math-pedagogy

[winged urchin]

But yes (a) was the (only) answer they gave

The unfortunate thing with these sorts of errors and pointing them out as a student is you probably just get penalized

[long pelican]

And then the student blames you for misleading them 😭

[winged urchin]

Like if you said something like "this question doesn't make sense because of x, y, z" as a student and just stop there I bet the teacher just marks it 0 because you didn't demonstrate what they thought you should do

[pastel sundial]

because of x y z more like because of x f(x) g(x)

[winged urchin]

I mean, that's obvious geezzzz

Oops I meant geeg(x)g(x)g(x)g(x)

[long pelican]

A student's response to the a question on the recent 3rd Calc II midterm

General question: at the end of a diffeq class, would you imagine everyone can answer this question at the drop of a hat?

or not even the end, let's just say 1/3 of the way into the semester

[winged urchin]

I could see someone maybe getting confused by the first bit

But if they can realize that that is the same as writing -cos(x) + C

[long pelican]

The intention is to NOT compute the integral!

It's to write down the definition, read the definition, compare the definition to that of the differential equation and realize they're identical

[winged urchin]

So then it depends on remembering the definition correctly. And most definition-based questions students will fail (in my experience) unless they are specifically told beforehand that they will need the definition

I'm just saying that I could seeee a student getting it with the -cos(x) + C approach since all they have to realize is what differential equations has that as the set of solutions

It's roundabout, sure

[long pelican]

Yeah except like with the indefinite integral and differential equations

I want to ask every student who gets this wrong or struggles with it, what in the world are you doing in this class if you don't know what an indefinite integral means or what a differential equation means

Those are like the two most fundamental things in the class

Obviously I know the answer, which is that they do the math by rote computation and procedure

It is also 0 memorization. Even if you have a computation based understanding of both things, you should be able to reason about what the computations are doing, and go from there

[winged urchin]

I think precise language is scary to some studnts

students*

I wonder if it were asked like...

"Think of what (integral of sinx dx) means. Now think of what a differential equation is. ..."

SOmething like that

Idk really, I guess I just think that it might give them more wiggle room in their brain.

Somehow..

[long pelican]

On homework, sure

On exams, it's too much spoonfeeding when part of the class is about having the skill to think precisely

[tawdry venture]

why?

is it because imprecise language gives them wiggle room when they are cornered?

[winged urchin]

So, my experience comes from tutoring, and when I tend to get more talk, more of an attempt to say what's going on with a problem when I ask for general statements.

"Try to describe what you think this means"

Or,

"I know you aren't 100% confident in what you're thinking right now but just try to put words to your thought process"

I think perhaps technical or precise language makes them doubt themselves more because they might not recognize one part of it

Of course I think what Icy has preached with regards to advocating for teaching more precise definitions would help this

Heck, even in myself, sometimes when I see statements that ask me to think of a definition I don't jump to the definition in my head. I look it up, immediately, because I know I doubt my recollection and know sometimes there are little fiddly bits of definitions that are often the crux of whatever the question is asking me

And when I encounter something that asks to prove something for all cases or these really, complete technical results I am always worried there is some exception I'm forgetting

It's a simplification but Id rather have a student speak imprecisely but honestly than been too intimidated or just go for whatever precise-ee looking thing they can remember

Perhaps they themselves know their answer is nonsense but are just trying to write something that looks like an answer

I do encourage students to speak up when they believe they're wrong. Like if they get to an answer in a test and can tell it's wrong and explain why it's wrong but can't (or don't have time to) correct it

I do tell them to write that. Because it is annoying as a marker to just read trash that the student seems to be buying

And it's more honest to speak up to their own doubts with what they committed to on the paper. And maybe they'll get pity points

[pearl onyx]

Public, I’m not teaching, it’s a class presentation (everyone does a mini-lecture for one topic as review for the final)

None of them understand it at all, they were taught how to do the proofs procedurally

They didn’t even use words in their proofs. It’s basically a strange-looking algebra problem for them

[pearl onyx]

Way too formal. Sometimes I think we get caught up with the formalities so much that we lose sight of why we’re doing it in the first place, and whether it’s necessary to actually learn math that way.

“Rigorous formal proof only becomes important when there is a crisis— when you discover that your imaginary objects behave in a counterintuitive way; when there is a paradox of some kind. But such excessive preventative hygiene is completely unnecessary here — nobody’s gotten sick yet!”

Children can absolutely reason about what numbers and functions are without learning their rigorous set-theoretic definition first. They’re human beings, not machines, after all.

Mathematics is (and always has been) motivated by problems; I think pedagogy should work this way too. It gives context and reason for everything, and it would make for much better mathematicians.

[pastel horizon]

Well yeah, Jo Boaler was mentioned here before. She doesn't advocate for precise language she just advocates for more emphasis on exploring numbers and spotting patterns

It's like learning to speak a language. You can explore the patterns and intricacies without needing to know the strict grammar rules

Plus with rigourous definitions aren't you just falling into the same trap of "oh the kids are just treating maths as rote learning and only memorising definitions"

[pastel horizon]

Tbf this is probably the fault with our exams. A few of our GCSE questions involve "Show that..." which basically just entails an algebraic manipulation into the form the examiner wants to see

This is something I struggled with too, I used to wonder why geometric proofs were considered as "valid" during my a level education

[astral laurel]

I'll admit: I don't know how to answer your question then.

[long pelican]

This sentiment is probably the inspiration for less attention to giving correct definitions in textbooks. I’m seeing the effects of what happens when you’re asked to do “creative” reasoning with vague, wrong, and limited definitions of fundamental concepts like function. “I have no idea what a function is.” Teacher: “Come on, do some critical thinking! What do you do when you solve for a function?” (Completely misguided attempt at instilling critical thinking in two ways)

[long pelican]

Hint: the integral of sin x is the set of all antiderivatives of sin x

[astral laurel]

Still no clue

[long pelican]

Hint 2: The set of all functions whose derivative is sin(x)

[astral laurel]

But you said to not compute it

[long pelican]

Yeah, don't compute it

Just take that and turn it into math language

The set of all functions g such that...

[astral laurel]

You mean don't write it as +C?

[long pelican]

No, don't compute what the antiderivative is at all

[astral laurel]

I'm only seeing 'set of all functions' as any -cos(x) + C where C is a real param

And so this set has 1 real parameter (not x lol)

[long pelican]

You're seeing the answer but not the question 😮

That's what it comes down to 😋

[astral laurel]

Sooo... -cos(x)=C?

That makes no sense either

[long pelican]

$\int \sin x,dx$ is the set of all functions $g$ such that $\frac{dg}{dx}=\sin x$

[burnt vessel]

Icy001

[astral laurel]

oh lmao

If I were a good student, I'd add 'nice' between 'all' and 'functions' but I see your point now

[long pelican]

🙃

[astral laurel]

You're trying to get them to look at the integral operation I guess

[long pelican]

Yep!

[pastel horizon]

Well the question said about differential equation so yeah you would have to write a differential equation at the bare minimum

[pastel horizon]

We used to have the "function machines" as a model for primary school. I wonder why that gets last when we try to formally explain a function

What is a function machine? You just provide an input, do a few rules and give an output. Notice how there's no formal definition yet

[long pelican]

Function machines is how I remember first learning about functions too

I also remember it wasn't easy to grasp, and took some time, but I also remember not ever learning the horrible "f(x) is another name for y" which would have just confused me and made me into a biology major probably

[pastel horizon]

You can generalise it further potentially and say "ok now we're going to say our input can be any integer and we will output another integer." Or "input any real number but only output an integer"

[long pelican]

Are functions part of the primary school curriculum in UK?

[charred silo]

I don't think it is. Or at least it wasn't. Majority still had problems doing multiplication table going into secondary school.

[pastel horizon]

Function machines are but they are never told that this is actually a 'function'

[pastel horizon]

No I definitely did them in primary

[pastel horizon]

So if you remember a couple of weeks back I asked about building up the area of shapes conceptually. Sadly it didn't quite work out in the end, all they got was the formula for a triangle and rectangle although some interesting discussions were had!

One takeaway I've had is you could teach triangles first? Most people say a triangle is half the area of a rectangle but what if instead a rectangle was double the area of a triangle?

Another benefit is you could essentially teach trapeziums and parallelograms as the area of two triangles combined. In the end it seems like actually conceptually triangles are very useful to understand the area of shales

[long pelican]

I'd say the unit square is the basic area unit, followed by rectangles honestly

[pastel horizon]

This is counterintuitive of course, the area is usually thought of in terms of unit squares, but triangles seem to naturally fit

[long pelican]

But hmm

Triangles are the basic polygon on 3D graphics

However, dA in integration is a square/rectangle

Riemann sums are rectangles

When thinking of the future, squares/rectangles as the axiomatic unit of area would be less confusing in the long run

[pastel horizon]

Yeah obviously that redefinition would just fly in the face of pretty everything

[earnest trail]

what's the current unit of area

[pastel horizon]

The unit of area is 1 unit square

[earnest trail]

oh ok

[pastel horizon]

The area covered by a square of side length 1

[earnest trail]

yeah

but how about a rectangle

a "unit rectangle"?

[pastel horizon]

That is impossible

[earnest trail]

yeah exactly

so how would that be defined icy

[long pelican]

Need Axiom 2: Any one-directional dilation by a factor of a multiplies all areas by a

[pastel horizon]

So my approach for a rectangle was you start off by counting squares

[long pelican]

Then an a x b rectangle is the horizontal dilation by a and a vertical dilation by b, of the unit square

So its area is ab by Axioms 1 and 2

[pastel horizon]

Then I used some variation theory

[long pelican]

Actually I misused the word dilation

[pastel horizon]

And finally got to a problem where we had too many squares to count

[long pelican]

A dilation is a uniform stretch in both dimensions at once

[pastel horizon]

So then we looked for a more efficient strategy

And then yeah that's how in my lesson we came to the area of a rectangle formula

[long pelican]

Axiom 2*: Horizontal stretching/vertical stretching multiplies areas by the corresponding factor

How do your students come up with the area of a rectangle of width pi and height 1?

[pastel horizon]

Essentially, you could teach that axiom without needing to formally state it

With the use of variation theory

[long pelican]

What's that 😮

[pastel horizon]

So you just say "ok I doubled one side, what's the area now?"

[long pelican]

Oh

[pastel horizon]

This time I doubled both sides, now what's happened

This time I've doubled one side and halved the other

[long pelican]

What's wrong with formally stating the axiom after a lot of informal discussion?

[pastel horizon]

Congitive overload

They are really weak

[long pelican]

One of the goals of math is to construct precise statements for things

Precise notions = strong foundations

Imprecise notions can pass the next test but provide weak foundation for building on

Aren't weak students the product of weak foundations

[pastel horizon]

See the problem is some are also weak because of behaviour so it's a risk you know

You can challenge imprecise definitions with careful choices in questions anyway

Like in my example, maybe they will fall into the trap that you can only multiply/divide lengths by integers?

[long pelican]

Maybe one can frame the situation as the question "What is the furthest math test you want them to pass?"

[pastel horizon]

So to counter that "this time I've multiplied a length by 1.5, now what"

[long pelican]

Oh yeah I didn't exactly mean precise in that way

I meant more like

Precise in clarity and understanding

No room for ambiguity in interpretation

It's the difference between saying "A logarithm is an exponent" and "The logarithm base b of x is the unique number a such that b^a = x"

A precise definition is something you can use to derive things

This actually goes back to the complaint about formal definitions you pointed out earlier. Formal definitions for its own sake is pointless of course, the point of formal definitions is to actually derive properties from them, so that students learn what mathematical reasoning is all about. That's something school textbooks fail to do, regardless of the quality of their definitions

[pastel horizon]

So you'd state the definition then derive everything after conceptually based on the definition?

That's interesting. I actually like the reverse idea quite a bit though, explore conceptually before giving something precise

[long pelican]

The two things you just asked are not opposites of each other. Conceptual warm-up and exploration is necessary in order to have the eventual precise definition make sense. Then you demonstrate how to use that precise definition to logically derive further properties in a way that is completely transparent from the definition

It's impossible to derive log(ab) = log(a) + log(b) for all a,b > 0 from the statement "A logarithm is an exponent" -- this makes it completely useless as a definition and should not be taught as such
On the contrary, one can (and should) derive log(ab) = log(a) + log(b) for all a,b > 0 from "The logarithm base b of x is the unique number a such that b^a = x"

[pastel horizon]

That's correct yes. I see

This was what I was on about before btw. I think it's quite powerful, once you know how to calculate the area of a triangle you don't need formulas for other quadrilaterals

Because there are always two parallel sides, the distance between them will always be the perpendicular height for both triangles. I wish I'd seen this when I was in school

[long pelican]

It's good for that example. Computing areas of regular polygons also involves triangles but coming out of the center, rather than from a triangulation of the polygon. But for curvy shapes, triangles are less helpful

[real mauve]

everyone working with numerics is offended

[pastel sundial]

Huh so now I'm wondering

I know and am on good terms couple highschool math teachers, and I wonder if I should send them some of the articles we've talked about here.

My neighbor across the street and my highschool's math teacher (who was never actually my math teacher since I transfered to that school in 11th grade and took math at a local community college my last two years of hs).

[long pelican]

Go for it!

Their thoughts would be pretty useful

[pastel sundial]

Do you have a compilation of articles I should link?

[long pelican]

#1 would be one of Wu's slides or articles on his homepage

Then maybe https://gowers.wordpress.com/2012/11/20/what-maths-a-level-doesnt-necessarily-give-you/

[pastel sundial]

What about this

https://www.maa.org/programs/faculty-and-departments/curriculum-department-guidelines-recommendations/teaching-and-learning/9-key-aspects-of-knowing-and-learning-the-concept-of-function

That you also linked a while ago

[long pelican]

Yeah that too

https://math.berkeley.edu/~wu/Bethel-5.pdf This one talks about variables

[pearl onyx]

Yes, that’s terrible. I think AoPS does a good job, it removes the idea that functions are just equations like y = polynomial of x

[long pelican]

Yeah, AoPS is generally very good

[pearl onyx]

This is the way mathematicians work. You start with a fuzzy idea, a concept, and then you define it precisely, and check if your rigorous definition is in line with your desires. For example, mathematicians didn’t just invent the derivative from f(x+h) - f(x) / h. They thought of a secant between two points, then what would happen if the distance approached zero? Then they got that definition. After that, you verify that your definition satisfies some of our intuitive notions, hence Rolle’s theorem, MVT, IVT, etc

[long pelican]

https://math.berkeley.edu/~wu/pspd4c.pdf The second half of this article is pretty illuminating

At this point, I want to take a side trip: two things related to the learning of mathematics proved quite difficult in the institutes because they were so unknown, and certainly not encountered in other mathematics staff developments, and both are basic in the field of mathematics. First, reading mathematics sentences or formulas in fully articulated English (or any other language for that matter), so that meaning is constructed, and second, the even more mysterious process of building meaning from certain specific assumptions and definitions and then using these assumptions and definitions to prove other conclusions (theorems). The definitions themselves seemed to have come from Mars! To remedy the first problem of reading mathematics for meaning, we partnered up and simply read aloud to each other from a randomly picked passage in Wu’s notes (cf. [Wu2001a], [Wu2001b]). The participants had many questions about notation, and had to be nudged to really see every mark on the page. Mathematics professors must be very familiar with this problem. The second problem of internalizing definitions and reasoning with them improved somewhat with daily practice but never became routine.

[earnest trail]

we need to teach students not to give these types of answers

[long pelican]

Damn

[pastel horizon]

They'd probably mainly agree with you. I know the teachers at my school do. The problem is there's not really enough time to put together a new curriculum

Tbf the curriculum used is pretty good but it's not ideal if the sequencing isn't something you agree with or you're not comfortable with the way it's modelled

[pastel horizon]

I disagree as well. That question should be written better! Could've just been a simple "what is the equation of this line" and don't label it

[earnest trail]

yeah you're right

[long pelican]

The question as written elicited a response that reveals the amount of understanding the student has, didn't it?

[earnest trail]

I don't make the pages tho so I have no control over that unfortunately

[earnest trail]

true

[pastel horizon]

Or better yet, maybe the graph wasn't actually y=5x+4 so you could ask "Ella thinks y=5x+4. What mistake did she make?"

[earnest trail]

so we gotta focus on giving a deeper level of understanding for the student

[pastel horizon]

You'll be amazed how many people don't understand what spot the mistake means

[long pelican]

In that sense it's an excellent diagnostic question

[earnest trail]

the student also answered the question incorrectly because it was supposed to be "no"

[pastel horizon]

Some for example would write "the correct answer is..."

That's not spotting a mistake though

And on a GCSE they get no marks for just writing the correct answer

[earnest trail]

ttue

true

[pastel horizon]

Obviously a mistake would be "she calculated the gradient incorrectly" or "the y intercept wasn't calculated correctly"

Maybe with more working shown you could put something more specific

But yeah that's the power of a well designed spot the mistake question

[long pelican]

What exactly is your objection to the current wording?

[pastel horizon]

Well if a student is in a rush under exam conditions it just seems like a bait and switch question

[earnest trail]

not an exam

[pastel horizon]

It feels like it's there to catch someone out rather than assess them

[earnest trail]

we give pages to students during their tutoring sessions with the company I work at

[pastel horizon]

Ahhh I see now

Well yeah maybe next time you could go over it but as a spot the mistake

[long pelican]

To me, an answer that shows understanding could even be as simple as "(0, 4) and (1, 9) are on the graph and 5x+4 is the only linear function passing through both points"

[earnest trail]

yeah I would have but I saw it after the session was over

[long pelican]

Asking "How do you know?" separates the students who were just going through the motions from those who understand what a graph is, what a line is, what a function is

[pastel horizon]

Ah that's unfortunate

[earnest trail]

exactly

[pastel horizon]

That's fair I suppose.

Another one that throws students off "state your assumptions"

[long pelican]

"Sources of error" is a very good meme

[pastel horizon]

See most people would implicitly assume things to solve a problem without even realising they did that

[earnest trail]

ohhhh

[pastel horizon]

Like e.g. an exam question on a bus timetable

[earnest trail]

interesting

[pastel horizon]

What's one big assumption you make? That the buses are all on time!

[earnest trail]

true

[pastel horizon]

I mean why wouldn't you?

But students don't realise that's an assumption

[long pelican]

Textbooks fail to state their assumptions in any proportionality questions because they don't mention that the thing is constant speed

[earnest trail]

thats especially useful when going over geometry questions

[pastel horizon]

Pretty much all real world problems have a form of assumption in them that's just the reality

[pastel horizon]

This too, people assume there's right angles without realising they're doing it

[earnest trail]

^^^

all the time

[pastel horizon]

Or assuming two straight lines are equal

[earnest trail]

or parallel

[pastel horizon]

Maybe it's needed for the question or maybe it's garbage

[earnest trail]

usually questions provide enough givens but then students make some assumptions and it's like bruh

[pastel horizon]

The only assumption you don't need to make is that lengths that are labelled as shorter on a not to scale diagram are in fact shorter

That's just treated as a given

[earnest trail]

yeah

ok gtg sorry

[pastel horizon]

See you

[pearl onyx]

It should be like “tell me the equation of this graph, and convince me that you’re right”

[earnest trail]

true

[pearl onyx]

Obviously you’d have to show students what “convince” means (ie a proof, or something close)

But it’s more in line with actual mathematics, and it has a conceptual understanding

[earnest trail]

yeah

[pearl onyx]

When you say “how do you know”, you’re really asking them to convince you, which is literally a proof anyway

[earnest trail]

yeah so it's still the same issue I guess

[pearl onyx]

Yeah

[pastel horizon]

Interesting that we have different interpretations for assessing the same understanding. I like that wording too it's great

I like spot the mistake because I think it's good to view mistakes as learning opportunities rather than just "oh look you got it wrong you suck". But then "convince me you're right" is more positively framed as well.

[pearl onyx]

Haha true. Personally “spot the mistake” brings in an extra layer of (1) reading and interpreting someone’s argument and (2) finding where they are wrong

If you’re testing their conceptual understanding, it’s much more straightforward to just ask them how they got their answer. Although being able to correct mistakes is a valuable skill, it should be done in its own question, instead of mixing it with a conceptual test as well

Also, “spot the mistake” can be solved only by knowing one correct step in the process. A student could understand nothing from the past week and get the right answer because she’s good at spotting a faulty argument. Whereas with a proof you really have to know everything there is to know (and you can pinpoint exactly what the student doesn’t understand).

[long pelican]

These are the "spot the mistake" exercises I'm putting out for next week's calculus assignment

The first time I did this (about 3 weeks ago), a lot of students had trouble and picked the wrong thing as the mistake, which shows their unfamiliarity with mathematical language. Hopefully they are more familiar this time around

[pearl onyx]

Wait if you’re working with a precise definition of domain, then shouldn’t every function be defined with its domain and range included already?

[long pelican]

Yes and no... the statement is wrong even for natural domain

And this is a non-rigorous Calc II class, so that's the no part

[pearl onyx]

Oh jeez

They know set theory too?

[long pelican]

Just set language

[pearl onyx]

Ah

Is the answer for (d) that it should be “is not defined for all values of x; it is only defined for x ≠ c”

Wait even that doesn’t make sense really

“All values of x” should be “all real numbers”, right?

[long pelican]

(d) is interesting, I would've have thought of it myself and I only came up with it because I saw it as the most common wrong response to an earlier homework problem

Basically there was a differential equation y' = y^2

The homework problem was to show that the claim "The solution set to this differential equation is the set of functions 1/(C-x) as C ranges over the reals" is wrong, and it's wrong because the solution set missed out on the 0 function

However, a lot of responses said that it's wrong because $C\in\bR$ is wrong and it should say $C\neq x$ instead

[burnt vessel]

Icy001

[long pelican]

Which makes no sense semantically, which shows they don't really think about what they read and write

Ok you've inspired me to make a change to it

That way they don't correct it by swapping x and c

although hmm

they might still swap x and c there

ok I went back to the old phrasing and added (It is true that the function $1/(x-C)$ has domain $\bR\setminus{C}$, but the previous sentence is not about domains.)

[burnt vessel]

Icy001

[tepid smelt]

I actually have not done a lot of spot the mistakes but its a fantastic idea that I am differently going to use it. I have two weeks to prep them for finals (with only a a few required lessons left) once we get back next week and I think I am going to use a lot of spot and fix the mistakes questions as my review.

[pearl onyx]

I think it’s cause they need to formally learn some set theory first. I understand why they would make that mistake; I don’t think they even know what an indeterminate vs a constant is

[real mauve]

that (d) is a weird question to ask, imo

i can see what you're trying to get at, but i guarantee you'll steamroll people with it

[pearl onyx]

I don’t get the C ≠ x thing, isn’t x an indeterminate? If you wanted to be more precise wouldn’t you write the set as $$\left{f: \bR \to \bR \mid \exists c \in \bR \left(f = \left{\left(x, \frac 1{x - c}\right) \mid x \in \bR\right}\right)\right}$$

[burnt vessel]

abs_0

[pearl onyx]

Where f:R -> R is a partial function

[long pelican]

The C ≠ x thing being nonsense is exactly the point

[real mauve]

i'll be surprised if more than a couple of people get it

[pearl onyx]

Yeah understanding that kind of nuance is not for calc 2 students

[long pelican]

I mean, no one can actually read and parse that in the first place?

If you read it and pretend to understand it, what are you doing

[pearl onyx]

You’d only see that if you had formal training in set theory, and you’ve seen and understand what an indeterminate is

[real mauve]

yes but the first reaction of a student will be that they are not understanding something

not "ah this was written wrong on purpose"

[pearl onyx]

He’s saying to correct the mistake

[long pelican]

It's mentioned at the start of the problem it's written wrong in one way

and the problem is to find it

[real mauve]

i'm aware, but still

[pearl onyx]

But I don’t really like that, because it feels a little, I don’t know

Cheap?

It almost seems deliberately confusing

[real mauve]

i think you should try showing that to some of your peers and see how they do it

i think you might be surprised at least at how long it takes them

i'm pretty sure this one seems easy to you cuz you made it up

[long pelican]

I didn't make it up actually

This was taken from one of many student responses on another homework problem

So my peers who have graded that have had the delight of seeing it

[pearl onyx]

I get what you’re trying to do, but the fundamental problem with this statement is about set theory

[long pelican]

Not really

Even without set theory, what is the interpretation of C ≠ x

that would make sense

I can't find any

[pearl onyx]

It would be that you can’t substitute the letter C with the letter x, because then you’d get the function 1/0

But that’s a nuanced concept that (imo) only someone who knows set theory would see

[winged urchin]

Yeah, division by 0 is bad

=p

Is what they knopw

[long pelican]

That's a failure to understand what's being constructed then

[pearl onyx]

Yeah it is a failure

But unless they already know set theory, it’s not really their fault

[long pelican]

A set of functions is beyond their understanding, you're saying?

[real mauve]

i'm pretty sure this will be interpreted as f: R x R -> R and steamroll people

[long pelican]

Perhaps we should stop teaching calculus to 18 year olds

[pearl onyx]

In some ways, yes

It’s not exactly obvious

I have friends who don’t even know what a function is

They think it’s a procedure where you plug in numbers

[long pelican]

That's a problem preventing understanding of calculus then

[real mauve]

my comment is more that you can construct less cursed scenarios that are easier to detect and correct

[pearl onyx]

No, you don’t need to know what a set of functions is to understand calculus

I’d agree if this were analysis, but it’s not

[long pelican]

I mean

What if I just said

collection of functions

[pearl onyx]

That might be okay, maybe

[long pelican]

The only sense in which I'm treating "set" is as a collection though

[pearl onyx]

But that whole C ≠ x thing is purely not understanding how set theory works

[long pelican]

I fail to understand where set theory comes in

A collection of apples, with the condition that the leaf of the apple is not equal to x

Makes no sense

Where's set theory in that?

It's more like order of operations

[pearl onyx]

They don’t understand that the second half is a condition on the first

Idk

There’s a lot of stuff they might not get

[long pelican]

eh, I know the syntax isn't going to throw them off because I've used it a lot

[pearl onyx]

Alright yeah

Idk what you’ve taught already but if you’ve used this type of stuff and they understand it

Then yeah that works I guess

[long pelican]

Yeah, it's not about the symbols

They would equally make the same mistake entirely in words

[pearl onyx]

Hm

[real mauve]

i'm honestly not sure myself what the issue with the x \neq C part is, it just defines the domain of one such f for a fixed C

[winged urchin]

What if they mention something about how the set is stated to be defined for all real numbers C, so the statement that it is restricted at all is in conflict with the definition?

[real mauve]

then again, i'm no mathematician

[long pelican]

So the problem introduces a family of functions, one for each real number C

[real mauve]

mhm

[long pelican]

And claims that this can't work for all C, you have to take out the one where C equals x

[real mauve]

is \neq not symmetric?

i would've interpreted it as restrictions on x

for each fixed C

[long pelican]

I mean it is but the only parameter in the set is C

x is a dummy variable anyway

[real mauve]

sure

[long pelican]

Suggestions for edits are welcome

[winged urchin]

I'm not entirely sure, but I think... would it then be correct to say that 1/x is a function even if we stated the domain was {0}?

[real mauve]

i honestly just read it as each f has a domain x that is a punctured set

[long pelican]

Ya if you read it like that then I should probably be more explicit about what I'm parameterizing

[pearl onyx]

Yeah this is what I’m getting at

[real mauve]

the question is indeed a good one, there must be a way to make it a bit more straightforward though

[pearl onyx]

That’s a nuanced thing in my opinion

To understand that c is a parameter, a variable being changed throughout the whole set

And that x is a dummy variable, an indeterminate

[real mauve]

i think something that goes in the direction of distinguishing a family of functions f:R -> R from a function f: R x R -> R

at least in my head that achieves something similar, although it is not the same as "finding the part that is written incorrectly"

[long pelican]

ok thoughts?

[real mauve]

yes, that is more evidently crank

[long pelican]

I wonder if thinking of arguments to functions as real life quantities (which is often emphasized in "real world applications") messes with people's understanding

f(t) : t is time, so if C is a real number, C could be time too?

[real mauve]

i think it could be made yet clearer by replacing the f(x) notation with $f: x \mapsto \frac{1}{x - C}$

[burnt vessel]

Edd

[real mauve]

just so that there is no doubt

[long pelican]

That would be clearer except for the downside is that they've never seen the arrow notation except for a handful of times and are way more used to f(x) = ... notation

[real mauve]

oh, all righty

but yeah, i think this already makes the point much more clearly

[long pelican]

Thanks for the feedback!

[pastel sundial]

I like this wording much better

I'm no educator, but I feel like it basically streams in the reader's face what's wrong with it, and that screaming will come across loud and clear iff they don't have the misunderstanding you're trying to address.

[long pelican]

Yeah, that's my goal with this type of homework question!

[pastel sundial]

it's actually way less subtle to me than most of the other ones

actually the only other one that isn't super clear imo is (c)

the point is it should say "the domain is all pairs of real numbers x,y such that x neq y" right?

[long pelican]

Yeah, all pairs of real numbers

[pastel sundial]

also I'm kinda bothered by the exercise statement way at the top

[long pelican]

(c) catches the people who treat domain as a procedural algebra exercise

[pastel sundial]

specifically the "then rewrite the statement" part

it's like, how are you supposed to rewrite the statement if the statement is nonsense?

[long pelican]

So for (d), you'd say "actually the function IS well defined for all real numbers C"

(e) I do mention rewriting is not necessary

[pastel sundial]

why do you make it necessary for the others?

[long pelican]

It's my co-instructor's idea tbh

[pastel sundial]

I'd say drop it

[long pelican]

I like that idea

[pastel sundial]

or add something like

Extra credit: for problems (a) and (c), use your best judgement to figure out what the statement was most likely meant to communicate, then rewrite it using precise and correct mathematical language.

[real mauve]

aha, that's what i was weirded out by, but couldn't put it into words until now

since the statements contain some amount of crank, it isn't always readily apparent what they were "supposed" to say

so you may find some creative responses that are technically correct

[pearl onyx]

This is great imo

[pastel sundial]

marking it as extra credit to not scare students, and I think only (a) and (c) are the ones that really lend themselves to being rewritten.

[pearl onyx]

Very direct and clear, and to correct it, they need to be able to articulate what’s wrong

[pastel sundial]

(a) and (c) are just unparsable, but what they're trying to say is clear and correct, but (b), (d), and (e) are just blatantly false statements.

[long pelican]

(e) can fall under unparseable too

f(g(x,y)) would throw a syntax error in a programming language

[pastel sundial]

edited to make my point more clear

[long pelican]

oo

[pastel sundial]

you don't rewrite (b), (d), and (e), you explain why they're crank

like for example imagine each of these problems was part of a second semester real analysis proof that you were grading

if the student wrote (a) or (c), you'd take a few points off and say "be more precise here", but the rest of their proof might be fine, but if they wrote (b), (d), or (e), their entire proof is likely bogus.

(a) or (c) looks like something I could see myself writing in an informal sketch of a proof when taking notes only I will read.

in fact I'd need to review my notes, but I feel like do stuff of that order quite frequently

to put it another way, (b), (d), and (e) aren't math, (a) and (c) are Hatcher math

[long pelican]

lol Hatcher math

I once asked the class one time what absolute value means and one student responded "It means that what's inside is always positive" -- she got one of the lowest scores on the first midterm and dropped after

[pastel sundial]

yeah I absolutely see why a student writing (a) or (c) is going to highly correlate with having massive holes in undestanding.

[pearl onyx]

Right, (b), (d), and (e) are impossible to steer in the right direction, so to speak, because they’re just totally wrong

[pastel sundial]

am I wrong though

[long pelican]

Exactly, I'm aiming to write statements that have exactly that correlation

[pastel sundial]

Hatcher be like "proof by 'CW complexes totally have this property trust me bro'"

[long pelican]

I don't remember having many criticisms of Hatcher except that it was very hard to learn algebraic topology from

[pastel sundial]

I actually like Hatcher, he just says super imprecise things all the time

[pearl onyx]

How do most people learn algebraic topology

[long pelican]

Granted, algebraic topology was my first ever graduate math class (concurrent with algebraic geometry)

So I was getting the firehose treatment during that semester

[pastel sundial]

did you use Hatcher?

[long pelican]

Yep

[pastel sundial]

out of curiosity, how far did you get?

Because I'm doing a reading course with it now and it would be nice to compare my progress with an actual graduate course

my prof things I'm doing fine, but I worry I'm going too slowly

[long pelican]

We did all of the first 3 chapters and a little bit of homotopy theory

Or wait actually

Hold on

[pastel sundial]

first 3 chapter seems like a lot

[long pelican]

I remember we were rushed on cohomology

So actually

[pastel sundial]

I'm likely going to just barely get first 2

[long pelican]

Maybe we just did all of the first 2 chapters and a rushed treatment of cohomology

[pastel sundial]

doing nowhere near enough exercises

cool cool so my pace seems reasonable

[long pelican]

I can't wait for a unification of cohomology and related ideas from a simple point of view

[pastel sundial]

like in a textbook?

or like in terms of research

[long pelican]

Wherever it first develops

Probably a paper

[pastel sundial]

I'm super excited to learn more about alg top

especially from a categorical lens

from what I've heard there's some really cool insights if you zoom back a bit.

like there's a notion of duality that's pretty central to cat theory right? And then is cohomology dual to homology in that sense?

[long pelican]

If you take the coefficient group to be a field and maybe some other assumptions, yeah

[pastel sundial]

Anyways this is getting off topic from #math-pedagogy and it is also getting pretty late so I'm going to go to bed.

[long pelican]

Every word of that sentence is quite true

[pastel sundial]

(though a discussion of how best to teach alg top in light of people's complaints with Hatcher seems like a potentially very interesting activity once I learn more alg top)

[long pelican]

at that edit

[pastel horizon]

For sure yeah. As I said before, some will just give the correct answer when the question asked specifically to state what mistake was made

Maybe it's a problem with the culture of maths education as well. That students see maths as just "right or wrong" and a mistake is not getting the correct answer

[gleaming flame]

I don't know how to phrase this thought in a polite way, I apologize. Since most (if not all?) of the members of this channel are in agreement that public maths education is bad, doesn't this channel devolve into a 'circlejerk' to poor public math education? What's the point of this channel?
Please don't immediately assume the worst of me for that question. I too would like maths education to change for the better.

[grand wharf]

For sharing methodology to improve maths education and news about math education; if you wish the consensus to be challenged then challenge it yourself!

[long pelican]

Agreement that public math education is bad isn’t enough; we have to diagnose what aspects are bad and exactly why it’s bad, the root cause or causes. People do not agree universally about that. America’s math wars are evidence of that. At the moment I think one of the core reasons is bad content in textbooks and online resources, specifically, content that leads students to believe math is a collection of disjoint facts devoid of reasoning. This seems like an easier thing to fix than mass-educating every teacher

[gleaming flame]

Is the discussion around poor maths education centred around K-12 primarily or university stuff as well? I ask this because I am soon to go into uni, and I have seen poor K-12 maths education, and I worry about what the upper level maths courses are taught like.

[pastel horizon]

I can't comment too much on university but I do know the prestigious institutions are prestigious for a reason and it's not because their education is poor

[long pelican]

I am confident that upper level math courses do not suffer from the same problem as K-12 classes; most if not all the professors are competent and teach math as a reasoning activity. However, their skills in knowing what a student does or does not understand, and their ability to “teach to their audience”, may be a somewhat weaker than that of a K-12 teacher, but this is usually offset by the self-motivation of the student

[gleaming flame]

What do you guys think of teaching a bit of Real Analysis before high-school Calculus?
This would lay the rigour in calculus so that such ideas like "taking a limit" isn't some "infinitesimal magic" to them. Of course geometric descriptions can be taught alongside Analysis but only really as an aid.

[long pelican]

So counterintuitively based on this semester teaching Calc II, I believe weakness in calculus actually mostly comes widely from misunderstanding functions and function/variable language, and not so much the notions of limit, derivative, etc. So your idea but teaching a bit of “proper” functions and how to think of them specifically, before high school calculus, would be pretty promising I think

[gleaming flame]

By proper functions, I assume you mean introducing them with the notion of getting an input and output, rather than some kind of formula?

[long pelican]

Yep along those lines. But the misconceptions run a lot deeper than just that idea. For example they believe that the x in f(x) is attached to a real life quantity that varies, that f(x) is just another name for y (this is actually taught explicitly in textbooks for some reason), and so on

[gleaming flame]

Hmm
Could you elaborate on how this would benefit students from a Calc II perspective? While functions are important and all, it feels quite unrelated overshadowed by the ideas of derivatives and so forth no? This is quite the counter-intuitive take

[long pelican]

Well, functions are a pre-requisite to understanding anything in calculus. If you lack it, calculus becomes reduced to manipulation of symbols

and a boatload of memorization

[long pelican]

For example a lot of people instinctively think f(x)=3 does not define a function because "there's nowhere to plug x into"

Conversely I've had the pleasure of asking someone why $y(x)=\frac{y^3}3+y$ doesn't make sense as a function definition, or is at best an implicit definition of $y(x)$ as a constant function, but she did not notice any of that. When pressed, she finally said something like "Oh, is it because when you plug in $x$, .... it doesn't help?"

[burnt vessel]

Icy001

[pastel sundial]

It's not a circlejerk since we have Icy

[stark pine]

I mena, just read through what goes on in this channel. most of the discussion isn't "how can we improve math education" it's "how do you teach this specific thing" or "how do you deal with misunderstanding [x]" or whatever, which can sometimes turn into a conversation about how the school system is failing students, but the purpose of the channel isn't to discuss that

[mint lark]

Hatcher is a very good book for learning algtop from a mostly classical perspective imo

Modern alg top is an entirely different beast but

[woven hedge]

Agree somewhat, I TA Differential Equations and the instant we depart from super standard calculus formulas and try to use any sort of "reasoning" as to why certain functions do/satisfy/are relevant to certain ideas/topics I get a lot of blank stares when I explain stuff.

Wonder how many of them got any conceptual understanding of what Laplace transforms or Fourier series do, the understanding of functions is shaky enough so I can't imagine what the notion of a functional space would be like

[winged urchin]

What do y'all think of subscript notation or like tilde, hat, prime, etc.. in K-12 education?

I think it's generally not advocated for but it is certainly more precise

And using the same variable to mean different things is technically a problem

[long pelican]

I'd say expose them to it all. Anything that is common in how math is used in math, science, economics, etc, should be exposed to them. If anyone can remember all sorts of exceptions to grammar rules, anyone can also pick up how to read/write real math notation and its grammar. They just need to be exposed to a lot of authentic math language in every math class and when doing homework

[earnest trail]

I agree with this since people may think $x=\hat{x}$

[burnt vessel]

gmod

[earnest trail]

but that just means you have to teach them the difference

[earnest trail]

because this

another thing is how people think lower case and upper case letters are interchangeable

e.g. using T and t interchangeably

like bruh

[pastel horizon]

I was thinking about changing h with h_p in the area of a triangle formula to emphasise the fact that it refers to perpendicular height

[winged urchin]

Ooo go all out and replace the p with the... I don't know how to do the symbol right now but the _|..

[pastel horizon]

The right angle symbol you mean?

[winged urchin]

I just remember it as I think superscript notation in linear algebra for the perpendicular space

So I always read it as perpendicular or perp

[pastel horizon]

I'd probably also say another problem with maths education is you could try your best to teach everything conceptually but ultimately you'll get some students who only want to see a method or a formula. There's not much you could do about that

[pastel horizon]

Oh yeah

But yeah I guess the challenge for teachers is how do you increase engagement? And I don't mean the wishy-washy "engagement" you see talked about by the senior leadership. I mean mathematical engagement, how do you force students to think mathematically so that they don't just rely on a method?

[long pelican]

I’m becoming more and more of the opinion that this mentality is acquired by their previous education and not innate. Basically they got stuck on understanding something and their teacher got impatient and told them (explicitly or implicitly) to just do the procedure and stop trying to understand

[long pelican]

Very good question. If what I said above is true, disengagement reflects weak prerequisite understanding. Try to think of times you say something completely obvious and it gets blank stares for some reason

[gleaming flame]

I find your takes interesting and very defined. Are you a maths teacher?

[long pelican]

Calc II in university, postdoc

My takes are interesting you say 👀 are yours different?

[gleaming flame]

I don't have a take, atleast not yet. The bit about a good understanding of functions is interesting what I found interesting.

[long pelican]

Yeah, I think teachers don’t detect it because their exams involving functions are not representative of how we actually use functions in “real” math

They ask about domain, range, vertical line test, horizontal line test, but not whether you can interpret things like “f(x) > f(y) for all x > y” and understand that to say that f is increasing

[pastel horizon]

I'm a firm believer in number sense exercises as starter activities they can really help get people thinking mathematically and everyone should have some understanding of the basic operations

If they don't then you have bigger problems

[long pelican]

Number sense including fractions, right?

[pastel horizon]

Yup

[quasi musk]

Isn't this exactly what pre-calculus is supposed to do

[long pelican]

Yep. But the

[quasi musk]

I'm not saying the execution of pre-calc is right

I'm just saying instead of re-inventing the wheel, why don't we try chipping at pre-calc and make that wheel run

[long pelican]

So rewrite precalc textbooks and in those textbooks re-teach everything involving variables, functions, and mathematical language?

[quasi musk]

No, that's too laborious a task

Just find a good pre-calc book and teach from that

Maybe supplement with your own stuff

The after school program, Russian School of Mathematics has good curriculum for variables

Full disclosure, I used to work for RSM and taught 4 classes for a year there

But I actually believe that their curriculum is very good

[long pelican]

Do they teach "for all [variable]" and "there exists [variable]" and maybe nested too?

[quasi musk]

I think so, I didn't teach pre-calc

I taught 5th, 6th, and 7th grade

And we taught it for sure

just in more of a game format

[long pelican]

Yeah that's good enough

[quasi musk]

Things like "I'm thinking of a number such that..." what's my number?

[long pelican]

I don't think any widely used pre-calculus textbooks in the USA explicitly teach variables as quantified numbers, correct me if I'm wrong...

[quasi musk]

I'd suspect that they do

At least the ones I looked at, like open stax pre-calc

I think instructors don't emphasize it too much

I mean the textbooks are so bloated, they contain so much information

[long pelican]

Yeah

Maybe we should add it to the common core standards and replace something else

[quasi musk]

More than likely is already in the common core standards

[long pelican]

"CCSS 1.1: Be able to interpret statements such as 'for all real numbers x there exists...'"

Hmm let's see

http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf

Ctrl+F "for all" and "quantifier" collectively yielded no relevant results

(3) Students understand the use of variables in mathematical expressions.
They write expressions and equations that correspond to given situations,
evaluate expressions, and use expressions and formulas to solve problems.
Students understand that expressions in different forms can be equivalent,
and they use the properties of operations to rewrite expressions in
equivalent forms. Students know that the solutions of an equation are the
values of the variables that make the equation true. Students use properties
of operations and the idea of maintaining the equality of both sides of
an equation to solve simple one-step equations. Students construct and
analyze tables, such as tables of quantities that are in equivalent ratios,
and they use equations (such as 3x = y) to describe relationships between
quantities.
That's the standard about variables

[quasi musk]

So now we have to compare the common core standard, and look at the books that line up with the standard common core curriculum

And see if they expand on it there

So if most textbooks have it in there, then it's just a matter of adding a line

"Students will know general and existential instantiations of variables, and how to nest them"

[long pelican]

Mm

[quasi musk]

My full time job is providing academic support like this at a university

and evaluating/re-evaluating, and helping with curriculum development

My hope is after I finish my PhD I can go change math in higher education

To not be bad

[long pelican]

You mean secondary education?

and/or middle school where they learn variables

[quasi musk]

Higher Ed is undergrad and graduate math

I work specifically in University and community college math programs

I tried the k-12 system, and it's just not my cup of tea

I prefer to work within the college & university system. You can actually make changes since things aren't as federally/state mandated

[long pelican]

Mm, higher education seems alright if not that it is not "caring" enough for people who lack fundamentals

Lacking fundamentals in the large scale would be K-12's fault

[quasi musk]

And that's exactly where I am!

[quasi musk]

There's also other things, like how do you increase access to research at an earlier level

Or get project based courses going in differential equations and linear algebra

Just because the k-12 system is the root, doesn't mean there's no value in working at the higher ed level

to resolve those issues

But there are a myriad of other issues with higher education mathematics, even for math whizzes

[pastel horizon]

So exercises for number sense basically. Sounds good

[long pelican]

In this context we were talking about introduction to quantified variables

[wispy slate]

guys

I know this isn't really related to math but I'm really worried about my academics

especially maths

I practice question everyday but I still get like 70 or 80% in my exams

and it's mainly due to silly mistakes

Is there any way I can fix this?

[grim spindle]

Recently my school's intro linalg course changed curriculum to introduce VS and LT before matrices and systems of equations. A friend argued with me that teaching VS and LT before matrices is bad because it's too abstract for a first semester student and they won't have any concrete computational skills by the end of the course. I argued introducing matrices before understanding what they mean is just going to make students rote the rules for working with them without understanding what they stand for and by the time they reach LT the correspondence will be lost on them. I feel that you should start introducing abstractoon while still appealing to concrete examples from the start.

Thoughts?

[long pelican]

In the computer age the only matrix skills you need to know by hand are stuff with 2x2 matrices. At least that’s true for me

[grim spindle]

I agree

The course used to be tooled towards cs majors

Who need to at least understand the computational side better

[woven hedge]

Dunno, but here every major which requires linalg here has it planned as 3rd semester and it has absolutely no features of VS, LT is barely covered (and towards the end AFTER matrices)

Department wants the long-term fix to be to have an engineer-and-other-major version of linalg which is the easy, standard version and then have a more math focused one that might be more code focused (MATLAB or something) OR more proof based, they just want some kind of difference

[grim spindle]

that

s what my department did

except it was always first year

well now there's 3 versions of the course

and engineering one, a CS one and a math one

each one more theory heavy than the last

[pastel horizon]

Well you need to know the theory first

[analog token]

What's LT ?

Linear transformations ?

[grim spindle]

yes

[tepid smelt]

Yeah students who have foundational skills don't really struggle to engage with the material. The only issue is my energy is pulled to the 90%+ who don't and the students who are fine get bored or distracted by students who are not engaged.

Most students like to do well and feel good when they understand something(some don't care about anything but are not as common as you think and they generally are dealing with horrific home conditions). Its just hard to make up years of learning loss at the hs level. Lack of engagement is not because we are not making math more like a video game but simply because they can't properly engage with the material and get frustrated and mad as a result

[long pelican]

https://www.reddit.com/r/learnmath/comments/cirj8v/is_it_possible_for_me_to_understand_variables/
This could be insightful to read to understand why some people have trouble with variables

[real mauve]

that is a bit painful to read

[pastel horizon]

So much easier to explain a variable from a programming perspective

[earnest trail]

interesting

[earnest trail]

nerd moment

but yeah sucks that some teachers cant teach for shit

[pastel horizon]

Don Steward is an actual MVP, just look at this

Interactive tool to discover pi

[real mauve]

i remember my teacher taking different spherical and circular toys and some measuring tapes to class

[pastel horizon]

Yeah that's actually a traditional method but the problem me and my colleague said is the kids aren't very accurate and you basically just tell them what it should've been anyway

Tools like these are great though, no inaccurate measurements but you can still play around and discover for yourself

Especially with a sphere like even I would struggle to measure around the equator

The pi as a limit concept was great and I am using it still but this I feel fills in the missing link between connecting diameter to circumference

[long pelican]

What’s traditionally confusing about pi? I don’t think I can remember seeing any egregious errors on exams that can be traced back to confusions about pi

[pastel horizon]

It's not that it's confusing. It's just that pi has great potential as a student led lesson but if you're gonna tell them what pi is anyway what's the point?

Fundamentally though it's the easiest concept in the world. Literally just two numbers on a calculator. Hardest part is rounding

[long pelican]

My thinking is that the subjects that are less likely to be confusing are also easier to turn into fun and interactive topics. Fundamental concepts like functions and variables are a lot harder to succeed in teaching properly and deserve more attention

They’re hard because you aren’t just teaching the chapter, you’re preparing them to read everything they might become across in the future that will use functions and variables in a nontrivial way

[pastel horizon]

For sure. My thinking is just that if you're doing a "discovery" lesson and you have to correct them afterwards, what was the point of them discovering for themselves?

Maybe there's an argument to be made that they've worked their problem solving muscles

[long pelican]

What would discovering pi properly entail hmm….

pi as an integral, pi as a root of a power series

Are the two things that come to mind

Both obviously very advanced

[pastel horizon]

Well yeah normally they would measure circles with a piece of string

[long pelican]

Oh I know!

[pastel horizon]

I'm showing the Archimedes approach I think it's actually the easiest to understand for their level

[long pelican]

You can use trig functions to express the perimeter of an inscribed regular polygon

Then let n go to infinity

The archimedes approach basically

[pastel horizon]

That's way too advanced given they haven't seen a trig function yet. But yeah the Archimedes could be hand waved as "he measured perimeter of the circumscribed polygon and the inscribed polygon". Then obviously as n goes to infinity to you get a better and better approximation

Yeah I think Archimedes used Pythagoras which is quite crazy

On a 96 sided polygon!

[long pelican]

Yeeeeee

[pastel horizon]

The Don Steward resource helps connect pi and diameter a bit better too. Since it basically unrolls a circle in a straight line, then you can see the diameter fits into that line 3 and a bit times. No matter what sized circle you picked

So that's gonna be the main visual to establish the link, then Archimedes is more "this is how we got a more accurate answer"

[long pelican]

How do you rigorously and age-appropriately prove that circumference is proportional to the diameter?

[pastel horizon]

Well that's simple with the Archimedes method right?

[long pelican]

Mm-hmm

[pastel horizon]

Double diameter means the polygon doubles in side length

So your approximation will also double

[long pelican]

That’s be a great way to engage mathematical reasoning

Might sound simple to us

But not to them

[pastel horizon]

Plus it's quite trivial to show if you double all the lengths of a square the perimeter doubles

Yeah you can easily scaffold a question like that

[long pelican]

In fact now that I think about it

[pastel horizon]

Then the last piece of the puzzle is they need to understand pi is irrational - there's an infinite number of digits.

The number on the calculator is just pi rounded to 9 decimal places

[long pelican]

It uses similarity and linearity of limits in a fundamental way

Do you mean take on faith that pi is irrational? 😉

[pastel horizon]

Proving that at that level is not a trivial task by any means

[long pelican]

At least one should mention that there is a proof and that it is not trivial, right?

[pastel horizon]

Yeah of course

[long pelican]

I remember the elementary school teachers completely not discussing that

when I was in elementary school

or middle

[pastel horizon]

Thing is then you get kids trying to memorise every digit of pi which is a fool's errand as we all know

[long pelican]

Hey I memorized 100 digits and don't regret it

[pastel horizon]

Best just to say "you could spend the rest of your life writing out digits and still not have finished"

[frosty flame]

Go

[long pelican]

-cheats-

jk

[frosty flame]

[pastel horizon]

Tbh after 10 you can pretty much make up any numbers

[long pelican]

3.141592653589793238462643383279502884197169399375105820974944592307816406286208998 eeps I forgot what comes after

[pastel horizon]

Nobody will doubt you at that point haha

[frosty flame]

I literally didn't know what comes after 3.14 until you sent it now

[pastel horizon]

I know up to 3.141592

[long pelican]

My TI-30 something calculator showed 3.141592654 every time I pressed pi so it's seared into my memory

ever since elementary school

[pastel horizon]

Yeah exactly

That's standard calculator precision

Plus, when you're a bored school kid... 😂

[frosty flame]

Should have memorized digits of e instead. Be at least somewhat original.

[long pelican]

I can memorize the continued fraction expansion of e to infinity numbers

[pastel horizon]

2.718(2?)

[long pelican]

[2; 1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,1,14,1,...]

[pastel horizon]

I know 1/e is roughly 0.67

[frosty flame]

Just add terms in the Taylor series until you reach the precision you want

[pastel horizon]

Time constant in physics is about 67%

Forgot ln(2), thought it was 0.691 but actually it's 0.6931. That's useful for half life

[winged urchin]

I think any sort of 'discovery' plan requires you to hook onto whatever pre-existing knowledge most of your class has.

So calculating pi with a series or something like that is best when they have the capability to understand and possibly even calculate the series for themselves

Otherwise it's again just like telling them what the result should be

[earnest trail]

when students think ln(2) is a function and get confused about why you don't need the product rule for something like d/dx e^x*ln(2)

[long pelican]

I might go as far as to say the root cause of that is not learning properly that logarithm is a function and only learning the procedures involving logs (convert to exponential form, etc)

[winged urchin]

Usually if I get a student who thinks (ln(2))' = 1/2 I usually have to try to get them to think of the graph of y = ln(2) since they're usually okay with seeing the derivative as the slope.

Pretty quickly they realize it's a constant function and thus the slope should be 0

I also like to remind them that unlike identities in the past, differential or integral identities cannot be so simply substituted into

Like you can say since sin^2(x) + cos^2(x) = 1 then sin^2(apple^banana) + cos^2(apple^banana) = 1

But when derivatives or integrals are involved it is more complicated. Sometimes if the student seems keen I'll explain the idea of an area element and show it in Cartesian and polar since that's usually the two systems they're familiar with

But it also really should just be as simple as seeing that ln(2) is a number aha

[long pelican]

Another possible explanation is that the student views "evaluating a function at 2" as a problem they don't want to work out unless necessary, meaning that ln(2) is some mysterious thing until they think about it

Unlike for us, where ln(2) automatically registers as a number

[winged urchin]

Ooo that's probably a pretty accurate take actually

[long pelican]

This is why I think it's good if teachers "speak math": use ln(2) in a sentence as a number, instead of only having them be exposed to plugging in a number as a math problem by itself

Analogy: if I am a French teacher and I spend all year on the 17 conjugations of the verb être and never speak it in a sentence, will my students be capable of using it in a sentence?

[earnest trail]

that's true

good point

[wanton walrus]

Would you guys accept symbolic answers to a calc problem that has an exact answer?
I have a student that converted a polar coordinate correctly to rectangular, except using (-4cos(3$\pi/2$),-4sin(3$\pi/2$)) instead of (2$\sqrt(2)$,-2$\sqrt(2)$)
Basically their answer is 1 step from being completely simplified

[burnt vessel]

Ping Warrior

[wanton walrus]

it's out of 5 points, I can't tell if I should take off 1 point for not fully completing the problem, or 0 because that's a little nitpicky

[light pond]

i would rage so hard if someone took a point off for that

did the question ask them to simplify as much as possible

[dawn walrus]

their answer is better than the other answer

[light pond]

yeah also what ryc said

[dawn walrus]

square roots are way less intuitive than the points that angles lie at

first answer is simpler

[real mauve]

isn't -4 cos(3pi/2) = 0

[light pond]

,w -4cos(3pi/2)

[burnt vessel]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by jesse#3659

[light pond]

looks like ur gonna need to do some remarking 😵‍💫

[real mauve]

everyone flunked rn

it does prove ryc's point lol

[kindred heart]

this just in jesse's grading decisions are based on how much he would hate himself as a teacher

[burnt vessel]

Ping Warrior

[winged urchin]

Idk... so are you okay with students not knowing their special angles?

In my experience I would take a mark off if they left it in any form that is just an 'obvious' special angle

[real mauve]

indeed

though in this case it seems to be more than just that. maybe the mistake up the pipeline is minor tho

[long pelican]

Has anyone else been thinking about the seeming contradiction in the way we teach interpretations of functions to students?

Example 1: If f(x) = x^3, then f(y) = y^3.

Example 2: If f represents the temperature in the room at time t, and x represents something unrelated like, idk, position, then f(x) is not defined?

Example 3: If y = f(x) is the temperature in a room, then y(t) is temperature as a function of time, while y(x) is temperature as a function of location. So y(t) and y(x) are different functions, apparently?

[winged urchin]

Just some thoughts on your examples.

Ex1: We write this yes but is it not a little hand wavey? If y is nonsense itself then f(y) is also nonsense, not y^3

Ex2: I would not necessarily say f(x) is not defined, but rather that it would have messed up units that are not an accurate description of reality

Ex.3: Again I think there is a units problem here in even talking about y(t) but ignoring that, they'd have to be different functions. The temperature when I wake up (t=0) is not necessarily the same as the temperature at wherever x=0 is. Also wait, there is a dimensionality problem here too since location is maybe a 2d or possibly 3d variable whereas time is 1d

[long pelican]

Clarification of example 1: If f is the function sending each real number x to x^3, then if for any real number y, f(y) is also y^3

What examples 2 and 3 highlight is that a real-world "function" of one variable is somehow not even a function in the proper sense, because a different letter for the input somehow creates a whole different meaning

In math, the "x" in a function definition like f(x)=x^3 is a dummy variable, but in real world applications, x typically means a real world quantity

I mean this has to be the underlying reason for the common misconception that if f(x) is defined, this doesn't say anything about what f(y) is when y is a real number

[real mauve]

i guess that can be confusing, yeah

this is usually handled in high school physics already though

[long pelican]

I doubt that high school physics helps them understand the abstract notion of function

[real mauve]

no, but it adds in the notion the there may or may not be units attached to what you plug into the function

[long pelican]

It's funny, it might also explain the heavy conceptual roadblock to viewing ln(2) as a simple number

Functions are this scary thing that show relationships between real life variables, y = f(x), and all that

and then you see ln(2) and you don't want to think about it because you know ln is a function and functions are scary murky hand-wavy relationships that you never learned properly about

[winged urchin]

Do you believe that there exists some ideal way to teach in which these misconceptions wouldn't occur? I know that's a pretty general thing and not well defined

Some way that if we explained it according to this 'ideal pedagogy' that a significant portion of any given class would not make simple mistakes?

[long pelican]

I've been thinking about that question for quite some time.... I think my answer is the following: stop teaching functions as relationships between real life variables, and teach them only as pure numerical gadgets: things that map an element of a set X to an element of a set Y

Don't even mind if X and Y are both the set of real numbers for the time being

Then give examples: square, square root, adding 1, subtracting 1, multiplying by 2, sin, cos, ln, exp, etc

Then when we talk about real-world examples like temperature as a function of time, that's what I'm trying to handle

[winged urchin]

I do like having addition, subtraction, multiplication, division as a function examples. For sure

Builds on already established base knowledge from elementary school

[real mauve]

i honestly think that might be a bit too challenging the first time though

you'd immediately introduce functions R^2 -> R in doing so

[winged urchin]

I just... I think I think that even with the perfect pedagogy there will always be misconceptions, possibly including well-used ones

[real mauve]

or if you try to circumvent it by fixing one of the summands or multiplicands, the immediate question is, which one do you keep fixed?

[winged urchin]

People mis-reading something, the teacher mis-speaking or a mistake in slides

[real mauve]

and idk whether such high level of abstraction would work well in school

[winged urchin]

A frustrated student who is doing homework and the only way they can see to make the problem work is by doing some simple mistake

They're like... (x+y)^2 has to be x^2 + y^2 otherwise I have no idea how this works

[long pelican]

Well there is a trick to how we manage to graduate people who can largely communicate in English in English-speaking countries

The trick might be quantity and variety of examples

[winged urchin]

Not to poo-poo discussing pedagogy of course, but perhaps, the existence of a well known misconception maybe doesn't necessarily imply a flaw in the pedagogy

[long pelican]

Well I can say things in terms of stats too: my students are probably top 25% of the country in math, albeit not math majors, and 90% of them have enough misconceptions that they can't do problems unless they are significantly similar to examples they are shown

Moreover, they get stuck on the "understanding the problem" part

[real mauve]

it could very well be that not everyone has the "ability" to learn functions well, for whatever reason

[long pelican]

I mean, I can agree that only certain people have the skill to learn math properly in high school classes where they're taught misconception-riddled mathematics by their teachers

I don't even know if I'm one of them because I never took a regular high school math class

[real mauve]

i meant in a more ideal scenario

even if the classes were "perfect", were such a thing to even exist

just like some people will forever be bad at a sport

[long pelican]

There's bits and pieces of evidence that causes me to doubt that hypothesis as the main factor

It's uncertain where the bar is but the prevailing attitude of many mathematicians and educators I believe today is that most everyone is capable of mathematical abstract thinking?

[real mauve]

i wouldn't say it's the main factor either, i agree there are several flaws in how the teaching takes place. just entertaining gemini's comment 😛

[long pelican]

So I would expect some misconceptions but the level of misconceptions I see in this class this semester is pretty overwhelming

[real mauve]

i do also think that the situation makes it seem worse than it is

like if you sat down one on one with one of these students and talked the problem out slowly, they'd get it right

test-taking is a completely different skill

catching important details, interpreting them calmly under pressure. "get at least X% or be a disappointment" is certainly not a good way to see someone's best side

[winged urchin]

I think the most damning evidence of math education's failure is the widespread idea of people admitting they are bad at math, being afraid of math, etc...

[real mauve]

that is certainly true

[winged urchin]

I guess I believe in the idea that if you can keep them at least not hating it then the misconceptions would decrease

Significantly

[long pelican]

My take on that is one of inverse causation

[winged urchin]

It is likely that the majority, probably large majority believes 'they are not good at math'

Thus the norm is to not be good at math

[long pelican]

The reason they hate math is because at some point their mental model of previous material was too broken to understand the new stuff, yet they were forced by their teacher (and standardized tests) to memorize math to pass

[winged urchin]

And so makes a reasonable thought in a students mind of "Oh most people aren't good at this, I guess it's okay if I'm not good, whatever who cares"

That's a good take too

The truth as with most things is probably somewhere in between

What's in between a function and it's inverse? =p

[long pelican]

I think my experience this semester really shaped this take

[real mauve]

was this your first time teaching this course?

[long pelican]

I've TAed an equivalent course

This is the first time I'm "responsible" for them though

[real mauve]

mhm, i see

[long pelican]

I can empathize with them if I pretend that math is a play in a foreign language I can't read

and I got here by memorization

[real mauve]

i think you're doing pretty well in trying to remedy the situation. the first time i taught a course was super frustrating for me and i couldn't understand where all the problems came from nor how to deal with/help the students lol

so i had ended up more complaining about stuff than trying to alleviate it

[long pelican]

Yeah my diagnosis of them kind of went from being the vague "these kids just template-match problems" to "these kids probably picked up the wrong idea of variables/functions in school"

(and therefore can't truly comprehend new problems, therefore being forced to template-match)

If I can go back to my original topic, I think maybe the difference between mathematical language and real-world language is that in math, everything is a local variable. "Let x be a real number. Let y = 3x" while quantities in the real world are global variables

Even a simple innocuous equation like y = 3x is taught in a confusing way

Because in the mathematical setting, y = 3x is just a true/false sentence that evaluates to true or false depending on what x and y are

but we teach algebra students that y = 3x is a (mysterious) relationship tying two everlasting variables named "x" and "y" together

In this latter view, "the set of pairs (x,y) such that y = 3x" wouldn't even make grammatical sense probably

You have to force yourself to shift viewpoint from "y=3x is a relationship" to "y=3x is a simple clause that's true if and only if y is equal to 3 times x"

[real mauve]

you think something like asking whether a = 3b and y = 3x describe the same relationship might help?

to realize the relationship is really not between these y and x things, but something else

[long pelican]

Might be even more confusing, because I don't even know how I'd answer that question

On one hand, if a, b, x, and y are tied to real world quantities, then they are different...

On the other hand, what does relationship mean

[real mauve]

well, more formally you'd say a,b,x,y \in \mathbb(R), and then use the definition of equivalence of functions

so i was thinking of going in that direction by first removing importance from the letters you use to describe the mapping

[long pelican]

Yeah if you expressed it mathematically as "are {(a,b) : a = 3b} and {(x,y) : x = 3y} the same subset of R^2?" then it's emphatically yes

However

{(a,b,x,y) : a = 3b} and {(a,b,x,y) : x = 3y} are completely different

😛

That's why I see it as a confusing question

[real mauve]

haha fair enough

[long pelican]

If we're completely formal, we can model the real world as some huge-ass manifold M and a variable like "time" or "temperature" is a coordinate function, so in fact a variable representing a real world quantity is a function from M to R

I think this view is favored in differential geometry

Anyway, yeah! Maybe we can spread awareness of how real world variables confuse matters under our noses and we should look at how we teach using the real world more carefully

[real mauve]

the annoying thing is that the distinction could easily be made by simply considering sets of reals multiplied by some "unit", but using sets so early might be challenging. maybe by simply calling them collections, as you have said before, that could be somewhat circumvented

[long pelican]

Yeah, honestly 99% of my set theory usage in higher math is simply thinking of sets as collections. There's absolutely no reason why sets are too scary to introduce to young people

The scary part of set theory deals with, like, ZFC axioms, the naturals as {{},{{}},{{{}}}}...

The amount of time I spend dealing with that is like zero

[pastel horizon]

Oh god, example 3 is completely dubious! Should be T(x, t) represents temperature as a function of location AND time

And even then we're assuming it only varies in one dimension

[pastel horizon]

Your explanation of 3 is why that function is "dubious". The temperature when you wake up, is the temperature when you wake up (t=0). Sure it could vary around your room but you have to interpret that as a snapshot when t=0. At that specific point x=0 temperature will fluctuate as time passed based on convection currents

Basically, that function should not be two functions depending on whether you mean t or x, it should be the same function applied to both variables simultaneously

[long pelican]

mm

Although I meant really the following Example 4:

If T = f(t) is a function of time, then if I plug in a real number a that's not attached to anything in the real world, then f(a) should make sense, right?

[pastel horizon]

Physically speaking, you would have to assume that temperature is uniform everywhere

[long pelican]

Ok I think the temperature bit is distracting

[pastel horizon]

Yeah it definitely is

The physical interpretation would just break down

[long pelican]

If P = f(t) measures some real world quantity P in terms of another real world quantity t, and then someone comes along and plugs in an unrelated real world quantity into f, what should the answer be?

we can even assume they have the same unit

say for example f(t) = t^2 + t for concreteness

[pastel horizon]

So let's say as an example, you first substitute t=10. Then for some reason you substitute x=t and let x=10

[long pelican]

Not that

Say t is "years since 2000" and then someone comes along and lets v be "years since 1963"

Is f(v) equal to v^2 + v?

Or is it (v-37)^2 + (v-37)

[pastel horizon]

You should have to apply a transformation yes

But in that context your function might not be the same model anyway

[long pelican]

So if f(v) is not equal to v^2 + v, this means that changing the name of a variable might change what f is

[pastel horizon]

For sure yeah. But it's not a hard and fast rule

[long pelican]

That's the opposite of the way it works in math, where f as a function is fixed in its behavior

[pastel horizon]

Depends on how you define your variables

[long pelican]

and as a mapping from \R to \R

and the name of the input variable does not matter

[pastel horizon]

That function is still mapping R to R

It's just that first you're applying a transformation

In a way, you could define it as a "composite" function I guess?

[long pelican]

So the function is changing, yeah?

So we cannot use the symbol f for (v-37)^2 + (v-37)

[pastel horizon]

In fact that doesn't seem right

Surely you could just define v = t - 37 and then substitute v directly

Either that or transform it into a function of f(t-37)

[long pelican]

Man I'm so confused now

[pastel horizon]

I am too now you mention it

[long pelican]

🤣

[pastel horizon]

t actually equals v+37 that's the mistake

The function should still be valid

It's still a function of t

Does that make sense

[long pelican]

I made sense of it in my own way

Mathematically, the time axis is a 1-dimensional affine space (vector space if you like), and the number t or v is just a representation of it in some coordinate system

[pastel horizon]

Basically we can interchange variables the same way, as long as in the physical world those variables are equivalent

[long pelican]

So if we say $f\colon \mathrm{Time}\to\bR$

[pastel horizon]

Obviously it wouldn't make sense to then substitute a positional variable

[burnt vessel]

Icy001

[pastel horizon]

Yes precisely

We can map time variables but anything to do with position breaks the function

[long pelican]

Then the variables t and v are actually secretly functions from $\bR$ to $\mathrm{Time}$

[burnt vessel]

Icy001

[long pelican]

and we're writing down a formula for the composition

[pastel horizon]

I mean you could do it, but then you just get garbage in = garbage out

[long pelican]

However, in my original setting

we had $f\colon\bR\to\bR$

[burnt vessel]

Icy001

[long pelican]

In this case I am forced to say that $f(v)=v^2+v$ by the rules of mathematics

[burnt vessel]

Icy001

[pastel horizon]

Well no because we have defined v = t - 37 explicitly

[long pelican]

Hm isn't years since 1963 37 more than years since 2000?

so v = t + 37?

[pastel horizon]

t = 0 means the year 2000

So v = 0 = 1963 which means 2000-37

Anyway point is you have explicitly defined what v should be

[long pelican]

ok so f(v) = (t+37)^2 + (t+37)?

This seems weird

[pastel horizon]

A better example would be "can you substitute altitude for t"

Then mathematically it would just have to be a direct substitution of a

[long pelican]

yea

[pastel horizon]

But physically it would make no sense

[long pelican]

yes

So in this case

If people get in the habit of thinking "f" is some "real world relationship"

They would be inclined to say f(a) doesn't exist

even if we say a = 30 beforehand (representing 30 feet or something)

[covert plaza]

If you define well your substitution corresponding to the equality you wrote between the two variables, why wouldn't it make sense?

[long pelican]

I mean let's not think of variables

f(0) : price in the year 2000 = 0^2 + 0 = 0

f(-37): price in the year 1963

all well and good

Now someone says ok since f is a function from R to R, let's remove the real world context, let v = 0, what is f(v)

[pastel horizon]

Well pedantically that car may not exist in 1963 ;)

[long pelican]

still 0, no matter what v is representing, surely?

[pastel horizon]

Yeah I was only joking

It's going to be 0 until someone actually makes the car

[long pelican]

Lol

My sentence was just a continuation of my previous sentence

I didn't respond to your joke yet

[covert plaza]

That's why domains exists, though

And if you were to set v=37-t, the domain would change with it too

[pastel horizon]

Sure, but it is valid to say the cost of a car that isn't real would be 0

[covert plaza]

Then piecewise function

Would be the closest to reality

I mean if you don't want to use a characteristic function lol

[pastel horizon]

Oh yeah as a sidenote, when you substitute x=y=z in your earlier examples. That should be valid for any physical 1 dimensional functions of space

[long pelican]

This is all to address why people might say "not sure" to the following:
Let f(x) = x^2+x. If y is a real number, can we say what f(y) is?

[pastel horizon]

Yeah that's pretty straightforward

[long pelican]

Would you just say yes, f(y) =y^2+y?

But in light of the discussion we just had, it might suggest that f(y) could have a different formula?

[pastel horizon]

I guess you implicitly assume x=y

[covert plaza]

So your question here is :
If we are not setting y=x, would that still be a valid answer?

[long pelican]

Yeah I'm not setting y equal to x lol

Otherwise f(y) = x^2+x would be an answer

Here's how I'm thinking of that question:

[pastel horizon]

But, that would be if we are working with spacial coordinates

[long pelican]

f is a function from \R to \R, sending each real number to its square plus itself. If I am now given a real number, named y, what is f(y)? It has to be y^2 + y, by definition

So in the mathematical interpretation, the x in f(x) is just a dummy variable and y is just a name for a real number

[pastel horizon]

Yup, we haven't connected the function into anything meaningful yet so the earlier arguments won't and shouldn't apply

I guess most people fall into that trap of y=f(x)

It's basically the same in programming

[covert plaza]

I take f as a restriction of an n dimensional space in order to getin return, from k independant variables, $f(x_1,x_2,...,x_k)\in\bR^{n-k}$

[burnt vessel]

Epsilia aka Epe

[pastel horizon]

You give the function a name, you define the variables it takes and define the output

[long pelican]

Programming view is mainly how mathematicians view functions too, with obviously the added restriction of no randomness and no state-changing involved

[pastel horizon]

In fact programming is a great analogy because inputs don't need to be numbers

You could assign a letter a number

Input: a letter A-Z Output: a number between 1 and 26

[long pelican]

Function from the alphabet to the integers

[pastel horizon]

It's still a mapping of one set to another

[covert plaza]

A variable could also be interpreted as a function is what you are saying?

[long pelican]

I think I did say that at one point but it's not the central point

[pastel horizon]

That's kind of implicitly taught anyway when people use stuff like y(x)

[long pelican]

In math, a variable is a pure thing, it's just a name of a number that might be quantified

In the real world, a variable is more like a projection function from a huge-ass manifold M encoding the state of the universe to the set of real numbers

[pastel horizon]

You use y(x) to mean y is a function of x

[covert plaza]

A variable is especially when a quantity is known to be in a certain range

[covert plaza]

Yeah everything in maths has a meaning because we wouldn't use it for so long

[covert plaza]

Well that seems like a big hammer just to kill a fly

[long pelican]

The idea of the manifold is mainly to explain why there are "pre-existing" relationships between real world variables

like... F = ma

[pastel horizon]

It's not a fly when you consider the abstract world of maths

[long pelican]

In math, a pair of numbers (x,y) isn't constrained unless we say so

In the real world, a pair of variables can already be constrained without us saying so

It's... like math class is trying to merge two different languages into one in a way

And probably favoring the real-world language over the language of "for all x, there exists y..."

In general I'd say my students don't have nearly enough experience working with variables as generalized numbers and it shows in their writing

ok time to sleep

I'm gonna be very tired tomorrow 💤

[pastel sundial]

I'm curious. Because most of the conversations here have been focused on highschool/advanced middle school math (how to teach functions, etc). But what is generally taught before that point?

How many years can you spend on fractions and how to multiply

[pastel horizon]

Well apparently in the UK it's 6 years spent

Something interesting that I've found today talking to a primary teacher. Apparently they are using very precise vocabulary (subtrahend and minuend) when it comes to arithmetic and this actually helps them understand more what's going on. So what happens in secondary?

[long pelican]

Subtrahend and minuend is pretty weird because it seems to focus on the wrong kind of precision

But if it helps understanding, by all means…

[pastel horizon]

Well the idea is the more "mathematically" kids can describe what they're doing, the stronger they become at reasoning. Apparently this helps primary school kids understand negative numbers

I would say the weakest areas when they leave primary are fractions, decimals and negatives

[long pelican]

I fear that a lot of teachers misinterpret the “attend to precision” mathematical practice in the CCSS, and become pedantic on vocabulary that mathematicians don’t actually use

While still teaching fractions exclusively by analogy (which is antithetical to attending to precision)

It’s only a fear though and I don’t have hard data myself about it

[pastel sundial]

what would you say the best way to teach fractions is Icy

because I do feel like analogy is a crucial part of teaching fractions well, especially considering the age they are generally taught at

like "a slice of pizza is 1/8 of a pizza" is a really good example of a fraction imo

but I guess maybe the point is you need to introduce some formalism

[long pelican]

I basically defer to Wu on this

Do note that if someone's primary conceptualization of a fraction is parts of a pizza, they will have a very tough time intuitively understanding multiplication and division of fractions and probably will only remember them by rote

[pastel sundial]

Right. The slice of pizza is a powerful motivating example, but not the definition

[pastel horizon]

Just the standard CPA model then?

The pizza scaffolds adding fractions with common denominators, but at some point you need to move into the abstract representation

[long pelican]

I have never heard of CPA

[pastel horizon]

Concrete, pictorial, abstract

So in this case. Maybe you start with something physical they can interact with. Then move to pictorial representations. Finally, get them to work out sums like 1/5 + 3/5

I imagine as well in primary they'd use some facts like 1/5 = 2/10 to scaffold into denominators that aren't equal

[long pelican]

Wu is advocating teaching one clear definition of a fraction and proving the properties using that definition

And the one clear definition is that m/n is the length of 1 part after a segment of length m has been cut into n equal parts

on the number line

[pastel horizon]

Ooo, that works perfectly with the bar model. A common pictorial representation

[long pelican]

No models, not even bar :P

[pastel horizon]

He thinks it would work at primary level?

[long pelican]

He says the current methods produce people who end up just memorizing fraction algorithms by rote anyway

[pastel horizon]

I mean tbh I've been quite amazed at the level primary kids work at currently so who knows

Just have a look at a recent SATs paper an 11 year old is expected to take

They're not hard questions in themselves. It's just basic arithmetic. It's the fact that they get such a short time constraint to do them and the amount of work they'd have to do

In other words, they're not just learning methods they're learning how to select which is appropriate so they can work more efficiently

I don't think they even learn column addition anymore

[pastel horizon]

I know some Y11s that would struggle with this 😂 let alone Y6

[long pelican]

Division of fractions is a particularly tricky point, but with a mathematical definition it's easy:
[\frac{A}{B}\coloneqq\text{the unique fraction $C$ such that $CB=A$}]

[burnt vessel]

Icy001

[long pelican]

Coordinate points and translation, damn

Well it makes sense that such a topic is given a very quick treatment

More emphasis on procedures

less on this stuff

[pastel horizon]

Like that's harder than a foundation GCSE question

[long pelican]

This is like me asking what does the integral of sin(x) dx mean, in the third exam of a calculus 2 class

success rate of people answering it is like 50%

The percentage of people who can express it in proper language is like 20% or less

[pastel horizon]

This one stuck out to me as well

That could even make the higher paper

[earnest trail]

isn't all they need to say just "the area formula of sin(x)"

[long pelican]

"The notation $\int\sin x,dx$ means, by definition, the set of solutions to what differential equation?"

[burnt vessel]

Icy001

[real mauve]

were you thinking of something like dy/dx = sin x ?

[long pelican]

ya

[earnest trail]

oh is that literally it

and only few people got it

tf