also contextualizes the motivation, which can make it more effective in the end.
tysm for your perspective 🙂

[distant fractal]

Supposing I have 2 extra classes and no more content (and I am already dedicating time to review), what would be a good topic to discuss in a Calculus 1 class?

I could just do some calc 2 topics, like integration by parts

Or I could do something more fun

[shadow flower]

maybe show them that their intuition can be wrong with the divergence of the harmonic series

I was always interested in that when I was in highschool

[austere inlet]

honestly I see no reason for it other than "it's a small, easy to remember set of axioms that covers a lot of examples we care about" -- it can definitely seem unmotivated if defined without any context so I agree with that point of view. Documents that skip the fluff and go straight to a definition are good when you're an experienced mathematician but perhaps not so much for undergrad teaching.

exactly, in mathematics it is often the case that a definition appears precisely because there's useful instances or examples of it beforehand; in the case of point-set topology the "family of open sets with funky properties" wasn't even the first definition of a topological space -- instead the first proposed definitions modeled some notion of "closeness" through accumulation points or neighborhood bases: https://hsm.stackexchange.com/a/8415

in my experience some students benefit a lot from being given some motivation or context for the abstract definitions, though I do agree that such an introduction should be clear and concise

[vagrant meadow]

personally, i think Taylor series are "simple" enough that a calc 1 student can understand them. getting a brief exposure so that seed can grow between now and when they formally learn them is helpful imo.
||but I'm also biased bc i love Taylor series||

i watched the 3b1b Taylor series video before I took calc 2 and it made a huge difference for me. by the time we started them it was already super intuitive to me.

but i also second the harmonic series idea

[shadow flower]

taylor series was my next suggestion bc the idea of that when i was first exposed to it was also mind blowing

everything is an infinite polynomial

[pastel horizon]

Been a while since I posted here but what do we think about AI and the future it will have? It's definitely a very powerful tool for teachers, takes a big load off planning but we also have to consider ways we can ensure students understand the material without cheating

For me when Co Pilot drops that's just gonna be such an awesome tool to have for delivering lessons

[zenith slate]

I've certainly seen a lot of hype regarding the use of AI to help with planning but I've yet to see anyone actually incorporate existing tools into their planning workflow. I plan a lot of lessons for high school mathematics and fail to really envisage how I could leverage AI in any meaningful way beyond perhaps problem generation (but this is rarely a bottleneck for me). More important to me is content sequencing and differentiation - I'm not sure how amenable these types of problems are to an AI-oriented approach, at present.

What would be a real game changer would be actually good OCR tech that can decipher awful student handwriting

[long pelican]

I don't think AI is at the point where it's better than me at anything I do for my complex variables class, from planning a lecture to creating problems. Especially creating problems. This week (the final week of the semester) I'm planning lectures on the Gamma function and the Riemann zeta function out of Ahlfors and I would be very surprised if AI could give me something meaningful on those topics that I can't already come up with

[zenith slate]

A particular application I have seen that is effective is generating individualised homework problem sets - my school already uses a service that provides this and it's been remarkably successful, at least in terms of student engagement.

Yeah I think the likelihood that AI will make meaningful changes to university pedagogy in the near future is slim

[long pelican]

Is AI capable yet of generating problems beyond routine textbook exercises or vague mind-overloading real world problems?

[zenith slate]

No - but what it can do is provide problems at a difficulty level appropriate for the student

[long pelican]

up to the difficulty of the hardest routine textbook problem, I presume?

[zenith slate]

In terms of differentiation this is a massive improvement over anything remotely practical for a HS teacher

[zenith slate]

Yes

I mean the particular service we use also has a kind of gamification built-in - students can earn points for completing problems and studying in their own time that they can cash in for rewards at school

This isn't AI related, but a very neat application of tech

Homework completion rates have gone through the roof

While the problems students are getting may be fairly boilerplate, at least students who would have never done their homework in the past are now actually doing math problems regularly

I think it's things along these lines - perhaps you'd call them lessons learned from the nightmare of the Internet and social media and its conditioning of young people toward getting Internet points - that have the potential to have a huge impact

Take the addictive qualities of being online and apply it to education

[long pelican]

Repeatable content 😉

Repeatable but varying content, rather

[zenith slate]

Yeah

I mean just the differentiation alone is a game changer

Good stuff

No child left behind but it actually works for every child lmao

Also the fact that online homework just gets marked by the algorithm is nice 😉

[long pelican]

The one semester I used Webwork I had so many emails reporting wrong answers that were actually correct and I had to fix the code every time -_-

[turbid zenith]

That's why I've been coding all mine from the start this semester lol

I have one left to do

[quasi musk]

Any chance to sneak in dirichlet's theorem on arithmetic progression?

[tepid smelt]

I have not used chatgpt much recently but I enjoyed having students find errors it would make on problems. It can present solutions that sound convincing but have obvious errors that are not easily spotted by HS students.

[pastel horizon]

Only a matter of time before online homework gets replaced with an AI tutor

Obviously the GPT model wouldn't be enough but with a more specialised model you could have something that can potentially deliver the material, identify misconceptions, tailor questions to attack the misconceptions and also scaffold/ provide challenge appropriately

Also having something that could just scan written work would be insane too

[pastel horizon]

So one idea I would have is to make use of diagnostic questions.

If you can provide an AI with some generic diagnostic questions for some topics and then train it with appropriate follow ups based on the responses then you'll end up with a pretty accurate assessment tool that can rapidly personalise a student's learning.

Currently it's just a generic system that recommends questions based on what they got wrong rather than why they got it wrong

[zenith slate]

Yeah I think the tech has a lot of potential, but it's not clear at this point what the hard limits of language models like GPT actually are. It could well be the case that the kind of thing you are proposing is simply not feasible with current models

I think teachers would also be loathe to hand over the reigns quite so much, at least for the foreseeable future. Nevermind what parents would think about teachers handing their kids over to learn from some mysterious AI!

Doubtless things like this are being worked on, though

[long pelican]

Unlikely 😦

[quasi musk]

It's ok, my complex analysis prof. talked about it briefly. Then the next term when I did my MS, I was taking a class where the main goal was to understand the proof of Dirichlet's Theorem

It was kind of cool

[rain rain]

I know a lot if ppl hate Tao’s analysis books but I really like the way he teaches

[long pelican]

What are the top reasons some people hate them? I don't know much about them

[rain rain]

I think it’s painfully slow/obvious sometimes, even for me, but I love how he builds motivation and provides so many counterexamples to show why it’s important to build things up rigorously

Well afaik it’s because he uses a bunch of terminology that only he uses and he goes way out of the way to build things

Like to introduce the limit he makes up an operation LIM and spends forever working with that before he shows that it’s equivalent to a limit

But the only reason I’m mentioning this is because I recently started diffgeo and I hate how the material is presented

And I’ve had that problem with a lot of other math books too

It’s just definitions and theorems vomited at you and you’re expected to stick with them until they come together and you see why you needed them in the first place

Earlier today I was complaining about this and someone explained why we needed those defns and how they tied into our original goal/chapter topic

And it made everything so so so much easier to understand and appreciate

I just wish that was more common practice, motivating concepts and giving a precursor to what machinery & objects you’ll learn about and why you learn about them before actually diving into the material

[cold trench]

this is what makes math books unreadable to me

[quasi musk]

There's this idea in math education that you should just move as fast as possible from wherever you are to the more interesting stuff to research

[rain rain]

I think it’s important to get to the meat of the material without taking too long

I came to study analysis, not learn about the construction of the integers

But I think there’s a balance too where people go too fast and it just feels like you’re thrown into a maze and you don’t see the end until you spend a couple hours being lost in it first

[rain rain]

Me too

[rain rain]

Especially for books claiming to be introductions or undergraduate texts or whatever

Exposing students to these concepts for the first time

Like I don’t think a first-time learner gives a shit about all these weird specific definitions and concepts & abstractions used in later research, they’re not even going to do research for a while

It’s daunting, silly, and stupid. Just help me build the skills to get a foundation in the subject before I go on to more advanced texts in the future

[rain rain]

I feel like this is completely acceptable for advanced texts

The readers have the mathematical maturity, patience, and years of intuition & experience to see why these concepts are (or could be) important, and it would be a waste of their time to not get into the crazy stuff immediately (which is what they came for, what they’re prepared for)

But for a lower level book it just feels like I’m listening to someone just stroke their ego to how much more they know & understand than me

[fathom furnace]

definitely me when im reading through my abstract algebra book

[cloud zealot]

has anyone used Topology Through Inquiry by michael starbird and francis su in a classroom setting? if so, how successful were you?

[kindred stag]

Oh I had that used in one of my classes.

Terrible book actually one of the worst class experiences I have ever had.

granted this was for an algebraic topology class, maybe the beginning of the book was better

[cloud zealot]

may i ask why you felt the book was bad? also, i was more interested in an instructor's point of view. i'd be interested if their perspective was influenced by the resources from the book's website here: https://math.hmc.edu/su/topology-through-inquiry/.

[wide ice]

Maybe Lindelof's conjecture? Just something not so mainstream as RH.

[long pelican]

I'm not sure what you are replying to

[wide ice]

Oh, I thought you were thinking about something to talk about that would be relevant to Riemann's zeta function

[long pelican]

For context, these are undergraduates the majority of whose mathematical knowledge is pretty limited

[wide ice]

I'm an undergrad myself, and I'm pretty sure I don't know much either :D.

I mean, if you can construct analytical continuation and Riemann's zeta, then it's straightforward to give a few words on Lindelof, and how stupid-looking it is

[wide ice]

"monad is a monoid in the category of endofunctors"

That's the only phrase I remember from Functional Programming class. Still don't know what it means, it's crazy that we didn't have a class on category before

In this case, I think it's best to bring some examples from Functional Programming. Maybe some quick intro to lambda calculus.

Otherwise, I have no idea how one can understand what's going on with all these arrows

[empty gull]

I also highly dislike the book (as a student tho that probably says some stuff about how instructors might see their class react to it)
Personally I'm generally not a fan of the "all the theorems are left as an exercise" style because, while i understand where they're coming from, personally i see that reading through some important theorems is in of itself an important pedagogical step to growing as a mathematician because it allows you to see and understand how somebody else attacked a problem and managed to get it (i.e. this is how at least i learned a lot of proof techniques and approaches)
Not to mention that a lot of theorems feature highly non trivial approaches that might frustrate some students (as well it took the mathematicians who originally proved these statements who knows how long to crack the problem)

[cloud zealot]

interesting, and maybe it leaves less time for exercises that show facets of a certain theorem or extend them

[pure coral]

topology is one of those things that requires tons of examples and applications (talking about pure ones like zariski)

but too many books in general treat it as a game

and also it’d be WAY more useful to go over structures that use the notion of locality like sheaves, than to spend a bridge too many lectures on weird topological properties

[pure coral]

yeah all that gobbledygook makes sense only when monads are treated as (associative) algebraic theories whose terms are points within a category (like elements of sets, values of types, etc) - a similar example is how a monoid in the category of vector spaces is just a vector space with sensical enough multiplication, such as R^3 with the cross product

for example, the list monad gives the ways to build expressions for any monoid, a way to promote values to expressions (the unit map A -> [A]), a way to reduce nested expressions into a non-nested expression (the join map [[A]] -> [A]); then, a map [A] -> A can be treated as an evaluation scheme for a specific monoid

e.g., if A is a type of numbers, the map [A] -> A could just add those numbers together, or it could multiply, etc

what the laws say is that if you have a nested list, there’s a canonical way to use your evaluation map on nested lists [[A]] that is associative, e.g., sum of [[3,4],5] == sum of [3,[4,5]], just like how (3+4)+5 == 3+(4+5), and furthermore, you could also just unnest the list first and then evaluate, so sum of [[3,4],5] == sum of [3,[4,5]] == sum of [3,4,5], just like how expressions like 3+4+5 make sense despite addition being a binary operation

[wispy slate]

https://www.youtube.com/watch?v=p8GTItTCp50

[cloud zealot]

https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1234&context=hmnj

pretty interesting

[long pelican]

26 years later, we’re in the same place

[long pelican]

Something important I think the writer misses is that there are approaches besides the approach of rote problem types and the approach of theory. I see introductory calculus class as training students to be able to come up with (to us, easy) mathematical insights on one's own. You learn and remember math a lot more effectively if you can do something on it by yourself. Both the rote problem types approach and pure theory approach very much fail at that, especially if the theory approach encourages students to memorize proofs

[tepid smelt]

Can anyone give me a high school student friendly reason why techniques we teach to solve systems in three variables or more works? I feel they understood why elimination and substitution work for two variables but are not quite understanding with three variables. I think I just confused them today and ended up focusing more on just getting techniques down and doing a few applications like finding the equation of a parabola given three points and some word problems.

[winged urchin]

Wait what techniques are you talking about?

The parabola example you mention. Like you can give ax^2 + bx + c as the general parabola and plug there points into that and get a system of 3 equations in 3 unknowns. You can totally solve that for the unknowns using substitution or elimination

But I suppose you mean like... Filling in some of the missing pieces like noticing oh my y-intercept is 6 so the c is 6! Or hey I can find the vertex and then oh hey if I go between the vertex and this point I know I see the x changes by like.. 2 and the y changes by 8. Oh that's twice the change we would expect so a=2

Or maybe more natural than that last one would be to start out writing it in vertex form then once you know the vertex you actually know two unknowns and only need the vertical magnification parameter

If it is those kinds of ideas then I think it's mostly just shortcuts. I would point to how most students find the equation of a line. They usually find the slope and plug that in then plug in a point to find the y-int

You could totally just plug both points into your favourite general linear equation and get a system to solve

It's just we use the 'shortcut' of being able to tell what the slope is and where that goes in a certain form of the linear equation

[quasi musk]

You can visualize three planes intersecting

[cloud zealot]

has anyone ever uploaded lecture videos to go with spivak or apostol's calculus books? or at least, videos for a course similar in level to those books?

[devout fable]

unlikely, they don't see much use, if any

at least in the us

however, there are plenty of analysis lecture series on youtube

[tall jungle]

i havent found any, they prolly not popularly used

[tall bone]

If I might ask, would the books be suitable for someone who knows analysis but is just trying to brush up on the details? Or would they be too slow for that?

[devout fable]

in my humble opinion, if you want to brush up on the details, a good option is just to read higher level texts

They assume you already know introductory analysis, so you will be forced to recall (or revisit) the details.

[fathom furnace]

so this has been a burning question. i have some statistics students at rhe place i tutor at and theyre bright kids but they ask some questions about an intuitive meaning behind distributions and their shapes. i never was one to get heavily into statistics; do you guys have any tips on how to describe a “feeling” for certain probability distribution functions and why we can integrate pdfs times x over their interval to get the expected value and such?

or is it even worth to discuss with students about an intuitive feeling behind statistics

[shadow flower]

the sum of xf(x) is the expected value. think like rolling two dice and summing them. adding up every possible combination of xf(x) gives the expected value of that, 3.

that’s how it makes sense to me at-least, but i’m not sure if that’s completely sound or what you were looking for

[devout fable]

my feeling is that stuff like this is stuff you come to realize on your own after having more experience

which seems to be the mainstream pedagological view as well...not sure if it's right or not but that's my exp

[fathom furnace]

gotcha gotcha

i wasnt sure if i should try and lead them into some intuitive thinking, or assure them that if they come across it later on and have some more experience, then they'll just intuitively understand it from there

[devout fable]

i guess its always frustrating to do stuff and symbol mash

something thats always interested me is what on earth is an average?

[unkempt hound]

who can give me 2D or 3D

grath

[rose trellis]

I think of expectation as the average but when the process is stochastic

[viral parcel]

in teaching upper undergraduate/graduate classes (e.g., proof-intensive classes), what are the pros and cons of using a slideshow presentation (e.g., beamer) vs writing it out by hand (e.g., whiteboard, chalk, paper on projector, etc)?

[unborn osprey]

different probability distributions represent different things, so the answer to your first question depends on what they’re learning. as for your second, i assume they understand why the expectation of a discrete random variable X should be given by E(X) = sum_x Prob(X=x)x, so why not just point out that pdfs are just the replacement for “Prob(X=x)” when X is a continuous random variable?

[pure coral]

the algebraist in me would say focus on reasoning with random variables as well as the distributions themselves

the concept of a random variable comes from the elementary concept of "a value depending on a context", which is simply a function Contexts -> Values - for example, sometimes we might just want the context itself (identity function), we want to payout $x if we hit a blackjack, etc

and even though you might not know what that context might be, you can still do things on the value (since f(x)+g(x) = (f+g)(x), etc.) as long as the context remains stable

the thing that makes random variables special is just that we know the model generating the context

[unborn osprey]

what???

[pure coral]

well "know" as in up to how much we know about the overlying probability space

[unborn osprey]

which is typically absolutely nothing, no?

i don’t think it’s possible to overstate how different our understanding of the world would be if we knew anything about the underlying mechanisms that govern the “randomness” of the phenomena we can successfully model with statistics.

[pure coral]

that's an epistemological problem

probability spaces (and measure spaces in general) are ultimately a model that we can make certain deductions about, and it doesn't matter how real that model is or how that model came to be for the idea i'm alluding to here, which is that they give more information than a bare set so (measurable) maps from probability spaces are enriched with that information as well

it's just another example of the important theme of studying relative objects (maps to or from an object) in addition to just the objects themselves

and even if (all) models are wrong, it's still useful to have a formalism for working with values that depend on our model

[wispy slate]

Hello mathematicians! I felt like math pedagogy was the best place to ask this, but if it's not, let me know.

I'm writing a philosophy paper about the physics of motion, that I eventually want to submit to an academic journal. I've mostly figured out what I want to say, but if I can find a certain equation of motion it implies, the paper would become significantly more powerful. Unfortunately, my skills from university have more or less rusted to death: I can take derivates, and some integration, but I've mostly lost the rest. (To be fair, I was never a great student to begin with.)

I'd like to ask for help. However, I'm barred from having my work show up online if I'm going to be allowed to post in the journal I'm targeting. I'd rather not post the question in public on the help-forum.

Any recommendations on how I can go about seeking help? Or where on this discord channel I should go?

[cosmic ibex]

Definitely not #math-pedagogy -- perhaps #math-discussion, but if you can't show the mathematics you want help about, it feels unlikely you can get much useful response. (And if you can show stuff, it might still be more productive to find a physics server instead; see #old-network).

[cloud zealot]

What are your thoughts on eliminating calculus from high school altogether and replacing it with something like linear algebra, discrete math, elementary number theory, elementary graph theory, elementary combinatorics, intro to proofs, etc.? This could prepare students for more rigor in their university calculus class, which seems like a better place to do calculus anyway.

[quasi musk]

I don't think it's a good idea for a number of reasons. Calculus seems to be a natural culmination of middle school and high school mathematics. From what I've seen in helping students with "Math for Liberal Arts" or "Math for Elementary school teachers" they tend to struggle significantly more than if they just took a college algebra class

There are certain colleges and universities that teach discrete math before calculus; but the level of abstraction tends to be a lot lower

A lot of probability, statistics, permutations, combinations, and other discrete math topics tend to be taught in a way that emphasizes calculator usage over understanding

And this makes sense, since many ideas in probability and statistics are well-modeled by tools of the trade in calculus

[quasi musk]

Another argument is that the more you fragment the HS education system, the harder it is to cross check pre-requisites. Parents and students often get confused on existing math pathways ~ making multiple choices at each level can be confusing

You can see this with schools that implement an Integrated Math curriculum vs the Standard Alg. 1, Geo, Alg.2/Trig, Pre-Calc, Calc pipeline

In theory everything gets covered, but in practice it makes for tough going

[long pelican]

I'm not sure of the answer yet but what I do know is that the content being traded off wouldn't be my main concern. What I really care about is how likely students are to get genuine mathematical thinking experience in the class instead of being trained in procedures. Every class you listed has the potential to devolve into procedure training with the wrong incentives. Calculus in high school right now is procedure training, especially AP calculus (exceptions in a few schools here and there). An example where even proofs became procedures: I had a student in Calc 2 last semester who went to high school in France and the school she attended was big on proofs... except all that happened was that the teachers simply had students memorize dozens of proofs.

[hard fog]

i keep pronouncing this channel as purgatory

[pure coral]

i'd say a course in basic representational techniques, like sets, functions, definition of the different number systems, and then time permitting, maybe a bit on algebraic structures like rings

now there's still the problem of solving how awful some university calculus classes are

but while calculus is probably more useful in the short term, diving straight into calculus without even defining things like functions or multi dimensional spaces properly is a loan against your future as early as multivariable calculus class

[cloud zealot]

is it that calculus is a natural culmination of middle and high school math, or is it that the curriculum is designed to culminate with calculus? i'm not entirely confident in my knowledge of the history of math pedagogy, but wasn't there a push to get to calculus sooner to keep up with the soviets?

[quasi musk]

It's a good question, but I'm firmly in the camp that things like Trig, Geometry, Algebra have a natural culmination in Calculus

Algebra you measure slope a lot, in Geometry you look at Circles, Triangles, Tangents, Secants, Trig you look at sines, cosines, etc.

These are all of the basics of learning applied math. Most of the physics & chemistry is based on that

You can restructure the math curriculum to culminate somewhere else, but I don't think it will be as natural a progression as that package

Have you looked into new math of the 60s-70s?

[cloud zealot]

it sounds like it was a total failure

way too much abstraction

[quasi musk]

For whatever reason it seems that students do better with Calculus before Linear Algebra

[long pelican]

I get the impression teachers emphasized all the wrong things about abstractions due to not understanding what it's about or what it's for, more so than that the books themselves had too much abstraction

e.g. being pedantic about vocabulary

"Numeral for a number vs. the number"

[quasi musk]

Another potential issue with messing the current Math Curriculum is that if teachers are already struggling to teach the standard curriculum

[cloud zealot]

can you explain why this is specifically more of a concern for high school than with college?

[quasi musk]

I call AP Calculus fast-food Calc

[cloud zealot]

maybe in calculus you can still depend on intuitive pictures and heuristics and still walk away knowing what's going on

linear algebra you have to know what a function is i guess

[quasi musk]

What's more is that a lot of very important examples in linear algebra come from calculus

[cloud zealot]

most people at that point still think of functions as formulas of numbers rather than as rules

[long pelican]

I'm not sure exactly what you're asking, but I can say some things and hopefully I answer it somewhere.

1. Math classes are often better in college. A bigger fraction of college professors than high school teachers can show what it's like to genuinely do math. (Of course not all of them, sadly)
2. If you're taught that math is procedures for 12 years, it is harder to change your mind during 1 semester of trying

[cloud zealot]

moonbears have you ever taken a look at hung hsi-wu's books on pre-college math? what do you think?

[long pelican]

3. The reason I'm less concerned about content is because I find students pick up on requisite content extremely fast once they look at math the right way, and conversely people who look at math the wrong way suffer eternally in math no matter if they're very strong in algebra muscle memory or not

[quasi musk]

I haven't looked at this largely due to not having much time to do so. I already Teach roughly 40 hrs/week, and my free time largely went to learning new math

[long pelican]

Kiran Kedlaya once said in an interview that being a math major is like having a superpower (in how you think). The high school math curriculum should at the very least show that thinking mathematically is a superpower

[hard fog]

Would be nice. But the kids all seem to gravitate towards dash 3 math anyway

[meager bronze]

Point #2 is so unfortunate but so true :(

So many students approach each new math class as just “here is a new set of arcane rules to memorize”

[wide ice]

Some of my folks are graduating with a Math degree but still see Maths as a bunch of rules

Can't blame them. Every single test leading to uni (and GRE for grad) is on rule memorisation and application anyway.

It also doesn't help that outside Math major, other branches of science are taught math as sets of rules.

David Gieseker said

We've been having a lot of trouble with scientists, in particular life scientists. They are teaching calculus by radically dumbing it down. E.g. no trig, a half page on the chain rule, .... and very weak exams.
https://www.dam.brown.edu/people/mumford/blog/2014/Grothendieck.html

And I think I can say the same for Physics and Engineering major

Truth is, physics folks I know learned how to apply Green's theorem without knowing what it means. We learned how to solve ODEs 2 years before studying them systematically.

[winged urchin]

I think the lack of interactivity is a part of the problem here too

Like biology? You can go out and look at insects, animals, see how they move and all that.

Physics you can build bridges or egg dropping things. You can experiment! Same with chemistry. Do some cool experiments with safe chemicals ahah

But what does math have? Interactive applets? Students just... Fooling around on their calculator?

Most of the time they just have a teacher telling them they've veered off track because RULES

And I think it feels somehow more demeaning to 'explore' simple math

Like if a student forgets how to add fractions Id wager they will feel some amount of shame and might be discouraged from asking or even feeling like they can explore it

I'm sure we've all been too embarrassed to ask one question or another because it feels too simple or we should already know it

cough IMPOSTER SYNDROME cough

I also definitely remember times in my mathematical career where I didn't feel I had the 'time' to convince myself properly of something

Assignment is due in less than an hour and you found a way to get the answer they want but you can't find anything saying that that's 'ok'

So you just do it and hope its not logically inconsistent

And then have essentially just memorized something that might not even be right

[pallid night]

The only thing that changed my attitude to this was discovering maths ideas on my own (eg deriving Maclaurin series before it was taught, and writing a Conjecture in algebraic geometry before I realised what it actually was and realising none of my teachers knew either)

And I've seen the effects of the arcane rules approach when tutoring my cousin in stats recently

[viral parcel]

that reminds me of this talk:
https://youtu.be/UOuxo6SA8Uc

he presents some cool studies in here about productive failure or something like that

[tepid smelt]

I feel it's wrong but also know many people in engineering or other applied fields who think like this and do fine.

I feel this is true in software engineering also where many do fine memorizing techniques without understanding what is going on.

So is it wrong if it works for so many people?

[cold trench]

I personally believe that you should try to understand how things you use work, but if it's not necessary to do your job, then it doesn't matter

[vapid herald]

What's the best advice to be a better lecturer

In your opinion

[wide ice]

Whenever you learn something new, pay attention to the way you learn it

Profs usually forget how difficult it is to learn the topic that they're now so familiar with. They tend to explain the way they would explain to a colleague, not a student.

[pastel horizon]

I think this usually happens because the teacher is not confident enough in exploring alternative approaches

What would you suggest to make it more interactive or independent? This is something people upstairs are trying to get teachers in all subjects to do.

But it seems risky to have them self study and come to the completely wrong conclusions

[wide ice]

It's not easy actually. Self studying, especially maths, requires a certain level of maturity.

In programs for gifted students, what happens is students get hard problems, but not really high level. Number theory, Euclidean geometry, and combinatorics are great examples. It teaches how to think and how to be creative in maths. (yes, creativity is something that can be taught, to a certain level).

But not all math teachers are familiar with olympiad stuff though. Many are not even comfortable with combinatorics or inequalities.

[pastel horizon]

I'm currently teaching it to engineering students and a lot of them "just want to pass"

[wide ice]

Teaching what specifically?

[pastel horizon]

Calc, matrices, complex numbers, stats, etc

It's quite stripped down tbh but the thing is not many are even interested in the deeper theory

[wide ice]

You don't have to go that deep to show something interesting

[pastel horizon]

Well what you and I find interesting some of them won't lol

[wide ice]

Well, can you give a topic? and I'll try to find something very basic but interesting

[pastel horizon]

Like e.g. Explaining how matrices are used in video games

It's more like a "huh that's neat" more than anything deeply inspiring

[wide ice]

Students need a lot of motivation, or else they won't care about that topic. A normal student has like, idk, 10 subjects? If they can drop anything, they will

You kinda need to blow their minds first, before getting their attention

[pastel horizon]

That's true

I guess that goes back to using some kind of applet

[wide ice]

Well, true, matrices are not so interesting when you only wanna do video games :D. It's not the main uses anyway, even in CS.

To many, matrices are just a short-hand notation

[pastel horizon]

That's true

Ofc it has a direct engineering use anyway

[wide ice]

So do quaternions. Doesn't mean I'll care so much about them

😄

[pastel horizon]

I think a problem is there's a culture of "getting people to pass" which makes it harder to instil critical thinking

[wide ice]

Well, when you have 10 subjects to pass, you won't like pondering on any topic anymore.

[pastel horizon]

Yep

Engineering is like that

[wide ice]

I excelled when I did my thesis, because I had only one problem to think about, and I would learn anything I need to solve it.

[pastel horizon]

It's a vocational qual which means they're constantly hammered with coursework

[wide ice]

Now that I came back to classes, even Math is distasteful

[pastel horizon]

Another problem is you get a set of students that signed on expecting it to be 100% practical and then they get maths

[wide ice]

I think a good approach will be to give them one huge project in the beginning of the semester, then teach them everything they need to make it work.

That way, students see connections between subjects, and why they need all of them

[pastel horizon]

I'm hoping I can move up to the degree level course anyway which will be interesting

When I say degree level. It's basically courses for people that will struggle in a traditional uni environment. So smaller groups, more personalised support and more vocational rather than academic

[pastel horizon]

I think it would be a great idea but sadly i don't think it really fits with the qualification body

[wide ice]

Huh, but it's engineering, no? I thought ppl would like a more hands-on approach

[pastel horizon]

The hands on things they do is like super practical skills like welding

Even then the assessment for welding is just them doing a "simple" weld

[wide ice]

Ohhh, that kind of engineering

I was thinking of modelling/designing/simulation/etc. Even they don't like math anyway.

[pastel horizon]

Yeah

On the more technical courses they would have an appreciation at least for how it links

For a lot of the students it's like "yeah do I really need calculus to weld this?" lmao

Tbf a lot of the settings will probably have some sort of mathematical reasoning but they just read it from a table anyway

[wide ice]

There was a study I think, I heard it from a friend doing finance. When they introduced computers to finance worlds, they found out that traders and brokers did much better than those formulae and models, even though the theory predicted otherwise.

Reading it from a table enough of times, and you don't need calculus anymore 😄

[wide ice]

Totally get this. I thought the same about Humanities subjects. I'm still trying to appreciate them, but it's hard when it's not really relevant.

[pastel horizon]

Something about masters know when they can break the rules

Lol just imagine what English teachers are feeling. They can't even say it helps them to write job applications because "chat gpt can do it for me"

[wide ice]

This is why I hate Supply and Demand curve so badly. There are things that you cannot put into a curve.

[wide ice]

I guess we will find a way, but I don't see how

Well, we managed when calculators came around. Not like we would fail this time

[pastel horizon]

I'm not scared tbh. Some students you can literally give all the answers to and they still fail

[wide ice]

Yeah, but we don't take joy over failling students, do we?

😄

[pastel horizon]

The funniest thing is when they copy but they somehow manage to copy the most dubious answer that's completely wrong

It's like Chinese whispers

[wide ice]

I feel this convo is a bit off #math-pedagogy already 😄

[pastel horizon]

I think it's relevant 😉

How to develop their critical thinking so they don't copy stupid answers 😂

[tawdry venture]

in what order would y'all suggest teaching quadratics?

right now i have a tutoring student who knows square roots and the null factor law, and is thus able to solve equations where one of the terms has zero coefficient. so far i have decided to introduce him to completing the square.

i'm not sure whether it would've been better to talk about vieta first

[cloud zealot]

https://www.poshenloh.com/quadraticdetail/

https://arxiv.org/abs/1910.06709

you might find this interesting

[tawdry venture]

oh sure yes i did kind of come close to this

[zinc cipher]

Not sure if this is related to #math-pedagogy but I thought I'd try to start a conversation here. I'm currently taking integral calculus and I've noticed that lacking critical thinking skills as it pertains to the logic beyond the theory is causing me to struggle. I think this may be somewhat related to performance anxiety as when I sit down to work problems I must first calm my mind in order to focus.

I've managed to develop critical thinking skills in other areas, so I believe that my performance anxiety is mostly related to math. I struggled with math, but managed above average grades. However, I always felt that my logical thinking in math and general quantitative skills lagged behind all other abilities. Now that I have returned to study math, I'd like to know how to improve my critical thinking aside from working problem sets. Is there a way to develop the mathematical intuition that I lack?

[wispy slate]

Keep failing forward. Challenge yourself FOR yourself, not to be competitive or compare yourself. Just knowledge for the sake of knowledge. Get a book on a subject matter within math that you're personally interested in knowing.... something that's not too difficult for where you are, but still a challenge. Go through the book cover to cover reading everything & doing every problem. By the end of the book youll have more CONFIDENCE in your ability to work with math & understand math. The CAPABILITY is within you even now. The ONLY issue is your ability to see it & tap in when you need to. So work through a book over the summer (or whatever season is coming up for you) and gain some more confidence in it.

There's much too much math out there for any of us to truly become a master of it all. So add some focus to what you want to focus on into your life - make it enjoyable. That will allow you to relax your defenses to it so you can begin to build a new belief system around what you can & cannot do. Because again - the skills are already within you. You just gotta tap in. Hope this helped at all!

[zinc cipher]

Ok, I'll try the approach you suggested. I appreciate your reply.

[wispy slate]

No problem. It's what I did myself. I used to struggle with math and it felt so unnatural until I got a math book and went through it myself. I gained confidence & now I have a math degree.

[zinc cipher]

I'm going to be taking discrete math this summer, so maybe I'll find something related.

[zinc cipher]

That's great to hear.

[wispy slate]

Yep something to help you prepare for that class so when its time to take it, youll have complete understanding of the foundations and be able to keep up in your class and even excel if you want.

[zinc cipher]

Ok

[pure coral]

for integration... well, other than "funny rectangles under graph", a lot of the intuition really comes once you learn the different approaches to integration. for now, you want to expose yourself to as many examples of the two basic integration maneuvers of integration by parts and substitution as possible, and not just to compute integrals, but also to turn complicated integrals into simpler ones

this is for the future probably, but it's often worth revisiting the basics later on to see what new approaches you have for old concepts. for example, there's the lebesgue integral which is the definition of the integral most congruent to the "area under a curve" intuition (because to define the lebesgue integral, you first have to define areas, volumes, etc rigorously now), and there's also differential forms which give you an algebraic tool to work with integrands and put the dx stuff on a rigorous footing

[tepid smelt]

What I find frustrating is that students often don't like to be challenged. I find they get angry when they have to think through a problem that could have multiple approaches. I find that because they have been trained with math as a recipe of ideas they really don't know how to handle what doing math is really like..

[white forum]

This was a great response!

[white forum]

I think that this feeling also comes from the fact that most math students face in a classroom is graded. If they have to take multiple attempts to think through a problem, getting an attempt wrong and moving on to a different method isn't so easy when you might lose points from your grade if you started with the wrong method. So students dislike questions where the path isn't clear, because they could get them wrong a few times before figuring them out

[long pelican]

Fun and informative story: this semester I uncovered two cheaters who both used a service like https://www.monorean.com/en
I wonder if anyone here is familiar with students using this. No other math professor (except maybe one) in my department is even aware of this. It seems to be the next cheating rampage that is happening under our noses

They're both seniors who need this class to graduate as well 😆

[wispy slate]

How they gonna pay hundreds of dollars and still get caught 🤦 how did they make it so far enough to become seniors in the first place?

[long pelican]

I feel like they're both rich enough not to care about the hundreds of dollars

One of them certainly seems to get coaching in political threats because he mentioned he felt "racially profiled" in an email

[wispy slate]

I can't with these people, I don't think I'll ever understand a cheaters thought process.

[acoustic mulch]

what i have seen college seems to not care if you cheat kinda wish teachers care more

they are allowing kids who are not educated to go to the real world and fuck everything up

[cold trench]

in my university professors will likely fail you if you cheat and then you'll have serious repercussions with the university, I thought this was standard

[long pelican]

The sad thing about my situation is that hard evidence of cheating is hard to present when it's dictation from a professional rather than plagiarism or copying off another student

[cold trench]

guess that's the point

[shadow flower]

how did you catch him?

[long pelican]

(I had him sit in the front) never did any scratch work anywhere yet wrote clean textbook-like proofs first try, combined with multiple errors that could only be explained as dictation errors

also never came to class, and did no homeworks

[rain rain]

Do y'all think there's any benefit to just reading a book and attempting informal proofs as a way to build mathematical maturity?

[quasi musk]

You don't always have to work rigorously on proofs to learn something

[rain rain]

I agree but I’m also worried about overrelying on my intuition and not verifying that intuition is actually technically accurate (at least for the most part)

[quasi musk]

Many times I just read books or papers for background knowledge while only writing notes on the margin, or half-attempting problems

[rain rain]

So I just go on with the wrong understanding of whatever I read because I didn’t take the time to thoroughly explore the objects I’m learning about

[quasi musk]

There'll be plenty of times in classes to correct your intuition

The only way to get a better understanding or to make sure you don't have the wrong solution is to have someone who already knows help you out

[rain rain]

Yeah issue is I won’t be taking any classes (or at least not for a long while), I’ll be learning purely math on my own/on the Internet

So no classes to force myself to learn correctly, just gotta chat with others and learn as I go

[quasi musk]

I think that's the best that you can do

[winged urchin]

Try contest problems at your grade level

Those are fun and if you find yourself seriously stuck or not knowing notation then look it up

[white forum]

I feel like I am on my colleges watchlist after clicking that link

[cold trench]

been there done that

[tribal tiger]

This could probably be worked around by a student dedicated enough but would having students turn off their phones fix this?

I'm also curious as to how the person on the other end knows the questions I guess you say them into the microphone?

[long pelican]

I believe the technology is advanced enough that you don't need phones or even wifi to do this communication. The person on the other end gets the questions via a hidden camera in, for example, a mechanical pencil, a pen, or a button on their shirt

If done properly, it's basically undetectable unless you bring some high tech scanner equipment yourself or they make multiple incriminating transcription errors (which is what happened in this case)

[lethal leaf]

damn that's crazy

I'm also surprised it's just openly advertised as a cheating thing

I would have expected some attempt at a fake secondary use case

how did you find the second person @long pelican

ok the person receiving the info, you found diction errors

I presume the second person was the person doing the dictating?

how did you find that?

[long pelican]

I can't find the second person, it could be anywhere around the world

[lethal leaf]

what?

oh so you found two separate people who were both on the receiving end?

[long pelican]

Yeah!

[lethal leaf]

jeez

[shadow flower]

what kind of discrepancies did you catch

like the errors

[tribal tiger]

Did they both make the same mistakes?

[long pelican]

Nah

"renewable singularities" is one

[tribal tiger]

Oh wow

That's pretty big

Like that shows you know nothing about what you're cheating on

[lethal leaf]

renewable singularities, as opposed to singularities created from fossil fuels

[sour geode]

what is amth pedagogy?

[tribal tiger]

Read the description

[wide torrent]

at uoft and Ubc the consequences for that kind of stuff are very strict

usually a suspension for a year and a zero for that class + it goes on your record

[pallid night]

Is this supposed to be a complex analysis course ?

How does one expect to be believed that they know complex proofs off the back of their head but don't know what a removable singularity is.

Sincerely,
Student cramming for a Complex Analysis exam tomorrow

[long pelican]

Haha exactly, and good luck!

[pallid night]

Do you have a logic to how you seat students for exams ?

Like, do you have certain things you watch for that are indicators they are more likely to try to cheat ?

[long pelican]

I noticed things on the first exam (where I didn’t make any seating requests)

[wide torrent]

how were you able to catch them, if you don't mind sharing? Those could easily pass as earring studs

I saw you say that they couldn't write out their work but how did you manage to spot the device?

[long pelican]

No, I didn't catch them that way. It's pretty much undetectable unless I want to break some personal privacy laws

[wide torrent]

oof

They might be able to get away with it then since there's no hard evidence

;/

[long pelican]

That's a possibility

At least more staff in this school will know about it

[wide torrent]

how someone would get to yr4 math and cheat is beyond me

[long pelican]

Lots of money

[austere inlet]

• the pandemic years I guess, my impression is that a lot of people have normalized cheating even more since then

if they're seniors they probably passed their freshman and sophomore courses in '20 and '21 resp.

[austere inlet]

I say this purely because I've heard of cheating instances a lot more often from profs. and friends since then

[pallid night]

This exam was seemingly kinda hard 💀
The questions were either completely trivial, really hard or conceptually obvious but sloggy.

[pallid night]

Because they got through Y1-3 by cheating

[long pelican]

Agreed

[vagrant meadow]

was sort of talking about this in #linear-algebra yesterday about how most courses use real vector spaces and always use real scalars and then (only sometimes) halfway through surprise students with letting scalars be complex and then they have to redefine the dot product. but why not just use F from the start, let most of the examples use F=R, and then talk more specifically about R and C when you get to inner products? is F just too scary to lower division students?

i don't think you have to rigorously define a field to use the word. i feel like you can just explain it as "scalars we can add, subtract, multiply, and divide. like the real, complex, and rational numbers".

most students aren't overly concerned with intense rigor. maybe they'd appreciate not having to redefine things over and over just because you're introducing different types of scalars? idk what do y'all think

[devout fable]

My course completely ignored complex numbers

[vagrant meadow]

see and i really kind of hate that tbh. because i think the complex stuff in linear algebra (hermitian/unitary matrices) are really cool and interesting.

[devout fable]

I guess so far with what I’ve encountered in mathematics, I’ve never needed to work with complex vector spaces. Sure you can always talk about what should be taught in a course, but at the end of the day I don’t really think it makes much impact on students future academic careers

[light garnet]

Mine did too. Furthermore, there's a trend these days of only talking about R or the n x n matrix with entries in R and maybe superficially go through vector spaces (maybe verify something is a subspace, for example). I know plenty of courses that omit the inner product space material, too

[devout fable]

Yep that’s exactly what mine did

[vagrant meadow]

but at the end of the day I don’t really think it makes much impact on students future academic careers
what about people who want to study linear algebra and complex vector spaces 🙃

[light garnet]

I have personally found that intro LA textbooks that people widely use to try to introduce things in generality do a poor job at it... let me find that example

I didn't realize until I took abstract algebra that this problem I got in my LA class was just poor notation

[devout fable]

Is that me?… I do find myself needing to learn random matrix theory and some stat physics right now 🤷‍♀️

Unless you mean people who want to study it for the sake of studying it and not some ulterior goal. Then fair, I don’t have a good response

[vagrant meadow]

kind of both

[devout fable]

This makes my eyes hurt

[light garnet]

Also, one general comment I'd like to make is the reason I observe most people struggle with vector spaces is because they haven't been taught how to deal with axioms, theorems, proofs, and the like. When you've gone through the motions of just learning computational tricks for 3-4 semesters of calculus without really thinking about what your computations precisely mean, it's not surprising

[light garnet]

Yeah, this was a "challenge" problem that my LA professor gave us

[devout fable]

Well you do think about what your computations mean, just not deal with the abstraction

[vagrant meadow]

honestly that's probably my favorite example of a vector space. i like it quite a bit actually

[devout fable]

Even as someone who just finished third year of undergrad I still find abstraction awkward and not quite making sense until I get to use it a lot

[vagrant meadow]

but i get what you mean. usually LA is the first intro to proofs.

[light garnet]

But yeah, especially these days (keep in mind, I was in school 10 years ago), when someone tells me they've taken "LA," I have to dig to know if they've done a proof-based LA or a computation-based LA where they're just programming stuff into MATLAB

[vagrant meadow]

i feel like generally the hardest part of LA (why most people struggle with it) is a combination of it being their first intro to proofs and just a generally poor approach to teaching it. like there's too much of a focus on matrices and rref rather than linear transformations. students don't know what they hell they're doing, what a matrix actually tells them, or why it matters

[devout fable]

I didn’t even learn eigen decomposition from LA course

Now it’s the only thing from LA I use haha

[vagrant meadow]

"wife decomposition" LOL sorry that's just so funny to me

[vagrant meadow]

wtf you didn't learn eigendecomp in LA?

[devout fable]

Apple auto correct 😩

[light garnet]

I didn't either, was forced to learn it as well as SVD in grad school

as well as the Woodbury formula and the Cholesky factorization

[vagrant meadow]

wait like... seriously? no eigenvectors/diagonalization??

[light garnet]

They would teach you the definitions and end it there

That's it

[devout fable]

I mean they might have taught it to us…but I never actually learned it

[light garnet]

Maybe show some really basic results

I definitely didn't learn eigendecomposition, SVD, Woodbury, or Cholesky in LA, and it wasn't until grad school in stats that I was forced to learn them

[devout fable]

Well I don’t think cholesky or woodbury is standard anywhere

SVD maybe

[vagrant meadow]

i sort of get the latter three. that is sort of more suited to a numerical linear course. i was lucky and got to sit in and tutor for a class where the prof's goal was to teach SVD. that was an extremely interesting course.

[pallid night]

QR, Eigen and Singular are the ones I've done after two full linear algebra courses

[wide ice]

It is for me

I did almost everything in basic LA because of numerical stuff. Eigen, Cholesky, SVD, LU, Gauss-Seidel, Jacobi, etc.

But nothing suuppppeeeerrr theoretical, just basic stuff.

[vagrant meadow]

hmmm i suppose that is an interesting point. the main application of linear algebra tends to be in numerical analysis and programming. and those are really important concepts in those fields.
maybe i'm too caught up in a very theoretical approach to linear algebra.

though, my goal is to teach linear algebra alongside ODEs, so i guess the theory is kind of the main point in that context...

[wide ice]

It can be numerical too, and ppl like it better when they can do stuff with it, not just proving.

I am pretty sure the only reason I still remember all of this is because I once coded them myself

[pure coral]

i like going beyond R and C and considering finite fields as well

and even modules should be covered as well i think

[vagrant meadow]

yeah agreed! i feel like real vector spaces are just too small a fraction of what linear algebra can offer.

there are so many other fields out there

[pallid night]

I remember a feedback quote a lecturer of mine wrote ...

"Some of you still think the real numbers are more real than other fields"

[vagrant meadow]

LOL i love that so much

[pallid night]

He's infamous for writing up some rather savage general exam feedback scripts

[cloud zealot]

https://www.maa.org/press/maa-reviews/linear-algebra-9

meckes is an introductory linear algebra textbook that works over arbitrary fields

it tries to give attention to both theoretical and applied people

maybe some of you might find this book interesting?

fwiw john baez wrote a very favorable review of this book

[pure coral]

i glanced at it and it's basically better axler for me

[pure coral]

yeah, especially when not even considering real dual spaces which are incredibly important compared to how little attention they get

plus i think there needs to be more emphasis on studying the whole family of abelian structures in general (abelian groups, modules, abelian-valued sheaves), and when they are the right theory for a problem, such as when you are studying objects that can be "scaled" by some action of a monoid and gluing doesn't mean much more than just taking multiple objects together

an elementary example would be that when counting how many apples are in multiple buckets, you can count them in any order, and as long as you count all the buckets once, you'll get the same result always – no coincidence that Z is abelian. or as another example, what an eigenvector really represents is a subspace where the linear map is just an action of a field, which means that when studying the linear map on those subspaces, we can use the field's nice properties like completeness which allows us to do calculus, perhaps also algebraic closure, etc

[stoic radish]

How would you recommend teaching/giving advice for hs level math (mostly computational)? Sometimes i feel a bit frustrated not teaching the theory as i feel like it makes it easier but they don't really need to know it. I also feel like telling them to just do problems until they get it is a bit of a cop out although that is what helped me ace my classes in highschool. For example, a lot of people who come to me for help struggle a lot with basic lin alg concepts like matrix multiplication and row reduction (finite math class). Now i dont go all the way into linear maps, but they dont even learn the notion of a pivot and barely understand leading 1s past the fact that "there is a 1 there." I feel like once they get that row reduction is very simple but im wondering if it would be better to spare the 30 mins teaching that and instead just work through and explain sample problems

[pure coral]

it's a good approach to teach inductively rather than deductively: you first start with worked example problems that illustrate a concept, then you give the student a chance to induce the general concept you're leading them to with clever questions

the big issue isn't really general knowledge as much as how difficult it can be to apply the right knowledge to problems

as usually how things are taught is that we immediately start categorizing knowledge before actually filling that category with concrete examples, but that's an issue since to solve a problem, the student has to have associated the features of the problem to the relevant knowledge

[vagrant meadow]

yeaaaah i mean i totally agree and empathize with what you're saying. and if the student is super motivated, maybe showing them some advanced concepts can really inspire them (i know i personally would have loved that, but that's also why i send messages in this channel). but those kind of students aren't usually in tutoring very often.

most students couldn't care less about the theory or tricks that would make things easier. trying or offering can sometimes even spook them because they're worried they might get docked points on an exam for using a technique their teacher didn't teach them. most of the time, they just want to know the minimum amount of knowledge/practice so that they can pass. that's also personally frustrating to me, and that's why i don't like tutoring high school, and i don't plan to become a high school teacher lol

telling them to just practice might feel like a cop out (to both you and them), but that's partially because they're subconsciously hoping that there is some magic trick to becoming proficient at math. the hard truth is that competence comes with practice.

i will say there is value in conceptual knowledge. exclusively practicing, without knowing what the hell you're doing or why, is not super effective. but then there's the issue of "do they actually want or care to know". and you can lead a horse to water, but you can't make it drink.

i work as a tutor at a community college, and i've also done tutoring for high school students. what's interesting is that the college students are far more motivated than the high school ones. i think because the college students know they need help, and come in for tutoring. whereas the high school students usually don't even care or want a tutor, and their parents are just forcing them to work with one. so that's also a factor in how receptive or not they are to advice and help.

[drowsy otter]

I am a Maths tutor currently tutoring someone online. This student is 10 years old and having trouble with division below the 12 times table, and I would like to be able to teach them how to, for example, figure out 30 / 5. I was advised to teach it visually as opposed to mentally, but I have no idea how to. Do any of you have any ideas how to visually teach division? The student is reluctant to learn, but it's likely that they just do not understand. They see the point in multiplication, but not division. Thank you guys.

[quasi musk]

You can try phrasing it like a multiplication problem

5 times a number is 30, what's my number?

You can also try drawing out something of length 30, dividing it into 5 pieces

How long should each piece be?

etc.

[drowsy otter]

To do it visually, I could make like a circular pie, and attach a value to that pie.

For example, 30, so how big is each piece if all 5 pieces are the same, etc?

Hopefully, the student will realise that it's "5 times what equals 30" and answer 6. Does that sound like a good solution?

[austere inlet]

I think using the analogy of diving a group of 30 objects (marbles, candies whatever) between e.g. 5 friends is more natural than just assigning a value to a circular pie. The numbers and the need to perform division arise naturally in the first problem and not so much in the second -- a pie analogy might work best to emphasize e.g. what it means to take 1/5, or 20% of something. Just my thoughts

or things like if I'm splitting a $30 bill between 5 friends, how much should each person pay

[drowsy otter]

I think using these "natural" analogies are indeed a better idea than my pie idea. I shall incorporate those examples into the word problems (which I do already).

[drowsy otter]

Thank you very much, you two.

[vagrant meadow]

finding a homogeneous + a particular solution is a concept that appears in both linear algebra and in solving linear differential equations.
in a course that teaches both, which of the two is it more natural to come up in first? would introducing that concept be more concrete in diff eqs, and then that intuition serves the student when they are solving Ax=b in linear algebra?
or would it be better for the linear algebra concept to lend itself to intuition for why you need both to solve a diff eq?

i suppose in general, there are a lot of linear algebra applications to solving linear odes. originally, i thought learning the linear first and seeing them as applications would be the most helpful, but now i'm thinking that there's far more motivation in a concrete differential equations application that could then serve to make the more abstract linear algebra concepts stick.

[meager bronze]

starting with linear equations makes more sense to me. students are already familiar with the idea of solving a system of linear equations; solving 2x2 and 3x3 systems is a skill typically taught in high school algebra

[wide ice]

I would do both in parallel

I would first teach simple ODEs first, then transforms the higher order ODEs into matrix form. That gives a good motivation to study LA.

Then you can teach LA while occasionally coming back to ODEs.

The idea is great, but a bit too much just to solve linear system. Doesn't feel natural to me.

Also all these eigenvalue thing-ies, if you will cover it, make best sense in the context of Hartman-Grobman theorem

And seeing how dynamical system "flows" is far more visually appealing than watching some abstract characteristic equations.

[empty gull]

I hold the view that when students learn how proofs work in math it would be immensely helpful to perhaps first have a mandatory first order logic course where you learn how proofs work in a more friendly environment where things are more streamlined

I'm unsure if this is how intro to proof classes are usually structures (but some people mentioned learning how to prove from Rudin which seems inhumane)

I had a first order logic class my Junior year of HS (mandatory too!) and I believe it helped me a lot to have better intuition and just generally keep track of things when doing proofs

[winged urchin]

I definitely feel like first order logic could be taught at the highschool level

I even show it to some of my brighter elementary school kids I tutor

[hard fog]

You could help him find appropriate ways to approach a problem. As 30 is a multiple of 10 he can start there. 30/10 = 3. If 102 =5 then 30/5 = 23

[covert hearth]

I definitely got FOL taught at the hs level

well at least predicate calculus

[cosmic ibex]

It's common to teach the symbolism of quantifiers and common logical connectives at a fairly early stage, but I'm somewhat skeptical that you were also taught either a formalization of FOL semantics or a formal proof system for it.

[empty gull]

We didn't develop FOL semantics rigorously (tho we did talk abt FOL semantics) and we were taught natural deduction which is ig a formal enough proof system?

[cosmic ibex]

Right, I'll accept that as a formal proof system. :-)

Teaching that in high school sound uncommon, though.

[empty gull]

(tbh I'm not too much into foundations so I don't really know what would constitute a formal proof system lol)

[empty gull]

Yeah here's an example of like a solution sheet for a hw we did

I feel like it was logic methods class rather than developing everything like super rigorously like I image mathematical logic classes do it

[cosmic ibex]

Ugh, Fitch notation.

[empty gull]

lol I got too used it now everything else looks awful to me

[covert hearth]

yes we used this notation as well

but we didn't go into FOL semantics

[pastel horizon]

Copying the answer for another assignment which was reprinted with different numbers because that person copied has got to take first prize for the dumbest thing a student has done

[long pelican]

I've had that too 🤣

[pastel horizon]

And yet they insist "oh no I didn't copy or collude with anyone" lmao

[pallid night]

What about using an online Chinese Remainder Theorem calculator and writing down the exact steps given by the calculator ?

Or ... copying a stackexchange proof ?

(All in open book online setting)

[pastel horizon]

Guessing you never taught the things they regurgitated lol

I wouldn't even care about that provided they could actually explain the steps. Usually they can't and it's like "right do it my way then"

[pallid night]

Oh this person couldn't explain the CRT steps.

They'd have got away with it if they hadn't written down the steps as well cos they didn't have to submit rough work, only a general outline of their final solution

[vagrant meadow]

#❓how-to-get-help ...

[pastel horizon]

Also this is about pedagogy

[dreamy moat]

ayuda

[meager bronze]

!help

[coral copper]

Please read #❓how-to-get-help

[tawny laurel]

how useful is a graduate math degree if you dont intend on teaching? (finance or something as a career as well? <assuming u have some finance background like a minor or some internship experience>)

[empty gull]

the higher the degree the more hireable you are generally

i don't think you can do much with just a math bachelors

[tawny laurel]

i see, thanks

[lethal leaf]

You can if you have other skills

I would also say that if you get a math graduate degree you also need to still have other skills

(not really a pedagogy discussion tho)

[winged urchin]

I do believe math is one of the more flexible degrees in that it applies to many other different areas and even just learning more of how to critically think is important

But I don't have the most experience with the job market even though I do have a PhD. I just chose to be a tutor after school and am happy enough aha

But that falls into the teaching category I suppose

[light garnet]

If you intend on entering the job market and even somewhat using your skills that you accumulate during your degree, you cannot avoid programming

Note that I have a very US-centric perspective on this, so this may not apply if you're outside the US

[light garnet]

My view of the US job market is the following: employers aren't paying people to "think." They want people to hit the ground running on day 1 and deliver.

[turbid zenith]

Not so much math pedagogy as much as teaching practice in general... What do y'all think of the idea of letting students see their letters of recommendation I write for them?

[meager bronze]

is there a specific issue you're thinknig about? like ethically? i don't see any problems with that

as long as the fact that they are going to see it doesn't influence how you write it

[covert hearth]

100% true

If you want to think you have to take the risk yourself

If you want to found a startup on VC money, then you can think about cryptography theory all you want

But if you want to work for example for google, what you gotta do is grind similar problems to the ones you'll work there

🙂 🙁

[shadow flower]

I would have loved to see the letters that were wrote for me but my advisor (who was writing one of the three) said that if i had requested to read the letters that it may alter what is written in it so i chose not to read them

so i agree i think you should let students read them but be really cautious on writing anything different because of that

[wide ice]

Maybe ask the student first. I was also asked if I wanted to see the reclet for me, and I said no, saying I trust the prof to write their best.

Imo the practice for reclet is you only ask profs whom you know for certain that they can and will write good things about you. If a prof can't write good about you, they will say so and say no already. Writing what good things is up to prof's call, and seeing the reclet doesn't change anything: you can't just read the reclet and say "Oh btw, I want to mention this and that." It's ordering the prof, quite disrespectful imo.

Reclet is supposed to be prof's impression of the student. If they forget something, so be it. If they want to recall impression because it's a long time ago, they will ask. Some profs require a one-to-one meeting/sending resume for this.

[cold trench]

"may alter what is written" is a respectful way of saying there's something he wants to say behind your back

[unkempt glen]

.

[shadow flower]

what do you mean? something that he would put in my letter that he would leave out if he knew i was going to read it?

[cold trench]

sounds like it

obviously don't know for sure

and I'm no professional

[shadow flower]

I hope he was being general with it. I never considered that though. I asked him why program would let the writer know that the letter would be read by the student and he thought about it and said that what he said 🤷‍♀️

[pale wasp]

Sometimes there's confidential information about other people (eg comparisons) in the letters

It is standard for students to not supposed to be able to see their rec letters

Employers reading the letter may view the letter as less genuine if they know the student could see the letter

[vagrant meadow]

i was able to see one of my letters and it really boosted my confidence and motivation to succeed. but i thought something like this would be the reason not to ask for them. i assume there's just more honesty if the writer knows the student won't read it. bc of that i've never asked to read any of my letters.
if i ever write one, i'd never let a student i couldn't write a strong letter for read it (though idk if i'd agree to write one in the first place if that was the case). but if i was writing a really good one, maybe i'd let them. but it also might be safer just to always say "no".

[turbid zenith]

https://twitter.com/solidangles/status/1661864327232512003

Bill Shillito - now at @[e-mail] (@solidangles)

I've been playing around a good bit with @OpenAI's #ChatGPT, and I'm continually astounded by what it can do.

So I've decided to fully embrace it and integrate it to help me become a better mathematics teacher.

🧵 [1]

Twitter · May 25, 2023, 10:38 PM

[light garnet]

This was a conversation I had with ChatGPT

[turbid zenith]

Also, GPT-3.5 or 4?

[light garnet]

Think it may have been 3.5. I'll give 4 a shot sometime

[vagrant meadow]

i think this is right. it's easy to point and laugh when it make a fool of itself rather than genuinely consider using it as a tool. for example, it has helped me realize a number of complicated coding projects, even though i'm new to python coding.

i think there is great potential to have it summarize, simplify, and explain mathematical concepts for students (with supervision). it's really quite excellent with language, the things it says sound good, but you just have to be able to keep an eye out for when the content is lacking or incorrect. with proper guidance--that is, not just trying to lead it into saying something wrong--i can see genuine applications to education.

that said, the danger of a novice using it can lead to a blind leading the blind situation. there have been cases where i've asked it math questions and thought "damn, if i didn't know this subject as well as i do, i would totally have believed that this nonsense proof/explanation was valid". supervision is key, i think.

[turbid zenith]

Already one step ahead of you.

[vapid herald]

Someone asked me to explain determinant. I'm going to explain how it's a signed scale factor of the volume of the parallelpiped with basis vectors for sides under the transformation. Should I include any equivalent definitions? This is a premed student in computational linear algebra course.

[cosmic ibex]

I don't think I'd consider it a full explanation unless it also mentions some other elementary properties of the determinant (a polynomial in the matrix entries; nonzero iff the matrix is invertible; determinant of product is product of determinants; invariant under basis changes; etc).

[vagrant meadow]

cofactor expansion i think would be good. but any of the other defs (permutation def, multilinear in the rows/cols or however Artin defines it in his Algebra text) are pretty unnecessary and confusing.
but it's good to do it in terms of volume/area i think.

[vapid herald]

I talked about the invertibility criterion and they were able to explain why it's true so I consider it a win

[wide ice]

determinant is one of those things that go infinitely deep 😄

[snow shoal]

Thoughts on what’s worked for a flipped classroom? I’m going to be student teaching 9th grade for fall and I’m pretty unfamiliar with a flipped classroom environment. Just wanted to ask if there’s anything that stuck out to y’all as working well and stuff that didn’t

Also they use “kagan cooperative strategies,” but I’ve never heard of him

[quasi musk]

You have to have a way of holding students accountable for getting students to actually do the work

Many students in 9th grade don't know how to read a math book

[vagrant meadow]

has anyone here had any success with a flipped classroom situation? i feel like this, or just general lack of motivation in students, results in it being a disaster every time. i don't think I've heard any positive experiences with it.

[zenith slate]

I've certainly never had any success with anything like the flipped approach

I could see it working for older students in fairly small groups

The model is predicated on the idea that students will self-study the material prior to the session and kids in 9th grade aren't exactly fantastic at this

As for the Kagan stuff... it's a nebulous set of "models" that are mostly communicated via expensive books and workshops

It's the usual "revolutionary" teaching bs that comes with a hefty price tag and absolutely no evidentiary basis

[snow shoal]

That’s mostly what I felt like I was seeing lol

[zenith slate]

Sure there will be studies that show it works but you'll invariably find that the only place it's been tested is in small classrooms of kids with wealthy parents at expensive private schools

Pedagogy is absolutely chock full of pseudoscientific garbage

[snow shoal]

Yep, that’s something I’ve noticed a lot lol

[zenith slate]

In any case a student-led approach can certainly work depending on the group but it is very important that the instructor still enforces a rigid structure

Why are you teaching a flipped classroom?

[zenith slate]

And what do you mean by "student teaching"

[quasi musk]

It worked really well in a graduate class I took with 2 other friends and the professor was teaching us his research area

but outside of that I haven't seen it work very well

[pure coral]

it's definitely going to be difficult for grade 9s, being able to effectively self study requires good executive functioning skills which grade 9s aren't going to be the epitome of lol

i guess the main value is that, provided the students review the material a few days before the class on that material, it gives them spaced practice on elaborating on it

and versus lectures, the students have time to process the information, whereas in a lecture where they might be more focused on copying notes down

but you'd need really good instructional materials as the students would not be able to ask questions, and also certain students might not have a great environment for studying at home

i think your main priority should be to enrich the content and provide new perspectives that their (probably) badly-written textbooks won't give - i assume that your grade 9 class is going over some basic algebra stuff, and i think you should try to make a link to geometry as much as possible, e.g., perhaps illustrate how solutions of equations relate to geometry, or how functions can be treated globally as transformations of space instead as locally on points

[snow shoal]

At the end of my degree is a semester internship where I work with a teacher full time (it’s in Texas if that helps)

I usually call it clinical teaching, but somebody yesterday said it was confusing lmao, can’t win no matter what I call it 😆

[snow shoal]

That all sounds good, for what it’s worth it seems like a fairy high performing school, so I’m not too worried about the instructional material being toooo far out of reach

What I AM really worried about has already kinda been mentioned. Do students have the proper foundation to learn how to be accountable for their own actions, and if not (which will almost certainly be the case) how do I support them.

I think that’s the main thing I’ll be looking at during the semester tbh

We’ll see the quality of the instructional materials. And differentiating’s going to be big because this school more than others has literally every combination of age, race, sex, socioeconomic status. It’ll be interesting

Sorry for off topic, it’s way past my bed time. Yeah, Geometry reach would probably be useful for students who work better in that kind of stuff or high achieving students

[snow shoal]

Not my choice. It’s the school I’m interning at to be certified

Didn’t get to pick the school either. I would have preferred a school that uses minimal technology but that’s just my preference

[pure coral]

i’d say set high expectations as it is too easy to be lax, the students don’t have the maturity to see that the stuff they’re learning is important yet even if you try to show why, so unfortunately you are going to have to be more of a drill sergeant and less of a coach

but don’t copy their screaming and swearing ofc, manifest it in the content and tests

[pure coral]

with a flipped classroom, a minor benefit is you could vary the amount of support given in the materials for each student

of course have a few things each student has to go over, but for the struggling students, you could give them more examples to study, while for those who aren’t struggling, you could instead give them exercises to apply and integrate their knowledge

frequent low-stakes practice tests are good not only intrinsically, but also to see how well each student is doing so you can tailor the material to their skill level

[tawdry venture]

does anyone else find the terminology "mathematical induction" vs. "strong induction" awkward

[wide ice]

Not really, no

"mathematical" is a redundant adjective, because induction has its usual meaning beyond math

But "induction" vs "strong induction" are clear to me

[tribal tiger]

I never really got why you wouldn't use strong induction if such a case exist I think that'd be good to teach as well

[snow shoal]

mkay, that's all been really helpful, thank you @pure coral, @zenith slate, and @quasi musk for your thoughts

[tawdry venture]

so what do you do when you need to contrast induction vs strong induction

do you call the former "weak induction" or "induction induction"

[wide ice]

When I say "induction", it means the classical induction, proving P(n+1) from P(n)

To me there are "induction" and "strong induction". If no one says "strong", i'll assume it's not "strong induction"

🤷

[vagrant meadow]

I'm curious about this too. wouldn't it be simpler to only have strong induction? i feel like that's a naive question, but the difference in assumptions doesn't seem large enough to me for it to be that big a loss

[wide ice]

It's a matter of taste :D. There are ppl who always use strong induction because why not, it's good to have something extra. I tend to use what is absolutely necessary: if the proof doesn't call for strong induction, I won't use it.

Both methods are equally strong. Just that one is more convenient in some cases.

Fyi, there's also something called "backward induction" 😄

It was the usual elementary, high school way to prove Cauchy-Schwartz inequality.

A very powerful idea though.

[tardy ember]

well there are some types of object where it's harder to define what "strong induction" actually is

but if we're talking about natural numbers then yeah strong induction does kind of just give you more so there isn't any case where induction works and strong induction doesn't

[wide ice]

Actually, there is one. I just remembered

It's the contrapositive form of induction. You show that if there's a counter example for P(n+1), then there's also one for P(n), i.e. !P(n+1) => !P(n).

As crazy as it sounds, there are problems where this is the only feasible formulation. I can't imagine doing them with strong induction.

[vagrant meadow]

i figured that was going to be the general reason. like "minimum necessary force" or something.

[vagrant meadow]

ah that's interesting

[tardy ember]

if there's a counterexample for P(n), there's also a counterexample for P(k) for some k < n

[wide ice]

I can't recall any exact statements rn, but I do recall it was very hairy to do that for those problems.

[tardy ember]

let k = n - 1

[wide ice]

I know, technically they are equivalent. I just can't recall why I recall it that way.

Must have been some reasons 😄 I forgot a lot of things already

[pallid night]

Good to ensure there aren't any inaccuracies

[vagrant meadow]

kind of a pedagogical question. in abstract algebra we use \langle and \rangle to denote generation. like <a,b> for the subgroup/ideal generated by a and b. in linear algebra, we have the analogous concept of span and subspaces, but we often reserve \langle \rangle for inner products. what would you think of a linear algebra textbook or course that uses \langle \rangle for span and something else for inner products (maybe brackets [v,w])?
one reason i kind of like the idea is that it's easier to write. i've seen linear algebra professors write things like F^2={e1,e2} when they actually mean F^2=span{e1,e2}. why not just do F^2=<e1,e2>?

[bright karma]

I like the idea of using something else for inner products. Square brackets sound off, they are always anticommutative to me. Maybe (v, w)?

This often is used for bilinear forms

g(v, w) or h(v, w) is very standard but its a lot to write

Angle brackets should be span and generation, yes

[vagrant meadow]

Square brackets sound off, they are always anticommutative to me
okay cool this is why i wanted to ask in here because there's a lot of math i haven't seen. which context are you referring to here for square brackets where they are anticommutative? my goal would be for the notation to be as connected to other fields of math as possible.
(v,w) could definitely work. i was a bit hesitant bc we already use () for vectors, but i'm open to it

[bright karma]

One thing to be careful about. Often angle brackets denote free generation especially in the context of group presentations

I like vectors to be column vectors, in square parentheses

Or even row vectors in square parentheses, I think that is quite acceptable

[bright karma]

Square brackets usually denote commutators of matrices (or other similar objects). For two matrices, A and B, [A, B] = AB - BA

See “Lie bracket”

[vagrant meadow]

oh DUH. yeah i can't believe i forgot about that i've spent so much time with commutators 😭

[lethal leaf]

there's a CS prof at my school who goes to the extreme of "if they use normal induction, rather than strong induction, take off points"

I don't get it but eh

[pure coral]

i think the most important thing is to qualify the notation with words as much as possible, saying “the inner product <a,b>” or “the ideal (p,q)” removes any ambiguity

it would resolve a lot of problems with readers understanding notation, if the author isn’t allergic to words and narratives

[vagrant meadow]

this. especially in linear algebra, i see it as a really engaging narrative. but it's taught in such a dry and unintuitive way most of the time. so much emphasis on computation and matrices, and students don't even know what the heck they're doing.

[cloud zealot]

but symbols save space on paper!

less space taken on words = more efficient communication!

[wide ice]

Sure buddy

[lethal leaf]

This is what I try to tell people in the discrete math class I'm a TA for

Also half the time they themselves (at this stage) are bad at parsing symbols into English

So idk why they try to go the other way

[pure coral]

yes exactly! linear algebra should be the story of how all the rules that seem overwhelming at first make reasoning with linear maps trivial, and it’s such a shame a lot of that is hidden under bad textbooks, unnecessary matrices, confusing notation, etc

[winged urchin]

I find a lot of students don't even realize what a statement like "the velocity v(t) ... " Is trying to say. Especially when it's even just a little more dense than that

Mind you when I was a student I wanted to put commas like "the velocity, v(t), ... "

Because it is kinda like that... In the same way youd say "the cashier, Bob, helped me with the bagging"

[pure coral]

and i think also linear algebra textbooks should be describing the absolute pain of solving problems that can be solved with linear algebra without linear algebra, and then show how linear algebra deals with it almost like magic

[winged urchin]

But my supervisors never liked the commas aha

[pure coral]

it’s only fitting that a mathematician came up with “eureka”

[winged urchin]

Yeah! I've often thought this kind of approach could actually be good for students.

Really show them why it's whatever technique they are learning is useful

Although it could backfire I think if the student thinks you just wasted their time or something aha

[pure coral]

haha sometimes i like the commas as well, it just seems off in certain contexts to not have the commas

[pure coral]

it could be slightly worse for the higher performing students but i think the average performing ones will appreciate a good story

just depends on the storyteller though lol

one thing from the little i read from grothendieck’s works is that he definitely knows how to tell a story, and how to convince people that the old ways are not it

[vagrant meadow]

absolutely! trying to think of some examples

[winged urchin]

Where else could we use awfully tedious calculation to motivate why we have the thing we're gonna talk about in class aga

Make em add 17's together 14 times before you introduce multiplication? Ahah

Oh oh if we have a bunch of temperatures in say F but want C you could make em do the conversion for each data point then show them how you could write it generally as an equation

Conveniently that also introduces the idea of inverses somewhat too

[vagrant meadow]

differentiate (2x+5)^6 by expanding and then teach the chain rule 😆

[pure coral]

i love the example of computing fibonacci numbers, where you can find the nth fibonacci number by diagonalizing the map (x, y) |-> (x+y, x)

can also show how it relates to two common techniques they'll see in their math journey over and over again: solving a problem by working in a bigger space with additional properties (even though fibonacci numbers are integers, in this case, we solve it by working in the reals instead), and solving a problem on the entire space by solving it on smaller spaces and then patching those spaces together (in this case, we take subspaces where the map is just an action by a scalar, then we "glue" these spaces together with the direct sum)

[devout fable]

linear algebra is one of those things you first learn for the sake of learning it. So many things are hard to learn and tend to become forgotten or learned poorly. Then as you progress in your studies, you are forced to pick stuff up

[pure coral]

haha, the darker the storm the brighter the sky though 😄

[pure coral]

to an extent yeah, linear algebra isn't the worst field to introduce to someone just starting since it's already so useful on its own, but it becomes even more useful with the tools of topology

(but a lot of math would be an isolated abyss without topology)

[devout fable]

i think you are just a topologist

[pure coral]

ehhh i mean a lot of math is about gluing nice objects in not so nice ways, manifolds for example are just smashing euclidean vector spaces together into one unholy amalgam

topology by itself is pretty lame though

[wide ice]

one of the prime examples is counting all-pair shortest paths.

aka Floyd-Warshall algorithm

You can really see how matrix multiplication stands out.

[wispy slate]

[meager bronze]

[barren plover]

I don't think Floyd-Warshall is matrix multiplication based? It does this with dynamic programming.
Are you thinking of Seidel's algorithm maybe?
I could be mistaken, there are so many algorithm names

[wide ice]

There are two interpretations. Dynamic programming one is what is usually taught first.

But there's also the matrix interpretation: basically you see the dp function as the power of the (weighted) adjacency matrix.

This has some applications. Some intermediate-hard problems in competitive programming arise from this interpretation.

I didn't know about Seidel's one, but it's pretty similar ngl

[wide ice]

With respect to the tropical ring, of course

Other than that, maybe the simplest example can be computing the number of paths of given length between any two nodes

[quartz lake]

#math-discussion

[tawny slate]

This is a question that I have found very meaningful to answer relative to pedagogy, so I will post it here, but let me know if it can also belong anywhere else so I can post there too for exposure.

What are examples of higher math informing more foundational math?

Examples of what I'm talking about:

• fundamental theorem of arithmetic
• fundamental theorem of algebra
• "If x>y>0 and a>0, then x^a > y^a."
  These ideas are used everywhere in not only direct applications and problem solving, but also help shape intuition as we teach them. I prefer these answers because they are implicitly used far more than we may realize, and their proofs are a lot more complex than we probably expect. The third one is particularly subtle in my opinion.

Technically acceptable but not great answers:

• Pythagorean theorem
  While it is often introduced before students are able to formalize an understanding of a proof, there are many very elementary proofs that require minimal prerequisites, and a proof of this can be given to even very young students in just a few minutes.
• "pi and e are transcendental."
  It doesn't require super high powered math to prove, but it is certainly not trivial, but what makes this answer not great is the fact that it is not applicable in many problems.
• Fermat's last theorem
  Better than the previous two, but still not a commonly used concept. However, it is very simple to explain and can trivialize many hard problems, so it's fine

Other answers I will also accept include a little bit of math history, where some realization about math changes the way we approach and teach math, such as the way we write numbers or our selection of an axiomatic system. This kind of stuff ultimately still does shape our understanding of math from a pedagogy perspective. HOW we select our axioms is the kind of answer I would like, but in the context of this question I'm not interested in the particular axioms individually (unless there is some broader context that makes it interesting)

please ping me if anyone has any responses

[cloud zealot]

maybe completeness of the reals <=> intermediate value theorem

for calculus anyway

[wide ice]

Pigeonhole principle, easy

This one alone starts the whole study of degree of transcendence. It's strikingly simple, intuitive, and powerful

Other than that, maybe Jordan's curve theorem. It's the butter of topology, and the proof is much more complicated than anyone would imagine. The intuition is clear tho.

[winged urchin]

What do y'all think of this question and 'answer'? Ahah

I'll refrain from commenting to see what you notice

From a student I was helping today

[tawdry venture]

i can't tell if that symbol is meant to be an a or a 9

t_10 = 10 + (a)(5) or t_10 = 10 + (9)(5)

[winged urchin]

Ahaha it's a 9 but that's hardly the worst part of this question/answer duo imo =p

I mean I suppose saying it's a 9 is obvious and infact i don't 'know' that the teacher meant a 9

Now I'm more curious whether anyone will see the same problems with it that I do. Maybe I'm crazy aha

I'm out for the night but I'll say what my comments on it are in the morning and see if anyone says anything more by then ^^

[tawdry venture]

i mean theres the elephant in the room of the student confusing (in their notation) t_10 with \sum[1,10] t_i

[tardy ember]

the question is... pretty ambiguous

first question: does "10 cm in the 1st second, 15 cm in the 2nd second, 20 cm in the 3rd second, and so on" mean that's the amount added in that second, or the total amount by the end of that second

you can tell it's the total amount because 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 + 55 is too large for the given answers but still

second question: is the initial 10 cm included in these totals or not

if it is, then no rainwater happened in the 1st second; if it isn't, twice as much rainwater happened in the 1st second as all the other seconds
the answers this produces are 55 and 65 which are both in the list

if you interpret it as being included in the totals, and to explain why the 1st second is seemingly an anomaly assume that this 10 cm reading is from the beginning of the 1st second, then you get an answer of 60, which is what there is at the beginning of the 11th second, which is after 10 seconds

[rancid hull]

is it a good idea to avoid using "scary" terminology for people new to the subject at hand? scary as in, terms that either make a very simple concept sound deeper than it is, or things that have an unfair reputation for being more complex than they are

at least until they're comfortable with the idea the word refers to

ex., Monads in Haskell are a lot simpler (to me) than i felt they were while i was still trying to understand them. they are also infamous for being one of the first hurdles of learning Haskell, and i feel like hearing that as a beginner builds up the idea to be way more difficult than it is. in general, i've been trying to sneak past possible preconceptions that people may have.

[wide ice]

It's good to present the terminology, but you also should say that it's not as fancy as it sounds

"Monads are monoids in the category of endofunctors" is a good meme. I passed the course, but I'm pretty sure I don't know monads, mainly because no one explained what it is essentially

[winged urchin]

Woo! I think your points pretty much cover my grief with the question and answer.

I didn't even think of your last point about the beginning of the nth second versus the end.

I differed in that I assumed that the 10cm initially there and the 10cm of rainwater that accumulated in the first second were different. The 10cm initially there isnt necessarily the rainwater that's falling in my head. But that is additional confusion!

My main grief was the insinuation of needing to sum up the terms rather than just get the 10th term. And it's such a subtle difference in the wording but unfortunate

10cm of rainwater falling in the first second seems to talk about how much is falling.. well.. within that time interval.

They could've phrased it as... 10cm of rainwater has fallen by the first second. Then 15cm of rainwater has fallen by the second second... This would be more accurate to the answer imo

And then yeah the initial water present really is confusing too.. it's either forgotten by the question creator or it is somehow really just that 10cm that falls in the first second? But then why even mention the amount initially there. You'd just say... 10cm falls in the first second..

By the by, the question before this was a falling rock problem where it falls a fixed amount every second ahah. Another bad problem in that it's actually teaching the linear fallacy most people assume where if something takes x to do y then it takes 2x to do 2y

And INITIALLY when I read that problem I also thought it might be a sum problem by the wording (and it felt better since it's at least more realistic) but unfortunately it was just a rock falling like... 5 ft per second every second onwards. No mention of terminal velocity or that of course

On a teaching note, I did find my tutoring was perhaps a little confusing for the student in these questions because I was communicating these errors and giving them the different perspectives on what it could be asking

What would you do if a student you are tutoring gives you a question like this where you're not even entirely sure how to interpret the wording?

Do you try to explain yourself to the student and show them how you're reading possibly different interpretations? Do you just give them an explanation for the interpretation that seems most likely at their grade level?

[tawny slate]

i would just say it's badly worded and move on

[pure coral]

it's a lot easier to introduce a bunch of examples and ask the reader to induce any similarities between them

than to have them deduce the meaning of your definitions

before haskell, monads were already a vast generalization of algebraic structures (even more general than lawvere theories, which encompassed stuff like groups, rings, modules, etc), and no wonder why the average haskell student will be tearing their hair out haha trying to understand the full definition

[winged urchin]

Hey everyone, just curious if anyone knows of any pdf textbooks you think are good for a student going into grade 9 next year but wanting to get ahead of the curve. I have a couple I'm aware of but figured I'd check in here to see if anyone has anything they think is particularly stellar

This grade 9 in canada I suppose I should say

[cursive elbow]

Depends what they want to get ahead in I'd say. If they just want to get ahead curriculum-wise I'd probably just recommend them to search for the most commonly used book in their province/territory and just start reading that. One small caveat if they're in Ontario is that they probably won't be able to find a pdf for a textbook corresponding to the most recent curriculum changes moving to a destreamed grade 9 course, but I can't imagine they'd be severely disadvantaged from reading a slightly older book.

If they think that mathematics is something they might like to study going forward and want to get ahead in the bigger picture, then perhaps they'd want to check out "A Concise Introduction to Pure Mathematics" by Martin Liebeck. The intended audience are those who already know some high school mathematics, so it may be slightly out of the student's range right now. If they're a driven student though, then it could be something to dive into.

[pure coral]

https://algebra.axler.net is a good book if they’re really advanced

it says “college algebra” but for non math majors that just need a review of high school algebra

grade 9 should only go up to the 3rd chapter

you could use this same book all the way up to grade 11 i think

or even grade 12 if you supplement it with a bit of stuff on derivatives

[next relic]

If you want to go a lil adventurous I'd recommend grade 9 textbooks from Quebec. Totally not biased :p

This is because I've noticed Quebec likes to throw more involved questions on exams and this in turn influences the questions they include in textbooks.

[vagrant meadow]

is grade 9 math in canada standard? in the US the math class that 9th graders take is not always the same. like at my high school some go into basic algebra, geometry, or algebra ii depending on how they scored on the placement exam.

[next relic]

Each province has their own course (or courses) but iirc we start having different courses in the same grade starting grade 10.

(Apart from Ontario, but they're moving away from applied stream in grade 9)

Starting grade 10 some provinces start splitting - we have, in Quebec, for example - CST for the weaker students, TS at some schools and SN for the strongest peeps who plan to do maths-related programs in cegep.

[robust mauve]

Has anyone here tried teaching via online videos (e.g. YouTube)? I'm trying to plan out into-level Calculus and Physics series with the Manim animation software. The plan was to use the AP CollegeBoard curriculum to guide content, but I was wondering if anyone has some advice for content development (or even original curriculum design), style, or really anything in that area.

[vagrant meadow]

oh interesting. yeah sometimes we start splitting as soon as grade 7.

[next relic]

A typical split you'd see is, say, precalculus 11/12 and foundations 11/12.

Precalc is targetted at students who want to pursue STEM, foundations for those who just want to do some further maths and non-stem majors (say, business?)

(ofc many students opted for precalc anyway bc they don't know what they'll study in uni)

We also have workplace maths 11/12, aiming at students who really cba with more math and just want to know enough to be a functional adult.

Frankly some topics in workplace should make its place to the more traditional precalc course, but eh.

[winged urchin]

Thank you everyone for the recommendations! They helped a lot. I don't quite think the student is advanced enough for your suggestion lucid but I've saved it anyway in case I have a talented student aha

And thanks for answering that question Andrew. That's what I see in BC as well and of course the Canada government site explains it similarly ^°

[silver marsh]

Hi, I'm currently in the running to teach an upper level HS math course this fall in the US. I'll have to be working on my certification during the year, and I was wondering if there were any posts/pins about resources for new teachers? I've been researching various comments/advice from reddit teachers and was curious if the math discord had any tips on what to expect/what to avoid.

[cold trench]

tips (not a professional): include as many activities as possible to keep students engaged and let them see & experience the importance of what they are learning; also don't hand feed them everything, have discussions with them and lead them to their own conclusions through socratic questioning

also as a professional student myself, please don't give excessive amounts of homework if you can control it, give the amount that's necessary and you can provide extra (optional) questions so people can study from them

and relate to them as much as you can, little comments like "guys I know it's monday but just bear with me for a bit" may not be the most professional but students will probably like that, idk how to explain it but it shows you're a person too

(the last one is minor but things like that can go a long way)

also rehearse your lessons but do not expect them to go perfectly, always have some extra time dedicated to answering questions or giving additional instruction in situations where the students still don't get it

and don't lecture for long periods of time uninterrupted, include those activities I mentioned periodically throughout the lesson and try to make your lessons more heavily activity-based when possible

definitely depends on the course and topic but some experimenting will help you get there

to everyone else, pls add more or correct me as I'm not a professional

[pure coral]

your hands are probably going to be tied due to the curriculum and varying student interest

[silver marsh]

The principal gave me the impression she wants me to “reinvent” the curriculum and that I’d be pretty hands on with it (state standards for the topic are written well and it doesn’t seem that awkward, but I might be naive on that end)

Ill make sure I’m prepared for that though and lower expectations on how free/involved I can get

[gleaming flame]

Throughout the history of this channel, I've seen alot of debates/discussion on framing questions, talking about education standards, and the curriculum and so forth...

Has this channel actually found and agreed upon certain things are pedagogically useful?

A compendium or list of such findings/agreements might be worth making if this is the case.

[silver marsh]

The students are probably going to be seniors. I’m thinking about doing a weekly format where the front of the week is lecture on topics (max 30 min at a time), the middle is discussion on example problems/problem solving/applications, and the end is a quiz to gauge where the students are with the topic we just covered.

On HW’s, I’m thinking of adopting a thing my physics dept head would do on his HW’s where students would write their own question and solve it using methods learned in that section

[silver marsh]

Thinking about doing biweekly problem sets and having them due on Mondays. That way they have an extra weekend to work on hw if needed. During the week, planning on making any warmup type assignments being something that you could do in 15 min based on the last lesson and also giving them 5 min at the start of class to finish up the work.

If they can’t figure out how to budget time for 15 minutes to work on something afterschool, in the morning, at home room, or at lunch, then they’re struggling with time management and I’d rather them realize that early on before they’re juniors in college panicking about it.

My physics department tried this, but in a flipped classroom way where you had to read ahead to solve the problems and that’s assbackwards imo

[silver marsh]

I was surprised there wasn’t a pinned message about it here

[wide ice]

That'd require some creativity there, not everyone is that interested in Math, r u sure it's a good idea?

[wide ice]

Pedagogy is a, uhmmm, spicy topic. A lot of times there are wildly different approaches, and they are all right with good execution. I feel like it's a matter of taste.

[silver marsh]

The class is on calculus, so it “might” work out starting midway into the course. I know the principal mentioned she wanted applications to be a major component of the class

Based on what you mentioned, I should probably avoid having them do that until the end of the year as a bigger project thing out of a pool I provide

[gleaming flame]

Even subjective topics usually have some consensus on some general principles to facilitate good teaching.
At the very least, shouldn't there atleast be some written record of many of the more popular approaches?

[wide ice]

On the contrary, Idt you should leave it till the end. Making problems is a skill, and not an easy one to learn. I'd suggest to start small early and go bigger as time passes.

So from changing parameters, tweaking random hypotheses here and there, and maybe ending with making up some easy corollaries. You can make up a lot of easy "theorems" this way.

[fading aspen]

I totally agree with making the students go into journey of finding the solution/understanding a new concept is beneficial. I had bad teachers during high school and I wish I had teachers who were passionate about their subject and also care about their students and not just give them excessive amount of homework without further explanation.

[silver marsh]

I think that’s solid advice.

[silver marsh]

From what I’ve read online, a ton of homework also means a ton to grade so I definitely don’t want to overdo it lol

[fading aspen]

The funny part is they used to tell me that they will give extra grades for doing extra homework, but nobody did that really except for one student. And do you know who failed the class? Yup, the student who did all the homework (he did not get any extra grades whatsoever).

[silver marsh]

I almost had a stroke reading that lmao

If the class was like just 10-20 students, then making hw exceptions for extra points would be maybe ok. But if you’re handling ~100 of students, no wonder it didn’t happen

[cold trench]

oh this is really good, just make sure they don't write a trivial problem

[light garnet]

I'm honestly surprised that none of these points have been mentioned, so I'll mention them here: I used to teach at a community college and keep in touch with many of my former colleagues.

1. Do not assume that your students have the prerequisites down. Standards enforcement has essentially become nonexistent since COVID; at the community college level, for example, plenty of people are entering the college equivalent of AP statistics without understanding what a percent is (and I have seen these people hang out on homework websites).
2. Do not assume that your students will follow deadlines. For example, it has become more common to, in practice, make "the day before grades are due" the deadline for homework assignments (without explicitly telling this to students). My former department chair had to do just this with several of his students for a relatively advanced statistical modeling course.
3. When switching roles in the classroom (i.e., student to teacher), you have to get used to silence when asking if any of the students have questions. Some people suggest rewording "Do you have any questions?" as suggested at https://lifehacker.com/what-to-ask-kinds-instead-of-do-you-have-any-questions-1828681418:

Instead of asking, “Do you have any questions?” Sasser started asking “What questions do you have?” That helped. But then she went a step further and gave the instruction: “Ask me two questions.” And that had a major impact.
I never had a chance to try this out, but something to take into account.

In general, do not be surprised if you find yourself having to spend time pestering your students to catch up.

I both taught in the year prior to COVID and a year into the pandemic: it became very clear to me when the pandemic began that disengagement from my course increased, students were waiting until the very last minute and would not ask me questions on their assignments even though I was kindly pestering them to do so about 2-3 times/week, and students frankly fell off my radar. I tried contacting them every way I could think of, short of knocking at their door. Retention at the college level is a massive issue; at the high school level, a similar concept is reflected in dropout rates, but both are substantially lagging indicators (usually 1-2 years before data are reported).

[silver marsh]

The implied longer deadline for assignments seems like a good idea in practice and I'll definitely bring that up with the principal/mentor about if that's something the school has done in the past.

Something I've been considering is doing a pretest on skills that were prerequisite for the course. I don't think I need to have the students do geometric proofs, but algebra 2/precalc level material before entering the calc course and maybe spacing out time in the first month to address weak points

[pastel horizon]

Yeah you always want to assess them to start with in some way

For assignments I would take the approach of having strict deadlines for whatever they need to pass and let them do higher grade stuff in their own time

And hammer them from the start to have passed it. that way even if they leave things until the last minute at least they've already passed

And you've done the bare minimum your management team wanted

Not sure how the US system works but this approach is best for the way it's structured in the UK

[pure coral]

if you have the freedom to deviate from the curriculum, that’s good

i feel like a very very underrated topic is going over how to think about math in general

how math is taught, there’s this false impression that math is just about problems and definitions and theorems, when there is a ton that goes into representing mathematical ideas

for example, concepts like affine spaces did not just fall out of trees, but instead came after serious thought about what our natural view of “space” is

[light garnet]

Be very careful about how this is received. It's one thing to do this for a college course: students may simply unenroll and enroll in a lower-level course. But it's a completely different thing when you're doing this for a high school with a more-or-less extremely tightly structured curriculum

(and you'll have to deal with the parents in your case as well)

[light garnet]

My experience is that this is an issue with math pedagogy in general; people are more than happy to present definitions, theorems, etc. - but very few (and you can see this in the textbooks) care to talk about historical developments and the way these concepts were developed

With fields like statistics in particular, you get people citing rules of thumb without knowing where the rationale for them came from

[pure coral]

yeah exactly! and a huge example of that has to be how point-set topology is taught, what should be an engaging narrative about the meaning of “close” and “local” is instead a cacophony of definitions that don’t make sense if one doesn’t understand the idea in the first place

[light garnet]

I learned basic point-set topology over a month because I had to between semesters. I wish I could be more interested in the subject, but I don't have time to read through Munkres

I annoy my former academic colleagues in computer science and data science about the same thing. Analogously, you can't just teach someone how to code by just presenting clean code to them and expecting them to replicate it.

[pure coral]

yeah a lot of cs papers tend to be borderline unreadable

[pastel horizon]

Is it not standard to do a baseline assessment if they're transitioning into high school?

[light garnet]

In my experience, no

[pastel horizon]

Fair

I think it's a bad thing the US system can be mixed ability in itself also

[light garnet]

It should also be emphasized that it is generally not in the interest of high schools to hold students back, even if their performance is low: the primary thing I notice schools love to brag about at every level is the graduation rate

[pastel horizon]

In the UK they get setted which means you can provide a proportionate level of support or challenge to each group

[pastel horizon]

Of course, you want to boost stats by having a high pass rate

With ours you get externally written exams. So some schools will kick out students that are at risk of failing if it's sixth form or recommend they drop down to a lower level qualification in high school.

For courses that are practical/assessed internally it's almost impossible for a student to fail unless they literally make no effort

[silver marsh]

Would it be wise to keep the results of the pretest to myself and then adjust the course accordingly? I’m guessing that this could end up being a massive blow to a student’s ego/parent lashing out

[light garnet]

That's one possible strategy, I suppose, but you do have people inevitably asking how they will score. You could try basically giving participation points

[pastel horizon]

There's nothing wrong with giving scores back. Just emphasize it's an informal test that doesn't count towards the grade

[wide ice]

It's not for the final grade. And sooner or later they will have to accept with how much they know and work from there anyway.

[pure coral]

you'd do some students favors by knocking their ego down a peg

kidding aside, frequent low-stakes tests are incredibly effective not only to see how well students are doing, but also to make them recall the material a second time

you might not only wanna do a pretest, but perhaps even a short weekly one worth around 1% or so

at the cost of being a bit disingenious, you could even frame it as "bonus marks" then just make the final exam super hard or something (personal opinion is please don't do this, but that's my emotions talking LOL)

[silver marsh]

That sounds like what quizzes should be tbh

Thinking of having extra credit + dropping lowest scores so that the average student who takes initiative to do more/do better gets full points for it/has a better shot at an A

I’d much rather have a student have a quiz average of 80 if that means they can get 95’s on exams, and I’ve always felt like quiz questions I missed always stuck with me prior to the exam

[vagrant meadow]

i really like extra credit opportunities. i have a prof who would often cover stuff and say "for extra credit, prove this", or include tough but cool proofs on hws. i dont see why that can't apply to HS

[lethal leaf]

I think dropping lowest HW + lowest quiz is a good policy

or at least allow for the retake of one quiz

[inland ferry]

Most of the math classes I've taken have dropped the lowest HW

For people who take way too many classes this can really remove a lot of stress which is really nice

[vagrant meadow]

replacing a lowest exam/quiz score with the final is also a good policy. maybe even better than just dropping imo

but you can use both

[cold trench]

dropping lowest exam score altogether is something I've seen in one or two of my college classes but imo that's excessive

[meager bronze]

If you dont want to drop, you can also add more weight to exams with better scores

For example

Lowest exam: 10%
Second lowest exam: 15%
Best exam: 20%

This is kinda in between “drop the lowest” and “all exams are weighted equally”

[empty gull]

for an anecdote on this subject
My grad algebra class had an incredible grading scheme
HW: 25%, midterm: 25%, Final: 50%
Or you can drop the midterm and have a 75% final lmao

actually the most stressful tests I ever took knowing how much they play into my grade

[cold trench]

oh yes I've seen this too, I think that's a good idea

[tawdry venture]

got a student who's looking to touch up on math up to and including 1st year of his uni. from talking to him today i decided that we would do well to begin with trigonometry

anyone got some example worksheets they could share, or maybe some pointers for how to design homework for him?

right now we're talking like, Basics basics. soh-cah-toa and shit.

[pastel horizon]

Corbettmaths is your go to

[winged urchin]

Honestly I've spent an hour or two with some students just reviewing the unit circle

I'd show them the idea that cos/sin/tan represent the horizontal/vertical/slope of the point with that given angle

Ask them a few simple questions like if my angle was this then where would the point be? Gooddd then what is the horizontal/vertical/slope blah blah and try to really cement that understanding of what the trig functions can represent on the unit circle

Then get into the idea of multiple solutions and perhaps talk about simply solving trig equations and again tying in the unit circle

[twilit arrow]

What is the best place to learn math for a high school student?

[timid topaz]

What kind of math are you looking to learn?

[twilit arrow]

I'm a first year high school student so calculus and trigonometry and such

[timid topaz]

oh nice, it kindof depends how you like to learn stuff, personally if I'm learning something brand new in math I look on Youtube for actual course lectures and follow along with a pen and paper like I'm actually taking the class. There's also Kahnacademy they have really great intro/high school level math stuff. When it comes to books or lecture notes I typically google actual courses online and see if they have lecture notes/pdf available, exercises with solutions, stuff like that, and even better if there are lectures on Youtube you can follow along with

[twilit arrow]

thank you for the advice

[timid topaz]

np!

[sharp nimbus]

I asked this in math educators stack exchange but didn't get an answer so I'm going to post this here as well.

I am a pre-university student who wants to help students with Algebra 1 and 2 in high school. I am curious to how the curriculum was built and what the goal of teaching both algebra 1 and 2 might be. I am comfortable with Algebra, I just never understood why the two classes are structured the way they are. I would like to know this more (and review the curriculum of the two) and see how I can help with students taking the course. My math background is college algebra, trig, and calc 1 and I am fairly decent at solving algebraic equations.

[pastel horizon]

Haha, yeah usually when you dive deep you'll realise it doesn't actually make sense

[sharp nimbus]

For me, I found that college algebra alone did most of the work when it came to teaching algebra. Alg 1 and 2 felt more like practicing how to solve algebraic equations than anything. Seems more like algebra prep since I didn't truly understand why algebraic graphs looked the way they did until college algebra. It just tied all the loose strings together from alg 1 and 2.

[pastel horizon]

College algebra exists because they know what high school algebra is like

So they reteach it to their standards

Sometimes they have a good reason for doing things a certain way if you ask someone more experienced but what I found in my institute was the people that originally designed the resources really didn't have a clue how to sequence it so I just did my own thing

[sharp nimbus]

So you teach at a college?

[solar sparrow]

im a 16 year old tutor, i teach highschool students uptill 12th grade. Does anyone with more experience have tips for me. (note: i primarily tutor struggling students)

[twin shell]

can you be more precise about your situation

[solar sparrow]

of course, which pieces information are you seeking specifically so i can directly answer to your needs?

[twin shell]

what subjects are your students learning
what topics are they struggling with at present
have you observed a history of difficulty with certain kinds of problems
do your students have trouble focusing on their work

[solar sparrow]

1)ussually trigonometry, algebra 2, calculus AB, and calc BC with one student

2. ussually algebra 2 is the most challenging for them

3. problems which require modern day applications (in our school we do math in german which isnt our native lang so there is a language barrier for some students which makes these types of problems harder for them.)

4. becuase im eigther the same age or even younger in some cases sometimes they dont respect my authority and try to distrupt the lesson.

[twin shell]

I would recommend not thinking about “authority”
socioemotional learning tells us this isn’t a good way to get your students to listen to you
you need to build trust with your students
try asking, for example, “how’s your day?” “have you been doing anything fun lately?” “how’s your family?”

[solar sparrow]

i mean authority by they would sometimes just directly ask to not do a lesson

or just keep it half as long

should i accept that?

[sharp nimbus]

I have some experience helping with Algebra 2. Usually the students I worked with will just get simple stuff wrong like distributing and working with rationals so I work them with their. On the note of factoring quadratics, I try to approach the subject with understanding what it means to factor and why we set things to zero (but this was my situation since our teacher didn't teach anything about the intuition of factoring, and just taught us how to do it).
I like to talk about the history of certain things we cover in math to make it more interesting and engaging but it usually makes the session longer and some students won't care in the end.

[sharp nimbus]

^This. U need to build a relationship with ur students (not in like a loving way) or else you are just another tutor for them.

[solar sparrow]

ussually they are already my friends, but thats why they dont take me serşously

i tutored a 19 year old once and it was very challenging beeing 3 years younger for example

[sharp nimbus]

Its up to you. Why do you tutor those students? What are your goals?

[sharp nimbus]

Ah I see

[sharp nimbus]

Im in a similar situation. Im finishing up calc 1 right now and everyone at my college is older than me and I try to help them but its kind of frustrating that they say "Oh ur just really smart I wish I were like u" comments.

[solar sparrow]

firstly financial needs, secondly teachers highly encourage me to do so because they dont have time to go that slow to be able to explain in a way some student can understand

[solar sparrow]

it really is

if you need help with calc 2 or 3 im here btw

[sharp nimbus]

I see. For me, I kind of gave up on the financial needs (I wanted to tutor for money) but soon realized that people here aren't really willing to pay and its hard to find someone wanting a tutor for alg 2 or anything highschool. I honestly loved algebra 1, and 2 because I started watching 3b1b and Eddie woo and I loved the way they taught maths so I wanted to do the same with my peers. I don't worry about time much (a lot of the time they either fail or succeed anyways) so I just encourage them with where they currently are, point out their critical misses in their calculations, and show them why.

I was helping out an Alg 1 student and he kept missing the part where you isolate x by dividing the coefficients to the other side. I would suggest looking for where they make mistakes and help them their as well as show them a few examples going through the problems and they should be able to finish the homework etc on their own.

[sharp nimbus]

Thanks that's cool to know since I will be taking calc 2 in a few weeks

[solar sparrow]

makes sense

[solar sparrow]

its nothing someone interested in math cant handle, cant say the same for calc 4 tho its a nightmare

[sharp nimbus]

Calc 4? Is that diffeq?

[solar sparrow]

vector calculus

[sharp nimbus]

Ahh

Didn't know it was called calc 4

[solar sparrow]

first time in my life i literally needed to give up bcs it was so hard to comprehend

[sharp nimbus]

I want to move onto real variables/analysis after calc 3 honestly

[sharp nimbus]

Wow. Sounds hard lol

[solar sparrow]

tbh i dont know calc 3 in full detail never saw it in school

just some tutorials

[sharp nimbus]

Same with calc 2. I just went through Prof Daves series on calculus 2 and just did the problems as I went.

[solar sparrow]

wait ur in college right, how old are you?

[sharp nimbus]

18 I just graduated highschool

[solar sparrow]

thats nice!!!

[sharp nimbus]

Im taking summer classes at my local community college.

Thx :D

How bout u

I find it interesting that u are doing calc 4 at 16 since that's when I was doing algebra 2 lol.

Actually, that's when I first looked into limits and got interested in calculus which has been my dream since. Happy to see myself taking it now.

[solar sparrow]

İm not really DOİNG it i watched the first tutorial 4 times and then gave up (we should move to general btw)

[sharp nimbus]

You are right or we can move ot DMs

[pastel horizon]

Sixth form

It's the same problem no matter what level you teach at. You always have to assume the previous teacher was wrong and so you reteach it "your way"

In theory the UK curriculum is cutting edge research wise but in practice there's simply not enough maths specialists

[sharp nimbus]

Hmm

Interesting

[cloud zealot]

https://math.meta.stackexchange.com/questions/1652/what-do-we-do-with-users-who-post-numerous-unlabeled-homework-questions#1661

thoughts?

[vagrant meadow]

totally agree. honestly, attempts to stop cheating or getting help will only result in students finding more creative ways to do it. you might as well let them get outside help on your own terms.
may say more when i get home.

[pastel horizon]

You can rely on the good faith of the maths community in that they'll help someone figure out the solution rather than give the answer directly

Teachers aren't stupid either they'll just make them rewrite their working out or explain the method if they suspect it's copied

[lethal leaf]

The algorithms class here has a policy of "you can use any resources you find online to solve your questions but please cite it + use your own words"

So like if you find some god awful algorithm published last week that solves your problem by shaving off a log factor

You're allowed to use it as long as you can explain the algorithm

Good policy

Wish that prof was in this server so that he could comment on this stuff

[tacit adder]

That’s the same with my algorithms course, we encourage students to discuss the assignment problems provided they submit their own write up

[pastel horizon]

Do they get extra credit if they can find something like this

[lethal leaf]

yea

so in that class the solution has a target run time

so if the target run time is O(n^3) and they get O(n^4) they'll lose some points

on the other hand if they get O(n^2) they'll get extra credit

but like you can't say "just go read this paper"

the grader has to be able to read what you wrote, understand it, and see that it's correct

[tacit adder]

the details are left as an exercise for the grader

[lethal leaf]

I think one time someone wrote that and I left "the rest of the points are left as an exercise to the student" or something along those lines as a comment

[tacit adder]

oh yeah I teach an advanced level algorithms course at my uni and the amount of times I've seen someone write that

insane

[lethal leaf]

I think the only time we allow less details is like

exams

but HWs they get a week so we expect a good explanation

[tacit adder]

sounds about right

I think we're transitioning from assignments to weekly homeworks starting from next year

[lethal leaf]

but yea back to the original question this is the correct take

assignment vs hw?

what is the difference?

[tacit adder]

the main issue I've seen with assignments is the turnover time for feedback

[lethal leaf]

no no like

assignment and hw are synonyms to me

[tacit adder]

ah

[lethal leaf]

so transitioning from one to the other means nothing to me rn lol

and I'm asking for clarification

[tacit adder]

ah right right