math-pedagogy

[ancient shore]

String theory uses much different math than economics

So study algebra and geometry

[proper dragon]

Yes I have some linear algebra, that's what I was asking for, in terms of subjects beyond that what is the logical order to learn, I read that abstract algebra, differential geometry, algebraic geometry are required

[ancient shore]

(disclaimer: I am not a string theorist [nor do I want to be]) the courses in order would be something like abstract algebra -> topology -> differential geometry -> algebraic geometry -> parts of analysis

[proper dragon]

And what parts of analysis real and complex?

Yeah don't worry your advice is very useful for me, I want something more theoretical to study

[ancient shore]

for analysis, im pretty sure you just need real+complex (maybe some measure theory as well)

[proper dragon]

Thanks 👍

[halcyon light]

lmk if this is more appropriate elsewhere but im wondeirng if I need to teach about compound interests where the rate is not 5%p.a. but 5% per quarter or per month

this is different to the compoundingn periods which i know about

[magic minnow]

Working in the ring ℝ[ε]/(ε²), we have f(x+ε) = f(x) + f'(x)ε for any polynomial (or even formal power series) f(x). Division by ε is of course a no-no, but you can do this:

(Bertram, "Differential geometry over general base fields and rings")

You can get k'th-order derivatives from the ring ℝ[ε]/(ε^{k+1})

What do you think pedagogically of teaching calculus via infinitesimals (implemented with nilpotents, not non-standard analysis)?

I told my linear algebra students a little about this on our last day (you can construct a + bε ∈ ℝ[ε] as the 2×2 matrix [[a b] [0 a]] similarly to how you can construct a + bi ∈ ℂ as the matrix [[a -b] [b a]])

I said if you visualize a real number as a point on the number line, a dual number is like a point with a velocity on the number line

but I've never tried this in a calculus classroom

(also, where would be the appropriate channel on this server to share a video series I'm making on Introduction to Algorithms, following CLRS?)

another example I like to use is: although x² ≠ 0 for any nonzero x ∈ ℝ, on a computer it's possible to square a nonzero number like 0.000000001 and get 0

[pastel horizon]

Yes they should know about it

[halcyon light]

okay thank you!

[pastel horizon]

Try not to make them over reliant on the compound interest formula, make sure they understand the principle

[cosmic ibex]

First, I would be skeptical an approach where higher-order derivatives are not by definition repeated integration.
Second, any approach with formal infinitesimals (such as your nilpotents) seems to depend on a function always being defined by an arithmetic expression, such that we can figure out what it "should" do to inputs that have infinitesimals in them. That's essentially different from the modern concept of function, which students_ should end up being able to work with. Since your ring doesn't admit division in general, it seems to me that your approach would have trouble differentiating even 1/x. You might start by developing the function as a power series, for which nilpotents behave acceptably well (I think it was Lagrange who used power series rather than infinitesimals or limits as the basis for teaching analysis). But you'd have to find these power series first, and you'd end up with a theory that can only differentiate real analytic functions.

For example, I think you'd have problems with functions such as $$f(x) = \begin{cases} 0 & \text{for }x=0 \ x^2 \sin(1/x) & \text{otherwise} \end{cases}$$ which is differentiable at 0 with the standard definitions, or $$g(x) = \begin{cases} 0 & \text{for }x=0 \ e^{-1/x^2} & \text{otherwise} \end{cases}$$ which is $C^\infty$ everywhere.

[burnt vessel]

Troposphere

[cosmic ibex]

The hairiness of nonstandard analysis is not there just out of spite, but in the service of making sure there is a unique canonical way to extend any function defined on the reals to a function with a value on every hyperreal, such that we can apply the nice-looking infinitesimal concept to it. Simply adjoining a nilpotent element to the standard reals won't do that.

(Hmm, actually nilpotents do allow division by a+eps as long as a is nonzero: 1/(a+eps) = 1/a - eps/a^2. So I withdraw the 1/x objection).

[vast zenith]

Does anyone happen to know what is the meaning of the circle

[blissful latch]

Interior of C, I think

[vast zenith]

Thank you very much

[magic minnow]

you can differentiate arbitrary C^1 functions using the approach in my screenshot. But that is just difference quotients / limits in another language. If I were to try this in a classroom, I think I would start with differentiating only polynomials / power series to build intuition, and introduce the full definition afterwards. Unfortunately there may not be enough time for this approach in a typical semester class.

[cosmic ibex]

I was responding to what you actually wrote about using ℝ[ε]/(ε²).

[cosmic ibex]

Your screenshot can indeed be taken as defining the standard derivative on C^1 functions; it doesn't have any infinitesimals other than 0 and seems to really use limits except for sweeping them under the carpet in an unseen definition of "continuous".

But ℝ[ε]/(ε²) doesn't satisfy the assumption in the screenshot, at least with order topology under the "obvious" ordering. The invertible elements are not dense in it: there are no invertibles strictly between ε and 2ε.

[tawdry venture]

1.5ε?

or am i missing something

[magic minnow]

ℝ[ε]/(ε²) has the product topology, the group of units is (ℝ˗{0})⨯ℝ, which is dense

[cosmic ibex]

1.5ε is not invertible.

[cosmic ibex]

Okay then. That doesn't feel much like an infinitesimal extension of R to me, but that might just be me.

[magic minnow]

the standard definition of an infinitesimal extension of a ring R is a surjection S -> R whose kernel is a nilpotent ideal of S

this got buried:

(also, where would be the appropriate channel on this server to share a video series I'm making on Introduction to Algorithms, following CLRS?)

[narrow nest]

how to promote the connection between word, symbol, and idea?

[normal frost]

Is this correct?

[native hemlock]

Can someone help me with math?

[tawdry venture]

https://cdn.discordapp.com/attachments/541382507656904704/971997475378458624/unknown.png

[earnest trail]

wdym

[wispy slate]

are some people just unable to grasp (even basic) math concepts? I'm currently tutoring one 16yo that struggles A LOT with just basic algebraic operations. To illustrate how badly, for example given 3x+4=5 he tried subtracting 3 to get x alone. Or given 3/2 * x = 7 he would try to multiply by 3/2 to solve for x. Or when adding/multiplying by something an equation he would only do it to one side. Or even when he was solving some linear equation and got to the point -2 = 2x he would divide by 2, write it as (-2)/2 = x and then conclude that x=-1 (seems like in this case there were too many steps). Today he also struggled to understand that (x-5)/5 is the same as (1/5) *x - 1. Is it the case of some kind of disability? Maybe someone here had similiar problems when tutoring?

[cosmic ibex]

I think it must be that he hasn't grasped what arithmetic expressions mean. We know an expression encodes a particular combination of operations, including that the outputs of some operations become the input of other operations in a fixed pattern that doesn't depend on what the numbers actually are. It seems that many of the students who really struggle haven't learned this way of thinking. At best they view evaluating an expression as a successive rewriting tasks: to them the symbols don't in themselves mean anything, but you can chant "PEMDAS" and crank the handle, and eventually it becomes a number that you draw a double line under. That's just a formal game with a mystery result, and they might not even have noticed that you can predict what the next thing to do will be before you have done this operation and substituted its result into the expression so you can consult the PEMDAS oracle about the new string of symbols.

Or even when he was solving some linear equation and got to the point -2 = 2x he would divide by 2, write it as (-2)/2 = x and then conclude that x=-1 (seems like in this case there were too many steps).
What's wrong with that, though?

If you have plenty of time, it might help to convert the symbolic expression to an explicit pictorial representation of a tree of operations. That lets you speak in terms of e.g. "see, a number comes down this pipe, and this box adds 27 and that box subtracts 27 again, so what goes further is just that thing up here, and we can cut out both operations without changing what comes out". The main point would be to get away from the conventional representation that he has trained himself to see as a matter of magical rules ("is this allowed?"), and get back to see it as an actual recipe for calculation ("will this produce the same results?")

[wispy slate]

Not 'wrong', but I really don't think its normal for a 16 year old now in high school to take like 20 seconds solving -2=2x. It kinda seems like a case of dyscalculia or something similiar at this point since I've been trying to explain these concepts in few ways, with examples, and every session same mistakes happen.

[abstract grove]

They may be able to learn to grasp it if they were to sincerely work on it every day

But then again, maybe not

If they're over a certain baseline of ability in quantitative/abstract reasoning they should be able to eventually get it with work

but it's overly generous to believe every single person is over that baseline

[long loom]

maybe have them use those old blocks/grids that at least I had as a kid and make him manipulate those as he's solving.
it seems like he has absolutely no intuition for the operations/concepts of equality/algebra, so maybe you can try and motivate that

[abstract grove]

I should add to what I said, I don't mean to write off this kid's abilities

many students have really bad misconceptions about algebra

and will try to do things like subtract 3 from both sides of 3x + 4 = 5

but once you correct them on it and they practice it the right way a few times, they can get over it

but the literal answer to the question "are some people just unable to grasp basic math concepts" is unfortunately yes

[long pelican]

I kind of wonder if things like

and will try to do things like subtract 3 from both sides of 3x + 4 = 5
are exclusively a result of being forced to do mathematical procedures that you don't understand over the years

As opposed to, say, being the natural way someone approaches a problem

[narrow nest]

Math is language.

lots of the kids i tutor struggle greatly with connecting the equation as written with the way you'd say it aloud

make your student verbally reason through things

practice reading equations out loud

and going back and forth between words and symbolic notations

[cosmic ibex]

Don't forget alternative graphical representations. They're important for understanding.

[native hemlock]

What would it mean to understand the procedure of solving a linear equation?

understanding that a bunch of operations have been applied to x in a particular order and you want to apply their inverses in the reverse order?

[long pelican]

At the most basic level, what the legal "moves" are (like in chess) are and why they're legal

My point above was that I was wondering if a natural problem solving situation would get you to err on the side of not trying everything or not knowing what leads to the solution, not err on the side of making illegal moves, and maybe that erring on the side of making illegal moves is the product of your previous math education

(Even if your previous math education didn't teach you anything incorrect)

[cosmic ibex]

Yeah, seems like the only way you could even form the idea of subtracting the 3 in 3x+4 would be if you have seen a procedure with "subtract numbers to make them go away" but not understood why that procedure works.

Which could be the result of being told "do these steps because they are the legal ones" rather than "do these steps because they work").

[winged urchin]

I like to take the students instructions and "perform" them myself and show them they aren't getting what they expect to get

Oh you want to subtract 3 from both sides?

What would (3x+4)-3 become then? I would ask

If they still get confused then pick a particular value for x and hopefully that makes it clearer for them

When I'm feeling particularly philosophical that day I'll describe solving equations or working with expressions, algebra really, as trying to find a path to a point in a forest or mountain

There isn't just one way in basically any case. You can always deviate from the path or use a weird combination of 'paths' but you have to still have to obey the rules of the world you're trekking through

If I'm hiking to the top of a mountain I can't just start flying or just teleport there of some nonsense

[haughty creek]

Jk I saw an opportunity to use this meme

[native hemlock]

How to solve equation

[pastel sundial]

@long pelican I think I've asked you this before but what would you say are Hung-Hsi Wu's best articles?

[long pelican]

https://math.berkeley.edu/~wu/ He posts his articles on this website, and they're all good. Pick titles that interest you! The best one in my opinion is his latest one, titled "Learnable and unlearnable school mathematics" where he takes a perspective shift away from viewing the problem as a problem in failing to teach students math, and more as a problem in teaching students what he calls "unlearnable" math

[brittle igloo]

IM DEAD NOT AGAIN FS

[pastel sundial]

lmao my final paper for my education class has basically turned into an advertisement of Wu's work

[long pelican]

Interesting lol
I'd be interested in seeing it

[pastel sundial]

maybe I'll post it but I'd need to remove my name and read it over to see that it's not otherwise a dox

and also I'd need to be confident it's good enough to share publicly lmao

[shadow basalt]

You should also make sure that the text itself isn't a dox

like its likely your university keeps the pdf in a place where it can be found if you google the text of your document

[earnest trail]

orrrrr just send it to specific trustworthy people

[pastel sundial]

I don't think they do? but yeah still worth checking

[pastel sundial]

this is probably the best idea. Like I trust Icy

[earnest trail]

and you should definitely trust me

totally 100%

jk lol don't ever trust me

[pastel sundial]

don't worry I won't

[earnest trail]

LOL

[pastel sundial]

I tried a text search of a paper from last semester and the one before and got nothing

and I'm not aware of any such archive

but yeah still I wouldn't post it publically

[shadow basalt]

Yeah I just know that a lot of theses end up getting publically posted

in very easily googleable ways

and sometimes profs keep them on their websites or something

[pastel sundial]

yes this is very much not a thesis

it is a final paper for an undergrad class in a subject I'm not majoring or minoring in lmfao

[cosmic ibex]

I read through it, and it seems to me that his concept of "unlearnable mathematics" looks rather disingenuous. The mathematics he denounces as "unlearnable" is exactly the mathematics we all know and love, except modified by "... but taught without any of the explanations". As regards the extent of the subject matter it doesn't look like he argues for any major upheaval, other than including the explanations that were excluded from his "unlearnable mathematics" concept.

[pastel sundial]

I think the point of calling the mathematics itself unlearnable is to put the blame on the textbook and curriculum not the teachers themselves

it is a bit weird though that he says it's the math itself that's unlearnable because yeah I don't think that's an entirely accurate way of putting it

and like, "but taught without any of the explanations" is a pretty major modification

oh Icy how did you ever do on the student evaluations you were talking about a while back

I think it was around last December?

[long pelican]

I think I did mention how they went

5/14 responded in the first class and those are the ones who complained that I was hard to understand (bad room acoustics or my deaf accent)
Everyone in the second class responded, none of them said I was hard to understand
They had zero complaints about the exams, but a lot of complaints about the difficulty of the homework

I remember that I also shared some comments here

They commented the homework required you to truly understand the material, but that there was not a lot of guiding on how exactly to do that because they were not used to having to do that

Something I realize in retrospect is that having to follow the day by day calendar (it's a coordinated class) really harmed the connectedness of my lectures from day to day, and the structure basically told the students that math is a set of topics to cover

as opposed to a new way of thinking

[pastel sundial]

I assumed you had mentioned it somewhere, I just forget if I had read it at the time. Although now that you repeat it it does sound familiar.

also I finished and submitted my paper but I don't think it really contains anything insightful beyond just a succinct summary of Wu's idas

[long pelican]

Does that mean you won’t be sharing your paper 😦

[pastel sundial]

I mean if you want to read it anyways I might

[humble aspen]

What would be a good way to help students struggling with algebraic manipulation? For instance, like taking an equation mp = q^2 * r and solving for q. If there's any articles or whatnot as well feel free to lmk. I'm working to find ways to make teaching or tutoring people easier.

[tawdry venture]

i would try to go back to basics

make sure the student understands things such as "doing the same thing to both sides"

[abstract grove]

yeah. Start with solving things like x - 7 = 10

then 3x = 15

then put those together with 3x - 7 = 8

finally introduce x^2 = 25

x^2 - 7 = 18

if they can't do those they're not ready for mp = q^2 * r

probably have them drill multiplication and division tables too as they're most likely inadequate

[winged urchin]

Random discussion spawning question:

How would you explain functions, equations, and their interaction when solving problems, to like... a teen or younger student who is confused by solving sometimes? Anyone have any decent analogues or more... intuitive statements rather than explaining it in a more 'mathy' way so to speak

[zinc wigeon]

An equation is simply a mathematical statement that says that 2 things are equal

A function is a “machine” that eats things and spits out other things.
f denotes the machine itself and f(x) - the thing spit out by f when you give it x.

[surreal warren]

How to make notes? I end up taking a long time to finish a book because I basically copy everything...

How should I study through math textbooks?

[earnest trail]

don't copy every little thing, at that point you're rewriting the book

but at the same time write what you feel is necessary

maybe you don't need to write down all the details of a proof but maybe just a proof sketch or smth

[topaz scarab]

I like this, the brackets ( ) looks like a mouth.

[surreal warren]

Oh. Thank you so much.

[burnt quail]

Hello, I want to ask what algorithm do you use for verbal problems? For example with a system of equations. Modeling with a system of equations. What approach would you take to explain to someone how to deal with these problems?

[winged urchin]

I have invented a word, "Mathematicalization", that I often use in these situations

I'll point to a sentence like "Sarah makes $6 for every banana she sells and her banana cart costs $20" and ask them to 'mathematicalize' it. Which I explain means turning that into some sort of equation describing the situation

The coming up with the equation is, I think, more of a thing learned through practice and the difficulties that come up are quite varied and I generally have to just step them through the problem

Let them come up with what they think the 'mathematicalized' version of the sentence is then ask them questions about it

"If Sarah sells 6 bananas how much money will she have? Does that match with your equation?"

It's a fair bit of back and forth

[teal moth]

would the word "abstraction" instead of mathematicalization be wrong?

[zenith slate]

I think abstraction is a perfectly good term for it, but not necessarily a term you'd want to use with middle-schoolers, say

[pearl gate]

i would disagree

you're not making anything more abstract

,wolf abstract

[burnt vessel]

Results provided by WolframAlpha

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

Requested by koala#7005

[pearl gate]

it means to make "more general"

but ur not really making anything more general

ur just rephrasing a scenario in a different language

the language happens to be mathematics

[zenith slate]

I'd say that formalising a real-word problem as an equation is necessarily an abstraction

The equation itself is more abstracted than the original problem

And indeed may describe innumerable other similar word problems

But really it's just semantics at that point

[teal moth]

ok so mathematicalization it is then

abstraction involves variables 😛

mathematicalization could just be about literals

1+1=2

trivial, but thats what middles schoolers crave

im so glad i'm not a middle schooler, because that would drive me crazy

[winged urchin]

In my mind abstraction is basically when you finish one problem and then ask a more generalized version of that problem

"Oh I've found the area of a regular pentagon with side lengths s. I wonder what the area of a regular hexagon is... Or shudders with curiousity what's the area of a regular n-gon! =OOOOO"

Mathematicalization is, to me, more about viewing math as a language and converting a statement in English or whatever other language and writing something in 'Math'

[abstract grove]

Mathematicalization is simply mathematical modeling

[winged urchin]

You are orthogonal indeed

[abstract grove]

I'm not sure what you mean by that

[winged urchin]

Lol yeah I'm sorry I originally thought I was using it semi-synonyously with 'right' but then it occurred to me that it could also be interpreted as you being somehow completely wrong since being orthogonal to something is in some way going in a completely different direction

Then I thought that was a cute double meaning with opposite meanings so I stuck with it lol

I agree with you ahah

[abstract grove]

ahh gotcha

[teal moth]

we call it modeling too in programming

[deep jewel]

would you guys say it's better to actively take notes during a lecture, or instead spend that time trying to make connections and understanding the material, and then taking notes right afterwards?

[winged urchin]

150% would spend time just thinking of what's being said, writing down rough work if it helps you better understand, and asking questions

[deep jewel]

hm

would it better to just purely try to understand the material being taught and save the questions for later tho?

to avoid running into a rabbit hole

[winged urchin]

Hopefully the notes are available after anyways. If the professor wants you to take notes as the only way to have them afterwards then you're kind of stuck and need to take notes

[deep jewel]

yea

[winged urchin]

In my opinion questions are important to understanding. Of course you don't want to stall things too much but one or two questions isn't ridiculous

If you really feel shy or otherwise inhibited from asking questions during lecture you can always write them down and ask afterwards

[deep jewel]

i can understand the social aspect of it

[long pelican]

I’d even say professors desperately want questions to gauge how fast or slow they should go and feel comfortable they’re being understood

[deep jewel]

but sometimes for me personally it's really a question of how much time i wanna dedicate to a question

yea

i suppose im veering more towards the more ambigious questions, or rather the ones that you'd need an hour to explain to a good extent

i guess that's really just a question of whether you should wait to see if the prof. explains it or if you should ask it

[long pelican]

That internal battle happens all the time

[deep jewel]

yea

it's really annoying sometimes

spent a week trying to figure out what a slope really meant

felt weird that it simply is what it is to an extent

[long pelican]

Slope of a line or slope of a vector bundle?

[deep jewel]

of a line

[winged urchin]

lol Icy, ahah

I love it

[deep jewel]

i don't know

i just guess i never truly realized that it's to an extent just a transformation

[long pelican]

Slope isn’t really transformation

[deep jewel]

well i guess more a transformation of x to y or vice versa

[long pelican]

It’s a number indicating steepness/angle in a numerical way

You could do the same thing with angle but that would use trig

[deep jewel]

yea but if you're assuming there no's c in y=mx

i personally see it as just a mapping at that point

i guess i was just looking for a more discrete way to see slope

more tactile

and that also brought me to slope point form

[long pelican]

Can’t really understand your second line

[winged urchin]

It sounds like you're trying to make sense of your material in whatever way makes sense to you right now. Sometimes you just need to munch on what you're learning over and over and hopefully over time your understanding will get more refined

[shadow basalt]

I would say the exact opposite

My personal opinion is that you should try to spend the lecture actively engaged and thinking

Especially when everything being said is already in the textbook

which is true for basically all undergrad classes

[deep jewel]

yea i have to agree, specially if the material is written somewhere else

also

[deep jewel]

when i say mapping i mean mapping from x to y

im not completely sure if im using the terminology correctly

but basically just transforming x or y to the other with m/slope

though im not really accounting for c in y=mx+c

which is something ive thinking about recently

[long pelican]

OK I understand it a little more now

[deep jewel]

also yea, ive been trying to get less abstract about calculus lately so ive taken some time trying to learn the basics more

i suppose i just don't like taking things for granted

[long pelican]

The phrase "mapping from x to y" is not really a standard phrase in mathematics though

In math, we map from a set to another set, not from a variable to another variable

(a variable is supposed to stand for an element of a set)

[deep jewel]

i guess i was thinking of it kinda like this

[long pelican]

Well in that representation, slope is even less of a mapping

because the slope can be calculated by, for example, (6-3)/(2-1)

It's a difference of two numbers, divided by a difference of another 2 numbers

[deep jewel]

i was thinking of it like the transformation between the two sets x and y

[long pelican]

So the two sets are actually $\bR$ and $\bR$

[burnt vessel]

Icy001

[long pelican]

x and y aren't the sets, they're variables

[deep jewel]

ah i see

well what if we have a function which has a specific domain and range

something like

y=x^2

where you can't have y be below 0

could you then call it a mapping?

[long pelican]

Well remember how when you graph, you start by drawing two axes

[deep jewel]

ye

[long pelican]

The entire paper represents the Cartesian plane

Each point on the paper represents an ordered pair of real numbers

When you graphed y=x^2 you drew the set of points (a,b) such that b is equal to the square of a

Here I used a and b to represent names of a general real number

[deep jewel]

yup

[long pelican]

So the phrase "Graph y=x^2" is shorthand for

``Draw the set of points $(x,y)$ in $\bR^2$ such that $y=x^2$''

[burnt vessel]

Icy001

[long pelican]

Anyway see in that sentence, x and y aren't sets

They're quantified numbers (aka variables)

[deep jewel]

are we assuming that the points are in r^2 because we are in the cartesian plane?

or rather the set of points

[long pelican]

Yes, $\bR^2=\bR\times\bR={(x,y):x,y\in\bR}$

[burnt vessel]

Icy001

[deep jewel]

interesting

also

I know this is a bit of an arbitary question which I could just search for

but is there a set which contains both r and c?

just asking here since we're chatting about it anyway

[long pelican]

$\bC$ contains both $\bR$ and $\bC$

[burnt vessel]

Icy001

[deep jewel]

ah i see

last question since i need to sleep and this is a bit off-topic, but what notation would you use to denote that your dealing with a non-euclidean plane

[winged urchin]

non-euclidean plane? Are you trying to talk about like spherical or hyperbolic geometries?

[deep jewel]

yea

[long pelican]

The 2-sphere is denoted $S^2$ and the hyperbolic plane is denoted by $\mathbb H$ or $\mathcal H$

[burnt vessel]

Icy001

[deep jewel]

ah alright

i see

one things for certain, i def. need to do a lot more research on terminology and notation

thanks for all the help guys

[winged urchin]

Don't feel like you need to run or sprint through the material

Just keep asking questions and steadily going through the material as you do

[deep jewel]

yup

that's includes a lot more then mathematics too

thanks again

[tawdry venture]

so it appears that beginner linear algebra students, when faced with a problem that asks them to apply some basic definition, like that of linearity or that of a subspace, are often confused because they don't know how to introduce variables for things in the definition

like for example if they were given the set W = {(x,y,z) in R^3 | x+y+7z = 0} and asked to show it's a subspace of R^3 they would not think to take two vectors (x, y, z), (x', y', z') in W or they would ask "where did x', y' and z' come from?" when shown the solution

does this problem have a name and is there a way to fix it

[long pelican]

Unfamiliarity with the methodology of proofs?

[abstract grove]

yeah I'm not familiar with a specific name for that weakness

it's basically choosing a representative for some structure or concept

or declaring variables

[cosmic ibex]

Lacking an intuitive sense of use and purpose of variables, and falling back to attempting to follow some kind of external authority for which ones to have.

[shadow basalt]

Yeah I think most early math education follows a pattern of "You give me explicit stuff and tell me what to do with it, and then I do that"

It is a sort of nontrivial jump in abstraction to go from that to inventing your own approach to solving the problem.

I think it mostly just takes a lot of practice / examples. Make sure you avoid over-explaining the results when they ask for help, and instead lead them through the steps you yourself take when you approach such a problem. It might be hard to actually break down your own thought process on something that is now so automatic to you.

I guess you should also make sure they actually understand what it means to be told to show that X implies Y. Like make sure they understand that if I give you a statement of the form If X then Y, what I expect you to do is assume X and show Y. It feels tautological when you already understand it haha.

[cosmic ibex]

Perhaps for students with that problem, the first answer to "where did x', y', and z' come from?" ought to be "oh, I made them up!" rather than jumping directly to an explanation of why inventing those variables is helpful in the situation.

[shadow basalt]

I think students often aren't like used to that level of agency?

The closest thing in highschool is probably getting to "choose" u and v for integration by parts or whatever

u subs

but even that comes from the problem

[abstract grove]

not really true

students are told to give names to unknown quantities of word problems all the time

it's step 1 of solving a word problem

[shadow basalt]

I think that is substantially different. Naming a given quantity by letting "water" be "w" is not the same level of abstraction as saying let "v" be an arbitrary vector in this set.

Students are used to the former, and it is not really any different from being given a name for water by the problem

[abstract grove]

I don't see how it's fundamentally different. If a vector space problem involves showing some property of a vector which isn't given, you give it a name

also note I said giving names to unknown quantities, not given ones

though both are important

[shadow basalt]

I am not really interested in arguing about this haha, it is my experience as someone who has taught a lot of math over the years. Students tend to struggle, as Ann said in the original message, with the agency involved in letting "v" be an abstract concept against which you can only leverage axioms rather than stuff you actually know. They are used to being asked specific questions about specific quantities or terms.

[abstract grove]

you're welcome not to discuss further (I don't think we're arguing). I just think this situation is much like a word problem like "two ships start at (10,5) and (3,6); the first goes north at 2 mi/hr and the second goes east at 3 mi/hr. How far apart are they two hours later?" And the student is expected to bring in the variable d to represent the distance, though nothing about that is given

it's basically a problem of precisely modeling the situation with symbols

[long pelican]

Does anyone remember a time in their life when a proof of "Prove that _ for all x in S" starting with "Let x be in S." was not completely transparent to them?

[abstract grove]

barely

I took a course on proofs called "intro to higher mathematics" my first semester as a physics major where we learned those kinds of things

I'm sure that wasn't completely transparent at the time

but it's 15 years ago now

[cosmic ibex]

If a student is sufficiently beholden to the "solve problems by following rules" paradigm, one of the things that frustrates them is that they feel they're missing the rules for which of the unknowns to give names. For example, the problem says that Alice is 2 years older than Bob, but does that mean that we should

1. Introduce two variables A and B, and write down an equation A=2+B.
2. Introduce the variable B and represent Alice's age as B+2 when interpreting the rest of the information in the problem.
3. Introduce the variable A and represent Bob's age as A-2 when interpreting the rest of the information in the problem.
   Each time they ask someone more knowledgeable for help, the helper will pick one of these in some inscrutable way (we know it's basically at random), and if the student asks "why did you do that" they'll get a long-winded explanation of why that works but which doesn't even begin to answer "why didn't you pick one of the other ways". The students shuts up and doubles down on their conviction that they're "bad at math" for not being able to choose the correct way to proceed ...

[long pelican]

Even though it's hard for me to imagine how it's not transparent, I just have to tell myself that for a class of freshmen, it's most likely transparent to none of them

[shadow basalt]

I think it helps to instead reframe the question @long pelican to something more like

is there any subject in which other people around you quickly understood something but oyu did not

For me there are even math-related things where this happened to me

So I just try to approach students who struggle with concepts I found easy in the same way I'd want someone to help me with stuff I find difficult you know?

[long pelican]

All the time in math if you're surrounded by grad students. (Lol)

[shadow basalt]

Yeah but like even simple things that just took me forever to click hahah

ofc i am surrounded by people who know stuff i don't at all

[long pelican]

It's still too easy to gloss over something difficult without even being remotely aware that that happened

You have to, like, prepare beforehand by thinking about what you're teaching extremely slowly

[shadow basalt]

Yeah for sure.

The weirdest part for me as a newer teacher is like

when i explain the same thing twice in basically the same way

and the student reacts super differently

even just changing variable names can have this affect and I have no idea how to predict it lol

[long pelican]

Could even happen if you say the exact same thing twice as well?

[abstract grove]

it totally can

sometimes they just need time to get it

[long pelican]

My most successful method of helping is the rubber duck method

Where the student does the talking and I only interject to say something they already know, in a particular way

[shadow basalt]

Right

[long pelican]

And they realize how to solve the problem from all the talking they do

[shadow basalt]

I wish I could do more of that kind of stuff

Maybe I should try it more in the classroom

but students are kind of reluctant to speak

[long pelican]

Students not talking is pretty rough. Either they're uninterested or are lost and think asking a question isn't going to work to unlose them

One excellent idea that I implemented last semester was simply to add the explicit question "How followable are the lectures? (with a slider, the left side being "I get lost" and the right side being "I can follow them in real time")

to the mid-course survey

And the survey is anonymous of course

[tawdry venture]

ought to be "oh, I made them up!"
or as i like to say it, "pulled them out of my ass" :P

and if the student asks "why did you do that" they'll get a long-winded explanation of why that works but which doesn't even begin to answer "why didn't you pick one of the other ways".
and this is why i usually try my best to say something along the lines of "of course you could've done this way or that way instead and it would make the calculations appear different but otherwise essentially the same. i'll just keep going with what i have on the board right now"

[abstract grove]

Ann your question reminded me a bit of this article. Not exactly the same thing but hopefully it's interesting. https://www.google.com/amp/s/gowers.wordpress.com/2008/08/16/just-do-it-proofs/amp/

[winged urchin]

I do personally make a point to be careful about using the word the wrong

Not even just in higher levels of math, but even like.. algebra

There's many ways to manipulate equations. Many ways to setup a problem

I only really use 'wrong' if they actually did an illogical step that breaks the equation or something like that

I'll use 'inefficient' if I think there is a simpler or better way to manipulate the equation or set it up

And I do think it's important to highlight that what I'm saying is just one way of doing things

Although I sometimes worry that showing many different ways to do something is not... Good for some students. If they're struggling sometimes I think it might be better just focus on what I think is the best way and not really go into other ways

[south raptor]

What is everyone's thoughts on just not assigning PEMDAS pages anymore?

I work at a tutoring center so it's not like I'm required to assign it because of standards or anything

[cosmic ibex]

"PEMDAS pages" sure sounds soul-crushing.
Are you talking about exercises that demand the evaluation of an arithmetic expression by rewriting a single operation at a time?

If so, I'd say it is relevant to train precedence conventions, but those exercises are not necessarily the best way to do it.

I like the "machine" model, where you understand an expression as a tree of operations, explained as a structure that the numbers move through from the leaves towards the root. (Though not necessarily explained in "tree" terminology, mind you). That way the structure of the expression stays the same while we evaluate it, instead of disappearing before our eyes while the calculation happens. This makes it easier to make the jump to algebraic expressions with letters in them later: the letters are input slots in our machine, and we can try the machine with many different inputs without destroying it along the way.

And the machine model makes more apparent that you don't need to worry about the permissibility of rewriting 3×4+x² to 12+x² even though PEMDAS seems to dictate that we shouldn't do the multiplication before we've been able to evaluate the power!

[south raptor]

The first couple pages have problems that just have two operations and requires them to identify which operation has to be performed first. Then by the end they're asked to take an entire expression and reduce to a single number.

So they don't have to rewrite it for every operation they take

[winged urchin]

Well I'd say they really shouldn't be skipping steps at that level. I find students try to jump too much when working out their algebra and it leads to misunderstandings of what they are actually doing or errors

It feels tedious and it is. But it's kinda like a writers license kind of vibe in my opinion. Someone learning to write should learn all the rules in detail and try to abide by them but as you get more experienced you can start skipping steps or breaking 'rules'

[tawdry venture]

jumping ahead means you can't debug your work later

[south raptor]

hmm you guys are making me think PEMDAS is more important than I thought even if it is pretty tedious. I'll keep assigning it for now. Thanks for the feedback everyone 🙂

[pastel horizon]

Some of the activities on NRICH are bloody brilliant

[tepid smelt]

I like order of operations puzzles.
Like (1+( 1 +(1 +1) * 2) * 2 )* 2. You can erase one open parenthesis and one close parenthesis or none. How many different solutions are possible? What is the maximum or minimum value you can make. Or like small competition like using numbers 5 4 3 2 1 in order and basic operations and parenthesis have groups race to create as many numbers as possible starting at 1. These are really games for elementary school level but make practice with order of operations less boring

[winged urchin]

Do y'all think we can kick the quadratic formula outta the curriculum? Ahah

Just focus more on completing the square

If the students make the connection between completing the square and what theyd probably have seen online or something as the 'quadratic formula' then all the more powerful and probably interest to them!

[tepid smelt]

Have you seen the approach poh shen loh has shared regarding the quadratic formula? I like the approach that 3blue1brown shared regarding it here https://youtu.be/MHXO86wKeDY

[shadow basalt]

No it’s useful

And completing the square is something someone should do like

Exactly once

[wise onyx]

we should teach kids Klein's icosahedral solution to the quintic

2F1 is the hypergeometric function

[long pelican]

228, 494, 522, 10005 👀

[pastel horizon]

I mean in industry I don't think they even bother using quadratic formula I think they just use Newton Raphson.

I think it's more interesting to be aware that there's a general formula

[cosmic ibex]

The quadratic formula is also a good example for "recognize that a problem has a specific form, and plug parameters from it into a pre-cooked formula".

[cosmic ibex]

That general skill is probably more important than the specific task of solving quadratics.

[tawdry venture]

how do you teach that skill though?

[cosmic ibex]

Practice?

[tawdry venture]

i mean tbh like, maybe this is me being close-minded to hell but i cant even see it as a skill, it's just something that ive always seen as second nature at best and something to "just do" at worst

[cosmic ibex]

That's good for you -- but you must have noticed that some students (or askers here) struggle with getting the concept -- both "in the small", with figuring out what the coefficients even are if some are ±1 or 0, and "in the large", with noticing that the thing they're looking at can become a quadratic equation by giving a name to a common subexpression and then rearranging.

[vast zenith]

That's a really great article I read it

[tawdry venture]

i thought all this time that lacking this skill would be like having the formula right in front of you, or you be able to recall it if asked, but unable to connect "V = L * W * H" and "L = 20, W = 30, H = 40" to get "V = 20 * 30 * 40"

[abstract grove]

Plugging into a formula that's given is one thing, but recognizing the form an expression is in and what formula applies to it is a harder thing

[tawdry venture]

do y'all think graph theory could be taught to students as early as middle school?

at least the basics of it, with plenty of pictures and whatnot

[native hemlock]

No lmao

Graph theory is so hard

[tawdry venture]

you think the concept of "a bunch of points connected by lines" is hard? i'm not imagining a formalist approach here jsyk

i'm talking shit like handshake lemma, connectedness, degree counting

[native hemlock]

Well I really struggle with it anyway

But I definitely struggle with easy things

[cosmic ibex]

Those are good ideas to teach. In an age-appropriate manner, of course.

They're very visual, and would probably be a good early introduction to concepts that will be useful later but are commonly only encountered in a very abstract setting.

[cosmic ibex]

Isomorphisms, for example. (Probably without the fancy word, though). There's a very physical sense in which isomorphic drawings of a graph are just the same strings-and-knots network laid down on the paper in different ways. But if we just have drawings of the graphs and can't pick them up and refold them, how do we convince ourselves they're isomorphic? Then we can learn about writing down a table of node correspondences and checking isomorphism as a purely local condition.
As a bonus, determining graph isomorphism is one of the few middle-school-accessible problems where (a) the problem is easy to understand, (b) it can be solved by eye in many simple cases, but (c) there is no good general algorithm to teach for it. That in itself can be a useful antidote to the feeling that "mathematics is about following rules" that school mathematics tends to create.

[austere inlet]

yeah I feel like some key ideas in graph theory could be taught through word problems (just like how most middle school math is taught), assuming the students are motivated and interested enough

[abstract grove]

absolutely graph theory basics can be taught to middle schoolers

I showed 7th graders the hand shaking lemma the other day.. not in a graph theory context, but most of them had no trouble following it

[abstract grove]

actually what I showed is the number of handshakes for n people is n(n-1)/2 which I guess isn't what's usually called the handshaking lemma but anyway

[earnest trail]

nooooo don't ruin it with word problems

kids fucking hate word problems

use pictures instead!

[tepid smelt]

At proof school here in the bay area they do graph theory freshman year but many start it in junior high. I think combinatorics/number theory/ discrete math is the start of more proof based math at the middle school level. I have coworkers who have kids at the school and my daughter is planning to go there.

With my daughter I have found that it's more useful to go deep on certain topics and math contests have really been the most engaging for her. The traditional curriculum for motivated students are garbage but contest math in general have good problems that really train you to think about math the right way.

[frigid crest]

Hello. Yes.

The last student I tutored started 6th grade at a public school last year at the normal age.

I helped him study proofs about numbers using Mathematical Proofs by Chartrand, Polimeni, and Zhang.

He was probably both the most gifted student and the least conscientious student that I have had.

He was definitely not a genius.

[tawdry venture]

is it good practice to teach kids a set-in-stone rule that if you introduce a new variable then you have to clearly define what it stands for?

[earnest trail]

I would think so

it's a good habit to learn early on

you can be like "okay so let's call our variable x. what does it mean here?" and let them answer you

in lessons and stuff

and maybe on the first few worksheets you can have a question that asks "what does x mean?"

because clarity in work is necessary to learn

[long pelican]

Y e s! And also its quantification (for all or there exists)

[tawdry venture]

what i had in mind was things like "Let x = amount of apples sold in one day, in kilograms" and things along those lines

[long pelican]

Oh ok. Make sure to emphasize the point is to imagine they’re sharing the solution with their classmate who will be wanting to understand it, rather than the teacher who is just grading whether it’s right or not

[tawdry venture]

the teacher needs to understand it too

[long pelican]

But the teacher already knows the solution so it’s just not the same from the point of view of the student

[frigid crest]

I agree.

[quasi musk]

I do that stuff all the time

I give "challenge" problems that are often times topological or graph theoretic in nature

I want to show students that math is more than just "solving for x" or "computing"

I want to show them that it's an intensely creative field

All my students did well on their practice final, so I'm replacing their final with their practice final. I'm giving a math puzzle day

Questions like "What does a circle passing through a 1D world look like? How about a sphere passing a 2D world? What would a 4 dimensional sphere passing through our world look like?"

Can you draw a 4 crossing knot? 5 crossing? 6?

When is a # divisible by 11?

You can give graph theory problems in terms of driving from town to town on different roads. But each road has a different toll. Can you minimize the amount you pay with respect to how far each road is to gas cost?

[earnest trail]

you are based

[earnest trail]

yesssss we need more teachers like this

[quasi musk]

Update: the kids got rowdy about half way thru

I told them if they didn't behave they'd have to take the final

They had to take the final after spending half the class on those puzzles

[long pelican]

Sad ending

[earnest trail]

awww that's unfortunate

how old are these students?

[quasi musk]

10-11

[earnest trail]

oh I see

rip

[tawdry venture]

hot take: the functions sec and csc should not be taught in school

[hollow kernel]

Don't know what US schools do, but here in Germany I have never heard of those before uni and never needed them anyways so far. I definitely agree, teaching sin, cos, and tan and their inverses as "the" trigonometry functions in school is absolutely sufficient. Better cover important concepts in depth than confuse the majority of students by doing a bit of everything.

[naive crypt]

Elo! I just wanna say that I have the advanced role because I have some friends interested in learning math and I want to help tutor them, and so would find this channel very helpful. I don't think I did anything wrong but if I did, do let me know, thank you!

I've always been told I'm a good teacher but I want to make sure my friends actually understand what I'm saying and aren't just memorizing it. Is there anywhere I can read up on teaching better?

[long pelican]

If you want to see if your friends understand what you’re saying, a good test for that is to give them a problem that’s not similar to any other problems they’ve done but which they have the tools to solve, and see what they try to do as they try to solve it

If you haven’t started yet, you should probably try it out and do the above to see your weaknesses, then find some readings afterwards

[earnest trail]

true, I think it's sufficient to just use 1/sin, 1/cos, and 1/tan when necessary rather than introducing 'new' notation

[winged urchin]

Yeah. Honestly why even introduce them? I know that sounds trollish but we're trying to give kids a basic idea of trig. Reduce the number of different things they need to understand. Introduce the unit circle with my the horizontal position of points given by cos, vertical by sin, and slope by tan. Let them reason from that and hopefully reach some understanding

Csc, sec, and cot are weird even from the geometrical perspective

Most tests I see as a tutor anyways don't test on the reciprocal functions anyway so who cares even there

[tawny slate]

i agree on principle but before i take away those functions I would rather first see a standardized notation that makes a distinction between exponents and function iterations

we write x^(-1) to indicate the reciprocal of x, but we do f^(-1)(x) to indicate the function inverse of f. and yet, when we write sin^(-1) it is an inverse as it should be, but we write sin^2 to indicate sin*sin, as an exponent. what gives?

by this logic, one would think that if we could standardize these differences, that we should also get rid of arcsin, arccos, arctan, but I think those functions can stay because they're not "true" inverses, since the domain is restricted in order to make the inverse possible

[tawdry venture]

i think it's arcsin, arccos and arctan that should stay

to emphasize that they are not "true" inverses

[tulip badge]

Yeah, when either teaching pre-calculus courses and below or doing formal proofs, I mainly use the arc notation these days, and reserve the ^(-1) for either calculus and above or particularly long and usually less formal calculations. When I was a student, we were taught with the ^(-1) notation and I didn't get the purpose of the arc notation.

[tawdry venture]

also i think the notation "km h^-1" or god forbid "m s^-1" should not be used at the school level to talk about speed

[shadow basalt]

I think ^-1 is a bad notation for inverse functions in general

At least for high schoolers

[naive crypt]

thank you so much!

[grim spindle]

@shadow basalt do you have an alternative suggestion? I don't think i've ever seen a good alternative to ^-1 for inverses (Besides division which is bad notation in noncommutative settings imo)

[shadow basalt]

Idk any other symbol

Dagger works

Maybe just put the letter i

Or just f^inv

Really anything else lol

[winged urchin]

I think it is primarily the problem of being confused for powers. So yeah anything that isn't a number gets around that problem.

But as with most things like this it would be awfully difficult to change everything. Calculator buttons, old textbooks, etc etc

Functional notation in general is pretty terrible for students imo

I'm sure there exists a significantly better notation for functions (for students) but I do not know myself

Maybe there exists a better notation that also allows a better notation for inverses

[ruby wharf]

@twin lichen for example when I learned differential geometry for first time,I was introduced to tangent vectors liek this

I had to spend like 3 days to understand this single sentence

it was so obscure, lol

[twin lichen]

is that a joke?

[ruby wharf]

no

[twin lichen]

Lmfaooo

[ruby wharf]

this is how I was introduced to tangent vectors

and he did not define what a jet is....

also,up to date,I still did not understand his definition of fiber bundle, lol

he never defined what a foliation is(transverrse direction?? saturated?)

this is one of the big reasons I am very detailed in my documents,it is a real struggle to learn mathematics for me this way

[gusty vine]

bruh

[twin lichen]

bro what are these notes

what the fuck am i looking at

[gusty vine]

that's like unironically defining group as groupoid w one object

[ruby wharf]

https://florianstecker.de/Skripte/

this is the differential geometry script (filled in with details), as much as the author could

but we had like way less details than this - i unfortunately found this script only at mid semester

[twin lichen]

i'm going to be real with you, i'm not a differential geoemter and i don't remember what a foliation is

i didn't learn what a foliation was before a fiber bundle haha

[ruby wharf]

that's the point, I think it is very very not pedagogical

[twin lichen]

get a different book on diff geo tbh

[ruby wharf]

I did that. Our curriculum was just too awkward

[upper solar]

I actually hate edexcel maths so much. They ask questions which are nothing like anything in the textbook and there is never enough time. So either you think about the questions logically and run out of time, or you blindly use the method and hope it’s applicable.

It encourages the bland style of learning that gives maths such a bad reputation

[long pelican]

Having to think about the questions logically sounds like a good math exam

Although this rant might be a common sentiment among lots of people, which could explain why freshmen calculus students don't like their math classes

[upper solar]

Yes exams should be like this. However I feel that in the quest for discerning between top students, cramming more questions into a paper isn’t the answer

[long pelican]

I'm all for more time per question

[earnest trail]

please

AP exams are awful bc you don't have much time to think

they are essentially testing you based on how quickly you can answer questions

like

huh

????

dat shit make no sense

[covert hearth]

I'm curious... I just learned about cognitive load theory and it leaves me a little confused

The things i understand best in math so far are things I've had to think really hard about

but it appears from the persp of cognitive load, if you want to learn something you'd want to break it down into tons of trivialities and analogies?

like I've noticed a marked improvement in my personal learning since I've come to a school where you're required to figure things out yourself/fill in the gaps yourself, but it appears cog load theory would assume the opposite?

oh and as a side note, i loved AP calc and the exams, it's true that answering questions as quickly as possible is a bit unfair in several respects i think, but doing them quickly was just extremely satisfying 🤣🤣

it took a lot of practice but if i can do it anyone can

[covert hearth]

(this could be placebo, it could also be that i actually like this stuff vs what i was doing in undergrad)

[zenith slate]

Cognitive load theory is not without its critics

Sweller's work on germane load might be of interest to you

You've also got to appreciate that the development of CLT was motivated largely by a need to understand how students learn in a classroom environment. Its applicability to individuals is kinda debatable, and it's largely used in the literature (at least to my knowledge) to inform the design of activities for students in a manner that reduces unnecessary load

[zenith slate]

Sorry, not Sweller. I meant to say van Merrienboer.

Sweller originated CLT, did some work on germane load (essentially beneficial cognitive load), then kinda fell out with the idea

[long pelican]

I’m not sure how much of these theories apply to math, seeing that math is cognitively about solving puzzles while most non-math subjects in school are just about learning information

[zenith slate]

It's actually of particular interest to math

Indeed the paper that kick-started the whole thing was, while not about mathematics, specifically in the context of cognitive factors affecting performance in problem solving activities, see Sweller, J., 1988. Cognitive load during problem solving: Effects on learning. Cognitive science, 12(2), pp.257-285.

It's often applied to mathematics in particular when considering the effectiveness of classic "applied" math problems you see in the classroom. Crap like "Jimmy runs a supermarket and sells 28,000 watermelons per day"

The idea being that the extraneous information is often detrimental to learning the actual core mathematical ideas

This is counterbalanced by the idea of germane load, which is associated with students exhibiting better knowledge transfer

There is no real consensus, however, in the literature, and observed effects depend heavily (like everything in education and pedagogy) on the studied demographic

[shadow basalt]

I am no expert but I feel like at some point you do have to learn to separate essential from extraneous

Like to solve actual real life problems

And I see a lot of students struggle with this sort of thing

Like going from real problem to minimal model

[zenith slate]

Oh yeah 100%

That's exactly why CLT without germane load is often criticized. While reducing extraneous loads can improve test scores, it doesn't really lead to better transfer.

[shadow basalt]

I see

Makes sense

I guess the fallacy is that they aren’t considering that separating wheat from chaffe is part of the original problem

So of course making the problem easier leads to better performance haha

[zenith slate]

Like I think CLT is very interesting, but it has (in my opinion) undue weight in discussions of pedagogy

[shadow basalt]

So what exactly is germane load

[zenith slate]

Mm it's kinda controversial and ill-defined

[shadow basalt]

So are all of my projects

[zenith slate]

Haha indeed

I mean it is defined (at least by van Merrienboer) as load that is not extraneous

Many proponents of CLT advocate for the unilateral reduction of all cognitive load in the classroom

Now, this does sound ridiculous but there is an evidenciary basis for this

Specifically, there is mountains of evidence from the late-90s for a purported effect which Sweller calls the "redundancy effect"

To be completely honest I think Sweller is a tiny bit of a crank. At the very least his ideas seem to have gained a lot of traction without much justification, in my opinion.

Anyway, the redundancy effect being the observed effect where the addition of extraneous information to any kind of instruction results in unilaterally poorer long-term retention in students

I may as well just quote the man himself. In his 1998 paper Sweller states quite definitively that he “knows of no experimental work demonstrating advantages of redundancy” and that “such a result only could be obtained under conditions where one set of instructional materials was so poor that any redundant alternative would inevitably confer benefits.”

My wife is currently working on a paper for a math education journal that is largely just "CLT is kinda a load of crap and can we start thinking about actually productive uses of applied, contextual problems in the classroom without getting worried about kids' heads exploding". To a large extent, I am inclined to agree with her

Certainly there is a lot of very interesting and important work being done in the pedagogical literature, but it does seem to follow this very predicable cycle of excitement and hype whenever a new idea comes along that seems to bear fruit. CLT is just the thing that's stuck around the longest, I think mostly just because it's easily testable in "laboratory" environments.

I'm particularly interested in Jo Boaler's work on math pedagogy at the moment. She's been very hyped in UK teaching circles and I think for good reason.

Though she is also kinda controversial because some of the old guard see her as very "radical" and "woke"

I think she killed accelerated math programs in California last year which people were not happy about

[earnest trail]

wtf why would she do that

[zenith slate]

Well, exactly. People get very mad about it without really considering that the underlying issues are legitimately tricky. Namely, given that both time and resources are sorely limited, is it better to spend them on accelerating the best students, or to spend them on addressing the staggering gap in educational outcomes between the most and least disadvantaged students?

There's also plenty of evidence that accelerated math programs are really quite overhyped in terms of their effectiveness. Indeed, they can lead to numerous unambiguously bad outcomes like the recently coined 'gifted child syndrome', though that's a whole other can of worms

Evidence suggests that by and large, kids who are good at math continue to be good at math regardless of whether they're on an accelerated programme or not. On the other hand, kids who are from socioeconomically deprived backgrounds almost universally get worse relative to their peers absent targeted interventions.

[earnest trail]

good point with the disadvantaged students

I see

makes sense

are accelerated math programs just like honors classes?

[zenith slate]

Yeah. Obviously there are strong arguments in both directions, but at some point money has to be allocated to certain things and a choice has to be made as to which students will be the beneficiaries of said funding

[earnest trail]

true

[zenith slate]

As far as I'm aware, boaler just happened to be the woman leading the charge on behalf of the kids who are, for lack of a better term, buggered without intervention.

I'm not really sure what constitutes an accelerated programme tbh. I'm in the UK and have no idea how the US school system operates. There isn't really an analogy in UK schools.

[earnest trail]

oh ok

[zenith slate]

In extraordinary circumstances kids can take their A level exams a year early, for example.

But that's really it outside of private schooling

[earnest trail]

yeah

[zenith slate]

There used to be a "gifted and talented" youth programme which was scrapped a few years ago for similar reasons

[earnest trail]

I see

gifted is kinda stupid tbh

they call everyone gifted

[zenith slate]

I was part of it myself. It didn't really accomplish anything except give me lifelong impostor syndrome lmao

[earnest trail]

yeah lmao

[zenith slate]

Yeah the gifted label needs to die imo

It's not of benefit to the students who have it

[earnest trail]

I think I was gifted but I didn't really pay attention bc I knew that everyone else was gifted too

[zenith slate]

And it definitely doesn't help the students who are struggling

[earnest trail]

yeah def

[zenith slate]

But yeah education at a national scale is super tricky. There are countless factors that need to be considered and weighed on an individual basis and unless the education sector starts getting unlimited funding there will inevitably be some group (or several groups) of students who are 'left behind' by the system

Also people like Boaler who are advocating for focusing funding on socioeconomic minority groups generally end up getting death threats and other crazy shit.

I think she had some crazies sending her emails containing (CW: SA and violence) ||graphic depictions of how they were going to rape and murder her daughters|| in the wake of the california accelerated programme thing.

[earnest trail]

oh damn

people be wack

[zenith slate]

Which is like... Absolutely unhinged lmao

[earnest trail]

yeah it's unfortunate

[cosmic bough]

dont suppose any teachers here have access to this years STEP papers?

wanted to have a peak

[tepid smelt]

Sadly being quick is useful for lots of important exams for students. The AP exams and SAT are classic examples of this for high school. I mean even the math GRE to get into grad school is awful in this respect.

[upper solar]

I guess I’ll just have to practice for faster algebra/more confidence. Annoying that exams are like this, on the plus side I can learn faster with better algebra

[tepid smelt]

I like some of her lessons on her site but https://stanfordreview.org/boaler-professor/ this article has made me revaluate her as a person. I don't think she is doing anything 'new' either and the curriculum she has created is not that good to be honest.

The idea of 'growth mindset ' is important but I mean that idea has been around forever. It is important to express this idea to students in that of course through practice you can get better and it's important to believe thatt

Tracking kids is a tough debate and I have heard good reasons for both approaches. I think for sure the rigor of the entire curriculum does need to be increased. It's been getting worse over time. I was looking at old Soviet/Russian k-12 textbooks and they are much better than what we have now.

I have ordered several texts from the Singapore curriculum for my daughter and I like the rigor a lot more than what we have in the states. AOPS is the only company producing high level curriculum for k-12.

[tepid smelt]

A lot of times it's remembering a more efficient method. This is a common thing with math contests. Really practicing to the test is the best way even if it's not necessarily the best way to learn math.

[summer ore]

at what level do you guys think math is the most relatively hard?

i was imagining that at the very beginning where you need to assign values to arbitrary letters and words, that it might be really challenging at that stage

[earnest trail]

depends on the learner

[noble hare]

That's a very hard question to answer, it depends on what type of maths you do and when, and like GMOD said what kind of things come easier or not to you

Personally the jump from 2nd to 3rd year university was the hardest conceptually, it built on basically everything previously learnt while introducing an even more abstract notion of thinking

But obviously "2nd to 3rd year of uni" is meaningless to anyone but me

[winged urchin]

Yeah the question is probably too ill defined to have an answer

For me my PhD thesis was the 'hardest' ahah. I personally was always pretty good at math but kinda drifted on my natural ability and wasn't the best student in undergrad. I could see a 'wall' so to speak in my path for awhile and there was enough faults in my mathematical substructure that by the time I got to PhD I felt out of my league.

I did manage to finish but that's honestly mainly because my results were novel enough, imo

I do not feel like the math itself was of a PhD level and it was very hard mentally and psychologically for me to pull that together

If you want to talk about topics that tend to trip up more students than others at the time they are learning them:

• variables and then functions I'd agree can be confusing. Students may use them fine enough but may just be symbol pushing and not really understanding it
• logarithms
• sine law (more particularly the ambiguous case)
• Integration (since it stops being so formulaic like it was when doing derivatives)
• linear algebra (because of the likely first introduction to more theoretical results. Tons of interconnected results altogether with a new algebra in a sense)

Oh! And Calculus 3 or whenever they get into 3d calculus. The visualization jump is very jarring for some

[zenith slate]

I think we went to the same undergraduate university so I very much agree haha

[zenith slate]

Hmm this is interesting. Wasn't aware of this!

Certainly her ideas are not particularly new, I agree. Nevertheless she has had a wide impact on current pedagogical thinking in the UK for whatever reason, so her work is what I'm most familiar with.

[tawdry venture]

i may have asked this before but i forgor

what is y'all's opinion on using the question mark as a means of distinguishing statements that we wish to prove from statements that are asserted?

such as if you're proving a trig identity, you would write it with $\overset?=$ rather than $=$ until you've proved it

[burnt vessel]

Ann

[noble hare]

I think its definitely good practice to differentiate between an equals you know is true and an equals you are trying to show is true. Students often get tripped up and confuse themselves when they don't make the distinction and end up basically just writing what it is they want to prove. My only confliction with that specific notation (which is totally fine I've definitely used it before) is that when I mark sheets, question marks often suggest to me the student wasn't sure if the step they took was legal, hence the question mark. But as long as they're clear in what they're using their notation for I think its perfectly fine!

[tawdry venture]

well i'd think this notation would be promulgated in class

[noble hare]

The alternative I see and often use is "LHS: bla = bla = bla.... RHS: bla = bla = bla... oh look LHS=RHS"

[native hemlock]

Maybe try amateurmulgating it first

[deep stream]

ye

[shadow basalt]

I think this is fine and helpful notation

[earnest trail]

I agree

as for the trig identities, cambridge uses the triple equal sign in their aice program

[long pelican]

After I started using the Question Mark, I can’t stop using it

[earnest trail]

lol yeah

[winged urchin]

Ooo the triple line equal sign is good notation that is left out unfortunately

Identically equal

I also do question marks above the equal sign when I'm writing math out for my tutor sessions if we don't know it's equal yet. Like checking a solution or something

I remember when I realized the importance of the distinction between equals and identically equals when I tried to differentiate a system of equations to get extra conditions to solve some problem

Or something like that, it was a little while ago but yeah ahah

[shadow basalt]

I'm not sure what you mean... I've honestly always been against the "identically equals" notation

Obviously I appreciate the nuance of different levels of equality but I've never seen "identically equals" used in a situation where "equals" was not also true

[winged urchin]

Well identically equals implies equals

What I meant before is like... In undergrad there were times in math we could differentiate both sides of an equation and get something that was also true

Like related rates problems, for instance. The equation you write to describe the situation or problem at hand can be differentiated and that resulting equation is also true so we can use it to find the rates

This is because, ultimately, the first equation was identically equal. Not just equal for some choice of the variables

[shadow basalt]

Oh I see, so people use it to distinguish "=" in the context of "find the x for which this is true" and "=" in the sense of two functions actually being equal

[winged urchin]

Like... f(x) = g(x) (the 'equals' equal sign) to me is essentially asking for an intersection point of the two functions

Some point or input or whatever that makes the two functions equal

f(x) 'identically equals' g(x) means that the two functions are the same. Are equal for any input

f(x) equals g(x) does not imply anything about the derivatives of them

But f(x) identically equals g(x) does imply their derivatives are equal

[shadow basalt]

Ah okay yeah I would never write f(x)=g(x) unless it were true for all choices of x

[winged urchin]

Not in that notation, yeah it is a little weird

But you would, probably, happily write x+2=3x+2

[shadow basalt]

Yeah I would not unless I had some accompanying english saying "We want to find a solution to..."

[brazen bluff]

Say you are asked to tutor a student as a supplement to some differential calculus (calc 1). How do you personally determine and plan how to tutor the student? Would you just do a bunch of practice problems for a session? Maybe review some definitions/theorems?

[winged urchin]

Ooo I can definitely talk about my experience with that but am busy for a bit. I'll return in about an hour to add my seventeen cents!

[brazen bluff]

It would really mean a lot to me to hear about your experience and how you went about things. Looking forward!

[shadow basalt]

Its important to start with a diagnostic i feel

some way of gauging where the student already is

what they are struggling with what they are good at etc

if you dont want to waste a session, you might consider asking them to do this ahead of time

a small quiz or something

[brazen bluff]

Noted! I was thinking of doing this in the first session with concepts from his algebra/precalc courses

[shadow basalt]

I think in general tutoring / private teaching works best when the students do some work ahead of time

and can come to you with what they struggled with with clear examples for you to work on together

This is why I always hold my office hours right before homework is due

[brazen bluff]

This is a good point. In terms of concepts from previous courses, the biggest things would be factoring polynomials and trig, right?

With limits, I feel like the algebra really trips them up.

And graphing maybe

[shadow basalt]

Yeah for differential calculus what you really want

is to make sure that they have all the manipulations super solid

not just stuff like trig identities

but also like

when do exponents add vs multiply

you'd be shocked how many mistakes of that level I see

just things that never got drilled down properly

a ton of the points people lose in calculus esp differential calc is just bad algebra

[brazen bluff]

Oh yeah, I have noticed that they're shaky on stuff like ln powers and stuff like that

[brazen bluff]

Would you say the structure also depends on whether the purpose of the tutoring is a supplement to a course they are concurrently taking or if it's in preparation for a big test?

[shadow basalt]

Its more about if it is supposed to be supporting the student in a class or just for learning for its own sake

[winged urchin]

Maxj has already chimed in some great points

In my experience I reallyyy like seeing the student work through problems. It really reveals how they think, what they think they can do, and I can chime in with possible variations of the problem that I might expose faults in their thinking

Now granted, I really only can do that when there isn't as much time pressure. If a student comes in last minute with the test the next day then sometimes the 'best' option I see is just working through practice material to show them how to navigate them. It is not ideal but for some students I feel it's the only way I can reasonably cover everything they need

I do not, in general, review notes or ideas. I will remind them of things as they come up in problems and sometimes this devolves into more 'lecturing' but again it's really only when I see it as something they are actively struggling with in a given problem

I would second maxj's remark that tutoring is really most effective when the student has done some problems beforehand and has a list of problems they are struggling with especially if they have their rough work too.

Kinda akin to my earlier point, being able to see their process through the question and ask them things about it, is just so valuable.

I think it's a little funny, but sometimes I find that as the tutor I end up asking more questions than the student. I ask them to justify steps, ask them to show they are correct, ask them to try and defend their mistakes (of course I don't say it's a mistake but if I notice a mistake I will ask them how they went from point a to point b and see if they can catch it)

[brazen bluff]

It makes sense that you would ask more questions sometimes. I feel like it's really important as a tutor to do this to check for their understanding, rather than you doing more lecturing

[winged urchin]

Yeah and it also can feel demeaning to the student if you're just always explaining to them. On the occasions that they notice their mistakes or at least notice something is amiss I think it feels better to them. They might get a little bit more confident in themselves. Which is nice ahah

Oh also I'd say if you aren't comfortable drawing things, try to get comfortable using some graphing software or something to that effect to be able to visually show them what's going on with a certain limit or derivative or whatever else they're doing.

That becomes doubly important once they move into anything three dimensional, sketching 3d out is honestly a pain

[brazen bluff]

Oh yeah, graphing multivariable functions is annoying

[brazen bluff]

If you don't mind me asking, how much do you charge when tutoring?

I guess the question can go for anyone who reads that

[winged urchin]

My usual rate currently is $40 an hour. (Canadian $ if it matters)

Though I will personally accept less if the student can't do that or if my schedule is more open than usual.

I've always felt bad about raising my prices personally. Ahah. I started tutoring when I was in third year undergrad and I asked for $5 an hour back then ahah. I still felt bad about that but have since risen it to more typical levels

[brazen bluff]

I think that's fair. If two people with PhDs and are professors at a university are asking you to tutor their son, would you raise that price?

[winged urchin]

I don't personally see why the parents having PhDs matters in that situation.

But no. I wouldn't raise the price I think in any situation. It just feels bad to me I guess

[brazen bluff]

I'm assuming you accounted for the drive you had to make, as well as session length/structure?

[winged urchin]

(I know there's some argument here someone could make since I would go lower for those who are more... poor. So in effect I kind of ammm charging more for those who are well off. But I have to just push that under the rug I guess ahah)

[winged urchin]

Nope. Mine is a flat rate. I am able to just use public transit to get around myself and I realize I could diversify my pricing through different levels but I just can't be arsed aha.

Id say if you wanted to do that go ahead I just a little self sacrificing I suppose?

Well session length is a factor of course. Since it's 40 per hour but ya I think thats clear

I'm also fortunate enough now that I basically don't need to prepare for most courses I get asked about. I know the material well enough I can answer anything most students bring up

If there is something that stumps me for whatever reason that's exciting to me ahah. And I'll work on that by myself and send them what I feel to be a good explanation at a later time

Free of charge, I suppose, but ya

[brazen bluff]

Wow, I really appreciate you for all of this

Thank you

[winged urchin]

No problemo. Come here anytime. We've got a number of great people here willing to answer your pedagogical puzzles! ^^

[slim reef]

anyone know where i can get students to teach maths to

i'm a university student in maths

[native hemlock]

Try Walmart

[slim reef]

wtf

[cosmic ibex]

Your question is a bit unclear. If you just want someone to explain stuff to for the purpose of sharpening your own understanding and expository skills (or simply out of altruism), then a straightforward plan would be to hang out in forums such as this server, and respond to the constant influx of students seeking help.
If you want someone to pay you for teaching them, then it's a different matter.

[earnest trail]

I think he wants to be a private tutor and wants to find people to tutor (for $$$)

@slim reef is this true?

[slim reef]

Yea

[earnest trail]

@slim reef irl or online?

[slim reef]

Online

[earnest trail]

oh ok

idk exactly, hopefully someone else can help you

[pale gull]

I might be teaching maths to an eight year old. Any tips? Whats the standard things that an eight year old is supposed to know?

Help is much appreciated!

[earnest trail]

good luck soldier

[pale gull]

I think I will have a lot of fun and learn a lot

But I should study what to do and what he knows etc

[noble hare]

Ask for some recent work from the student before the first lesson if you can, very helpful

[pale gull]

Yes

[winged urchin]

Honestly, I've tutored kids at that age before and unless they're really gifted or the parents absolutely want some hardcore tutoring...

[pastel horizon]

From what I've seen they just get loads of drill questions on reasoning/fluency/problem solving

[winged urchin]

I recommend just trying to have fun with it

If you're getting them to do problems without them hating it then that's like... Almost the whole pie right there

[pastel horizon]

Yeah I'd probably say split it up so have one session only focusing on reasoning. As an obvious example,

2 × 216

A) 432
B) 563

[winged urchin]

I'd agree though that a lot of what we did were just drills. With me reminding them of 'tricks' so to speak

[pastel horizon]

For a kid that age they should know it can't be B that's an odd number and anything times two is even

[winged urchin]

Trying to encourage them to attempt numeracy problems on their head

[pastel horizon]

Well yeah some kids might be able to do that in their head and that would be brilliant, but the main goal is to try and get them to develop reasoning

[winged urchin]

Yeah that's true

It takes some patience I think. I'm a very patient person but I could've seen other people get mad at the number of repeated errors I saw

But you just have to stay nice and encourage them

Again in my mind I wanted to absolutely avoid making math more of a boogieman to them

[pastel horizon]

That's just coming off 2 days watching primary school though I wouldn't say I'm an expert at teaching someone that age

[pastel horizon]

I agree, but on the flip side kids that age generally want to try and please the teacher so you can also use that to your advantage

What they tend to do in primary is have math first lesson and something like PE/Art/etc last

Maybe you can strategically try and get them in a morning slot while they have energy to burn? Haha

[pastel horizon]

Oh in terms of content, things like place value, fractions, money, measurements are good places to start looking

[teal moth]

why would canadian $ matter? am i missing something here?

[sour mural]

Value may vary quite a bit depending on currency (e.g. there’s also AUD and USD that use the $ sign)

[teal moth]

i still don't understand why it would matter to anyone, a totally unnecessary parenthetical

just use the peso sign is sufficient

also, any extra info added with parentheses is unimportant and can be ignored, i just dont understand the comment inside. it must be some kind of joke

[cosmic ibex]

It was probably an attempt to ward off readers criticizing their rates as being too high because they thought it was 40 USD.

They even indicated in the same post that they feel self-conscious about raising rates.

[teal moth]

just pick the $40 that pays the most, that is the power of symbols

kinda like which ever beer is the most full is mine

[cosmic ibex]

I'm not even sure which point you're trying to make.

[teal moth]

the point has something to do with math-pedegogy (if it matters)

[cosmic ibex]

Then I doubt you ought to be trying to make it here.

[teal moth]

thank you, both the teacher and the student can learn

[cosmic ibex]

It was explicitly a discussion of how much it is usual/reasonable/common to charge for one-on-one tutoring. When answering that question, it is obviously relevant to specify which currency the number one quotes is in.

[teal moth]

why would canadian $ matter? am i missing something here?

[cosmic ibex]

Because not all kinds of dollars are worth the same. This should not be a difficult concept to grasp.

[teal moth]

why would it not matter?

[cosmic ibex]

You are a troll. Go away.

[teal moth]

maybe because $ is ambiguous?

its mot to me, i would just pick the one i like if left up to me

anyways, im going to go away, i feel like i have learnt something today

[south raptor]

Hmm I wouldn't worry too much about what an 8 year old is "supposed" to know. Many kids have fallen behind the math standards because of the pandemic. Especially if the kids parents feel the need to get tutoring for them.

If you're going to seeing this kid more than once then during your first session I would focus on figuring out what problems they can and can't do and then go from there

Also if the parents are ok with it then you could try letting the kid have some input on what they want to learn

[winged urchin]

You know, on the topic of tutoring effectively. Of course there is a more... Verbal/communication based side to it too of course

Speaking clearly and slowly. Trying to use language they know and explain or check in with any language you think they might not know

"Well this is a subspace so... Wait. Just quickly, what do you think a subspace is?"

[earnest trail]

using the word subspace to an 8 year old?

i dont think they are learning linear algebra lol

[winged urchin]

Sorry I was mainly just rethinking about my previous discussion about tutoring. Not necessarily 8 year olds specifically though the advice I think generally applies ahah

Smaller details, I try to always sit on the right of who I'm tutoring since I'm right handed. If I'm on their right I can draw clearly for them while speaking. Although it can be tricky to both draw and speak clearly at the same time, it can help to cement things better

[earnest trail]

oh ok lol

[earnest trail]

lmao i do this too, never noticed

[winged urchin]

Nice! I've had some students who were left handed and then it's just perfect. If we sit like that then we can both see what the other writes ahah

Different colour pens! Also! Is more important than it seems I think. Differentiating ideas or different parts to the problem with color

[earnest trail]

lol nice

[winged urchin]

Visual aids can be good with forethought although I've definitely just used my three pens to try and communicate three dimensional ideas

[earnest trail]

oh true that makes sense

i never needed to distinguish a bunch of things since they were usually one problem at a time

idk if i will tutor in college

i work as an online tutor

maybe i can start going in person once i get a car lol

[winged urchin]

Even then within the same problem I do change up my colour. I couldn't tell you exactly why. My actual tutoring process is kinda more organic or something... With most things I do actually. My body/mind tells me to do something and I trust it and I seem to be a good enough tutor ahah

In zoom I definitely use different colours too!

Help distinguish different shapes within a composite shape perhaps

For younger students I think it might also just make it more interesting for them

[earnest trail]

thats awesome!

[winged urchin]

I personally don't find myself drawing figures very precisely though. I'm decent at drawing semi accurate sketches and it's faster/flows better. I don't use rulers or protractors or that

[earnest trail]

true true i do that sometimes to not make things confusing, esp if the page is cluttered

[winged urchin]

I like with Zoom you can select a bunch of the work and shrink it into the corner or something ahah

[earnest trail]

lmao yeah

[winged urchin]

I also save my slides as I go to send to the students so I can save and clear stuff as we go

[earnest trail]

yeah

the company i work at doesnt use zoom

we have our own program

oriented to turoring so a lot better to work with

[winged urchin]

Oh damn really? I wonder what features you have available

Graphing software? LaTeX style text?

[sour mural]

I have read a few studies suggesting that changes in color can help with memory, so I think that this is a good idea

I’ve started incorporating that too

[noble hare]

I recently discovered the term "socratic questioning" to describe lines of questioning like this thanks to someone on this server, very interesting

[winged urchin]

Interesting! I'm familiar with socratic learning or teaching in so far as teaching by asking questions

Actually in a sense it's even when I ask back the same question sometimes, ahah

Like if a student asks what an eigenvalue is or something, sometimes you can just ask it right back at them and as long as they attempt to answer, it can be valuable

Sometimes I'll tell them as well that they shouldn't be too worried about being super technically correct or even just describing their feel of what it is if that's what they can do

I find most students usually use kinda weird inaccurate language when describing new ideas but it's mainly just them trying to process it I feel

In any case, socratic questioning is awesome! =p

But I think a different style of socratic learning occurs when a student is trying to explain their solution or approach

Like if I ask them to explain steps in an algebra problem

They might try to use language that doesn't really describe formally what they did

"Oh I moved the 3 over to the other side."

That sort of thing, there I can ask them what do they mathematically mean by that? And if I have to I'll remind them of what we can do in algebra, do something to both sides, multiply by 1, add 0, and a small handbag of sometimes obscure tricks that you can ignore thinking about them in mosttt cases

And to me that's a different flavour of this socratic learning. Asking questions about their explanation or their understanding of an idea, in order to expose any flaws or... inaccuracies... or inefficiencies perhaps in it

Has 3/4 x 8/9 written on paper and the student then writes 24/36

I can ask them if there was anything else they could've done other than multiply the tops and bottoms?

Maybe they aren't comfortable enough with fractions to see that they can cancel the terms before multiplying

Of course in this example problem, I think it's important to be very careful to distinguish between them being wrong and them doing an unnecessary step

I guess in a sense there is a strength in socratic learning that you might learn things other than what you perhaps initially expected

Whereas if you just directly correct their misunderstanding or just tell them what they can do, you lose that opportunity to test their understanding

[cosmic ibex]

[Irrelevant tangent, but if you go read Plato, you'll find that half of the time Socrates's questions were, "And wouldn't you also agree that bla bla bla bla bla bla bla bla bla?"]

[supple wave]

Wouldn't you also agree that the teaching mathematics is directly correlated with translation between two languages?

It seems to me that the process by which humans shift between thinking something in, for instance, Latin to Klingon is precisely the same shift that happens between English and Mathematics. The key difference is the subject matter being translated.

Teaching mathematics becomes two primary skills.

1. Identifying what a student currently believes about a topic.
2. Contriving a story that modifies the understanding of the individual student on the topic as understood in Mathematics by using English.

[noble hare]

Would definitely agree, a big part of teaching maths is converting the logic into something digestible for the student for them to build intuition and see where things have come from and why we might want to do them

[winged urchin]

Absolutely! Math is a language and should be taught more like one!

Actually on that vibe, there's a noticeable change in how someone describes an equation or reads out math when they truly understand it vs. they don't

I noticed this with myself too, if I don't have the greatest sense of the math written, I'll read out symbols and the overall description will sound very awkward

But as you understand what the symbols mean and overall what the equation or mathematical statement is saying, you can state them much more intuitively

Like, {x \in \Z | (x =/= 0)mod2}

(Sorry I know we have LaTeX here but I'm on my phone and in a bit of a rush)

But that you can say super awkwardly by reading out every symbol

Or you can just say like... The set of all odd integers

[oblique bobcat]

So this is my first time grading; I have four kids I'm working with at a summer camp and I'm finding it time-consuming to grade all their work. Any advice on how to streamline/speed up the process?

[winged urchin]

Oh grading! Ahah. What grade level is this at? Or what kind of questions?

[winged urchin]

I think it's a little atypical to consider just four students. I'm curious as to how it is taking more time than you feel it should

[oblique bobcat]

I'm working at a summer camp for high-school students, though it's college level number theory

I'm spending over an hour on a single assignment. Given that each of the four is turning in at least one assignment a day, I easily have about 5 hours of grading. This is still the beginning of the camp, I expect it to pick up.

(These assignments can be a lot of pages - it's a dense page full of questions; most of my students are turning in like 6-8 pages per assignment minimum)

[winged urchin]

It sounds like you are giving great comments in their assignments

And unfortunately that may not be a viable option going forward

Couple things you could think of perhaps:

• Create acronyms or some sort of identifying mark for a similar kind of error and give them a 'legend' for those at some point. So instead of writing "You've made a mistake in the algebra here" you could perhaps just write "AM" for algebra mistake and circle the offending part of the equation perhaps

• You could skip over some parts of the work. This perhaps feels a little bad but if you are absolutely swamped it may not be your best use of time to look through a huge block of calculation when the end result seems to be coming out more or less correctly and their process seems to be sensible. Yes students can make mistakes that for some reason still give them the same answer, this is unfortunate but you are only one person

• You could try to compile comments you might write on their papers to a more... group oriented comment. If you lecture or speak to them you can just tell them about common mistakes you noticed and refresh them on those points. Instead of writing the same kind of comment on several papers

• If you're a little less familiar with the material they're working on perhaps, then take some time before grading to work on the problems yourself. Making sure you have as good an understanding as possible can help you grade their solutions more quickly AND it might eliminate some backtracking if you've ever had scenarios where you mark something one way but then later on realize you should've marked the other way

[supple wave]

Ask the student to explain what they are about to do.
Show them the thing you want to teach them.
Have them attempt to do the thing you’re trying to teach them.
Correct mistakes they make by asking them, given the information that they did something wrong, what they may have done wrong.
Tell them as precisely as possible what they did wrong.
Recur until they get it right.
Ask them, when they have done it correctly, how it might be used to actually do something else.
Iterate.

Anyone have any issues with the ideas or ordering above?

[winged urchin]

I think all of these things are good in certain situations. I disagree with the... formulaic nature of how you've laid it out or the 'order' you've put to it, personally

But I don't necessarily disagree with any point, I think...

[delicate spruce]

what kind of assignment are you giving where 6 pages takes an hour to grade? is it 6 pages of mostly proofs?

what's the format of the course, do you have the students for a whole summer session, or just a few weeks?

[winged urchin]

^ Great question ahah. I'm also interested but I have definitely had my times where I went wayyy too hard into marking and took far longer than I was supposed to

But the situation Abelian is in seems quite exceptional I agree

[oblique bobcat]

Entirely proofs, often proofs that require interesting ideas. And the kids are like 15, 16, so I really don't want to hurt them too much by being too harsh, but at the same time, I want to make sure they're doing the absolute best they can, so I'm not going easy on them, and it's really hard to strike the right balance.

[oblique bobcat]

But yeah, I tried doing this today. I meet with each of them for each assignment that's been marked, and every day I run an hour-long group session (today's I focused on 'how to write clear and concise proofs', largely stylistic things that make everything easier to read)

[delicate spruce]

perhaps it depends on which is more important for you to teach, proof writing or the number theory. in either case, if I were you I'd probably implement some more varied assignments, where they can still be working with proofs, but there's some questions for example like "spot the mistake" or fill in the blank, where you ask which theorem/property did the author use here, or what is this result equivalent to/what does this imply, etc

[oblique bobcat]

I mean the goal is to teach number theory. I just went over proofs so that they could write good proofs in number theory.

I feel like it's common to go over proof-writing techniques in courses of this level; as recently as the semester before last, my real analysis professor was also running proof-writing one-on-ones for people in our class struggling with proofs.

[delicate spruce]

then I think there's not really much harm in reworking assignments so that there's less full proofs, even 1-2 per assignment would probably be sufficient

[oblique bobcat]

I'm not the one who sets the assignments, I'm basically a TA

[delicate spruce]

ah, I assumed you made the assignments

[oblique bobcat]

No, I just do grading and some sort of mentorship duties for these kids and run group discussions

[delicate spruce]

I'm assuming also that the proofs are free write and not some unified format like two column?

[oblique bobcat]

Yeah

Two column does not work super well at this level

[delicate spruce]

well, unfortunate if you can't modify the assignments, 6-8 pages of proofs a day seems excessive, regardless of level

but with some time you will get faster and find your rhythm, part of it is getting to know the students and how they think

[zenith slate]

Is anyone aware of any literature or studies looking into the benefits/drawback of note-taking during lectures? I'm curious as I notice there seem to be several distinct groups of learners with very contradictory opinions on the efficacy of "live" note-taking. I suppose I'm also wondering these self-identifications amongst learners are, much like the VARK (and related) models, largely meaningless. Any pointers to relevant literature would be much appreciated.

[native hemlock]

Funny coincidence, there’s a math prof at my school who’s really into pedagogy and I just asked her about this and she sent me a paper on it

Lemme post it

[lethal leaf]

idk where to put this but this channel seems fitting

Any tips for running a reading group?

I'm running one on models of computation but really my concerns are

1: what if someone doesn't show up for a week and comes back? How much time should be spent on catch-up

2: partially as a resolution to 1, I am debating the idea of having a weekly scribe. Is this a bad idea?

3: how, if at all, should I pick problems to discuss?

4: how do you figure out how to structure how much we're reading a week?

[pastel horizon]

How old are they

[delicate spruce]

note taking generally has a positive effect, hand written having more of a positive effect than typing

VARK on the other hand is completely useless

more or less every study I’ve seen about it or similar models shows that generally what learners indicate as preferences don’t actually align with how they learn

i.e. everyone learns better from mixed-media text or presentation vs text or audio only

https://anatomypubs.onlinelibrary.wiley.com/doi/abs/10.1002/ase.1777

this is just one I found with a quick search

so, in general, it’s a waste of everyone’s time to go out of your way to try and support specific styles

the best way to learn concepts or information depends on the concepts or information in question

a great diagram to represent a complex relationship is going to help everyone, not just the visual learners

[tropic summit]

I'm not so sure how helpful "diagrams" are after looking a bit at algebraic topology💀

generally it sounds intuitive that note taking is supposed to be helpful. But very often I noticed it being a distraction for me personally. Some courses are too fast for me to keep up, sometimes when I take notes I have to rush simply writing stuff down and I do not understand the material I write or think about it at all. Which is why in some subjects I gave up on taking notes and I felt better with it.

[sour mural]

This sounds more like for a classroom setting

For individuals it most likely can vary greatly

[wispy slate]

Are there any standard, academic texts on math pedagogy.
Ideally ones that are exploring different frameworks and approaches to research in the area.
Perhaps developing various pedagogical theories.

Maybe something exploring like issues and competing theories of measuring mathematical skill.
Or even like what is "mathematical skill" in the first place.

[delicate spruce]

@wispy slate depends on what kind of mathematical skill you’re referring to, there’s a pretty big difference between mental arithmetic, memorizing facts, vs abstract reasoning

[wispy slate]

Right. If there are canonical texts that explore the variety of ways aptitude is even conceptualized and measured, that'd be helpful.
Or even something more broad than that would be helpful as well.

[flint wharf]

Hi! I'm going to be a tutor this summer for a math/stat program, and I wanted to know if anyone had any recommended reading on combatting microaggressions and such (in general, any recommended reading on pedagogy is useful tbh - I'll be tutoring for precalculus/algebra/basic statistics - but I'm particularly interested in this topic because it's not one I know much about).

[wispy slate]

You said that there's a "pretty big difference between mental arithmetic, memorizing facts, and abstract reasoning".
Do you know of any scholars in the field that try to highlight these differences and explain how and why they are different?
Ideally they do this in a systematic way.

[delicate spruce]

@wispy slate sorry, I was away last week and completely forgot about this reply

I'm not sure I can give you exactly what you're looking for, but I have some recommendations

"Adding it up: Helping children learn mathematics" (National Research Council, 2001) is a synthesis of research & book of recommended practices for teachers, and proposes a theory of mathematical proficiency in five strands

also "How students learn" (National Research Council, 2005), this report expands on the previous one and also includes other subjects

the NRC reports though include calls for more comprehensive and systematic research on math proficiency, so though there has been a lot of research on the topic, it has mostly been in the context of US grade schools and improving outcomes

the current prevailing influence in US math education is the NCTM, they have their own process standards to which common core is very closely related

"Principles and standards for school mathematics" (NCTM, 2000) and "Focus on high school mathematics: reasoning and sense making" (NCTM, 2009), might be good resources for you, the latter really develops reasoning and sense making and advocates for their place in the curriculum

[slim reef]

Any ideas for spaced repetition practice when helping high schoolers?

Like how to set questions and how many?

[quasi musk]

Are you teaching or tutoring?

[slim reef]

Tutoring

[delicate burrow]

right, so i am using the complete calculus course 10 book

as well as the notes from previous professors

[halcyon glade]

(disclaimer: I have never taught a math course before) you don't need to have all the problem sets you're gonna assign ahead of time, but it helps to know roughly what you're gonna assign like a week in advance

[delicate burrow]

How im used to being teached is that the professor goes through a set of problems and guides you through the solution

thats how i will do it as well

but its a bit indirect and easy to lose people

that course had a 75% fail rate one year by the way

[halcyon glade]

oof.

do you have lectures and recitations?

usually a TA goes through problems like that in recitations

while the professor covers material in lectures

[delicate burrow]

so the way it works is that the professor does classes 8 hours and i will do about 16 iirc

my task is to help people understand what the hell is going on

[halcyon glade]

dang 16 hours a week?

[delicate burrow]

the professor brings new stuff

its a big course

mind these are kids, i was 18 when i took the course

[halcyon glade]

yeah going through problems is one way, another thing is you can just open it up for questions and let the students sort of guide you

[delicate burrow]

they have never seen stuff like this before

[halcyon glade]

as to what you need to clarify/show them

[delicate burrow]

how do i not bore the students that understand though?

nothing is worse than being held up by that one kid

[halcyon glade]

you'll probably figure out over the course of the first few weeks what's most helpful for your situation

[delicate burrow]

you gotta let them go dont you?

[halcyon glade]

usually if one kid has a question, like 30% of the class will also be confused

another thing is you can have them go in groups and explain to each other the concept

[delicate burrow]

oh thats fine

but i was thoroughly confused when i took the course

i could go for hours asking questions on epsilon delta

im just curious how do you regulate the phase?

[halcyon glade]

hmm I don't think there's any hard-set rule, are you also holding office hours for individual students to come and ask questions

a lot of teaching is honestly just intuition

[delicate burrow]

they can send email

i live far away from the university, i work at a machine shop and not available

[halcyon glade]

ah got it

do you get any sort of training before teaching

[delicate burrow]

nope haha

[halcyon glade]

as to like expectations and guidelines

oof.

[delicate burrow]

yes guidelines

but no training

i know what i have to do

but how to do it? unsure

[halcyon glade]

yeah I would say try to make a plan ahead of time for each session, but don't make it too rigid

[delicate burrow]

yes

that is what i was expecting someone to say

how do you make a plan like that? what does it look like?

[halcyon glade]

hmm I'm assuming the professor gives you what topic they want you to cover, or something like that beforehand right?

also how long is each session?

and around how many students per session?

[delicate burrow]

roughly 2-3 hours

[halcyon glade]

dang

that is truly intense

[delicate burrow]

its a filter

to pick out the dumb ones as they say

[delicate burrow]

when i took the course it was roughly 20 a class

15 at the end of the semester haha

[delicate burrow]

yes, i know what chapters he will go through

and i know the subjects by heart

and i have all my old study notes for it

the presentation and pedagogy is what i never learned

i kind of hope i get rejected 😅

[halcyon glade]

yeah that's a ton of time to fill up

[delicate burrow]

i cant talk for 30 minutes straight let alone 3 hours

[halcyon glade]

you could just have the students work on problems during like an hour of it and go around from person to person for any questions?

[delicate burrow]

the way it was done when i took the course, they went through problems the whole class and took questions

i guess a bit more interactive way to do it is i write the question have them try for 3 minutes or so

depends on the question

then give some hints

then solve it after they get a few tries with more hints

then go through the solution

as well as different ways to solve it

[halcyon glade]

personally I would have them split into groups of maybe like 4 and have them talk through problems together

and whenever they have questions as a group, you can go around and answer

it makes it easier to manage

[delicate burrow]

yeah thats a good idea

i try to be careful with groups

in case people dont get along, or people arent motivated

but it cant be perfect

this majorly applies to group projects though

[halcyon glade]

yeah true, I think with this kind of setting it's less relevant

because you're not relying on the other group members for your grade or anything

and plus, if they're like freshmen or whatever, they'll probably be anxious to meet other people

so you could have the first class just going around introducing themselves

for the beginning

and then also ask them for ideas/feedback regarding how they want the class to be structured

after explaining your plans

[delicate burrow]

yeah, thats a good idea

[halcyon glade]

as long as you're nice and polite and stuff, they'll like you even if you feel like you have no idea what you're doing

[delicate burrow]

i really, really dont

[halcyon glade]

it's okay! one learns

I have no idea what I'm doing either

most of the time

[delicate burrow]

apperently teaching strategies werent a requirement