math-pedagogy

[long pelican]

You mean historically? Because nowadays the college track seems very much like it is taught with college calculus as the end goal

[zenith slate]

I absolutely disagree with the idea that the general high-school math curriculum should focus on preparing students for math undergraduate study

[long pelican]

I’m just talking college calculus/multivariable, the required courses in many schools

[zenith slate]

I'm not all that familiar with the US curriculum to speak to its specific merits

[winged oar]

What about even earlier differentiation then? So something like general calculus for high school, then another part which is closer to pure/uni math for some students who intend to go uni math

[zenith slate]

In any case one can envisage myriad explanations for why college professors might not be happy with the level of mathematics from HS graduates that do not involve a fundamental shake-up of established pedagogical principles

[zenith slate]

I think it is reasonable to start introducing optional classes specifically for prospective future math students yeah

The UK does this and I think it is very successful

(Though I have ample criticisms of the actual content of the curriculum...)

[winged oar]

O can you show some highschool/secondary syllabi more towards pure math?

[zenith slate]

Hmm?

[winged oar]

I've seen like basic decision theory at A-levels too, but I didn't notice anything closer to uni math

[zenith slate]

What do you mean "can you show"

[winged oar]

Uhh any subject codes/subjects in particular

[zenith slate]

It varies quite a bit between exam boards

The A level mathematics course is general-purpose and is designed for essentially any STEM subject

[winged oar]

Most of UK pre-uni is still calculus-ish right? By this I mean integration and differentiation

[zenith slate]

The a level further mathematics course is the one that is specifically targeted at math/physics/cs students

[zenith slate]

Unfortunately, yes

[winged oar]

Pockets of uni math I see in some pre-uni education would be basic number theory

[zenith slate]

From what I recall the AQA further mathematics A level is pretty good

[winged oar]

There's also some really unenlightening definitions IMO on function continuity for example

[zenith slate]

They introduce basic group theory and some other more rigorous stuff

[winged oar]

Oo, I will take a look

[zenith slate]

It is still not perfect preparation for a math degree because again the target audience includes physics and compsci students

[long pelican]

One thing I can see based on my experience is that “direct instruction in what types of skills” Is a very important question. 12 years of solving “problem types” did not lead to much growth in problem solving or mathematical thinking

(Could be US specific)

[zenith slate]

I mean generally my opinion is that professors who teach undergraduate mathematics need to take the stick out their ass, understand that high-school has a much broader purpose than to funnel students into abstract algebra, and appreciate that university-level pedagogy is fucking easy and HS-level pedagogy is fucking hard

The school system does not and should not revolve around potential future algebraic geometers

[long pelican]

Are you talking to me here? I actively avoid any theory (analysis or whatnot) in calculus classes

[zenith slate]

I was addressing the general attitudes of people who complain about the math abilities of freshmen undergraduates

[zenith slate]

I'm not totally sure what you're saying here

What do you mean by "solving problem types"

[long pelican]

Oh it’s all the things I shared and talked about like 2 years ago (and 1 year ago) this channel

I’ll try to summarize

[zenith slate]

Oh

I am not acquainted

I gotta run but will read later

[long pelican]

Basically there is a sudden and massive drop in mathematical thinking ability with many students when switching from a familiar problem (identical to textbook example or exercise with numbers changed) to an unfamiliar but still doable problem. Like, yes the latter is harder. But I’d hope I can still see evidence of applying skills they learned from practicing math problems all the years. And I expected lower scores but also to see good thoughts in their “shown work”. But there was little of it. I concluded (after more evidence than what I said in this paragraph) direct instruction of problem types in US high schools inadvertently helped students avoid what mathematical thinking is like, in order to improve scores

A totally not adequate summary but it’s a starting point

Just remembered a name that has written about much of the same effects: Alan Schoenfeld. Are you familiar with him?

[zenith slate]

I do not think this is an issue with direct instruction - I certainly do not dispute that it is very easy to do direct instruction poorly and I think poor implementations are unfortunately very widespread. When I teach, I will start off with an example of how to solve a problem, then have the students replicate these steps with a slightly modified setup (different numbers, etc). This is more-or-less the main idea behind variation theory and is very successful at identifying procedural weaknesses and misconceptions in students. It also has the marked benefit of making it very easy for the weakest students to access the material.

Ultimately, I think that drilling students on procedure is a necessary step toward building problem-solving ability. Students feel empowered and more able to tackle novel problems when they have confidence with using the basic, underlying mathematical machinery.

After establishing the tools, however, it is absolutely vital to encourage students to employ these tools in a variety of different situations. Encouraging problem-solving is absolutely a key component of direct instruction. It's simply that the means of achieving problem-solving fluency begins with drilling procedure, rather than hoping it all comes naturally.

My primary contention is with academics like Jo Boaler who goes around saying that she never learned her multiplication tables and that this has never caused her any issues - a statement that I find deeply problematic for many reasons. First and foremost being that it is an astounding demonstration of privilege - just because she is the type of person who can perform multiplication without needing to memorise tables doesn't mean that this is a universal skill; it is demonstrably and categorically not. I won't belabor the point lest it appear like I'm strawmanning, but this type of crap is why I tire of this particular brand of "progressive" educator (who are generally the staunchest advocates of inquiry-based learning).

[zenith slate]

As for Shoenfeld I've not read much of his work but I generally respect that he has quite a nuanced view on the matter, though I tend to disagree with his conclusions. (Most strongly with his claim that mathematics is a science but this is by-the-by).

Overall I think he has some interesting things to say on, like, epistemological concerns in education, but I'd struggle to really apply any of it in the classroom.

Ohh hang on I do remember his one paper uhhh

"When good teaching leads to bad results"

This was really interesting actually and a great example of how direct instruction can go disastrously wrong if the emphasis of said instruction is misplaced

Like, iirc the case study was about a math class where students had to do straight edge and compass constructions and they weren't given any reasons for the steps they were taking and weren't required to give any justifications in their exams

Which is just... a really fucking awful idea for a class that I think literally every educational theorist would take issue with lmao

However if one does set the goal of being able to construct a hexagon then I challenge inquiry-based learning advocates to come up with a lesson plan where all students manage to simultaneously, accurately, and with correct justification, 'discover' a procedure for constructing a hexagon

And like, this isn't even a difficult one. But I've taught dozens of students who can't even replicate the procedure to construct a perpendicular bisector, let alone figure out how to construct a hexagon.

Perhaps if they had been encouraged to discover the perpendicular bisector for themselves..........

[twin shell]

I believe my ed professor said something like this, that for math in particular direct instruction appears to be more effective than constructivism discovery learning
of course you have to ask “what does math mean?” in this context

I don’t really doubt that pure discovery learning is bad but my impression was that guided discovery outperformed both

but I don’t know much about elementary math ed

there’s probably a good reason for that, since math takes way longer to fully understand than most other core subjects

(biased but in my experience)

あと “I never learned my multiplication tables hahaha” is especially stupid considering it’s objectively the most important skill to grind in arithmetic

[long pelican]

Yes that’s the paper I was thinking of too. Don’t forget that the point of inquiry based learning is that it is more difficult to learn the current task (if you just cared about the current task, why NOT direct instruction in the style of that geometry teacher? Don’t see any downsides) but easier to approach the future tasks for which no example will be shown. So the hexagon example you gave is not surprising.

I don’t like the rote vs inquiry learning debate because I find it too reductionist. “Pure discovery” isn’t the logical opposite of direct instruction.

Do metrics in the direct instruction study you cited involve solving problems similar to examples?

[vagrant meadow]

what kind of requisite knowledge do you need to learn about collatz? being able to multiply/add/divide is generally expected of 8th graders. at what point is someone "allowed" to learn about collatz?

[paper sierra]

Does anyone here have experience being a grader for an analysis course? I just got my assignment for this semester as the sole grader for a section and have only graded more computational courses in the past; wondering if anyone has tips.

It's a standard intro to analysis course for undergraduates, blind graded online. Looking largely for advice in constructing rubrics etc.

[valid trout]

I trust this is not the way you want to build relationship.

This is what I would say to math elitists. This is not StackOverflow (!)

We want to build and cultivate supporting community over here, I hope.

[zenith slate]

I don't think anyone should learn about collatz. Collatz is an infohazard and exposure to it is a significant risk factor for developing crankism

[winged oar]

Learn about Collatz when you can read Tao's paper on it.

If not no shot

Without even an intuition on dynamical maps (in Z) one should not touch it

[vapid herald]

You guys are exaggerating quite a bit the "harm" I caused these students, let alone the impact I had at all, given it was just 15 minutes of class to kill time

[vagrant meadow]

tbh that just sounds silly to me

[twin shell]

seems a bit excessive to me
if you come across it while reading wikipedia I think it’s fine to learn about

‘tis what I did

[vapid herald]

It was practice in following directions more than anything

[vagrant meadow]

it's an amazing example of how simple problems do not always have simple solutions. that's one of the interesting things about mathematics. the thought that an 8th grader is going to have a crisis and forever swear off mathematics because they can't solve the collatz conjecture, or become obsessed with it in an attempt to solve it or something is just absurd to me.

like what would you all say about fermat's last theorem in 1994? would that be something "dangerous" for 8th graders?

[vapid herald]

Im scared to say it now because I seem to be getting dog piled lol but I had nothing to talk about first day so I did cantors diagonalization theorem, and you don't have to believe me but there were a lot of "aha" faces in that room

[vapid herald]

I mentioned it too lol. They were surprised it has infinitely many solutions for "2" and none for "3"

[vagrant meadow]

that's awesome! yeah i mean if you can explain it well enough, i think you can get the general idea across. as for if they'll be able to really understand the full consequences... well probably not. but i still admire the fact you were able to at least some students to get the gist (:

[winged oar]

Unfortunately we see too many Collatz cranks here

[vapid herald]

I just think different sizes of infinity is a really cool fact and they thought so too

[vagrant meadow]

i think they need to be a bit older than 13 to develop the hubris necessary to become a collatz crank 😆

[vapid herald]

Tbf we see plenty of divide by 0 cranks too

[vagrant meadow]

but yeah i think this is great too. that's just a cool fact that you have infinitely many pythagorean triples, and then absolutely none for any higher power. and that's something i think middle schoolers can definitely understand, even if there's no way they would be able to really understand why. andrew wiles himself was inspired by the problem, and decided he wanted to prove it, when he was just 10 years old.

[vapid herald]

Oh cool

I don't spend a great deal of time on these topics which I think is the impression some people got. I spend my time on run of the mill algebra 1 stuff

[twin shell]

maybe so
but that seems a bit biased [statistically, not psychologically] to me since most kids these days™️ don’t go out of their way to join math discord servers
unless I’m really out of the loop, which is possible

[vapid herald]

I wonder how many people actually understand the why lol I'm guessing not too many

[vagrant meadow]

one of the reasons i think bonus topics are great for that age is that math at that level can get pretty boring. so even the students who are bored silly by math normally probably appreciate breaking the monotony by doing something they aren't expected to have to really learn or memorize. i've even seen some students who actively refuse to learn for the simple reason that they have to.

[zenith slate]

I don't think a handful of short sessions on this is going to cause harm

But it is not a good pedagogical foundation

You need to remember that you and all the people on this server are good at math and enjoy doing math. Many students are not like this, and examples like this are generally not going to be "inspirational" to them

I have personally taught dozens if not hundreds of students for whom being made to work on problems with no end goal or hope of solving them is indeed going to be detrimental to their confidence, enjoyment, and learning

My comment about collatz was mostly just a joke. Nevertheless, you should really consider whether "sometimes easy-looking problems are borderline impossible to solve" is actually a good message to be communicating to weak students who already struggle to solve problems that are actually simple.

[zenith slate]

8th graders regularly have crises and go off math forever because they cannot grasp the fundamentals of trigonometry or even the pythagorean theorem. So no, it is evidently not absurd.

What is your background anyway? Are you familiar at all with any pedagogical literature? Have you taught high-school students?

Like what is your justification for what you are saying

[twin shell]

i agree that i would prefer assigning problems that your students have a realistic chance of being able to solve

[zenith slate]

Mhmm

[twin shell]

builds confidence

[zenith slate]

Lile I struggle to see anything that you can say your students have actually learned if you let them loose on collatz for a whole

Collatz is hopelessly impenetrable

And I'm not going to apologise for saying your justification is bad if it amounts to nothing more than "wow look at this problem! Very difficult! Much mystery!"

"Some problems are intractable" just does not seem like a worthwhile message to teach students to me

Like I do not see that getting kids to compute collatz sequences or whatever is any more mathematically enlightening than like, playing scrabble lmao

Yea you get to practice multiplying by three and dividing by two

What else

[tardy ember]

you also get to practice adding one

[zenith slate]

What else can one possibly learn, practically speaking, from collatz

[zenith slate]

Fuck

All the ingredients for a division ring I suppose

[vapid herald]

You need to slow your roll. You only have one year of teaching the age group. You claim to have 6 years teaching undergrads, I have 2, but we both know that's not transferable

[zenith slate]

I do have six years teaching undergrads

[vapid herald]

Ok. It's not transferable

[zenith slate]

But in terms of the pedagogy, no, not a huge amount is transferable

This I agree with

Teaching high-school is a totally different can of worms

In any case, regardless of the personal experience I may have under my belt, pedagogy is a scientific and academic discipline and I am well read on the subject

What personal anecdotes I may have regarding individual students and what works for them does not, in fact, a pedagogical theory make

[vapid herald]

I think the literature is all ivory tower

But I'll admit i don't like my job because I feel like a baby sitter more than a teacher.

[zenith slate]

I mean this is just false

Plenty of leading academics in the field of mathematical pedagogy have decades of experience both with teaching and conducting actual, you know, experients in the classroom

Sure there is a lot of shit out there that talks primarily about the epistemological and broader philosophical foundations of what learning is

But this is not research i particularly care about

[vapid herald]

I know people with degrees from music colleges who don't write good music. I think it's similar

[zenith slate]

So the fact that some people with music degrees aren't good musicians is a justification for the field of pedagogy being a useless ivory tower? Forgive me if I don't follow your reasoning here.

Many of the most foundational educational studies are longitudinal studies conducted over decades with thousands (if not tens or thousands) of participating students

[vapid herald]

What little I've read suggested sarcasm doesn't work in the classroom but plenty of effective well liked teachers roast misbehaving students and get respect for it. I think reading a book about teaching doesn't make a good teacher

[vagrant meadow]

"don't bring pies to class on pi day because the literature does not indicate doing so will actually teach students anything about the circle constant"
i just think it's silly to say there's harm in making class fun by mentioning interesting math topics for like 10-15 minutes at the end of a class. the idea that teaching young students is a pure form of pedagogical science seems like a very cold way to view education. especially with younger students, i think connecting with them and making class fun and interesting is important.

[zenith slate]

What evidentiary standard would satisfy you? More anecdotes from teachers who think they've cracked the code?

[zenith slate]

You seem to be misunderstanding what the pedagogical literature actually has to say. Building and maintaining positive relationships with students is a critical component of literally all pedagogical theories, even those that I totally disagree with.

[zenith slate]

Sarcasm and humor in general can be a very useful tool in the classroom, but one does have to be wary of the possible harms that can result if these techniques are applied incorrectly

[vapid herald]

So there are experts you yourself disagree with

[zenith slate]

Yes

I can't think of any experts who I 100% disagree with on all counts but yes, naturally I am more convinced by some arguments over others

[vapid herald]

Then let's just agree to disagree

You hate the Collatz lesson. I think it went fine

[zenith slate]

I'll agree to disagree insofar as it seems I am unlikely to make any headway here, sure.

[vapid herald]

I think you just like to argue

[zenith slate]

Though I make no claim that your collatz lesson in particular went badly

[vapid herald]

One student even wondered allowed why they care enough to pay a million dollars, which is thinking I like to see

[zenith slate]

If you think I'm being contrarian for the sake of being contrarian then you're suggesting that I am some way being intellectually dishonest

[vapid herald]

I don't understand the doomsaying over something pretty innocent

[zenith slate]

You attack not the substance of my argument but the fact you don't want to engage. If you don't want to engage, just say you don't want to engage.

[zenith slate]

What doomsaying

You're using a lot of borderline pejorative terms to characterise what I am trying to communicate and I don't appreciate it.

It is quite straightforward.

[vapid herald]

You kicked off this whole thing saying the lesson was pedagogically toxic

Continued on saying I'm fostering little cranks

I don't agree with you

And don't understand the explosive reaction

[zenith slate]

I do not, will not, and have not said that I think your lesson went poorly or that it caused harm to your students. I wasn't there and it would be completely incorrect of me to make any judgements. However, you came here before giving the lesson and specifically asked members of this channel to give feedback on your intended approach. I gave my feedback.

[vapid herald]

All right. Fair enough

Really I just wanted to know if the Zeno paradox was readable

[zenith slate]

Note that I haven't said anything about how the lesson went. I completely believe you when you say that the session went well because I respect your judgement to make that call as the person who actually did it.

Also I did clarify the collatz crank thing was a joke

[vapid herald]

I must have missed that then.

[zenith slate]

It's a bit of a meme in math circles that working on collatz will turn you into a crank. I don't think it's actually true (though this server has seen its share of collatz weirdos....)

[vapid herald]

That's why I thought you were serious

[zenith slate]

Fair enough

My bad

[vapid herald]

I shouldn't have said you liked to argue. I was defensive

[zenith slate]

Tone is difficult to convey on discord

I wanted to lighten the mood by saying something over-the-top

No worries and no hard feelings I hope

I'm not actually angry or think you're a bad educator or anything like that

I'm just passionate about pedagogy and like discussing it. I'm also majorly autistic so can come across a lot more... aggressive than I intend to at times lmao

[vapid herald]

Well I appreciate that. You are obviously passionate about what you do and that's commendable because it's a very difficult job

[fallow siren]

I enjoyed reading this discussion and it has given me something to think about. I'm not a fan of Collatz, but I like telling my students about things like Goldbach's and twin primes.

[vapid herald]

I too am on the spectrum and can be overly sensitive to criticism

[zenith slate]

It's funny this is actually one of the reason I love working with kids. I find it way easier to be less....... autistic when working with them lmao

Don't know why

Perhaps because I was a deeply troubled teenager I find it very easy to relate to many of them

[vapid herald]

I admit I think I have a great command of conveying the subject matter but classroom management I need a lot of work on

[zenith slate]

Classroom management is tricky

I also need to work on it lmao

I tend to be too lenient and rely a little too much on being friendly and likeable

Which is great because my students like me a lot but also behavioural issues can be tricky to stamp out

My wife is the same though she has a few years teaching behind her now

She's 6 years older than me so has much more experience

I've been reading Bill Rogers lately and I think he has a lot of great tips and practical examples on classroom management

[vapid herald]

I'll look it up

[zenith slate]

Though it is very much UK-centric so I don't immediately know if it translates

It basically boils down to being friendly, fair, but also firm

And avoiding getting yourself into those traps where students try to argue you into a corner

Which they're pretty good at doing... little shits

I have pretty solid beliefs and ideas about instructional styles... behaviour though this is a whole thing of its own that I regularly find myself out-of-my-depth with

Ho hum live and learn

I do think that behaviour management is much much more difficult to formalise onto a theory or framework just because the problem is like 100% context-specific and really relies entirely on your interpersonal relationships with students and each individual student's quirks

I think teaching very much becomes more art than science at this point

[vapid herald]

Little shits ain't that the truth

[zenith slate]

Yup

I gotta love em tho

[vapid herald]

I agree

[fallow siren]

For sure you can have two different classes back to back and have to manage them completely differently.

[zenith slate]

I was a little shit at that age lmao

[zenith slate]

Yeah exactly

I see kids arguing with me in the exact same way I used to argue with my teachers

Game recognise game

[vapid herald]

They're such heathens it's a mystery they become functional adults

To me

[zenith slate]

I think being a little shit is an important part of learning the boundaries of acceptable behaviour

[vapid herald]

That's an interesting take

[zenith slate]

Sometimes the best way to learn a thing is to learn what not to do

I don't have any evidence for this just my general feeling lmao

[fallow siren]

I had one student that I could tell was specifically testing boundaries of how little with he could show. It was interesting to notice in real time. Not that I'm very perceptive or anything, he would ask explicitly: is this enough? Is that enough?

[zenith slate]

At least for me I feel like this is how I learned to be a functional adult

It was certainly how I learned basic people skills as a child struggling with autism

[fallow siren]

(I started typing that slowly on phone before you mentioned boundary testing, btw.)

[zenith slate]

Yeah I've seen this lmao

[fallow siren]

It was greatly amusing too because he put a lot of critical thought into it.

[zenith slate]

Yup

Kids optimising for the minimal amount of work necessary to complete a task

Gotta respect the hustle

This is what most adults do lmao

[fallow siren]

It was interesting because I don't think he was ever trying to be difficult. I have him again a few years later and he's figured it out and is still very perceptive.

[molten urchin]

I volunteered to teach some basic set theory to newly-enrolled undergraduates at my institution. The audience consists of a lot of students who never did calculus before, know almost nothing about sets, and have likely forgotten the elementary trigonometry they once learnt. The goal is to get them up to speed with just enough calculus and differential equations that helps them survive the mandatory physics courses (classical mechanics and electromagnetism). Do you all have any suggestions on what all I could/should cover as a part of the basics? I was hoping to at least introduce the basic terminology of sets and functions, but it will still be quite a leap to limits of functions from there I believe.

Also, anything that you believe is worth emphasising on early up (things that otherwise adversely affect students in something like calculus)?

[tight jay]

I think if you give an intuitive introduction to limits you can start teaching epsilon delta proofs right after sets and functions. What my high school teacher did was he gave us exemples of limits and then made us try to guess how it should be defined. Each time one of us did give a definition, it was wrong and he showed us a counter exemple (a function that fit the dfinition given by the student but that doesn't have a "limit" in the sens that we would like it to have). Then he asked us to think about it until the next lesson, that started with the actual definition. I thought it was pretty effective. I also think you should force them to do some basic exemples by hand (with polynomials for exemple) by going back to the definition to acustom them to the formalism.

[tight jay]

I'm a little biased as a french student, but I also think it is very important to put an emphasis on the computation of low-order taylor expansions for physics in order to be able to simplify quickly and painlessly expressions that are hard to reason about from a physical standpoint

[molten urchin]

Interesting point, I will bring this up with other instructors for this thing. I am still conflicted about introducing epsilon-delta arguments though. How did your class (as a whole) respond to the epsilon-delta approach? Did it consist of people who voluntarily chose mathematics as a subject of study?

[vagrant meadow]

yea taylor stuff is useful for physics. i basically got through kinematics almost entirely through my understanding of taylor series

[vagrant meadow]

this is a very cool approach. though i wouldn't trust my ability to come up with a counterexample on the spot. idk if you remember what kind of things the class suggested that were wrong

[vagrant meadow]

my HS teacher taught us epsilon delta. it was AP calc so most of us weren't going to be math majors. idk how many of my classmates really understood it, but it didn't really sink in for me. but i was glad i had been introduced to it before when i formally learned it later. if you're just prepping them for physics classes, i wouldn't bother with epsilon delta, though.

it's cool to know how mathematicians formally define the idea of a limit, but it's useless to people who don't want to study math.

[molten urchin]

Right, that's the concern. I'll still try to at least shove in epsilon-delta in disguise, even if not in its fully formal fashion.

[vagrant meadow]

sounds interesting. i'd be curious to see how you do it once you've figured it out

[molten urchin]

First big hurdle is to not let the number of attending students dwindle too much. In prior runs the attendance dropped rapidly, so I guess I will try to keep it engaging but not intense as far as possible.

[cloud zealot]

i never saw epsilon-delta proofs in my AP calc class and i ended up fine

to the extent i hear of epsilon-delta proofs being commonly taught in university calculus classes, it is limited to verifying very "obvious" limits, like finding lim_{x -> 1} 2x+1

but the limit definition is never applied in contexts where students might genuinely see the need for the formalism over intuition

the reason the formal definitions of the limits were invented in the first place was because it was very difficult to intuit how fourier series would behave. but since you're probably not going to be throwing any particularly pathological examples at them, it's not really worth it to dedicate some time specifically to learning how to handle epsilon-delta proofs

[vapid herald]

What about a guided proof of Euclid's square root of 2 irrational proof for algebra 2 students? I'm sick so I made a Google form of it for the substitute to give them, since it feels a little impossible for me to trust the sub to cover ground.

[zenith slate]

Yeah I would be extremely wary of introducing epsilon-delta proofs to this group of students. If the purpose of the class is to equip students with the tools necessary for undergraduate physics then I'd probably start with motivating calculus via kinematics or something similar.

It is entirely possible to learn calculus without even knowing what a limit is

Sure it isn't a mathematically rigorous understanding but this isn't required for just computing integrals and diffeqs and the like

As for sets - do they need to know about any sets other than, like, connected sets in R^n?

[zenith slate]

To be more specific, an exercise I have used before to motivate the idea of integrals is using a piecewise linear velocity-time graph and asking students to calculate the final displacement. Then you can show them a random (continuous) velocity-time graph and ask them how they might calculate the same quantity. This quite naturally leads to the idea of an integral as (very informally) the 'limit of the trapezium rule'

Teach them how to compute elementary antiderivatives and you've got the FTC

Honestly I think restricting the study of calculus to polynomial functions is a good way to ease students into the important ideas of calculus (how to work things out when you only know instantaneous rates of change and vice versa) without getting into the weeds of trigonometric integrals and derivatives and all that crap

The main mental block I see in students on first contact with calculus, especially those who are not mathematically minded, is why one would ever care about slopes of graphs and areas under graphs

Given that the defintion of velocity and Newton's second law are fundamentally statements about derivatives, these examples can be a great answer to this question

Also deriving all the different kinematic equations students generally have to memorise in high school (at least in the UK) using only F = ma and basic integrals + derivatives is a fun exercise

Like it takes the mystery out of shit like s = uv + 1/2 at^2

[normal aurora]

suvat

[zenith slate]

If the end goal of the class is prep for classical mechanics then seeing how the whole field (modulo anything to do with rotation) is basically just calculus built from F = ma is probably a good way to get them invested and to make their future studies as straightforward as possible

[vagrant meadow]

i've seen a lot of people say they were shown that proof in high school. it's not particularly advanced, but the hardest part is usually explaining the whole n^2 is even means n is even thing. otherwise, they may not fully grasp how the proof by contradiction actually proves it. it's often hard for students to see why a proof by contradiction really works.

the main problem being a lacking knowledge in logic, mostly.

[vapid herald]

The n^2 is even means n is even I think they'll be ok with.. the part I wonder about is if the will understand the part where I rewrite p as 2r

As a preamble I explain the method of proof by contradiction using "assume you can divide by 0" and showing a contradiction as an example

[zenith slate]

Yeah I can imagine this factoring out of 2 being a hiccup

[long pelican]

Is the whole "how logical arguments work" part of the UK high school math curriculum @zenith slate ?

[zenith slate]

Hmm tricky question actually

I'd give a soft yes

The curriculum has been moving in that direction in recent years

Students are expected to be able to produce coherent proofs in euclidean geometry

Not working directly from the axioms, mind you

But being able to string together facts about angles and congruence and parallel lines and the like is needed

If one takes A level further maths then you'll see induction proofs and possibly proof by contradiction depending on exam board

Some go up to proving Cayley's theorem in group theory, for example

But yeah there is a big focus in the curriculum on "making coherent mathematical arguments" but these arguments don't necessarily constitute a formal proof in a mathematical sense

[long pelican]

Ya, euclidean geometry and proof by induction are, on paper, two things I can think of in the US curriculum, but students still graduate mystified by the chain "There exists k such that n = 2k" -> you can write "Let k be such that n = 2k" next in the proof

[zenith slate]

Yeah I think this would also trip up UK students

Like it's not a massive barrier and can be explained fairly quickly

I'd probably fall back on a hand-wavy justification along the lines of "this proof relies on the fact that n is an even number so we want to rewrite it in a way that really highlights its even-ness"

[long pelican]

It's funny because anyone who sees that a student has this specific gap would infer that the student has some preparatory work to do in logic before learning mathematical induction, but instead we just force-feed them mathematical induction to be forever understood incompletely because of it

Actually that particular chain also comes up in things like the zero factor property in algebra

I think

I mean make no mistake, logical basics are a prerequisite to everything in math for sure. Just that the curriculum kind of designs everything around not needing it for some odd reason

[zenith slate]

As an aside my favorite proof of the irrationality of sqrt2 is to assume it is rational and consider the smallest n such that n * sqrt(2) is an integer. Then set m = n * (sqrt(2) - 1). This gives 0 < m < n for m an integer. But m * sqrt(2) is also an integer. Contradiction!

[zenith slate]

I don't think I would make this inference

Like I have seen plenty of little hacks used in proofs that I only really understand the reasoning behind once I've seen it used and applied enough times

[long pelican]

I find it hard to call "Let k be such that n = 2k" a little hack rather than a fundamental technique

[zenith slate]

Yeah but kids know what even numbers are

This is just a reframing of what it means to be even in a way that is a little novel if you've not seen it done before

[long pelican]

Oh I'm specifically referring to "there exists" -> "Let...", not the definition of even

[zenith slate]

Right

I mean I still think that these little logical steps are something that can quite happily be learned "on the job"

The problem with teaching formal logic is that it is a layer of abstraction which I strongly suspect would be inaccessible to a lot of kids

Like I think trying to explain "there exists" -> "let" in the context of a specific instance of doing it in a proof would be much easier than trying to get kids to wrestle with quantifiers and implications and the like

[long pelican]

For sure it can be learned "on the job," I am just making the multi-step inference "gets confused about that" -> "probably hasn't seen too many proofs using that technique" -> "probably hasn't seen/grappled with too many proofs in general" -> "needs more exposure first"

[zenith slate]

Needs more exposure to what?

[long pelican]

Edited

[zenith slate]

More exposure to proofs in general?

[long pelican]

yeah

[zenith slate]

Fair

[long pelican]

Induction is pretty daunting for people who haven't been exposed to many proofs in general

Would be interesting to see research on what percentage of students graduate high school understanding it

well, what percentage of students who have it as a topic in their math classes

[zenith slate]

Sure but I think the issue with induction is that the axiom of induction is kinda subtle and tricky to wrap your head around in the first place

I don't think exposure to anything other than inductive proofs is going to prepare one for inductive proofs

[long pelican]

Ya, and it's ideal if that's the only thing you are grappling with rather that along with "How do proofs work?"

cognitive overload ;)

[zenith slate]

True true

Yeah I do think the fact that inductive proofs are generally the first kids see is silly

Proof by contradiction would be a much better starting point imo

"This thing has to be true because if it wasn't a whole bunch of other shit would break" is a fairly easy sell in my experience

Like somebody said earlier basically every kid asks why you can't divide by 0 and the reason for this is essentially a proof by contradiction so they're generally happy with this idea

[long pelican]

I agree proof by contradiction is a much better starting point

[molten urchin]

Noted, all this makes a lot of sense.

[lethal sierra]

what happened to the server vc btw?

[pallid night]

It got shut down because the server become so large that they entirely couldn't monitor safeguarding on the vc's

[pallid night]

Proof by Contradiction is standard Maths

[zenith slate]

Oh is it now

Which board?

[pallid night]

Definitely AQA

I'm pretty sure it is technically on all of them

[zenith slate]

Hmm

I wouldn't be surprised

I'm more acquainted with the further and core maths curricula

The standard A level not so much

[pallid night]

Standard Maths has a fair amount of proof.

Notably contradiction, counterexamples, implications, equivalences and direct proofs.

Some of the counterexample proofs are very awkward.

That being said, this is definitely not generally a main topic of focus.

They do focus a lot on making people give commentary of what they are doing , and that is something I entirely agree with.

[pallid night]

Opinions on pedagogical benefits of a problem like this ?

[burnt vessel]

Ama Dablam

[pallid night]

Without calculating any derivatives, find the nature of the critical points of the function $f(x,y) = x^2 + sin(y)$

[burnt vessel]

Ama Dablam

[tawdry venture]

turning points on a function of two variables...?

[pallid night]

That's why I had to resend it.

It was too late to unsend the first TeX rendering 😔

I will forever curse TeXit bot for not allowing the delete option for longer 🙃

[pallid night]

turning point on a function of two variables ?

[turbid zenith]

I do like it. It’s a good way to potentially have students think spatially instead of algebraically

I would totally ask something like this, have students guess what it’ll look like, then go to math3d.org to see if they were right

. . . I may actually steal this problem

[pallid night]

💪😁

[pallid night]

Yeah this is the aim

Especially given how much of a challenge thinking spatially in 3D can be. Building some spatial intuition would likely help students sense check answers that they reach after a bunch of algebraic bashing.

[cloud zealot]

https://math.berkeley.edu/~wu/

anyone have thoughts on hung hsi-wu's books for k-12 math teachers?

[cloud zealot]

https://math.berkeley.edu/~wu/AE2020A.pdf
https://math.berkeley.edu/~wu/teacher-education.pdf
https://math.berkeley.edu/~wu/lecture.pdf
https://math.berkeley.edu/~wu/wu1999.pdf
https://math.berkeley.edu/~wu/ICMtalk.pdf

i found these articles very interesting

[vagrant meadow]

curious to get some thoughts on something. in the context of a linear algebra+ODEs course, how would you introduce/teach constant coefficient linear diff eqs? specifically, I'm not sure how to order the following concepts.

• linear differential operators
• superposition/homogeneous+particular solutions
• the characteristic polynomial
• matrix systems (x'=Ax)
  I'm worried going in this exact order would start too abstractly. I guess mostly I'm not sure when I should start bringing in the operator/linear algebra perspective. like it's nice to know that finding homogeneous solutions is still just finding a basis for the kernel of an operator, but that seems too abstract to start out for such a concrete topic to me. but I also don't want to wait until the very end to mention it, because I think it can make a lot of the motivation for the methods more intuitive.

[devout fable]

its doesnt sound abstract at all to me

and i have basically zero exposure with ode

the only exposure i have is running a brownian motion to solve a pde

[pallid night]

It's the linear algebra that's cause for concern about abstraction

[pallid night]

Characteristic Polynomial definitely before Matrix systems

I think this order is fine if there's an introductory lecture where you speed run what the aim of the course is and spell out how the linear algebra links to matrix systems and differential equations

[fallow berry]

When doing the linear algebra part I think doing a side by side column comparison of eigenvectors vs eigenfunctions is a good way

Because you want to extract as much intuition as possible. Linear algebra is a very intuitive subject so you want to invoke that intuition as much as possible

In terms of the order, I think getting comfortable with the concept of a differential equation is important before introducing linear comparison

I say that because not all differential equations are linear so you don’t want to introduce false hope that all differential equations can rely on linear algebra to solve all the time

But the types of problems we can solve analytically have nice linear forms which we can utilise linear algebra for. For me that’s the main point I think.

But in order to be able to understand that point you need to know what a DE is and what a solution is

[tepid smelt]

I have heard good things and it's been brought up before. The issue is at least at every public school I have taught you are given a curriculum to follow so you don't get much choice. I wish the books were cheaper or I could preview it.

I'm curious though I have a lot of freedom teaching a summer class on algebra to advanced middle schollers. I wonder what order would you choose for topics given 6 weeks and 3 sessions a week for about 3 in a half hours. I generally have started with general functions and moved into linear equations/systems/inequality then exponentials and; ending with quadratics. It just has to cover algebra 1 topics. I generally assign similar problems to the algebra book by aops along with whatever topic we cover using Alcumus for HW. The kids are strong and answer hundreds of problems on Alcumus along with weekly written HW and projects so any ideas on how to push them I'm open to. This is generally their first exposure to high school math.

[near oriole]

can bad teaching at an undergrad+ level be good? A lot of people are forced to self learn this way

[normal aurora]

[near oriole]

yh idk. if ur overly good with some stuff that might mean your students don't gain as much independence

but i dont want good teaching to be bad

[twin shell]

good teaching imo really revolves around making the content interesting enough for your students to want to pursue studying it

which is as hard as it sounds, but that's the goal i'd strive for

[zenith slate]

Teaching poorly to "make your students more independent" is a horrendous idea

[near oriole]

Of course. My point was more so the converse.

That having a bad teacher might not always be so bad

in terms of results

But I think I very much agree with what Bladewood said

[zenith slate]

I'd need to see some serious evidence to back that up

Bad teaching will bring grades down and this'll be felt more by less able students

But yes making content engaging is a good thing

[near oriole]

My thought process, is for example - if you've seen sloppy notation before, would you

• appreciate better notation when you see it
• better understand other sloppy notation
• be able to/better at creating your own notation

Maybe this is completely untrue... but idk.....

Like, if you've only ever seen good notation, you probably won't appreciate/realise that is good, and won't understand as much why it is?

[zenith slate]

I don't really see what you're getting at here

Bad notation is bad

Being forced to deal with bad notation is bad

Making your own notation is seldom a good idea as an undergraduate and you're much better off sticking to standard convention

Convention exists for a reason and deviating from it - even if it is 'bad' - is going to cause more headaches than it solves

[near oriole]

By the time you hit undergrad, I think there rarely is "one" convention for most things you are exploring

If you check out different texts

[zenith slate]

Sure

There are often multiple ways of denoting the same thing

That doesn't make it a good idea to invent another way

[tardy ember]

i think "maybe bad teaching is good" is kind of bad terminology
if there's some class of teaching methods that might actually cause good outcomes then you need to come up with a better name for it

[shadow flower]

i am still a believer that we should notate the limit as f tends to infinity as f(infinity)

[twin shell]

feel free to do that in your own notes

I use nonstandard abbreviations all the time as long as I don’t have to explain it to someone

[winged urchin]

I guess that might lead one to believe that the limit as f tends to any number x can be written f(x)

[late schooner]

what are some self-contained, 'nice' results that would be sufficiently accessible (when built up) to high school students for workshops that are a few hours long?

more asking out of curiosity than necessity. things i've done in the past which have worked well:

• diagonalisation and countability (suitable even for primary school; can adjust upwards)
• combinatorial invariance, e.g. tiling, processes, games (chessboard tiling)
• hamel bases and non-linear additive functions for tiling a rectangle (a la dehn invariant)
• a taste of variational calculus through the brachistochrone
• error-correcting codes
• cool applications of IVT, including basic chaos theory (period 3 implies chaos)

recently i've been thinking about introducing some basic analytic number theory (i.e. exposure to what simple bounding/density arguments can accomplish), or dirichlet approximation (leading to the cool result that any string of digits can begin a power of 2)

[wide ice]

diseases build immune system. Doesn't mean diseases are good.

[near oriole]

growing up in a greenhouse

[wide ice]

my thought process is: learning is hard. Good teaching should therefore make learning feel easy. Doesn't mean learning can be easy, sometimes things are fundamentally hard, but students should feel assured that they are guided, that there's an adult in the room to accompany them.

[near oriole]

anyways idk, its self-fulfilling in all likeliness - whether u try to avoid it or not. No one can learn math without eventually running into not-so-good teaching

[wide ice]

This is similar to "no one can appreciate life without eventually running into death"

Of course not-so-good teaching exists, and it's unavoidable. Doesn't mean it should be advocated or not to be removed

[near oriole]

in terms of construtivity, my point (if true) can only be more practically realised by advocating a more independent approach with teaching but eh

as in getting ppl to figure things out more on their own, but this isn't suited for everyone... idk

[near oriole]

This musing stemmed out of nowhere in particular

[wide ice]

think of it this way: I'm saying good teaching is letting a student struggle by themselves a bit while learning how to swim

what you're saying is basically throwing them into a sea, and either they learn how to swim, or they drown

[near oriole]

yeah and maybe the ones who dont come out stronger. ok this is terrible

[wide ice]

did I figure things out on my own? Yes. Do I expect everyone should be able to? Nah

I did because it was life-or-death matter

[near oriole]

but that is what happens in those classes with notsogood teaching

hmm

[wide ice]

either I figured it out, or I failed the class, got kicked out of the school, and lived under a bridge

[near oriole]

i just have anecdotal memory of good outcomes in a few of such cases though

like students banding up and teaching each other

that kinda thing

i also have plenty of bad experiences remembered too ofc

[wide ice]

not out of "love for the subject" most of the time, but only because the teacher sucked, from my experience

I remember learning measure theory in one week just to understand Borel-Cantelli lemma because the prof assumed we knew measure theory. I survived, but many didn't.

[near oriole]

On another note, the way the system is rn. It seems the 'norm' you get taught each thing by at most one person.
If you get more from other means, you might consider yourself lucky.
But this puts the burden of good teaching on one person. And the same teaching that suits some might not suit others

Just an observation.

[wide ice]

Lmao, I am about to take Algebra, Topology, Differential Calc, and Measure theory again, and I don't find myself lucky. Annoyed even: I passed the courses, and yet I am not allowed to proceed on something harder.

Of course at the end of the day, profs are not saints: they can only do so much, and there are always someone who struggle.

But that's different from the profs being inherently bad tho

[near oriole]

I haven't figured "the" way that best suits myself to learning tbh. I gather uni is meant to transition you more towards self-learning, because that's how it will be on research but I just haven't picked that up

I've still to go through the experience of picking up an entire subject from reading a text

My learning speed really stagnated past hs

[wide ice]

it's normal. Hilbert was also very, very slow when it came to learning. When he wanted to learn about quantum mechanics, he asked a student of his to read the papers, then explained to him very, very slowly. (ref: Constance Reid, Hilbert)

[near oriole]

what do you think about gaining intuition:
is it better if the teacher gives it to you, or for you to come up with it on your own

Like. In my experience, it takes ages to come up with it on my own, but its probably an essential skill worth developing.

I dont know if being lazy and googling "intuition for ???" has been bad for me in the long run

[wide ice]

there are two things here: intuitions, and the skill to come up with intuition.

[near oriole]

When I explain something to someone, I want to just feed them my intuition thinking it'll save a lot of time, but I think about this question after.

Or maybe I think after a good while (probably near the end of a course): wait, why couldn't it just have been explained this way to me.

[wide ice]

it's good that teacher gives it to you initially, especially at the beginning when you have no idea about the subject. Terence Tao said himself something along the line of "undergrad is about making good intutions and destroying the bad ones"

But at the same time, you are expected to pick up the process of coming up with intuition. Good researchers are fluent at this (not good students tho, not always). And no one can teach you this: you see profs doing that enough times, wonder how they do it long enough, then it'll come. Hopefully.

[near oriole]

One example I can think of is basis for topological space. I remember the more practical definition being introduced:

1. every point is in some basis element
2. every point in the intersection of 2 basis elements is contained in a basis element within the intersection

This is how it was introduced to me each time I learn it (twice) afair. But to me, this is far less intuitive and harder to remember than just wanting your basis to generate your topology under unions.

I just couldnt understand why would anyone want to introduce it this way, even if it is the defn used more in practice in proofs.

[wide ice]

I have an intuition of that in my head, by coming back to canonical topology on R^n. But tbh, idk how others manage.

[zenith slate]

I've done this hundreds of times and see no reason to believe it has harmed me

[pastel horizon]

I am looking to design a baseline test for an HNC course. Are there any "challenging" question ideas I can use to try and differentiate the strong students

Not like Oxbridge level btw

I'm thinking of perhaps giving some sort of unseen topic before with a guide and see how well they apply the rules of the guide

[wide ice]

HNC?

[pastel horizon]

Errr like a degree but not quite

[pallid night]

What topics are in the baseline test ?

[long pelican]

Past contest problems (there's probably tens of thousands of those available on the internet to draw from) and AoPS book exercises are where I'd start looking to get inspiration, then I'd make problems based on similar ideas

@pastel horizon

[pallid night]

If you want to do this you could guide them through finding the centre of mass of a 3D object with constant density.

Assuming they are expected to have covered basic calculus that is

That would be quite tough though

[pastel horizon]

It might work as an applied maths course

[hot pagoda]

Why don't university seem to teach non-standard analysis anymore? It was quite a curious thing. I found it out while goofying around the internet.

[twin shell]

did they ever teach it?

[wispy slate]

Do you use any cognitive theories for guiding how you teach (say constructivism)?

[plain pebble]

Because it’s harder to put on a fully formal footing iirc

[pallid night]

I think this

I expect it's also not as straightforward or as widely useful elsewhere in an undergraduate curriculum either

[hot pagoda]

oof. Yeah it make sense it needs tons of abstract algebra and rings. I will probably take the time and know more about them. Cause it's cool to know that maybe engineering way of doing calculus may be legit after all.

[zenith slate]

To an extent

I am generally wary of pedagogs who lean too heavily into a particular theory

Like this can lead to some quite "extreme" styles of teaching being advocated for

An obvious example being "pure" inquiry-based learning which is largely a result of people taking constructivism and running with it to a somewhat absurd conclusion

[wispy slate]

Yes

Do you think there are types of learning ? (visual, auditory...)

And, do you think there are types of thinkers, if that were different, that mostly think of images, for example ?

[zenith slate]

No, this has been thoroughly debunked in the literature

[zenith slate]

This I have not given much thought to though it seems plausible to me

[wispy slate]

yes, I think some of us tend to confuse those two

[zenith slate]

I don't really see how it might be used in a classroom environment, though

[wispy slate]

I think some people are visual learners in the sense that they produce images, not that they learn from images

[zenith slate]

The problem with that type of thing is that for it to be of use you'd need some way of not only classifying the different "types" of "thinker" but also identifying each type in a classroom environment

[wispy slate]

their models of ideas are imagery

[zenith slate]

Yes, visual intuitions are important to some people and not to others

[wispy slate]

yes, i know that i was rather slow down by this, in some areas

( but in chemistry it was useful )

[zenith slate]

I mean the general consensus is that a multimodal approach is best

[wispy slate]

yes, makes sense

[twin shell]

fyi there are people who literally cannot visualize anything in their heads https://en.wikipedia.org/wiki/Aphantasia
doesn't mean you can't get gud tho

[zenith slate]

I am one of them

[wispy slate]

yes, i'm aware of it

[zenith slate]

Total aphantasia for me

[wispy slate]

i was watching huberman talking about that, also about synesthesia

when did you find out @zenith slate ?

and how

[zenith slate]

A few years ago

[wispy slate]

can't you imagine a face, for example ?

[zenith slate]

I think I saw a post about it on reddit to the effect of "TIL some people don't have a 'minds eye' and can't visualise things in their heads"

To which I was like wait what

I always assumed "visualising" things was a metaphor or something

[zenith slate]

Nope

[wispy slate]

when you remember your parents or dogs, what appears your mind ? I mean, not literally but perceptually

in my head, is their face, even for my old dog, that's dead now

[zenith slate]

Very difficult to put to words

But there is no visual information there

[wispy slate]

interesting

[zenith slate]

There is a lot of information but none of it anything you could call "visual"

[long pelican]

I first read about aphantasia in the context of visualizing a 3x3 grid of numbers in one's head. So @zenith slate do you have trouble visualizing 3x3 grids of numbers?

[zenith slate]

Moreso "spatial"

[long pelican]

More specifically, let's say you put 1-9 in reading order. If you had to find what the middle column is, would you have to resort to thinking logically?

[zenith slate]

Like, for example, if you move your hand behind your head to where you cannot see it, you have this deep feeling of awareness of where your hand is in space

You can't see out of your hand nor can you see your hand but you understand viscerally where it is

This is the same sensation I get when remembering somebody I know

It is, I suspect, impossible to explain

[zenith slate]

Yes

More-or-less

[long pelican]

Nice

If you ever encounter commutative diagrams in math, those must suck as well

[zenith slate]

I can imagine the numbers as being distributed in space in some ordered manner

But I can't "see" anything

Like if I imagine a die with numbers on each side and imagine rolling it around

I can quite happily """visualise""" what number is facing upward

But again it is not visual

It is more that I can imagine the die as being a physical entity over which I have an awareness of its location and orientation

[long pelican]

Are your dreams entirely un-visual?

[zenith slate]

Interestingly they are visual

This is very common in aphantasia

I don't know what the mechanism is

[long pelican]

So you'd dream visually but when you remember the dream while awake, it's entirely nonvisual?

Interesting stuff

[zenith slate]

yup

[wispy slate]

what about montessori method

is that a common read for teachers ?

[zenith slate]

I teach ages 11-18 so not really

I am aware of it

I am also sceptical of any educational method that has only really been tested on the kids of wealthy white folks

But other than that I don't know anything about it

[wispy slate]

i see

do you know tom sawyer ?

[zenith slate]

only the rush song

[wispy slate]

i think that is the best book on teaching ever, but i am not sure how would a book a suits good to my visual head would be to you, and viceversa

probably there isn't an obvious way to know

sorry, ww sawyer lol

[zenith slate]

which book

[wispy slate]

for example https://www.amazon.com/search-pattern-His-Introducing-mathematics/dp/0140211020/ref=sr_1_3?crid=1QD7FJXWP40ZT&keywords=ww+sawyer&qid=1694918833&sprefix=ww+sawyer%2Caps%2C172&sr=8-3

he's the author of prelude to mathematics and mathematicians delight as well

[zenith slate]

I see

[wispy slate]

this is the only comment, not mine:

Sawyer is the kind of math teacher you wish you had. Recommended for insight and motivation to middle school mathematics

however, it is quite visual imho.

[long pelican]

Every non-bad math teacher gets that kind of comment by someone (look at Professor Leonard's comments on youtube for the majority of those), so it's not that much information tbh

[zenith slate]

true

[wispy slate]

well, i can say i've read many books at least

[long pelican]

Oooo this book is libgen-able

I feel like skimming it

[wispy slate]

nice

[long pelican]

The first group is very
high-powered; it consists of mathematicians themselves and of
research workers in fields such as physics and the more abstruse
parts of engineering, which are so permeated by mathematics as
to be in effect branches of mathematics. It seems unlikely that
any of this group (except perhaps at a very tender age) will be
readers of a series entitled Introducing Mathematics;

Look at me go

[wispy slate]

It's techniques for teaching

[long pelican]

The irony is I'm in the group and I'm currently reading it

[wispy slate]

maybe, but i don't think in terms of how to teach anyone is guaranteed to be good, anyways; clearly in terms of material to learn, though.

[long pelican]

I'm not making any point whatsoever, just saying something funny

[wispy slate]

understood

[long pelican]

ok I skimmed it, but unfortunately I didn't think of any conclusions or judgment from my skimming

I googled Tom Sawyer though, and it turns out he's a fictional Mark Twain character. The author's first name begins with W!

(Mark Twain was who I thought of when you said Tom Sawyer, then later messages made me doubt myself)

[wispy slate]

yes

that character draws a counting system, i think

story is told somewhere in the first pages

[long pelican]

I like this excerpt

It is often a little difficulty of communication; some symbol or sentence suggests to the pupil a meaning different from what the author intended, and progress is halted until this misunderstanding has been cleared up. It is almost impossible for an author to foresee these minor misunderstandings. This, perhaps, is one reason why teaching machines are unlikely to replace teachers. A word from a well-informed friend is often the quickest way to resolve such a difficulty. When no such help is available, one can try reading the same topic in a variety of books; there is a chance that one of the books will provide the needed clue.
Holds true for everyone of all ages, including me reading textbooks and research papers

[wispy slate]

nice, yes. some teachers use peer instruction (can't remember how it is called now.)

im currently stuck with variations principle, and can't quite find good sources

it's interesting to see how we/others react when getting stuck

certainly i do not ask in stackexchange sites for a pedagogic answer

[cloud zealot]

@zenith slate have you ever looked into hung-hsi wu's work? do you have any thoughts about his work?

https://math.berkeley.edu/~wu/

[zenith slate]

I've heard of their work but have never looked into it

AFAIK they write predominantly about the US math curriculum

I teach in the UK so it has very little relevance to my own teaching

[tawny slate]

i have some questions regarding inequalities

(assume all variables are real)

If a>b, and b>c, then a>c. (transitive property)

as far as i know, without going into really obtuse foundations that are not appropriate for middle school level, this is axiomatic, correct?

next

If a>b, then a+c>b+c is also not really provable, you can intuit it but it's also axiomatic "at this level", right?

the last one ive been having a little bit of issue with is

If a>b>0, n>0, then a^n>b^n.

really not sure how you would prove this using first principles in any way that is clean and easy to follow

[robust glade]

I think you get into the wrong channel. #❓how-to-get-help

[tawny slate]

i'm not sure this is the wrong channel

this is a question not about rigorous foundations but about how to teach inequalities

specifically at the middle school level

[robust glade]

Ah my fault sorry for that

[tawny slate]

of those three questions, I think the first two I'm mostly just sanity checking or seeing if someone had some ingenius method i never thought of

the third one is the one im most interested in

im even happy if n was loosened to be a rational number instead of real, because rationals are dense anyways

[robust glade]

I reall there is a factorization of a^n-b^n, like a^n-b^n=(a-b)(a^{n-1}+something+b^{n_1})

This is purely algebraic operation so I think it should not be too hard to follow.

[long pelican]

For the third one, repeatedly apply the property that if a>b and c>d and everything is positive then ac > bd

This gets integer exponents

If you prove the same thing for <= instead of > then use the contrapositive, you get reciprocal integer exponents

Combining gets rationals

[long pelican]

This is probably clear as mud so I'll expand:
Prove $(0<a\leq b\text{ and }n\in\bN)\implies a^n\leq b^n)$.

The contrapositive of this is $a,b>0\text{ and } n\in\bN\text{ and }a^n>b^n\implies a>b$.

Apply that to $a^{1/n}$ and $b^{1/n}$ to get the desired result for exponents of the form $1/n$

[robust glade]

For the first one, a>b and b>c ---> a>c. Transitivity is assumed axiomatic even considering more foundational ideas. Transitivity is an important characteristic of binary relations, and it is the reason we have interesting results about numbers.

[burnt vessel]

Icy0

[tawny slate]

and then i can use a limit argument to extend it to reals

brilliant! thanks so much

that's pretty clean

this has been bothering me for a while when im teaching inequalities

glad to finally have resolved it

[tawdry venture]

this is a minor grievance in the grand scheme of things but

does anybody know why some schools/places/etc. teach people to simplify $\ln(e^x)$ by applying log laws to make it into $x \ln(e)$, as opposed to recognizing $\ln(e^x)=x$ as the definition of log?

[burnt vessel]

Ann

[plain pebble]

What’s the point of that?
Because you then immediately see ln e = 1 and it falls out

[tawdry venture]

yeah but it's still an extra step that to my eyes looks unnecessary

[long pelican]

I’ve had the exact same thought numerous times before, and it’s unfortunately a small example in the larger pattern of the whole of math education doing a disservice to logical reasoning, probably because they don’t trust students to be able to do logical thinking. One of Hung-Hsi Wu’s fundamental points actually!

[timid topaz]

Assuming they know the sum of positive numbers is positive, you can just say if b - a > 0 and c - b > 0 then c - a = (c - b) + (b - a) > 0, I think this is at least understandable for a middle schooler

[tawny slate]

You have some typos there but I get what you're saying and I like this a lot, thanks

[timid topaz]

fixed, lol sorry about that

[timid topaz]

The simplest argument I can think of is that 0 < b < a => 0 < (b / a) < 1 so 0 < (b / a)^n < 1 => 0 < b^n < a^n. The only thing you'd have to argue is that if you have 0 < x < 1 then 0 < x^n < 1 for all n, which you could explain as "multiplying a number by another number less than 1 makes it smaller, so if 0 < x < 1 then x^2 < x < 1, x^3 < x < 1, etc"

[tawny slate]

ah, that's pretty clever, though it also has a lot of limitations and issues. I want to prove this for all positive n, so the thing you mentioned that I have to explain is actually slightly trickier

but nonetheless that's very good, i like that a lot and that approach has its applications

thanks again

[timid topaz]

np, I think the only thing you couldn't prove rigorously is that it's true for all n, since that would probably require induction, you'd sort of have to hand wave that part

but imo that's a fine thing to hand wave lol

[tawny slate]

I agree that the schools are doing a disservice not teaching it this way, and i think the reason why the schools teach like this follows a certain commonality. once the schools decide what concepts are important, they try to minimize the number of concepts required for the students to solve a problem, whereas we try to maximize the number of concepts learned to provide the students with the best tools

that particular method is easy to intuit because it's simple "symbol manipulation", the rule is more syntactic than it is semantic, making it easy to explain the rule and how to use it. obv in this case i think it's silly but i think that's the trend with public schooling

[zenith slate]

Interesting

Over here we teach both "methods" for simplifying logarithms

Generally I think using and understanding multiple different methods for reaching the same result is good and important

[zenith slate]

I don't see this as being particularly antithetical to "logical reasoning"

You could make the argument that the method Ann mentioned requires more logical reasoning since it requires additional steps and involves applying a more general result to simplify

I am extremely wary of making broad condemnations against mathematical education on the basis of a single choice made in teaching a thing

For example, following Ann's method allows students to tackle problems like ln (3e)^x in a way that is arguably more straightforward than expanding to ln (3^x e^x) etc

[long pelican]

I think you're missing that the property ln(e^x) = x is the first definition in the development of the definition of logarithm, from which all the other properties (power rule included) follow

[zenith slate]

Not necessarily

[long pelican]

What do you mean? The definition of logarithm is the inverse function to exponential, and f and g are inverse functions iff f\circ g and g\circ f are the identity

A similar phenomenon (and I don't know if this happens as often, but I suspect it does) is: why is -a + a = 0? Oh, easy, add a to both sides, we get a = a, and that's an equality.

The real reason is that -a + a = 0 is the definition of -a in the first place

[zenith slate]

One can equally define log_a as the unique increasing function on (0,infty) such that f(a) = 1 and f(xy) = f(x) + f(y) for all x, y

There is no "correct" characterisation

[long pelican]

Is that what appears in UK textbooks?

I'd be shocked if so

I'm under the assumption the working definition all students are under is the inverse function one

[zenith slate]

One can argue that the fundamental property of a logarithm isn't that it inverts exponential functions but that it "turns products into sums" and vice versa

It is not in UK textbooks no

I am simply pointing out that the fact that log is the inverse of exp is not necessarily the only way to learn about and use them

[long pelican]

Log is actually redefined as an integral in calculus class for "no reason"

(the reason being that the theory is easier to develop from that direction using what the students know)

But this is a digression

Do you agree the roundabout justification is weird, under the development of the theory of logarithm and their class is working with?

[zenith slate]

I'm under the assumption that what Ann was implying is that US students or whomever aren't taught about the fact that log is the inverse of exp

Or have I misunderstood

[long pelican]

Ehh it's in all textbooks

[zenith slate]

Oh

Then what's the issue

[long pelican]

Well being in textbooks doesn't imply students will learn it properly

[zenith slate]

Right

[long pelican]

Just pulled a section from libretexts as an example

[long pelican]

Also doesn't imply teachers learned it properly either 😔

[zenith slate]

Yeah this is basically identical to UK texts

OK well disregard what I was saying I was trying to justify why one might not initially teach logarithms as the inverse of exponentials

Lmao

[long pelican]

Lol

[zenith slate]

I certainly don't teach it that way myself

I prefer to present both characterisations

[long pelican]

Do you feel UK classes in practice also suffer from too much emphasis on procedures?

[zenith slate]

No

[long pelican]

Interesting, because the couple of UK (non-math) friends I have seem to confirm that

[zenith slate]

Do not mistake me for saying that there is not a massive emphasis on procedure

There is

But I think this is a good thing

Students who are mathematically inclined likely disagree strongly

But I am an educator which means I am not tasked with only teaching the mathematically inclined

[long pelican]

Thinking that non-mathematically inclined students can only learn a shell of what math is, namely, procedures, is a big misconception though

[zenith slate]

I'm not saying that they can't

Only that it isn't a priority

For context I teach at a rural school in very a deprived area. Most students here are not going into STEM. Hell, most aren't even going to University. A staggering proportion of our students have specific learning difficulties and the single most important thing I can do for them as a teacher is to set them up for success so they can achieve a passing grade in their maths GCSE

My job is not solely to create mathematically insightful students (though I do emphasise this with groups and students for whom it is appropriate) it is to equip students with the basic skills needed to get the GCSEs that their parents, in many cases, didn't get

It would be great if I could instil a love of Mathematics and insight into its structure or beauty or whatever into all of my students but this, plainly, is a pipe dream

Leaning too heavily into this simply disadvantages students who are already disadvantaged

[long pelican]

Is the GCSE part more or less a goal decided for you from up above?

[zenith slate]

It is an important metric yes

Though I should note that the actual metric isn't X number of students getting their GCSE

[long pelican]

That metric is already problematic, the pressures you just communicated are the same pressures most teachers face in USA thanks to No Child Left Behind (and its successors)

[zenith slate]

Rather it is how many students get X grade versus what the cohort data predicts this number should be

The metric attempts to measure improvement as opposed to raw attainment

I do not see it as problematic

[long pelican]

There's been a lot of discussion about it in USA

[zenith slate]

These are not values that i have been inflicted with from the powers that be

I wholeheartedly agree with the stated aims of the curriculum

[long pelican]

I think the general consensus from what I've read is that test-based metrics, including improvement based metrics, have been debunked

[zenith slate]

Debunked how

Do you mean to say they are unsuccessful at achieving their stated goals?

Like what do you mean by "debunk" a metric

[long pelican]

I'm gonna try to find something concrete to link

It's been a while

[zenith slate]

I have seen the arguments that these types of metrics have the effect of narrowing the attainment gap by essentially reducing the grades achieved at the upper end of things

I certainly see this as something to be wary of

A big part of my teaching practice is adapting my teaching to the strengths and weaknesses of my students

The aim is to elevate both the higher and lower attained simultaneously

How good am I at doing this? Idk, I'll let you know in a decade lmao

[long pelican]

I found something

2003 to 2018 USA PISA score has been flat despite NCLB dramatically increasing pressures to teach in a way that maximizes scores on standardized tests

(Link) [page 30]

You might wonder about plausible explanations of this, I'll mention that a bit over a decade ago (at which point NCLB had been in effect for 10 years) I was still in high school and I got to watch how students help other students with homework, I got to see math teachers at my school teach, I got to see how math teachers explain things, and based on everything I saw, it was already abundantly clear to me then that students were pretty good at passing tests (class tests and state standardized tests) without being good at math

[zenith slate]

I don't think this is necessarily evidence of why tests and attainment-based metrics can't be effective

I think teachers and educators get a lot of flak for their teaching methods and, in particular, teaching to the test

Like there is a very important variable that is often missed in these discussions - namely what is actually on the tests

Teaching to the test seems to me an inevitable conclusion of the fact that education is formalised through standardised testing and while this continues to be the case I think we need to start holding the organisations that actually publish the tests and exams accountable for producing sub-par tests that do encourage thoughtless repetition

[long pelican]

What you're saying sounds like 99% of the difficulty of being successful with a metrics-based approach is choosing good metrics, which I agree with

[pastel horizon]

So the approach I took with the baseline test I mentioned a while back was some more problem solving style questions to try and see how they approach problems rather than what they know.

I even had a GCSE question lol, which is what a 16 year old would sit

The question is something like
Given x (a+bx)(a-bx) = 25x-4x³ find b^(-a)

Will be interesting to share the results when I hand out the assessment

[long pelican]

b^-a looks like a typo

[zenith slate]

Yes otherwise what on earth

[pastel horizon]

Yeah -a in brackets

Made the correction

[long pelican]

There's also a y on the RHS but not on the LHS 🤔

[pastel horizon]

That should be an x

[long pelican]

and final issue, you should put "Given there exist a and b such that for all x," in front

I could do a rant on how the ubiquitous practice of leaving out quantifiers in math theorem statements and problems contributes to weaker students (here, meaning less able to guess or infer patterns) failing to understand math

[pastel horizon]

Yeah I'm not sure if the original question included that and I just forgot it

A level exams are usually well defined not sure about GCSE

[pastel horizon]

I also added the stipulation that b and a >0

[pallid night]

Could write it like this

[long pelican]

Don't forget the "for all" quantifier on x

[pallid night]

True

Show that there exist integers $a$ and $b$ such that $$x(a+bx)(a-bx)=25x-4x^3 \text{ for all } x$$

Calculate the exact value of $b^{-a}$

[burnt vessel]

Ama Dablam

[pallid night]

Could extend with a second part like this

[long pelican]

Part b) is pretty cool because it's well-formed, but at the same time not a procedural problem -- they have to think, on their own, to reduce the question to an unsolvable linear system

Also a good opportunity to practice introduce basic logic such as how to negate a quantified statement

[pallid night]

Part b has at least two approaches
One of which is a one liner

[long pelican]

I see the 1-liner, since you said integers, you can simply plug in x = 0 to reduce the equation to a^2 = 2

[pallid night]

That's not the one liner I was thinking of actually

But that's very neat.

My one was plug in x = -1
Then LHS is 0 and the RHS is not

[long pelican]

oh man that's even nicer

[pallid night]

For a baseline test this kind of question gives an insight into how the student thinks

[long pelican]

Probably the most common 2 responses will be not being able to start the problem, or doing something logically wrong such as misunderstanding what the question is saying

that's my guess

[pallid night]

Following the method of part a) leads to a contradiction

Not clear how many would find that though

In any case one could hint it's meant to be short by giving it very low weight

[long pelican]

Not saying there is anything wrong with the problem. I think more problems should be like that, not fewer, and properly explained too (i.e. logical reasoning instead of handwaving and procedure). Then there won't be so much of this kind of issue

[zenith slate]

I certainly hope this isn't question isn't intended to be an introduction to logical negation

In any case, what is the "target audience" for these questions?

[long pelican]

Where is logical negation in the math curriculum? Actually is proof part of the UK math curriculum outside of geometry?

State standards here have 14 instances of "proof" but all related to geometry (specifically high school geometry), so it's a no for the US

[zenith slate]

Yes we do teach proof

Though not in a particularly formal setting

I am currently teaching what is called in the curriculum "algebraic proof" to my Year 11 (15-16) class

[long pelican]

Is the extent of proof in that "algebraic proof" class simply chaining equalities?

That would be in alignment with the precalculus trig identity proofs unit in the United States

[zenith slate]

It's a topic that spans several grade levels so covers quite a lot in terms of difficulty

For example, the simplest question in the unit would be something like "prove that the sum of two odd numbers is even"

[long pelican]

Oh yeah, I am not at all in favor of teaching "formal proofs" or "formal logic" at all. Actually, what I would like to see is proof being integrated into every single piece of content, which is much better than having proofs as a separate topic as is done now

[zenith slate]

Whereas a question I set for my top students was "prove that at most one of the numbers in a pythagorean triple can be even"

[zenith slate]

Yeah big agree from me

We do have a lot of proof scattered throughout the curriculum which I think is good

But it is mostly restricted to geometry

AFAIK this is similar in the US

[long pelican]

It's precisely the same yep

[zenith slate]

Like I think there are a lot of places where "informal" proof can be naturally included but often isn't

I certainly think these skills could and probably should be developed alongside algebraic manipulation skills

So I do like this "algebraic proof" thing we're doing at the minute

Like sure it seems basic and mostly revolves around proving whether things are even or odd

But there is a lot of subtlety to be explored even with this

[long pelican]

Ohh, even/odd proofs are what you call "algebraic" -- there's none of that in the US

[zenith slate]

Right

Yeah the "algebraic proof" unit i'm doing at the moment covers even/odd proofs mostly

It does include some equality chasing and some geometric arguments rephrased in algebraic terms

But they are expected to understand how proof by contradiction works

[long pelican]

It's really good that your students get to grapple with how to use the hypothesis "there exists k such that..." and how to use multiple such hypotheses and understand why you need to use independent variables for each, and so on, well before university

[zenith slate]

Hang on no I'm getting muddled up we teach this at A level (usually near the start of Year 12)

[long pelican]

ok so that's pretty late

[zenith slate]

Yes

Algebraic proof is toward the end of Year 10

Mind you this is not necessarily representative of the UK as a whole

This is just how we sequence the content for this particular exam board

While all exam boards teach basically the same material they all do it in slightly different orders (and slightly different ways)

But yes "there exists" and "for all" have been surprisingly difficult for students to pick up

It is my first time teaching this material so it has been a learning experience for me lmao

[long pelican]

Very cool!

[zenith slate]

One of the biggest hurdles was convincing them as to why a single counterexample is sufficient to disprove a "for all" statement

Which, like, was kinda surprising to me

Since to me this seems trivially true

[long pelican]

That's not surprising to me now but would have been in the past

Although I'm still not completely sure why it is

One guess is that they believe "Exceptions don't disprove the rule" from real life

[zenith slate]

Perhaps

I couldn't quite get to the bottom of it

The example I used was the statement "all prime numbers are odd"

For which 2 is the counterexample

I think what was happening in that case was that students were like "ok well if you just ignore 2 then it's still true"

Which is obviously correct and valid but they seemed to struggle with making the link back to this fact disproving the original statement

In cases where counterexamples may not be so obvious I think this problem is compounded

[long pelican]

Something very related is my observation that very few students actively understand that you can show an algebraic rule cannot be right by plugging in a number

[zenith slate]

Yes

That's a good one

[long pelican]

This could be avoided if the curriculum was designed so that they properly learned algebraic rules are "for all" statements as well as what counterexamples are very early on

and would benefit weaker students a lot, since the stronger ones are the ones who can infer these things or learn them from outside school

[zenith slate]

I think the biggest difficulty with teaching "disproof by counterexample" is that finding counterexamples is oftentimes very hard

And there isn't really a general logical procedure one can follow

[long pelican]

Mm-hmm, counterexample-finding is guess and check but somehow still 100% rigorous

Certainly goes against what they are taught

[zenith slate]

I think another very important lesson with counterexamples is the fact that being unable to find a counterexample is not the same as proof

[long pelican]

That's a good one

[zenith slate]

The way I explained this the other day was by demonstrating an example of a hilariously large counterexample

Statement being that a^4 + b^4 + c^4 + d^4 = e^4 has no integer solutions iirc

[long pelican]

Ah the famous one page Euler paper

[zenith slate]

Yeah

[long pelican]

Oops not by Euler, but by Lander and Parkin, about the Euler conjecture

[zenith slate]

That's the one

Lesson being "unless you can prove something is true for literally every single one of the infinite possibilities then you can't prove anything"

I think this could lead into a nice discussion about the use of contrapositives etc to make statements easier to prove but I haven't tested this

I don't think that would be accessible for most of my students

[long pelican]

You mean they can't grasp the logical equivalence of P => Q and not Q => not P?

which is contrapositives

Or that they haven't been exposed to it

[zenith slate]

Both, really

When I used to teach undergraduate I saw a lot of math undergrads struggle with contrapositives

So I don't think it would be appropriate for my mixed-ability Y11 class

[long pelican]

It's funny to me that year 11s and even undergraduates are not expected to understand contrapositives, tbh

Because they are probably used in a lot of proofs of things they had learned earlier

whose proofs are inaccessible to them, because they did not have exposure to logic

[zenith slate]

I do not think it is a matter of expectation

My undergrads were certainly expected and indeed required to understand contrapositives

Nevertheless, it is not a trivial thing to understand

[long pelican]

I'd consider it a very foundational thing for math

[zenith slate]

So do I

Doesn't mean students will necessarily be able to grasp it

[long pelican]

I know, it'll take a lot of thinking for anyone, even me

I remember a lot of times learning math in school where I had to seriously think hard about something for up to an hour to get it, contrapositives was definitely up there

the definition of limit? 3+ hours

How often does the average student get the chance to really ponder something for an hour though? Homework doesn't ask them to do that

[zenith slate]

I think the difficulty with the contrapositive is that while you can fairly easily illustrate it with simple examples

[long pelican]

That's one big difference between my experience and the average student's

[zenith slate]

Actually demonstrating why it is always logically equivalent to the original statement is kinda fiddly and honestly pretty unenlightening

[zenith slate]

When you're talking about highschool it's less a matter of having the chance to ponder and much more getting the fuckers to actually sit down and do it lmao

[long pelican]

Hard to ponder without logic in your toolset, can't blame them 😉

[zenith slate]

I mean how do you prove the equivalence without resorting to truth tables

[long pelican]

Truth tables were pretty strong for me... there's also that thought experiment about drinking age which is very intuitive to anyone

"Say you're caught drinking. If you're under 21 you are violating the law"
"Now suppose someone is caught drinking and was found not violating the law. Logically they must not be under 21"

[zenith slate]

Yeah, like I said, it is easy to illustrate

But proving it

That's the difficulty

[long pelican]

Let me see how it's proved in Lean, one sec

[zenith slate]

Truth tables are indeed very powerful but they're like two levels of abstraction above the usual math content in the curriculum and honestly you will have a large proportion of students who will genuinely never be able to grasp it in the time they have

Like heck I know the contrapositive equivalence but would struggle to actually write out the TT for it

lmao

I don't remember how that shit works

Because despite having a PhD in math I have never actually needed to sit down and write a TT since like my first year of undergrad

This does raise the question, I think, as to how worthwhile it would actually be to teach this to students

[long pelican]

Hmmmmm

Found the Lean proof, now time to understand it

[zenith slate]

[long pelican]

Oh yeah they define "not a" as "a implies false"

I think they boil it down to a proof by contradiction

[zenith slate]

interesting

[long pelican]

Me saying "I think" when the screenshot literally says byContradiction

[long pelican]

One of my opinions which is very out there is that teaching the basics of logic (this may or may not include contrapositives) and then integrating this logic and proof into mathematical content will help it make sense, especially to students for whom things like algebra make no sense and feel like a bunch of arbitrary rules. Currently we rely on handwavy intuitive explanations, or just rote, for understanding in a lot of places

I just realized I also have a lot of anecdotal evidence from helping people in #prealg-and-algebra to back this up!

[zenith slate]

Yeah I agree that logical reasoning and deduction should be a big focus of the curriculum

But I do draw the line at formal logic when it comes to highschool

[long pelican]

I mean yeah, no need for formal logic as is taught in undergraduate courses, just the logic needed to understand and comprehend mathematical proofs

[zenith slate]

sgtm

Relevant quotes from the UK national math curriculum for 13-16 year olds if they're of interest:

All pupils should reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language

Pupils should be taught to make and test conjectures about the generalisations that underlie patterns and relationships; look for proofs or counter-examples; begin to use algebra to support and construct arguments and proofs. [...] use vectors to construct geometric arguments and proofs

[long pelican]

That's also more or less the langauge used in the common core. What I think is missing is the explicit integration of these into actual content

[zenith slate]

Yeah

[long pelican]

e.g.

• the development of fractions and their rules
• the development of exponential laws
• the development of algebra
• the development of logarithm laws
• etc etc etc

[zenith slate]

Not sure how it works in the states but the national curriculum I quoted above are simply the statutory requirements that any examining body must comply with to be accredited to award GCSEs etc

As a result the actual implementation of the stated intentions of the curriculum are, to a large extent, determined by the exam boards

Some do it better than others

[long pelican]

Hm a common failing among people who use these standards (at least here) is misinterpretation of terms like "understand" and "prove" because the people involved don't have sufficient mathematical experience to know what is actually intended

For example "understand" should mean "have complete mastery of, and be able to work problems involving it", but it inevitably gets interpreted by less-mathematically-inclined educators to mean "be able to roughly explain intuitively" or simply "be able to show evidence of exposure"
Which is why you hear "I understand the concepts but I just can't do the problems!" as the #1 complaint by students

[pastel horizon]

The problem is students in primary education get told very precise terms with definitions and it seems to just get lost in high school

Like you could literally ask a ten year old "explain why 2×3=3×2" and they would give a perfect answer

[long pelican]

What's the perfect answer for that?

[pastel horizon]

They would say that because multiplication is commutative you can do the sum in either order and get the same answer

Rather than a woolly "it doesn't matter what the order is" or "2×3=6, 3×2=6"

[long pelican]

Ah yes just recently I witnessed someone helping someone on algebra and there was a whole slide about various "the _ property of equality"s

[pastel horizon]

Ironically I think not having maths specialists helps them

[long pelican]

I don't like that at all

[pastel horizon]

Well hear me out

Those non maths specialists don't know the terms initially

So they take the time to study them and understand it

And then pass that down to the kids

[long pelican]

No, I mean I don't like this at all

[pastel horizon]

Whereas a specialist might just gloss over it

Oh fair

So in other words I think because primary education doesn't have specialists they put more effort in trying to explain the concepts which I know seems like a paradox

[pastel horizon]

Well I saw a post on quora that completely changed how I saw equalities

[long pelican]

What I don't like is that textbook publishers don't understand what is meant by "use precise language and logical reasoning" and think that it merely means using the "correct terms" for things

[pastel horizon]

So the initial stipulation is you have

f(x) = f(x)

And to solve x you are applying functions to both sides that still preserves the equality

[long pelican]

and then turn the whole logic aspect into more shallow memorization

[pastel horizon]

I mean you should also show them visually why multiplication is commutative through grouping

[long pelican]

tbh can't think of anything more simultaneously rigorous and age-appropriate than "draw 3x5 array of dots, rotate your head 90 degrees"

That explanation is also pretty fascinating to me as a mathematician for a very peculiar reason

[pastel horizon]

You could formalise it via matrices

And say if you transpose an n × m matrix you have the same amount of elements

Therefore n × m = m × n

At least for integers

For other types of number you probably need to go deeper

[long pelican]

I don't think that adds anything in terms of rigor, it's just using matrices as arrays of dots but in fancier language

[pastel horizon]

Yeah not really

There's also the traditional method using set theory

[long pelican]

Mathematical induction on the definition of the naturals is how Lean does it

To explain what I mean by incorporating logic in algebra, here's an example:

We are told that to solve something of the form x^2 = y^2, we can't take the square root of both sides. What's the usual explanation? "Because it misses solutions" is what we're told. "You have to do plus or minus square root" is another explanation commonly given. Both are unsatisfactory.

Imagine how much clearer it is when students come in understanding implication and you can simply say "For real numbers x and y, x = y implies x^2 = y^2, but x^2 = y^2 does not imply x = y." And maybe for slightly older students you can add "This is because the squaring function is not injective."

[pastel horizon]

That's where the function definition helps too. You can't apply sqrt because it's not bijective

[long pelican]

So "can or can't do" goes from being seen as arbitrary rules with handwavy explanations, to straightforward examples of logic

[pastel horizon]

If you restrict the output to only positive solutions you can apply it

O

It also reminds me a bit of Gaussian elimination. You're generalising the standard method of solving simultaneous equations with "row operations" which is effectively applying linear transforms for each function

[long pelican]

Wait, what's the connection? I'm talking about middle school algebra here

[pastel horizon]

Here

So you treat solving equations as if you are applying functions basically

[long pelican]

o, I see

[pastel horizon]

In other words you'd be saying there is no such thing as squaring both sides

You are applying the function g(x) = x^2 to both sides

[long pelican]

Confused again

[pastel horizon]

I wish I could find the quora post it was interesting

[pallid night]

I think it is :
\newline Suppose you are solving the equation $$f(x)=g(x)$$
You can apply $h(x)=x^2$ to both sides of the equation to obtain $$h(f(x))=h(g(x))$$ but only if you appropriately restrict the function $h$ to the positive real domain

[burnt vessel]

Ama Dablam

[pallid night]

And sometimes it's more obvious what the \newline solution of the equation $$h(f(x))=h(g(x))$$ is than for the equation $$f(x)=g(x)$$

[burnt vessel]

Ama Dablam

[pallid night]

So one can treat solving equations as a successive application of functions to each side to eventually proceed from $$f(x)=g(x)$$ to something of the form $x=c$ for some $c$ in our domain of interest

[burnt vessel]

Ama Dablam

[cosmic ibex]

I don't think "because multiplication is commutative" is a very good answer to "why does 2×3 equal 3×2?" That just says the student knows the fancy name for the phenomenon they're invited to explain, but giving the rule a name is not the same as answering why it holds.

[long pelican]

Oh you trying to guess what the quora post said?

It sounds like what a top 10% math teacher in school would say to explain the concept, nothing I’m too unfamiliar with. But I can imagine getting lost at the line “but only if you appropriately restrict the function h to the positive real domain” because that seems out of nowhere, since it is just an assertion. And I think many students do get lost there in terms of understanding, in practice, although not in terms of being able to memorize the procedure

And as I said above, teachers do try their best to explain assertions like this but the best tool they have in the absence of logic is handwavy analogies to real life

[long pelican]

That’s exactly what I wanted to get across!

[pallid night]

Yeah I ... may have failed to complete that part of the explanation 🙃

[long pelican]

Actually, for something like this even an amazing explanation will leave many students lost. Everyone has to ponder for maybe an hour over this, reading the explanation 15 times, to get it if they’re seeing it for the first time. Best to assign that as homework! (Only in the universe where they come in with a logic underpinning. Without logic there’s nothing to ponder really)

[cosmic ibex]

The easiest way to lose students with an amazing explanation here is probably to keep the amazing explanation fully generic and abstract. It needs to be paired with a simple concrete example of an extraneous solution arising ("suppose we attempt to solve 2-x=1 by squaring both sides...") so students have a chance to see how the general discussion applies to that example and match up the f's and g's to actual expressions.

[deep kindle]

Developmental learning theory 🔥🔥🔥

[vapid herald]

Advice for teaching adding and subtracting rational expressions?

[long pelican]

Generally the same advice for teaching anything else mathematical. Preparation is very important, work painstakingly through why everything is true the day before you do the lesson, so that you can impart the same kind of thinking to the students instead of merely drilling procedure

[zenith slate]

What do you mean by rational expressions

Like p(x)/q(x) where p and q are polynomials?

[vapid herald]

Yeah

But

For kiddos

[zenith slate]

How old

[vapid herald]

8th and 9th

Advanced 8th graders and 9th graders specifically

[zenith slate]

Right

Yeah so I'd probably start out making sure they can definitely add and subtract regular fractions and then build from there

There's a kinda natural progression in complexity

And you can use variation to kinda simplify things

Start with [\frac{1}{2} + \frac{1}{3}] and you can easily move to [\frac{x}{2} + \frac{1}{3}] and [\frac{x+1}{2} + \frac{1}{3}]

[burnt vessel]

primordial rat

[zenith slate]

With something like this you want to introduce the complexity gradually so they don't get lost

Depending on the ability of the students you can go fast or slow with this

Kinda have to play it by ear

[vapid herald]

Thanks!!

[zenith slate]

From this you could go to [ \frac{x+1}{2x} + \frac{1}{3} ]

[burnt vessel]

primordial rat

[vapid herald]

Yeah staging it gradually

[zenith slate]

The idea of "variation" is that you keep as many things the same as possible so that they're only really doing one new thing at a time

It's a very well tried and tested technique and there is plenty about it in the literature

The nice thing about it is that you can always move forward or backward as needed

Unfortunately this topic isn't really one where you can develop a strong intuition as to why the procedure works without just... doing it a bunch

But done carefully it is very straightforward

By building up gradually you turn it from something that likely looks scary and impenetrable to just a mundane procedure with fractions that they already know how to do

I taught this exact topic like two weeks ago haha

[vapid herald]

Yeah I realized it's not so easy to teach!

[analog fern]

For me, at least, it helped a ton to recognize that, to maintain equality without changing the other side of the equation, you're simply multiplying by F(x)/F(x) (and F(x) is a linear operator), which doesn't technically do anything to the expression since it's just multiplication by 1

that explanation might not have made a ton of sense, but it's been a long week, so that's about the best I can do lol :P

you could also maybe introduce that with a question like, "Now, why can we perform this operation to only one part of the equation and still preserve equality?" so that they have to poke around with the mechanics of what they're actually doing

[light wasp]

I hope this is the right place to ask this question: I've been having this problem where I forget old concepts and when I go back to learn them I find that it's not just the knowledge I don't know, but also my brain has completely unwired itself to where my reasoning isn't there anymore. I am really concerned and I wanted to ask if anyone knows about this / should I see a doctor / etc?

[cosmic ibex]

It's better to take that to #math-discussion. This channel is for mutual support among people who teach.

[light wasp]

Thanks

[zenith slate]

Yeah multiplication by 1 or adding 0 is one of those techniques that crops up all over the place

It's not always obvious where to apply it though

[loud plaza]

any good idea how to introduce integration?

also I’m teaching a level math, if anyone interested to help becuz I do need help then just dm 🙏🙏

[wispy slate]

e

[ornate otter]

Hello, you beautiful people 🙂
I'm a math tutor for university students, with no formal teaching background, so I would greatly appreciate the opinion of people who do have such a background. I have an idea for teaching derivations. So there is a way to kinda sorta defining derivations without introducing the limit, which is by using hyperreal numbers.
In case you don't know this niche concept, those are the real numbers plus "infinitely small" and "infinitely big" numbers. The construction is a bit complicated, but I feel like you could skip that entirely and simply introduce these very big and very small numbers and teach the students to do arithmetic with them. I feel like most people would find it intuitive, and the arithmetic is really not different then for standard numbers at all. For example, you can divide by an infinitely big number and get an infintely small one, or double such number with the expected results.
It's possible to teach derivatives by simply doing such arithmetics, using the equation f'(x) := std(f(x+dx)-f(x) / dx).
If you are interested in this concept, you can find more information and an example here: https://en.wikipedia.org/wiki/Hyperreal_number#Differentiation
Why do I think this is a good idea?

1. I believe it is actually more intuitive to most people then the limit method (even though the calculations remain more or less the same
2. The element dx gets a specific, well defined meaning and can be viewed as a single object (where in the standard method, you can only give it meaning in a context involving multiple objects at the same time, including the concept of a limit wrapped around them)
   However, what is intuitive to me is not intuitive to everyone, so I would really like a reality check. Thank you all for your time 🙂

[tropic grotto]

rotate a 2×3 square, it becomes a 3×2 but the area maintains, so 2×3=3×2

[vagrant meadow]

gonna be a TA for lower div linear algebra and odes this quarter. was thinking about what I should cover day 1 of linear this Thursday. atm I'm leaning towards a basic intro to logic and proof writing. anyone have any suggestions for some good proofs to cover which include many cases?

I personally like "x^2 even if and only if x even", since it requires doing both directions, and the forward direction is a nice case to use the contrapositive. but I'd like an example that only works one way too. one thought was "x integer implies 2x integer". any suggestions or notes? this is my first time as a TA so I'm not sure how it works really

[zenith slate]

I don't really see the benefit of this approach

You're teaching students who don't have good foundations with regards to limits etc that they can just treat infinities and infinitesimals like normal numbers

[ornate otter]

And they can! It just has to be explained well.

[zenith slate]

It doesn't make the calculations any easier and has the distinct downside of making it very likely you are going to teach them bad ways of working with limits

[ornate otter]

I wanna skip limits here.

What do you think of the two advantages I cited?

[ornate otter]

Although I'll think on that, I see what you mean :/

[long pelican]

To be fair, the theory of hyperreals allows you to develop a theory of calculus without using limits

[zenith slate]

Yes but it doesn't generalise

[ornate otter]

To what?

[zenith slate]

Nonstandard analysis is weird and nobody really does it

[twin shell]

i don't think skipping limits is a good idea since limits are universally used and well established in the language of math

[ornate otter]

I know no one does it ... but is that an argument?

[zenith slate]

To formally define the hyperreals is a nightmare

[ornate otter]

Fair

[zenith slate]

Given that the construction of the reals via dedekind cuts is practically a canonical exercise in a first course in real analysis, how do you explain the construction of the hyperreals?

[ornate otter]

Yeah, but you don't have to. Also, you can just define them as all monotonous sequences and redefine "=" and "<", then do all the operations pointwise and you're good.

[zenith slate]

Limits are fundamental

And they can be applied to far more than just the real numbers

You also lose a lot of important properties and constructions when working in the hyperreals

Most obviously you lose the metric topology which is kind of a big deal

Trying to use nonstandard mathematical hacks to avoid learning about limits is going to have very troublesome consequences when students come to look at anything other than R

Hell can you even construct a meaningful definition of a vector space (R*)^n

Perhaps you can but I've never seen it

The hyperreals aren't even complete

Cauchy sequences don't have unique limits

Honetly the whole thing is a disaster

[ornate otter]

I'm not doing it too avoid limits. You can teach limits in addition, especially as a method to get back to the reals from the hyperreals. I feel like being able to have dx defined as an independent mathematical object with clear rules has a lot of understanding advantages

To be clear, I'm proposing going to the hyperreals to calculate derivations and back when we have a nice expression, basically using limit rules.

I'm not advocating for skipping limits entirely

[zenith slate]

But it doen't even simplify the calculation

It's basically identical

[ornate otter]

It doesn't

That's why I didn't list it as an advantage

[zenith slate]

Just teach limits

Like I cannot see what benefit this could possibly confer

[ornate otter]

I've listed the benefits clearly. You can say that I'm wrong about 1) and that 2) has no value, that's valid

[zenith slate]

I do not think there is anything intuitive about infinitesimals

This approach has been done before FWIW

But like, nobody really uses it and for good reason

[ornate otter]

Fair

[long pelican]

I feel like in practice, it's not actually a question of "derivatives via Cauchy definition of limit" vs "derivatives via infinitesimals" vs "derivatives via hyperreals", it's more a question of "derivatives via handway intuitive concept of limit + rules" vs "derivatives via any logical definition of limit"

Unfortunately, anything of the second type is inaccessible to most students due to the simple fact that math classes give them zero training in understanding logical definitions