Not interested in math is fine, I think. Not everybody needs to learn it. I think what is more important to the education is teaching math non-mechanically and having a compulsory course in logic, in increasing order of rigor/difficulty, for everybody (not just science majors).

@Daminark true, but there is really no good argument I've ever seen on why it should take more than one semester to learn integer arithmetic fully and understand it. I mean, in some countries they have children doing real analysis younger than we have children in school (granted, those are pretty rare). Surely, there should be something in the middle whereby the idea of real numbers can be taught within a year or two.

or even fractions for that matter

i know the idea of adding them is tough, but multiplication and the basic idea of defining them can be set forth immediately

Real numbers are a bit much, fractions maybe

Basic maths is hard for a child, you don't necessarilly want to put that much stress on children and teach them "complicated stuff" at a young age, especially when it'll be really usefull to only 5% of them in the future. What's more it is essential to make sure they completely master integers before moving on to fractions IMO

There is absolutely no way to teach fractions to first-graders; one must be out of his mind to plan such a course. You don't see a fraction in the night-sky. You need to be able to count, learn numbers, do basic operations with them, be able to divide. That itself takes a nontrivial amount of time.

But my point is more, the typical level of a child is such that abstraction is hard to grasp

Fractions are an abstract concept.

@BalarkaSen pizza. pies. glass of water. etc.

and let's just forget about real numbers.

fair enough

i mean the greeks didn't even have natural numbers. Their idea of numbers was just abstract lengths.

@TheGreatDuck Those are all whole numbers. The idea is a fraction of "stuff" out "stuff".

That is nontrivial to understand.

Greek mathematics did not develop in a milisecond.

Some of us are lucky enough for these concepts to appear natural to us. Others have a way more trouble making sense of that.

I meant that real numbers can be taught as being equivalent to length

:p

@BalarkaSen um... dude. You cut a pizza into slices. You cut a pie into slices. you can drink half a cup of water. etc.

fractions occur all the time if one watches.

@BalarkaSen Eh, at this point we might as well teach them integral domain and fraction fields, why bother with special cases when we can have the results in full generality :P

Can you drink four thirds of a cup of water ?

@TheGreatDuck I think you have no real experience with teaching first graders. :P

Lol @Alessandro

@BalarkaSen Like I said. The math teaching majors take wildly different classes than the math majors.

@Alessandro Infinity 2-topoi. Nothing less than that.

im just saying that it seems like the current teaching process goes quite slow.

Yes, it does go slow, but I wouldn't let you plan for a faster course.

It accomplishes something. Yours would accomplish nothing.

Are there complex structures on $S^n \times S^n$?

@BalarkaSen im not saying teach it all in one year. We were all complaining about a lack of proofs in pre-college math. I'm merely saying that perhaps the issue is not just proofs but rather a mildly inherent slowness in the way it is all taught. Obviously arithmetic isn't trivial to teach but that doesn't mean that one cannot teach fractions at an earlier point than grade 5. From what I remember, all of the standard arithmetic operations were finally nailed down at grade 2-3.

On an elementary school level you need to give a good reason for defining something before doing so, is what I'd respond

If you just say "Yo, so remember how square roots can't be negative? Let's see what happens if we can do that!", that won't work

How old is grade 1 ?

grade 5 is fine. one should know fractions by then

i did

Ted is pretending not to be here, too bad I noticed

hey everyone is there any general method for finding triangulation of a topological space? I see triangulation of cylinder and torus in books but have no idea how to create them myself.

I also know from experience that there is a really large amount of time wasted doing 'practice' in long multiplication and division. I know that in 3 different grades (4-6 I believe) that there was a whole chapter (about a quarter of the school year) devoted to 'reviewing' everything from previous years from doing arithmetic all over to multiplication all over along with roughly 30-50 practice problems involving 3 digit (or greater) numbers.

Now, lack of proofs is not even totally the complaint. It was that right now, the curriculum for math in high school, at least, is such that you have only the option of going through the rather technical and symbol pushing-y curriculum of high school algebra and calculus

Perhaps I pick up on it faster but it seemed like those days were just wasted time because other people were too lazy to remember their previous years of schooling.

especially when it wasn't so much of learning as it was just useless practice to supposedly 'become faster'

Many people will not do well in that particular kind of environment, so they will feel discouraged from math and grow to hate it unnecessarily

Even if they might be fine at other things. That's my point, there ought be some route which does things, slowly if needed, with an eye toward ideas

I rarely see a need to be able to do that many operations. I'm guessing that there was a 'test' that the state mandated and it was probably part of why it was so emphasized.

@TheGreatDuck Why do you say this practice time is "wasted" time ?

@Astyx because by then everyone already knew multiplication and division. This wasn't just practice in one year. I was 3 years in a row starting a school year by reviewing all previous grades of math by literally reteaching them and spending roughly half the year doing so.

this continued up until high school

@zed111, I think I tackled that problem by researching - simplifying and subdividing a mesh.

and it was definitely poorly motivated because I remember students asking why and the teacher said "because the county has this test you have to take that is roughly 600 problems on it and if you don't pass it you immediately fail"

from what I understand, that test only lasted about 3 or four year before going away

I honestly believe this three years practice is very efficient to make one sufficiently confortable with these operations. Doing it for a shorter period of time would probably make one forget about those very quickly

no no no

that's not including elementary school

@zed: No, there isn't. There are lots of topological spaces that you don't expect to have triangulations (like, say, the cofinite topology the reals). But there are even spaces that you'd expect to have triangulations which don't (e.g. not every manifold has one). On the other hand, there are general methods for surfaces, which I don't know but @MikeMiller could probably tell you.

there was a year or two before that in which we learned fractions and root taking and stuff

Oh then yeah sure

literally, we went into the class and we all laughed cause the teacher said "There's a big test coming up and so you will be doing nothing but multiplication for the first half of this class"

@Astyx america has a bad history of overtesting to the point that the subject isn't being taught. Rather everyone is being trained to take some ridiculous test.

(in this case roughly 1000 problems in about an hour)

So the thing is, understanding well how to do the arithmetic operations is one thing. Spending a lot of time doing things to huge numbers in order to do a test is another

it was a pretty large 20-30 page test

Thanks @user16839 and @EricStucky. I don't want to go into very complicated spaces. I just expect to be able to create triangulations for simple ones like cylinder, torus, RP^2 etc.

You can always triangulate open subsets of Euclidean space, say, by just sort of progressively expanding your triangulation. For surfaces (or other manifolds you can "see"), you might do the same thing - draw some big triangles and then keep drawing until everything is covered.

In general for computational topology you don't triangulate things. You start with the objects given (or at least amenable to finding) a triangulation.

@Daminark true, but I'm referring to those sorts of wastes where a lot of stuff could go into those areas. It'd be different if it were just homework but it's different when the actual class time is devoted to it.

It's often a hard theoretical question whether or not a space admits a triangulation.

i mean, our teacher actually apologized and flat out said it was a ridiculous expectation.

1000 problems in an hour is less than 6 seconds per problem, are you sure about that figure ?

but that it was government 'idiots' getting involved in things that don't concern them

@Astyx we were expected to do them mentally.

perhaps it was less

Well, the problems are just one digit at that point

but it was definitely so many that if you wrote your work, you'd take too long

@Daminark no. All 3 digit.

That sounds far too suspicious

There's a lot of ways to triangulate surfaces. Triangulating little disks and then making it progressively larger as Mike says is probably the topological way to do it (which i don't know the details of).

@MikeMiller I see, thanks.

@Daminark that's why it didn't last very long. it was only once but then subsequent years did it because (and to quote the teacher) "because we don't have enough to teach you without wasting time"

Duck, a big problem here is that when a student says "everyone already knew multiplication and division", it's hard to know whether this means anything different from "me and my friends already knew multiplication and division". The teacher has the responsibility to teach everyone in the room. (and, yes, to ensure that students pass their standardized exams)

I know for a fact that I was completely unaware of the wide range of abilities in my class, when I was in high school.

And I was even a little socially conscious by that point; I imagine that middle school was a lot worse.

@EricStucky the teacher actually apologized day one because here were a class full of 7th graders being told that the entire first half of the class was nothing but arithmetic practice to deal with an exam that the teacher said "was a complete waste of time". I mean the teacher actually took points off for showing work or using a calculator during that time because she said that we would quite literally not have any time to write out work.

@Daminark I was never told it wouldn't work. I only got that from a calculator and the teacher said that was just a bug. All they said was that it wasn't a real number (as in falling into that group, not that it was nonexistent).

Here's another thing, Duck. The problem that the teacher has is even worse when you scale it up from a classroom to a school district, not to mention a state. It's certainly possible that the exam was a waste of time for everyone in the room; this doesn't mean it was a waste of time.

No what I meant was teaching complex numbers to middle schoolers doesn't work

Or at least elementary schoolers

I'm definitely being a bit contrary here. I agree in general that the US has a bit of an unhealthy fetish for standardized exams.

They may know how to push the buttons but it's kinda pointless to see those

But the issue is a lot more subtle than I was inclined to understand when I was a student, and right now you're sounding a lot like me when I was younger :P

And I mean, so the thing about these kinds of exams is that in order to drop them you need a kind of insurance regarding the competency of teachers

They're all really well trained in teaching

The standards and training that are in place right now don't seem to allow for that

@EricStucky the test was only mandated by our county and it has 3 digit multiplication or division for every problem and consisted of roughly 500-1000 problems. I don't remember the exact number. From what I remember, many of us didn't do too well but apparently the actual exam had to be changed to requiring only a 33% to pass as basically everyone in the county failed it.

There's also a cultural emphasis, so even if everything was set up such that you no longer needed such exams, they wouldn't go away

@EricStucky Idk if you come from America but I actually did research a while back (probably 2 hours worth because someone else's English project interested me) which basically corroborates what you said. Many things over-emphasize tests to the point that the classes become the test preparation.

Damin— standards: agree. training: disagree. Try looking around at #MTBoS on twitter; there are a lot of people who are trying really hard to make the pedagogy better.

(I do, Duck. I assumed you did too because you mentioned the US testing situation, but it seems not :P)

i come from america as well

:p

arrighty then >.<

@EricStucky what're your thoughts on trying to teach about number sets earlier in school? Good? Bad?

I mean, I've heard that at least in some states that the training individual teachers are provided is not necessarily the best. In any case, making sure that students are at the appropriate level makes testing a bit of a necessary evil

History already tells us the answer, which is: disastrous

Hey guys

Nobody loves me

well

;P

Well yourself!

hi dod

@EricStucky I simply mean trying to teach fractions and maybe real numbers a bit earlier than they already are.

we were thinking of ways to make math more "interesting"

I think that functions in Canada should be taught earlier and better

Maybe even calculus

So getting rid of a kind of technical element in the earliest stage of math is debatable and couldn't really succeed in implementation

Fractions could be presented somewhat earlier in the right context but it requires people who really understand how kid's minds work

@Dodsy I'm suspicious of calculus even in high school to be honest

Fractions are pretty .... I can't even think of the word

Duck: here's the problem with making things more interesting. The culture still associates mathematics ability with intelligence, and intelligence with innovation potential, and innovation with GDP. This is why the "math wars" are a thing— the gut-level reaction of most voters is that changing math classes has large and unpredictable effects on the US's economic stability.

I think fractions are intuitive

But also I don't remember much of my math learning until this year

This discussion stemmed in part from my complaints about how there was no way to do math in high school aside from the whole algebra-calculus train

How true is it? Well, not false, but of course, but in reality math has been used as a sort of proxy for 'logic', which, in principle, could be taught in many other settings

Maybe college. I remember doing fractions and I basically didn't go to class because it was too easy

Since I thought it did a huge disservice to anyone who wasn't engineering/science minded, which included many math types

Only showed up to write the tests

Oh yes

(personally, I think mandatory CS at the high school level is a lot better idea than mandatory math beyond algebra)

@EricStucky we were simply talking about how high school math has a distinct lack of truly correct proofs. Then we were talking about how to make people more interested in math in high school. My argument was to maybe put a bit more higher level math into the regular curriculum. Maybe perk some interest in it.

Math needs to be much better

Math shouldn't be manditory

Yeah, Duck. I definitely sympathize.

But it should be taught in such a way to be interesting

I wouldn't say proofs so much as ideas. Argumentation at the right level

Yes ideas are important

So for example, this is a bit of my own personal taste being injected, but stuff like basic number theory or graph theory

Formal proofs shouldn't be taught but interesting ideas should

@Daminark two column proofs in geometry are the laughing stock of proofs. Geometric proofs should be written in a paragraph with diagrams where appropriate.

I think early elementary number theory maybe

^^^^

That's not overly technical, even perfectly valid proofs are rather accessible before you have the kind of "mathematical maturity" needed to handle nested quanitifers in analysis proofs

But the curriculum is basically in place so that everyone is at the same level

basic concept of appending numbers to the integers should be at least taught

maybe have a day or two explaining the idea of rings and fields

Which means that it makes it so that a student who may not have learned what a square of a number is learns it

For instance in grade 11 we learned trigonometry

The main reason why I'm not too happy with the idea of doing things too structurally is that I think at an early level you need notions to be motivated

@Dodsy true, but there is no need to teach a square root 5 different times.

But trig should be second nature by that time, the early stuff is so intuitive

Like, to understand the importance of complex numbers you need to see the importance of polynomials

ehhhhhh, disagree.

@TheGreatDuck well they do that because most students don't grasp it the first time

Which means more segregation

Then there's 7 different math classes

(you can package complex numbers as a framework for algebrizing plane geometry; i.e. talking about 2D linear algebra without admitting this is what you are doing)

@Dodsy wrong. 11th grade is when trig is introduced if memory serves right. Also, the idea of polar and hyperbolic coordinates need to come in at some point (the latter not at much). That might be where your confusion sits.

Potentially

Lmfao

Though I haven't seen any amount of linear algebra taught either. Honestly I'd be down for that

I literally just said that

@EricStucky that may be true. I actually learned complex numbers and the complex numbers on my own with no verification of it being 'correct' until this past years geometry class.

Something like

I literally just said 11th grade is when it's introduced

It should be earlier

granted, I made the mistake of assuming +i and -i were in the same equivalency class

The basic elements of linear algebra, number theory, and graph theory being one potential path through the high school curriculum

XD

Anyway, this has been fun but I should do real work :( cya folks in a few hours

cya @EricStucky

See you @EricStucky!

@Daminark matrix algebra is taught as a means by which to solve linear equations.

the echelon forms and stuff

@TheGreatDuck I said it was introduced in grade 11, it's so intuitive and so easy it needs to be introduced earlier

Meh

Every prodigy knows it earlier

@Daminark My algebra II class (ages and ages ago) taught matrix manipulation and a few linear algebraic facts (det A = 0 implies the system may be inconsistent, etc.)

I'm thinking linear algebra of the more, transformation geometry type thing

The latter is interesting, though how'd they define the determinant?

@Dodsy I think that it makes sense where it is at. However, I've always been intrigued by the idea of a 'coordinate systems' class whereby parabolic coordinates, spherical coordinates, and polar coordinates along with trigonometry as the basis for them can be taught as one whole construct.

@Dodsy I've always been of the opinion that the human brain is ready for calculus and trigonometry by middle school, but the pedagogy we have right now doesn't allow.

^^^^

really it's just a matter of having a class teaching of different mathematical objects

but math is so convoluted in setup that everyone messes it up

@Fargle I'm slightly nervous about that, calculus has an idea but it's a lot of symbol pushing. I guess it's best to have two branches floating around

@Daminark it aint that bad once you explain what an operator is

@Daminark Yeah, but I think at that point in a kid's mathematical career, they're not necessarily going to be clamoring for rigor. I don't mean make them do chain rule, just introduce the idea of the "slope" of a curved thing.

i.e. function upon a function

well and product rule and chain rule aren't that bad

I mean, it depends on what types of functions you're doing

The notion of limits is alright

@TheGreatDuck They can be if you're a sixth grader. Recall that algebra I is 9th grade in the standard pedagogy.

@Daminark at that level they wouldn't know trig or exponentials. So basically just polynomial functions.

But if you're calculating derivatives then I wouldn't be terribly happy with more than polynomials or something

OK that might actually make sense

My point is mostly that I don't want to at least force all math to be mechanical so early in the game

i actually know a guy in high school

a ninth grader

he likes programming and once in a while we chat about math

he doesn't know any extra stuff but I showed him some calculus and he understood the geometry behind it

Reminds me of my high school days.

his main complaint was "why are there so many integral identities to memorize"

Even in the vague sense, I want the focus to be ideas. That, I think, people can benefit from, and it would help a lot for everyone.

of course my response was pretty much "you don't. You print them out and refer to them when you need them"

Showing random nerd friends of mine stuff like the Abel-Ruffini theorem and the Weierstrass function.

btw, why don't we teach parabolic coordinates?

they're quite trivial to convert to cartesian

@Daminark Speaking of Weierstrass, if you taught the ideas of differential calculus to a young student then showed them that function, I think that'd generate a lot of interest in math.

weierstrass doesn't intrigue me

what really stimulated my interest was being told to figure out how to integrate the floor function.

totally useless integral but man is it a tough one to do

(unless you know to split the integral but i wasn't taught that yet)

My account will be deleted in 23 hours, bye everyone, good luck!

@JasonBourne what did you do this time?

@Duck Having done math, and especially analysis, I don't really see the point in overdoing coordinate stuff

@TheGreatDuck You're also not in sixth grade. I just mean showing kids weird stuff will be a way to generate interest.

@TheGreatDuck Nothing, as usual.

@JasonBourne then why are you being banned?!?

Not banning

He's doing it voluntarily

@Fargle true but I think a fractal won't intrigue people if it's just an infinite sum of absolute values or something similar. I'd rather see a fractal with a closed form.

and something easy to write down/remember.

then i can show to friends

XD

@TheGreatDuck But the intent was to avoid symbol-pushing. Just say "we could conceive of a function where, if you zoom in, it's still bumpy, etc."

You don't need to demonstrate a fractal to kids as a sum

Again, no one was asking for technical rigor in analysis

Almost the opposite, make sure that kids are at the right level of technicality but then get them to focus on things that are cool, but that they can latch on to. And if you're not of a very specific disposition, infinite sums almost are in the complement of that

Anyway I think I've more or less already made my case, at some point the details of when you do what can't really be talked out

It requires a lot of pedagogical research

where $C^\infty$ is the class of smooth functions and $C^\omega$ is the class of analytic functions on $x>a$.

@SimplyBeautifulArt can you render the latex and link it to some image pastebin?

xD You can just copy/paste to the main site to see it render

good point!