math-pedagogy

[prisma hearth]

Agreed, Soviet textbooks are truly unique

[lapis bone]

Hi, i’m new to tutoring. How do I track a student’s performance in math?

[lapis bone]

So far i’ve been asking the students where they struggle, but I dont know how accurate that information is

[shadow basalt]

I think it is much better to observe and look through work

and figure out for yourself where they struggle

students often lie to themselves (so do experts)

[lapis bone]

Dangit, i havent developed those skills yet

[pastel horizon]

They'll either pretend they struggle with something they're good at to boost their ego or just say "I don't get any of it". Kind of rare to encounter a student that accurately reflects on their own weaknesses before university

[lapis bone]

that makes sense, yeah

[tepid smelt]

What age and class is the student getting help in? In my experience don't really trust what they say. Like if your showing an example and ask "Was this clear" don't expect most students to respond honestly.

In tutoring generally most time is spent doing HW or reviewing for an assessment. Resist strongly the urge to help much early on. Observe how the student does on a particular assignment. Use Socratic type approach to help guide them if they are stuck. Don't give in to awkward silence or start doing anything for them.

I would also request looking at old tests they have done especially if your getting the student mid way through a class and they have likely struggled on earlier tests. I often will get a good feel for problems to focus on based on their previous work.

In general focus on letting the student do most or all the work and have open dialogue with their thought process. In general especially for younger students their parents are forcing them into tutoring so be patient also and don't neglect some relationship building like interests they have. Also don't neglect general study/organizing habits. You likely have a lot of general study knowledge you can share also.

Lastly don't be too hard on yourself. I have had students that were difficult to help because they were really behind or had other issues preventing them from making progress.

As far as tracking them I focus on the grade in the class and test performance since that is what the parents generally care about.

[pastel horizon]

Yeah instead of asking if they understood it's good to get them to repeat the steps. For example:

"What type of equation is this? Ok it's a quadratic so what's the first step?"

[lapis bone]

Thank you, this is super helpful! My student is preparing for the SAT, so i’m supposed to make sure they understand all the high school math concepts.

[slim reef]

Yeah I ask the students what they think they're weaknesses are and how they found certain questions. I think being able to self assess is an important skill

[humble aspen]

I'm taking on a wild, batshit crazy task as part of a self-guided project to propose a re-write of high school math curriculum (more of an argument to look into alternative math pathways other than the calculus pathway, but the absolute shock of it is intended to get discussion started). To start, I'm looking into starting off by discussing functions, mappings, and linear transformations. I've seen a few aspects of function-first teaching, and I'm kind of wondering if it sounds at-all pedagogically sound.

I think if students learn the concept of mappings/functions/linear transformations/etc, the rest of math can branch out from it (i.e. elementary row transformations for solving systems of equations are linear transformations, functions can be used for modeling and analyzing data, or their properties and features can show continuity and change, etc.)

However, there are parts of math that aren't function-based, but the intuition could possibly help?

I'm not sure if I'm articulating what I'm trying to say right, feel free to ask questions (I'm undergrad, very new to the whole teaching thing, and I'm learning along the way by doing this)

[remote brook]

wild, batshit crazy,

you seem like you have an idea, and although I don’t know if the calc path needs to be done away with, I do agree it’d be interesting to see math go quicker

What if we started algebra in 6th grade as the norm? What if we made it to calc in 10th grade and then got to do things like statistics or linear algebra or other cool things later on?

[humble aspen]

I think that it might be nice to teach math without the purpose of calculus because you can do more with mathematical thinking

and explore why math is so cool to so many people

instead of just rote learning and applying formulas

[remote brook]

But you should remember that calculus is such a critical thinking based class

You get to apply things

Which is why I think we should move math faster at the lower levels

People are babied in the beginning unfortunately

[humble aspen]

Definitely, but I think you can apply things with graph theory, linear algebra, statistics, geometry, etc. without needing calculus

or too much advanced math

students still should learn some sort of basics, but I think giving a why? as to "why do we do addition, subtraction, division, multiplication?" or "how does scaling or moving something work mathematically?" might be nice

abstract algebra, graph theory, lin alg, all take levels of math expertise, but you could spend time doing that in high school instead of calculus and it would be about the same level of difficulty

While calculus is really good for a lot of widespread applications, forcing the large majority of students into a calculus pathway doesn't seem to be doing much benefit when most of them won't apply it in their careers later in life

The underlying thought of mathematical thinking, whether it be with or without math, is a really useful skill

and it sorta goes hand-in-hand with critical thinking too, and I think that is more important to develop than calculus

[earnest trail]

just to put this out there: a lot of stem majors will use calculus regularly in their careers

but not as many will use abstract algebra

perhaps yes graph theory and some others

and all of these different fields have the ability to strengthen the problem solving skills of students

so I dont see much of a benefit of doing away with calculus, but I agree with wumpus - try to expedite the curriculum in some ways (possibly by removing the less important stuff) so that there's time to learn graph theory and linear algebra among other things

and discrete math

as well as calculus

[tepid smelt]

My school is moving into a data science type pathway. They are offering basic data science that jo boaler came out with. The main benefit of the class will be learning to use some good programs like tableau/R. I think the curriculum is ok and not really rigorous but I like the idea of a data science type pathway with more coding.

I still think calculus is one of the best math classes around when taught well. I can't see it ever leaving and most of my best students want to study engineering and calculus is essential for them.

If you want to see what I think is the ideal curriculum I would look at what they are doing at proof school. It allows for the expedited approach a long with a ton of variety in math electives. Many of those kids are doing calculus in middle school and graduate level math by the end of high school.

[plain valve]

being a textbook and having been made in soviet Russia

[prisma hearth]

Yeah

They are much more direct and don't beat around the bush

It's also worth noting that more math was taught to an average student then in other places

Pereleman has some nice books

[prisma hearth]

More focused on proofs as well I'd say

as can be seen by introducing proof-based real-analysis before the more computational calculus (for certain select students)

[prisma hearth]

Interesting

So was there no introduction to the intuition of limits and the like before learning about cauchy sequences, construction of real numbers, etc. ?

makes sense

[tepid smelt]

The quality of the problems for k-12 are much more rigorous than in the states. AOPS books are the only ones I have seen at a similar quality. I like the clean exposition also they don't fluff up the material with long winded explanations.

[turbid zenith]

Just got caught up on some of the interesting discussion happening here 😮

I agree with the need for an overhaul of the curriculum, and I agree that students can handle algebraic thinking much earlier if framed correctly

Where I have an issue is the idea of "getting to calculus" as if that's a (or the) goal everything should be building toward

[prisma hearth]

Yeah the focus on reaching getting to calculus

by local school officials

school board members and the like

has skewed the conversation to just reaching a certain class that may not be helpful to everyone equally

[prisma hearth]

Part of the issue is that the people who are setting out the standards and the path way on math, don't know math very well

[turbid zenith]

This is also true.

Calculus kind of has a place on a pedestal that it should at least share

[sour mural]

I’ve read a few articles about how some understanding of elementary statistics should also be part of the capstone, which I generally agree with

[tawdry venture]

absolutely

[pastel horizon]

They do it in China from basically the age of 5

Just basic things like

2×3=6
2×4=8
2×n=2n

Is how they get introduced into algebraic thinking from a young age

Essentially it's about providing a tool to describe patterns more generally so why not start at a young age while they're already learning basic patterns in maths anyway?

[prisma hearth]

I also think an earlier introduction to proofs and proof-based thinking would be invaluable

[turbid zenith]

Depends what you mean by "proofs"

If you mean the two-column magical incantations of high school geometry, those can take a hike

[earnest trail]

nobody likes those

I think introducing some form of logic and proof is good but not in that way

[tawdry venture]

i remember seeing a blog somewhere propose three-column proofs for a formalization. the cols are "statement", "reason" and "prev steps relied upon"

that makes it more in line with like, formal logic proofs

[earnest trail]

interesting, so like a way to bridge the gap between each step?

[turbid zenith]

I'm just not sure that rigid a framework is really what students need at the high school level

[earnest trail]

why not

a rigid framework shows them exactly what they need to do so that they can use that knowledge more creatively later

but I do see the argument to encourage the creativity earlier on

[turbid zenith]

I would rather ask students to "explain their reasoning" and then make that reasoning more precise gradually as they learn more math

[earnest trail]

true

[turbid zenith]

Rather than try to make it logically airtight right from the beginning

The way proofs tend to be taught in high school geometry it comes off as needlessly fussy

∠A ≅ ∠B ... given in diagram
m∠A = m∠B ... definition of congruence of angles

stuff like that, where if you don't phrase something juuuust the right way it's wrong

Sends the message of "proofs are where you have to say the right things in the right order to establish an obvious fact that I can clearly see from the damn diagram"

[prisma hearth]

Yeah those need to go

[prisma hearth]

Yeah this is what I meant by proof-based thinking

They don’t need to jump right into rigorous proofs

An informal justification of why something is true works great to start with

[light garnet]

I'm a former DS adjunct with an MS in stats. The issue I have with math educators is that they act as if calculus and data science + statistics is an either-or situation. What is quite annoying is that once you get past the very basics of DS, you can't do much more without a calculus background. Most DS programs try to avoid this problem by just teaching people how to write code to an execute algorithms without explaining how the algorithms actually work, because they simply can't due to the lack of an optimization (calculus) background.

Not to mention, pedagogy in DS is generally extremely inefficient. Back when I taught, I managed to cover nearly 4 years of DS material in one semester and had my community college students compete and place against graduate students in DS.

Given how many stats classes are taught by non-statisticians, including in math departments, and the many complaints that come out of such classes, I would not trust your typical math-oriented teacher or professor to be able to teach DS well.

Perhaps the most insulting aspect of teaching DS, in my experience, is that (more, but not all) graduate-level programs seem to have a better handle of what is required than undergraduate ones do. We had an undergraduate program that didn't even require Calculus 1 of our students; yet, the graduate program in DS required both Calculus 2 and linear algebra as minimum prerequisites. It was ridiculous to me that we could have students who could graduate with a BS in DS yet not even meet minimum requirements for a master's degree in the same university system.

I would also argue that beyond a very basic level, stats is essentially useless without calculus despite the perception that it's more "applied," but I will avoid writing more of an essay here.

[turbid zenith]

I don't think there's anything wrong with making sure more people see that basic level.

[light garnet]

To your typical math pedagogue, most don't. The issue is when people practice in academia and industry with just that background in stats. Many DS programs only require one semester of stats, for example. It is a particular struggle to work with health science and social science researchers who only have that background.

As a simple example, it is hard to explain to people that a p-value is not the probability that the null hypothesis is true (or some other magical interpretation) when those people have no idea what integration is.

I can go on and on with misconceptions that come out of that first-semester class of stats: thinking that all data sets are normal if n >= 30, thinking that all data must be normal (or implications thereof: e.g., thinking that mean = median = mode in all cases, thinking that the empirical rule holds in all cases, etc.), thinking that if your data don't follow some arbitrary property of normality that there's something wrong with your data, etc.

and then it makes my (thankfully former) job as a statistician much more difficult when you have to work with others who are very confident in their statistics

[turbid zenith]

I mean my AP Stat class addressed most of those things

[light garnet]

Yes, I've taken AP stats and have TAd for equivalent college-level classes. People don't learn anything else other than how to use 1-Var Stats and execute hypothesis tests in their TI-83/84

[turbid zenith]

That's about equivalent to saying that in AP Calculus, people don't learn anything else other than "take the power down and subtract one"

[light garnet]

A former colleague taught a stat modeling class which came after that class, and asked the room what a confidence interval was. Absolutely no one knew.

[turbid zenith]

What people are supposed to be taught and what they end up getting out of it are sadly two different things in a lot of cases

It's in the curriculum, if people will take the care with it that needs to be taken

[turbid zenith]

That sounds like less of an indictment of AP Statistics and more of an indictment of how students have been conditioned to learn math in general

[light garnet]

Sure, no disagreement there

I'm just of the opinion that if one is going to really learn statistics, that dirty work of working out the calculus is going to have to be done

because I've seen way too many situations where someone with an AP Stats background is overconfident and just executes hypothesis tests, misinterprets p-values, etc

[turbid zenith]

Depends what you mean by "working out the calculus"

You don't have to learn how to integrate with error functions to understand the "area under the probability (density function) curve" idea

[light garnet]

Unpopular opinion, but I'm honestly of the opinion that if you can't write out the integral representing a p-value for one of the conventional hypothesis tests, chances are, unless someone has corrected you many times, you're going to interpret that p-value incorrectly

[turbid zenith]

"Unless someone has corrected you many times" is what a teacher should do 🙂

[light garnet]

In an ideal world, sure

[turbid zenith]

No more ideal than a world in which using an elongated S makes you understand the meaning of the statistics concept better

Full disclosure: I'm about to teach a non-calculus-based statistics course in about a month. So this is literally my job.

[light garnet]

I guess you can say that from my years working in ed and academia, I've gotten really sick of academic publishing

[turbid zenith]

So correcting them many times is absolutely what I'm supposed to do.

[turbid zenith]

So have I, maybe even for some of the same reasons I suspect

[light garnet]

I'll keep this short and sweet, but I've been in the private sector for a year now. Was planning on going down the research director track until COVID completely derailed those plans (I won't go into those reasons), so I dropped a statistician role, an adjunct role, and quit doing a PhD

[turbid zenith]

Ahh... I'm just about to start the PhD myself

[light garnet]

Moving away from stats pedagogy for a bit, I think it is hard to find pure uses for statistics outside of academia - i.e., the stuff that is generally taught in schools

[turbid zenith]

What do you mean pure?

[light garnet]

The stuff generally taught in schools: hypothesis testing, p-values, etc

In my academic job, that was my life: work with others to gather data, clean it up, figure out hypotheses, compute p-values, write a paper, get it published

[turbid zenith]

Seems useful in business ... market research, etc

At least that's one of the things that really stuck out to me when I took AP Stat ... nothing seemed contrived, every example seemed believable and useful

[light garnet]

In industry, it's more like: work with others to gather data, make a dashboard, figure out how you're going to use said dashboard to drive strategy

(for at least the first 5 years of my career)

[turbid zenith]

Interesting

[light garnet]

I am of the opinion that "data science" in academic institutions misses the main reason why DS is so highly valued in industry

It's not because it's an off-branch of statistics. It's that people want professionals who can guide them on how to use data strategically.

The type of training one needs to do that effectively, quite frankly, I have never seen in a STEM degree

[turbid zenith]

Seems like it needs to be better developed then.

That gets back to pedagogy again though.

Because we're still stuck in this modality of "get them to memorize all the things, and applications and interpretations get pushed to the end of the section"

[light garnet]

Right

Which is why, when I taught DS, I elected for a drastically different approach

It ended up compressing about 4 years worth of material in one semester, but I think my students appreciated it

The approach was simply: show them how I would approach a data analysis project step by step, and have them do a daily workflow as I would as a professional. Then be sure they can do it by the time they're done with my class and that they can communicate it to others.

[turbid zenith]

Seems reasonable

[light garnet]

I've seen plenty of syllabi for DS which will spend one semester on R, one semester on Excel, one semester on Python, one semester on machine learning, etc... and I just thought the world is moving too quickly for one to spend so much time not understanding what a realistic workflow looks like

[turbid zenith]

But but but you have to learn every single theoretical aspect of R before you can move on

🙄

[light garnet]

Lol, that's precisely what annoyed me

So the week before I first taught that class - I was a new adjunct, keep in mind - I realized I had thrown into one semester of a community college class the equivalent of what the transferring institution taught in almost their entire Bachelor's degree, so I thought this was going to be a disaster

But it worked. I'm happy with it. Left after 2 years being proud of what I did and glad I tried it.

[turbid zenith]

Still have your syllabus from when you did that?

I would be interested in seeing it.

[light garnet]

Let me see if I can find it

[prisma hearth]

To be fair, these still do have their uses in industry (though they are more specialized)

[turbid zenith]

Wow. Looks solid!

[tawdry venture]

does anyone think using "whole numbers" to mean N \cup {0} is just stupid

[tropic summit]

I do

"whole number" should intuitively mean integer

but no

idk whose idea this was

[turbid zenith]

Yes

I refuse to use it

[tawdry venture]

as in refuse to use it altogether, or use it to mean Z?

[turbid zenith]

As far as I’m concerned the definition of integer is whole number

[turbid zenith]

I use whole number to mean Z

[tawdry venture]

in russian the word for integer is "целое число", which literally translates to "whole number". and the fact that "whole number" sees use as anything but Z in english is just idiotic from my pov

[turbid zenith]

Tsyeloye chislo?

[pastel horizon]

Whole number implies it can be positive and negative

[tawdry venture]

first e is just e. it's only ye at the start of a word or after a vowel or hard/soft sign

if i were to transliterate that more in line with latin orthographies for other slavic languages it would be celoje čislo

[turbid zenith]

Ahh okay

[tropic summit]

In german we also use "ganze Zahlen" for Z, which word for word translates to "whole numbers"

[earnest trail]

yeah that's where Z comes from

[sterile dune]

this is a big debate in the pedagogy literature

[shadow basalt]

whole numbers seems fine to me

its a way better name than natural numbers

also, N\cup 0 = N

[earnest trail]

N \cup {0} max smh

[turbid zenith]

Is it really?

I would really love to see other stuff in the literature talking about this

[meager bronze]

i think pedagogy researchers have better things to think about than whether or not 0 is a "whole number"

[shadow basalt]

I think the question was about terms like whole number entirely

But I agree I have no idea how or why one would really study this

[turbid zenith]

I remember when I was in prealgebra and learned what "integer" meant I thought it had something to do with being negative, like you had to be negative to be an integer

If "whole number" had been used to describe it I might not have had that misconception maybe

[shadow basalt]

But the suggested use of whole number doesn't include negatives

like whole number isnt a replacement for integer

I think that its more important to focus on how students learn best to associate new terminology with their correct definitions than it is to try to totally optimize word choice

like obviously there are bad names

but I think integer and whole number and natural number are all fine, the sticking point is teaching students to learn the definitions correctly and giving teachers tools to recognize easily when students do have such a misconception

[turbid zenith]

I'm saying I think those are related 😛

Obviously it's not the only way to fix it, but I do think it's a component

[shadow basalt]

Sure, my point was more that this is a minor component

Like theres nothing about "integer" that suggests negative, you just had a misconception

I think trying to optimize word choice among options that don't have obvious flaws is probably unproductive and also hard to study

[turbid zenith]

I've seen a number of students with that same misconception though!

[shadow basalt]

Probably an issue with teaching moreso than the words then

I think that if you go from N to Z using any term

thinking that the new term only relates to the new numbers

is going to be a common misconception

(you can see this by looking at all of the other places this happens)

like plenty of people think complex numbers can't be real, real numbers can't be rational, etc etc

[turbid zenith]

Fair points.

[shadow basalt]

My guess would be that the best solution would be 1) make sure to explicitly say that the new term encompasses the old one and 2) say it more than once in case people weren’t paying attention

[cosmic ibex]

It doesn't make it easier that much of the time actual working mathemaricians say things like "but if the roots are complex, then ..." and rely on the listener to understand "not real".

[earnest trail]

is it normal to not tell high school students that the pythagorean theorem and distance formula are slightly different versions of the same thing? I was never explicitly told this when learning about these things

[austere inlet]

doesn't the "distance formula" follow directly from the Pythagorean theorem

sadly, not teaching where the "formula" comes from nor why it should be true seems commonplace (in e.g. HS courses at least), and I think it's really harmful for math ed

[paper glen]

dont think most people care tbh

and the ones who do find out anyhow cos theyre interested

[turbid zenith]

I kinda hate this :<

It's way more common than it should be

Though to be honest I'm not sure I see much use in a distance "formula" in the first place at the time it's usually taught

[earnest trail]

I mean yeah it is mainly for coordinates but you could very easily use triangles and apply pythag

[earnest trail]

yeah I agree, very unfortunate

[earnest trail]

thats true but it does give some motivation and connection that sort of condenses the amount of formulas there are

[noble hare]

well this is exactly how you derive the distance formula for points

[earnest trail]

yeah

[paper glen]

in the uk we'd tend to do distances when covering vectors

so then you get the connection to pythagoras

[earnest trail]

haha imagine covering vectors

[paper glen]

lol

this is in the last 2 years where people narrow down to only 3 subjects though

so its different

[earnest trail]

oh interesting

right thats your university or smth

or college

idk y'all make the distinction

[paper glen]

lol

[earnest trail]

we don't

[paper glen]

college is 16-18 and then uni is 18+

[earnest trail]

oh ok

[paper glen]

yeah always have to double take when i hear college

[earnest trail]

yeah in america seems like most people don't see vectors until calc 3 or linalg

[paper glen]

that's probably the most natural place for them to be used tbf

we do vectors and they're basically pointless

[earnest trail]

yeah true

makes sense

haha pointless, ironic

[paper glen]

lmao

[noble hare]

they're like barely vectors in a maths sense

about as hard as it gets at GCSE ~16yr olds

[paper glen]

i suppose but we were discussing this in context of explaining distance function and pythagoras being same tihng

[long pelican]

Deriving the distance formula from the Pythagorean theorem has proven to be a mindblowing experience for 100% of the students I have shown it to

[plain valve]

Is it usually just given without proof?

[turbid zenith]

Pretty much from what I’ve seen

[plain valve]

Ah fair enough. In my school system (UK) we were shown the distance formula as coming from Pythag theorem but Pythagoras' theorem was never proven

[cosmic ibex]

Not proving it at all is a shame. Euclid's proof, for all its elegance, is a bit of an acquired taste, but everyone deserves to have seen the proof-without-words with four right triangles in an (a+b)×(a+b) square.

[earnest trail]

it certainly was for me

[plain valve]

Yeah.

[turbid zenith]

I personally like the proof in terms of similar right triangles

I find that students who are more comfortable with algebra like that one

[tawdry venture]

i wish there were easier ways of conveying the difference between mechanical and motivational confusion

i.e. "i don't understand what 'adding stuff to both sides' means" vs. "i don't understand why you chose to add 5 to both sides, instead of doing something else"

[turbid zenith]

One of my coworkers asks her students “is this legal” vs “is this helpful”

[tawdry venture]

that's similar but not quite the same as what im going for

[shadow basalt]

I feel like you can just ask, no?

[winged urchin]

I am cognizant of the same sort of idea when I tutor. I am careful as to whether I say what they have write is just incorrect, illogical, or inefficient.

Don't just say they are wrong if they are just doing something a little less elegantly. Acknowledge the correct logical steps they use but highlight a more efficient way to approach the problem

[quasi musk]

Holy shit ~ I've seen you on here a lot DM Ashura

And I just realized that you run the best math subject GRE website

Good stuff man. You've been a big help in my push to take this stupid thing this year

Completely unrelated to the pyathogrean theorem distance discussion

[earnest trail]

pyathogrean

[quasi musk]

pythagorean

Me be tired

I work from 9 am till 8 pm with an hour break

[earnest trail]

damn that's crazy

[quasi musk]

yUh

Gotta get that $$$$

[earnest trail]

wait I just looked at dm ashura's wikipedia page and I realized he does so much, what an inspiration

[quasi musk]

Yeah ~ I had seen him on here, but now I'm studying for the subject GRE

Never put the two together

[turbid zenith]

Eyyyyy

Glad it’s helpful!

[quasi musk]

I was considering contacting you for private tutoring

but I think I'll be ok

[quasi musk]

You should add a donation link on your website

That study guide you wrote + the practice test. That's a lot of work, I'd gladly donate to your cause of creating math education resources

[slim reef]

I was tutoring a kid today but they were being very quiet and not interacting how do you make them interact more

[shadow basalt]

Depends on the kid

Might be shy

[tawdry venture]

why are students in discrete math made to learn about relations in great detail when these things are not really given much attention later unless they are functions

[turbid zenith]

Because that’s always what happens in math textbooks … you spend a whole bunch of time learning the full generality of something you’re only ever going to use in special cases 😛

Think about introductory calculus books going through the whole rigamarole of convergence tests for all sorts of random series and then being like “Okay NOW we get to tell you what a Taylor series is”

[turbid zenith]

Sitting in the IBL seminar at MAA MathFest btw!

[slim reef]

Ye like when I read a maths textbook I skip to examples then do the routine work and then go back to look at the theorems generalisations and proofs

[turbid zenith]

The examples and applications should be first

Depending what the topic is

[earnest trail]

I agree

I think schools should mention relations briefly but spend more time with functions

[tawdry venture]

i mean... fourier series require some of those more obscure ones right

[turbid zenith]

My point is you can play with them first and add the nitty gritty edge case rigorizations later

[turbid zenith]

As for relations … equivalence relations ARE pretty important

[earnest trail]

true true

[turbid zenith]

Relations in general … eh

[earnest trail]

oh I didn't see ann say discrete math, I thought she was talking about middle school students

which I don't think need to know much about relations anyway

[earnest trail]

why do we teach addition to students this way (left) instead of the one on the right?

imo the one on the right is more intuitive and makes more sense

[cosmic ibex]

It's the same computation; the right way of writing it down just uses more space for all the redundant zeroes

[drowsy night]

also might be easier for kids to comprehend and do

[cosmic ibex]

Students show know that the more verbose expansion is why the usual one works, of course.

[earnest trail]

yeah true and I guess the left one is less time consuming but you get lost in the mindless mechanical computation so idk

[turbid zenith]

It’s because we confuse efficiency with proficiency

[cosmic ibex]

I don't think it follows that the computation is "mindless" just because the notation on paper is abbreviated.

[turbid zenith]

No but that’s a risk it runs

And one that plays out decently often unfortunately

[cosmic ibex]

To the extent kids should be taught to do arithmetic with pencil and paper at all, I think it does them a disservice to demand that they keep writing all those redundant zeroes in every computation.

[earnest trail]

I agree, good point

I think it should be shown to students as like a "this is why the method works" kind of a thing

[shadow basalt]

I think there is a generally useful strategy of like

showing someone the slow, conceptually clear way

making them use it until they are convinced its annoying

and then introducing the faster, trickier way

This shows up like, all the way to grad school lol

[earnest trail]

just like with the definition of the derivative

[earnest trail]

what exactly are u talking about here

[shadow basalt]

I mean that this pedagogical approach is useful basically until people are ready to do research themselves

Its the right balance imo between pretending there is no easier way and forcing people to do awful stuff like @grim spindle 's algebraic topology prof

and never telling people the original or more conceptual way and just forcing them to go straight to the algorithm

[earnest trail]

oh ok makes sense

I agree

[turbid zenith]

Regarding derivatives you could do what I’m doing this semester in Calc I … introduce them on the third day of class using them conceptually, numerically, and geometrically; have students learn the derivative rules by observing graphs and doing some derivations; then introduce limits at the end of the semester and only then do the limit definition of the derivative

[tawdry venture]

do your students actually learn to differentiate functions symbolically?

[turbid zenith]

Yes, pretty quickly

So yeah we do derivatives of power, polynomial, exp, ln, and sin/cos on day 4, then the other four trigs by day 5 and non-natural exp/log by day 6 in this schedule

But like for example to first figure out the derivative of sin x they just look at the graph.

[earnest trail]

wow nice

[quasi musk]

I'm conflicted on this issue: On one hand, I think having more time to understand limits is better because students do find them so confounding

On the other hand, they can learn everything operationally without missing any meaning if you show the geometric, numerical, and conceptual ways of manipulating derivatives

Like if you look at your schedule "Limits & Continuity, then limits and differentiation, then limits and integration" in a 3 week span is brutal right before finals

Whereas if you front load, or ease into it throughout the semester, there's not a huge spike of difficulty at the finals

(When students can't get a refund or transfer out)

Do you have any experience with that Ashura or others?

[turbid zenith]

I think one reason they find them confounding is that they’re so abstract and students don’t see the purpose.

I did something close to this last semester, leaving SOME of the limit stuff until the end, and students seemed to like how it tied things together, so this semester I’m doing it all the way

[shadow basalt]

Maybe an added idea is that you can introduce the limit formality and provide a satisfactory bow to tie the story together, without necessarily including that material on the final

Since the real necessary result is that students can compute and use derivatives etc

That also reduces difficulty as students can relax a little for the week before finals insofar as they might be learning material that won't end up on the exam, allowing them to review the other material in the meantime

just 2 cents

[earnest trail]

I agree with that in general, have a little buffer so students don't have to worry about new information to study for last minute

[turbid zenith]

Hmm. That’s a thought, but I’m not sure if my department would allow me to just not assess students on limits 😛

[earnest trail]

lol makes sense

[quasi musk]

I think I more or less agree with this approach given Ashuras structure for the class

[flint wharf]

is there a sort of 'standard list' of soviet textbooks for hs/early undergrad level math? interested in looking at these as a reference

(i've been tutoring college algebra/precalc over the summer, and hopefully will be doing so again in the spring alongside some calc tutoring)

[quasi musk]

Not soviet books, but Art of Problem Solving has good books

[slim reef]

I'm getting a lot of 'not serious' students from a tutoring site, ones that don't pay or ones who arrange a lesson and don't turn up and never respond again, my friends haven't had any issues with this site tho, but they charge much higher rates, im charging 20£/hr for maths lessons, so should i up the price too?

[earnest trail]

maybe, that way you get people all are more dedicated and motivated to learn

[quasi musk]

Yes, selling yourself short means that your customers won't take you as seriously

"Why is this person only charging $20/hr when everyone else is charging $40+/hr there must be something wrong with them"

So you only get the most desperate students

That usually can't afford to pay

[slim reef]

yeah i work for a tutoring company too they charge like 69£/hr lol

[tawdry venture]

nice

[earnest trail]

yeah 69£ vs 20£ is a big difference, raise your prices my guy

[tawdry venture]

as is 69£ vs 420£

[earnest trail]

LOL

[quasi musk]

First day of teaching for me

One student had his mind blown when I said I've taken many classes beyond Multivariable calculus

[earnest trail]

LOL

what course? @quasi musk

[quasi musk]

7th grade Geometry

[earnest trail]

oh ok then this is understandable

[native hemlock]

hey boyt

[wary ether]

Hey Josh

how's it going?

Here's something actually I wanted to ask here.

I'm probably going to help out with some problems classes next semester (I'm starting as a PhD student.) Any pointers to effective ways to teach in this kind of situation?

[grim spindle]

wdym by problems classes

[wary ether]

As in, students come in with worksheets and their answers, and we go through them and correct their mistakes.

[grim spindle]

ah

I think it's important to understand the way the student thinks from their answers and to try and cater the answer to their style of thinking. Or like, try to explain to them what's wrong with their style of thinking

I don't have any very concrete advice myself

I think you just need to get a feel for it

have you tutored before

wait do you mean you work with students individually

or do you solve the worksheet on the board for them

and correct common mistakes

[wary ether]

Yeah this would be the broad format... I want to try and make it somewhat personable though which is why I ask advice

I would also mark their worksheets

[grim spindle]

do you get to mark the worksheets before solving

because that way you get to address common mistakes and bad thinking patterns

[wary ether]

Yes

[grim spindle]

then i'd say try and look for patterns as much as possible, and rank the most common mistakes. Invite the students to participate and see if someone displays the mistakes while participating, and then correct them, or otherwise while solving comment on how a lot of students did x here which is actually wrong. I think the most important thing is to get into the mind of the student, try and understand why they could have made that mistake and comment on how their way of thinking could be corrected to prevent it in the future

this doesn't always apply but I think in proof-based courses this is critical

what classes are these going to be

like what courses

[wary ether]

I'm afraid I don't know yet really, but I expect I'll be asked to help with linear algebra

which isn't that proof-based, they're only first year undergrads...

[grim spindle]

My first year linear algebra class was proof based but I know most places aren't

Either way, there are common misconceptions that crop up in computations too

[noble hare]

In my tutorials I will always try to recap and explain some relevant content from that weeks lectures that will benefit them in the coming problem sheet. Giving vague hints a few days after they get the sheets has proved beneficial to my students

[grim spindle]

Just make sure to not soend too much of your time recapping

Be succinct

[south sigil]

dont spend too much time recapping, but spend al the time youd like knee-capping

[lost pebble]

personally, Id spend a lot of time going back to R^3, because that is like a model space ans where a lot of intuition comes from

if somebody gets stuck, show them how to ground their ideas on examples

a lot of early undergrads cannot deal with the simple concepts like linear dependence and bases even in R^n, let alone understand why you need the abstract definitions or what theyre trying to tell, even

[pale wasp]

Anyone have tips on having students work on problems in class and present their solutions on the board? e.g. what works well, how long to give them, how to encourage volunteers, anything else, etc.

[wispy slate]

You should give more context. For uni classes that kinda sounds like a waste of time in my opinion.

And also this approach might be discouraging, some students just don't do well under time pressure.

[pale wasp]

yeah it would be for uni, there is likely going to be extra time in this class

other ideas for keeping students engaged would be helpful too

I was also thinking it might be something I would have a sub do when I have to miss class

so the sub doesn't have to introduce a bunch of new material

[wispy slate]

In my experience the best classes were when the prof assigned a weekly homework (that was quite hard which forced us to try different strategies, sit on the problems) and during the classes once a week we went over them quickly, and then professor proposed problems to do in class, asking for ideas, and when someone gave even obviously wrong answer, that prof explored them to show what breaks. If it's a pure math course, preparing exercises whenever possible of the sort 'why is that assumption crucial?' seemed to give a lot insight and made me remember assumptions to theorems better. What I also found great in my first linear algebra course was one prof asking entire class "Who understands?" instead "Who doesn't understand?" after most problems and tried his best to make it clearer if substantial number of students didn't understand. But all of that is from a students point of view, I myself have never taught a university class, but these points made me more engaged in classes, especially in the beginning of uni, even as one of the worst students.

[quasi musk]

So I do this ALOT

What I find is that students often times get nervous or forget, so plan a structured way to ask them questions, or give them an out (as long as it's not part of their final presentation)

So like if a student is presenting a proof of the Intermediate Value Theorem, ask "What are the cases we can have?"

Ok, what conditions are we given on the function? How can we use that?

What are the relevant theorems from the book? what chapter are they coming from?

Go ahead and look, do you see theorem 6_2 in Spivak?

etc. etc.

Grad students are usually pretty good at presenting complete or nearly complete solutions

But undergrads and lower have a hard time

[pale wasp]

Thanks! I will work on asking the class for ideas on problems and why certain assumptions are needed, I definitely didn't try that last time I taught, though that time we had a very strict/fast schedule to keep which made interactive things difficult

[pale wasp]

How do you decide how much time to give them in class to work or how to encourage volunteers to present on the board if the class is shy?

[tawdry venture]

does anyone else think the terms "dependent variable" and "independent variable" are horrible bc of how easily confused they are phonetically

[pale wasp]

I don't know how much I see the term 'dependent variable', since normally it seems to be 'random variables with covariance matrix sigma'

[quasi musk]

Yeah, so each lesson I have a dedicated amount of time to each topic

I don't always complete the lesson plan, but in general if a student gets stuck I'll ask probing questions to guide them

If they don't get that, I'll either take over, or let a student that I know got it right take over

[quasi musk]

As far as the class being shy, I just try to break it with a few jokes/ice breakers

The other thing I do is set the expectations early that everyone will come to the board once every few weeks, and I keep a list of such students

It's a lot of work to keep track of, but it really makes a difference for some people

[tepid smelt]

I can't say what a standard list would be. Some authors I think are good I have read for the secondary level are I. M Yaglom and I.M gelfand. I think they both have high quality textbooks and I have gained a lot of great insight on the math I teach at the secondary level.

The AOPS books also but more so for the problem sets.

I am curious for those of you curve tests what system do you use?

[wispy slate]

Testing, Testing!

Alrighty.

[long pelican]

Here's an ad-hoc system that's worked for me in the past. Take each easy/routine problem and decide that the A student should score 0.9 of the points on that, take each hard problem and decide that the A student should score 0.5 of the points on that, add them up and that should be the A cutoff

e.g. a test with 50% easy problems and 50% hard problems should have an A cutoff of 70%

[wary ether]

I'm looking for an introductory book on mathematical thinking and elementary set theory, in particular one that discusses infinity at least a bit. I ask because I have a relative who is interested in learning about mathematics, but isn't by any means mathematically fluent, and is really quite busy as-is. I'm mostly just wanting to avoid giving him a textbook!

I was impressed by a quick reading of Stillwell's "Roads to Infinity" but I think it is quite technical and, while clearly well-written, doesn't take the time to explain things quite as much as one would have to for a non-mathematician. Maybe I'm wrong though! Any thoughts?

[molten urchin]

I think this might be worth looking into: https://mitpress.mit.edu/9780262539791/proof-and-the-art-of-mathematics/

Hamkins is a brilliant expositor, and this book probably fits as a good glimpse into mathematical thinking with some substance (so it's not a pop-mathy read either).

[wary ether]

This seems very useful, thank you.

[weak thicket]

Hey guys, i'm searching somewhere where i can perform my mathematics skills and i found a bunch of people talking about Brilliant.org , is it a good investment ?

[frigid crest]

Yes.

[weak thicket]

ty 😆

[grim spindle]

No

Imo

I find it quite lacking

But also this is not a question for #math-pedagogy

[weak thicket]

oh sorry, i didn't found where to post it

[tawdry venture]

for uni undergrads is it a good idea in general to do this thing where you call on somebody in the class to answer simple questions regarding the material being taught

during a tutorial

like "[studentname], can you tell us how this expression simplifies?" while pointing at it on the board

these are 1st years im teaching

[distant fractal]

I've found that TAs that have done this are nigh on universally hated and their students don't perform noticeably better. In my personal experience. YMMV

I'd try to find other ways to engage them

I've had moderate success in first getting them to talk to each other, then to answer my questions directed at the general audience. It's tough to prevent, then, the active students from always answering first through. I'm still trying to figure out how to get around that. Saying "anyone but X, Y, or Z" or "anyone who hasn't answered yet" feels a little shame-y.

I'd like thoughts on this as well

[noble hare]

agree with your first point, from experience being taught and teaching, the majority of people don't like being called upon.

[austere inlet]

there's also the issue of not everyone getting equally called upon so to say -- that's partly why it's so hated

like obviously there's some subjectivity as to who you pick

[noble hare]

and eliminating the subjectivity with some kind of random picker just puts everyone on edge

[austere inlet]

yeah

[austere inlet]

true engagement is hard to create and it certainly can't be forced

group dynamics might help

kinda why some people love kahoots lol

[noble hare]

Wow I forgot about kahoots, I might try one next semester

[meager bronze]

just to add on to this: I also don't "cold call" on students anymore. I will open the floor for student responses though. I don't say things like "anyone but X can answer" but I do say things like "I want to hear a new voice today; someone who hasn't answered yet, I'd like to hear your thoughts"

or "someone from the left side of the room" or "someone from the front two rows"

Another way I get more people to talk is by breaking up answers -- if I'm going to ask a question that requires an answer and an explanation, I'll ask just for the answer (and then even if it's correct I'll ask if anyone else had a different answer), and then if multiple students indicate that they had the same correct answer, I'll explicitly ask someone else to provide the explanation

A third trick I use which I like to call "warm-calling" (as opposed to "cold-calling") is that I'll have students discuss something in pairs/groups, and I'll walk around the room and listen in. when I hear something that I think is a really good point or a really good explanation or a really good question, I'll go up to the group and say something like "I just overheard X and thought that was a really great way to think about it. When we regroup in a minute, would you be comfortable sharing that to the class?"

They usually say "yes" and then when we have the full discussion I won't open it up to everyone, I'll say "and there was a group over here who had a really good way of explaining it -- would you share it?"

but they already know I'm going to call on them and they've already had my support/validation that they are correct (so no embarrassment of being wrong) and they've also had time to process and think about what they are going to say out loud

that last one has worked super well for me in the past

@tawdry venture

[distant fractal]

I really like this idea, wish I could do this effectively. Unfortunately the room is paced and doesn't really have room for me to walk around. Im impressed it doesnt violate some fire safety regulations

Today I got 20 white boards (for a 40 student class) and had them work on problems in pairs between the problems I present, and raise their answers when done. This worked super well, it's the most active I've managed to get them. I didn't do it the first class its the 4th) and I had them find partners on the first day, so they've been with them for 3 days now. Maybe this helped

It also obviously helps gauge the class' understanding and what mistakes they're making. I figured out that they haven't been distributing the negative sign when subtracting two rational expressions

[meager bronze]

nice!

yeah that sounds super active and engaging!

[noble hare]

What does paced mean?

[cosmic ibex]

In context, probably a typo for "packed".

[jaunty moon]

does anyone here have a higher degree in mathematics?

[meager bronze]

Yes, plenty do.

[jaunty moon]

yeah i see

can anyone help me?

[meager bronze]

a) I dont think you are in the right channel… are you going to ask about teaching math?
b) just ask your question. Dont ask to ask. If someone can help they will

[jaunty moon]

yeah idk if im in the right channel or not tbh

i just graduated from EECE and i wanna do a masters degree in math and there is a pre masters degree before i can get enrolled for masters so i need some help with the courses and their prerequisites

[meager bronze]

Yeah wrong channel

This is for discussing strategies for teaching math

[jaunty moon]

oh yeah i see

im sorry

[meager bronze]

You should just ask in one of the general channels, maybe #math-discussion or #advanced-lounge

[jaunty moon]

thanks bro

[tropic summit]

Maybe I am asking in the wrong channel here. But does anybody have good sources for riddles aimed at any of these categories of students (preferably all of them)?

Math enthusiastic middle schoolers (not competition people)

Math enthusiastic highschoolers (not competition people)

People in middle/highschool participating in competitions?

[distant fractal]

Riddles?

I have a neat problem that advanced high-school students might find fun, I'm not so sure what qualifies as a riddle though

[tropic summit]

I'm not looking for one riddle. I'm looking for consistent sources for these. I personally have a hard time coming up with riddles that fit into any of these categories. Either they are unsolvably difficult or get solved in about 5 minutes.

[tropic summit]

generally I look for something that takes time to solve, and usually involves a neat idea that isn't obvious, but also not impossible to come up with. Like a problem that remains stuck in someones head for a little while before being able to solve it

[distant fractal]

Gotcha

[molten urchin]

This isn't the place to shitpost. Keep memes to #chill .

[solemn moat]

That's fair

[steady stag]

as a student the last one is perfect

[steady stag]

and perfect reasoning for it too

thats exactly how i felt as a student

[void lynx]

what do you guys say as a teacher when a high school student learning math says "when am I ever gonna use this in real life"?

[wispy slate]

Never.

[meager bronze]

“Youre going to use it in 3 weeks when you take the exam”

[distant fractal]

If I can conceive of a reasonable use case, I'll provide that. If not, this

[tawdry venture]

"people who complain about lack of use cases will never use it"

[cosmic ibex]

"You'll need it in order to understand something else next year, which you'll need for being able to practice something yet else the year after, which you'll need to understand why a rule of thumb you might end up using in real life makes sense."

[pale wasp]

hs geometry can be useful if you're trying to move furniture and need to know if somethings will fit in your car

apparently people are quite bad at this

[tawny slate]

I have several approaches for this, though they may be dry and not things kids care to hear:

• Suppose I handed you a screwdriver but you've never seen a screw before. Of course you wouldn't know what it was for. Math can be viewed as a tool, but its more peculiar in that you need to understand the concept before the tool can be applied. It's not enough to just know what it's used for, because that's not enough to actually use it, and you can conceive of its use cases until you learn it first. I can't show you all the cool ways I can apply derivatives until you first know what a derivative is

• abstractly, learning math, critical thinking, and problem solving techniques change the way you understand how to solve problems and challenges by adjusting your intuition. You might be subconsciously using techniques without noticing or realizing. For example, say you're playing a video game and you want to level up faster, you could be subconsciously trying to maximize your exp gain rate, thanks to the fact that you understand how ratios and rates work. This subconscious understanding of concepts needs time to be developed and you probably won't notice it, it shapes your intuition

• Math and critical thinking help you not get scammed in life. Scammers are always looking for ways to make a quick buck off of vulnerable people, and the more you understand how the world works, the less likely you will fall prey. Scammers really are everywhere, from the ads you see online, to employment opportunities, etc. Not just directly to take your money, but also to radicalize people in insidious ways. For instance, a young male could radicalized into becoming an incel if they understand that boys say that they have on average 4 different sexual partners vs girls having 2. Not only is mean not a good measure of statistics to compare yourself against others here (use median), but the only way the discrepancy is this big is if men are having a ton of gay sex (compared to lesbian sex), there are WAY more women than men, or most everyone is lying. Mathematically it is possible

[void lynx]

nice answers! all of you! thanks!

[tawny slate]

This is why in the long term, I think kids need a good balance of "lecture math" and recreational math

Recreational math can really help kids see that math is not only elegant and beautiful and fun, but also can be applied in sneaky ways in unexpected places

If they consume enough recreational content of high quality and a diverse range

They would likely not be asking these questions in the first place

They would already understand to some extent

[tawny slate]

@void lynx oh one more thing to add to my first point, sorry for another ping

Students should also understand that they should have the right mentality when approaching viewing math as a tool. This isn't a prescriptivist thing where a teacher says derivatives are useful in rate problems so they are somehow useful for all rate problems or that's the only thing derivatives are good for

Students need to understand that they have to look for places where math can be applied. If they only apply math where they have been explicitly taught, they are not only going to get limited use out of it but are also missing the point entirely

I'm sure students have never thought about the fact that calculus is directly applicable to driving. If you want to conserve gas and improve the lifespan of your brakes, you should hit the gas and brake pedals as little as possible. This is not something school teaches you directly, but anyone who understands calculus can intuit this concept if they looked to think about it that way

[winged oar]

This is not necessarily correct and makes assumptions about actual car engines which should not be made.

Derivatives in the real world are fitted models for a reason. You should be able to see and use typical differential equation solutions, rather than extrapolate with simplistic statements.

[tawny slate]

I was talking more in generalities but yes you are technically right, thanks for the correction

[winged oar]

For example, bubble shapes and how wires hang are solutions to equations. These are natural things, and have easy assumptions (ignoring bubble-bubble interactions, though these are still solutions even if more complicated)

[tawny slate]

Even from a pedagogical perspective I think your correction is important to highlight because it shows that simple assumptions based on intuition may not be correct, but digging deeper into the math and science can help us understand much more nuance and help us appreciate the value of doing the actual math

[winged oar]

I think physics and civil engineering does provide a lot of good examples, and would likely bring about easier examples than those requiring calculus of variations

[feral swan]

Got a question about students coming into university
I used to privately tutor those students (they were coming into fields like geography, not very good at math), and whatever I did they seemed to be at total shock at taking initiative in maths.
Any time I asked them something that was outside of rote process doing they could not compute it. I remember geometry in particular being a wash. It was very frustrating and for the life of me I did not manage to cross that divide, and in the end I taught them the rotes as well as I could.
Is there any ways of helping students cross that line?

[ancient shore]

Math is way more applicable than any other subject

[devout fable]

Well I feel like the unfortunate truth is that most people only do math for the sake of getting a degree, and it is possible to pass gen ed math classes through memorization, and arguably requires the least amount of effort to do so. For me, if I had to take geography, I would do the bare minimum to pass. And if that was simply memorizing things, that’s what I would do too. As another example, most pre-medical students in the US do not care at all about calculus. They care about getting into medical school.

[vestal eagle]

Most 'riddles' work like a switch right? It either doesn't make sense at all, or (usually after one word or phrase) it all makes sense

[tropic summit]

I don't quite get what you mean by that

[vestal eagle]

But what you're describing sounds like math olympiad problems, eg:

[vestal eagle]

I was trying to get what you meant by riddle in math. With words it usually makes sense after a riddle, like 'what has two hands and a face, but no arms or legs?'

a clock

[tropic summit]

yes, I am looking for these types of problems, except easy enough such that people who aren't necessarily olympiad aspirants can solve them. (the one you posted as example looks a little too easy though

[vestal eagle]

oh? give it a try then

[tropic summit]

replace y=3^x

then just factor the polynomial

[vestal eagle]

fair enough

should be a good enough exercise for a middle schooler or high schooler though?

i'll send you the pdf i got it from

https://newb.kettering.edu/wp/matholympiad/wp-content/uploads/sites/13/2017/05/2016-soln.pdf

[tropic summit]

but maybe I need to change my opinion. it might be good for middle schoolers/ highschoolers anyway. because this trick is kind of hard to come up with if you do not know it

[vestal eagle]

right. it's just a clever substitution

but the problem is simple enough for someone to remember

the sum of 2 powers of 3, equalled to a power of 3, each raised to x

gets someone thinking

[tropic summit]

yes in fact. I do remember the trick from seeing it once in early highschool. I didnt know how to solve some equation involving exponentials and couldnt figure it out. that solution stuck with me for like 7 years now

I'll have a look and thanks for the pdf, it might very well be what I was looking for

[vestal eagle]

yeah i know what you mean

personally i dislike these problems for that reason. but to each their own

[vestal eagle]

no worries. again, they're called olympiad questions. plenty of them online. good luck with what you wanted to use them for

[tropic summit]

I always thought olympiad questions were only for really ambitious and engaged students. which wasnt my primary aim. but I guess people that want to solve riddles need some baseline engagement anyways

[vestal eagle]

It's hard to get anyone in middle/high school engaged in mathematics at all. Only in highschool do math classes split into levels that filter people with absolutely 0 interest

Worse, you'll have to explain the algebra to people who don't get how to work with exponents or logarithms.

word problems might be better, like "A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost?"

[cosmic ibex]

I'd be worried that could backfire in the long run -- there's no easily imaginable reason why one would know the just the difference in prices in a real situation, and word problems that are so obviously contrived might just reinforce an impression that math is useless in the real world.

[tropic summit]

what do you mean, I always have 200 spare watermelons in my basement

[vestal eagle]

only 200? i've got the the sum from 0 to infinity where i buy 200 watermelons on the first day, and buy 4/5 as many the next, and so on

[vestal eagle]

how much could it backfire if most disregard it anyway. the value of mathematics is difficult to appreciate in middle/highschool because it's often taught as if it's puzzle solving or memorizing tricks.

i've got a younger brother in middle school who's learning how to manipulate equations involving roots and quotients, and the book teaches things on a case-by-case basis.

one 'theorem' i remember that they showed was that if ab = ac, then a = 0 or b = c. this shouldn't even be a theorem, he was taught that if a*b = 0, a = 0 or b = 0 last year. the previous is just a special case after doing one line of rearranging. but the book doesn't mention that

[cosmic ibex]

Sounds like a perfectly good theorem to me.

[long pelican]

Haven’t heard my students ask this unless they were really not having fun in the class. So in my opinion the message the student is sending is that they are frustrated and not having fun.

If the student is having fun, they won’t ask this. The one exception being if they actually happen to be curious about where the current topic is used in real life, but then that kind of question can just be answered honestly, without much trouble.

Assuming this is a case of the student not having fun, my response would be “OK, I see this class has been frustrating and not fun for you. Let’s talk more about it sometime.” It’ll probably be the case that they lack significant background knowledge preventing them from making much sense of what’s going on

[tawny slate]

i typically use simple but misleading examples

for instance, say the birthday paradox, or the chessboard/rice problem

trick the students into something that seems intuitively to work in their favor, put some odds against it (even simulated)

and when they are shown to be completely and totally wrong, explain how had they known a certain concept, they could've avoided being tricked

if you want more examples of problems where you need abstract problem solving from a generic toolset but isn't clear about "rote" process

honestly any math competition could work, you could always scour the AMC/mathcounts/ARML/etc tests for problems

[turbid zenith]

So I've got kind of a weird question.

I know that linear approximation can be used to approximate values of a function.

But under what circumstances would anybody actually use a linear approximation to estimate the value of a function?

I'm trying to come up with a realistic example to use for my calculus class.

So far the only one I can think of off the top of my head is small-angle approximations in optics.

[winged urchin]

Whenever I help a student with that I tend to refer to the a value (where it's centered) as the 'good value' and the point we want to use the estimation at as the 'bad value'.

Sqrt(3.99) is 'bad' because you don't easily know the value other than a little bit less than 2. Sqrt(4) is 'good' because you do know the answer immediately. So if you want to know sqrt(3.99) more accurately we build the linear approximation from the good value a=4 and evaluate the approximation at the bad value x=3.99

So the answer to when you use it is when you cant easily evaluate the function with the tools you have at hand.

I guess I don't have specific examples but I always believed it happens all the time that some function is difficult to calculate at some point and we use an approximation in order to decrease the calculation time for a sacrifice in accuracy. Although I'm talking more generally I guess. We tend to have better approximations than just linear approximation but the core idea is there I believe.

Oh oh, also, blood scatter like CSI stuff I think I remember someone doing a talk and saying the norm is to imagine the blood droplets travelled in a perfect line from when they left the body. Obviously this isn't true but becomes more accurate the faster the blood was moving. This is essentially linear approximation then?

Perhaps don't focus on the linear part? There's plenty of things we use approximations for it's just those tend to be more complicated than linear approximation

[turbid zenith]

Sure, but I'm literally teaching linear approximation 😛

And I really try hard to have examples of the actual thing I'm teaching in action

[winged urchin]

Did people not use linear approximation before electronic calculators were available? Like I could imagine using it if all electronics just stopped working and I had to work out the details of something by hand then needed to figure out what some value in the work was

[turbid zenith]

Sure. But ... we do have electronic calculators now.

See my issue sorta?

I'm trying to come up with examples that are still relevant.

But I am fine with "okay this is a simplified way of looking at the problem, the actual solution would be harder but would use the same principles"

[winged urchin]

I guess the gap is kinda narrow... The problems you might use linear approximation on, you'd probably use a better approximation method in practice

[turbid zenith]

Like I used maximizing v² sin(2θ) / g to find the "best" angle to hit a baseball (45°), but then briefly discussed that the real answer is close to 30° but that the principles we talked about would still go into the more complicated model

[pale wasp]

according to https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2006/resources/lec9/ engineers used it to calibrate GPS transmitters

[long pelican]

(Higher dimensional) linear approximation is used in turning intractable differential equations into tractable ones, that's one application I have seen

[tawdry venture]

yeah like analyzing the motion of a pendulum

[cosmic ibex]

It seems to be hard to imagine many use cases where the linear approximation is used to compute a single concrete number and yet we know enough about the function that the derivative in the approximation could be computed symbolically in the first place. It must be much more common to take a symbolic representation of the linear approximation and use it in place of the original function for further calculations where the actual definition of the function would be difficult/impossible to handle. But that can't be shown in practice until the students know those further calculations...

[turbid zenith]

I ended up going with an example that's not quite EXACTLY what was done but at least justifies why we'd still use approximations

Specifically, the Quake III fast inverse square root

I figure needing to deal with a tradeoff between accuracy and speed is a really good reason to need an approximation, so I'm introducing the idea, using linear approximation to calculate 1/√(4.1), and then mentioning that Quake III actually did it in an even faster way that still deals with linear approximations and also uses tangent lines as part of Newton's Method

[prisma hearth]

Yeah that’s a good one

Though that algorithm isn’t really used anymore as CPUs have a built-in instructions for it

Off the top of my head, many video games use LERP aka linear interpolation

Which I think can count as an approximation depending on how you look like it

[late schooner]

Hopefully this is an appropriate question for this channel... do you think a ~1 hr session on infinities (why reals are 'bigger' than integers, why |Q| = |Z|, or what does it mean for two sets to have the same size) would be suitable for gifted students in grades 5/6 of primary school?

The past few topics we've covered have included modular arithmetic, invariants/colouring, proofs and they've handled these well

[tawdry venture]

1 hour sounds good but i would probably go a bit longer depending

[late schooner]

The duration is inflexible, so it would just be a brief exposition - I'm just wondering if it won't be found too abstract/incomprehensible

[grim spindle]

Given the previous topics I think this could be good

Infinity can be made very visual and accessible

eg hilberts hotel, cantor's diagonal, enumeration of rationals

[tepid smelt]

I have been assigning alcumus this year to my higher students in particular. I find it's time consuming to come up with good problems so when my strong students finish the assigned work I just have them do alcumus. It's worked really well so far to give them experience with more engaging problems. You can get a free teacher account to check their progress. If your a teacher you can also get a free brilliant account and many of the daily challenges are quite good also.

[late schooner]

Yeah, it ended up working really well!

[grim spindle]

Great i'm glad to hear!

what did you end up doing

[south raptor]

Recently at my job I've been getting kind of frustrated by parents who complain that they can't help their kids with their homework because "the math is different than what we learned as kids".

And I'm just thinking to myself "math hasn't changed in thousands of years, what are you talking about?"

I suspect it's things like this which are confusing parents which really frustrates me cause I feel it's good for kids to understand the method on the right.

I guess my question is should I be more charitable to parents who are frustrated by the new method of teaching or is it just a necessary part of making the curriculum more rigorous?

I also wanted to say I learned a lot of interesting things from looking up mentions of "common core" from this channel like how apparently it isn't even a method of teaching.

So maybe it's wrong to scapegoat common core as the reason why the curriculum is confusing to parents?

[spice snow]

I was confused reading this because I thought this conversation was 3 years old since you replied to old posts

[south raptor]

yeah even though I used the reply function I thought of it more like quoting some interesting things I found in this channel

[steady stag]

I wonder if that's just a way parents can shift blame away from them not understanding the material by using that as an excuse

[south raptor]

Hmm that could be part of it. Those types of parents are usually quick to admit they're not very good at math so it's not like they don't take any blame themselves.

[tepid smelt]

In my experience parents find any reason to complain if their kid is struggling by shifting the blame on the teacher/curriculum etc. Many could care less if their child knew nothing as long as they are getting a good grade.

Many parents don't really value understanding math and that is a cultural thing. My students who have parents who value understanding math the kids take more ownership in asking for help if they are struggling. It may be racist but many of my best students have parents who come from countries that value education like china/India. Kids who come from households who don't value math look for excuses for why they are struggling such as the teacher isn't helping.

It's unfortunate that many admin side with parents though.

[south raptor]

YES! That's what really frustrates me when parents shamelessly declare that their bad at math. It's like how do you expect your kid to learn about this stuff if you don't care.

I would avoid making generalization based on countries of origin. Being able to immigrate halfway across the world and navigate America's complex visa system requires a lot of privilege. Immigrants from those countries are simply not a representative sample.

Whereas immigrants from countries in South America are doing so out of desperation. So they are also not a representative sample but in the other direction.

[quasi musk]

@turbid zenith Rambo has a new edition to his practice test for the mGRE. I got it to practice

The new edition now has 2 practice tests, instead of one. The second one feels like a remix of the first one. That is if you learned the tricks from the Rambos first, then you can do the second. I haven't looked at your practice test yet as I'm saving that for the October 29th prep

[winged urchin]

Hey everyone, kinda tangentially related to teaching but a bright student who I'm tutoring (in grade 12 right now) seems interested in more pure math areas but had fears of job opportunities afterwards. I of course have seen my past grad student friends talk a little about applied math generally being more... attractive? on the job market I guess but of course that's just what I've seen and I didn't keep up with most of the pure math crowd after school.

So, what do you all think? Is a pure math degree less attractive than an applied degree on the job market right now? Also he was curious about what jobs a pure math degree could get into with that.

Just looking for some insight so I can give them better advice than just my own anecdotes.

[turbid zenith]

Yeah I've seen that one. The second Rambo test is essentially a Jimmy Hart version of 0568.

[lethal leaf]

I would say that if they are worried, they can pick up some CS stuff

that's kind of what I am doing

Alot of higher level CS (algorithms and complexity theory and programming languages) is extremely close to pure math (especially if they like algebra)

and so I'm ending up doing alot of algorithms and cryptography courses which involve very little coding and alot of proofs and theorems

and still have gotten software engineering internships (which has been nice cash in the summer and also is a good backup if academia doesn't work out)

But I'm not really doing any applied math classes either so I'm getting more than my fill of pure mathematics courses

As far as jobs, MSFT and Google have research positions, could go into cryptography and work at various places (Sandia, NSA, NCC Group, etc)

Quantitative Finance is an extremely lucrative and increasingly popular option and you don't even have to do software engineering there (trading and research positions exist)

and then also just normal software engineering is open to many many people

[tepid smelt]

This student is I am guessing 16 or 17 they should really just pursue what interests them. I teach kids in this range and know the extreme pressure they feel to find a good career. Yet they are also young and have plenty of time to explore and try things out to hopefully find something they are passionate doing. I would say that I agree that learning some coding is a great idea even better if they learn while trying to achieve some sort of project they are interested in.

[lethal leaf]

also yea they have a ton of time

very good point

[winged urchin]

Thanks for your reply! This helps a bunch

[wispy slate]

Any Webapps that can compute and display syntax-trees or polish notation for algebraic expressions?

[wispy slate]

This comes kinda close: http://ironcreek.net/syntaxtree/ , but unfortunately numbers aren't atomic symbols and you have to write [* [+ [zwei] [drei]] [vier]] And I'd like it to also be able to interpret python-style terms like (2+3)*4

[idle estuary]

@wispy slate sympy https://docs.sympy.org/latest/tutorials/intro-tutorial/printing.html#dot

The only thing left is to have Graphviz installed so you can render the dot output to svg/png/jpeg

You can also write a custom printer that traverses the AST and serializes it in a format of your choice

[wispy slate]

I'd like a bunch of highschoolers to use it to practice symbolic manipulation, so I guess it would have to be a little more polished, but maybe I can make that work.

Doesn't sage come with graphviz?

[idle estuary]

Sage probably includes sympy and graphviz yeah

[wispy slate]

Looks almost usable with some comments and explanation... https://sagecell.sagemath.org/?z=eJw1izEOwyAQBHtL_sN1HLHlIr3rPCB9JBKwgxQ4dBALeH2M7Gib1e7MwuQgFhcKWBeIE1z67mwrq_DebO27ljxCGaHC3PAnfSKKDAWqkO297fufn-705ZdBTSmw9QlNDsprxAzDbgxQ5eMq5eFNbLw2jAuxU2kWcVuF_AH08y5O&lang=sage&interacts=eJyLjgUAARUAuQ==

If I only could get the image to display immediately ;-9

[idle estuary]

I do all the symbolic math in Jupyter. I think Sage must have a way to be run through Jupyter

Jupyter allows displaying the result of a cell right into the document

[wispy slate]

It does, but I just want it for a quick exercise on people's phones.

[idle estuary]

Google Colab is a nice hosted variant of Jupyter that I use a lot. Onlu requires a browser, so nothing else installed on my machine

[wispy slate]

Not worth it to go to the lab and manage all the commotion...

[wispy slate]

Needs a login.

[idle estuary]

I was never able to do math on a phone. Id require at least a laptop

[idle estuary]

Yeah.. I do have a gmail account so that’s how I use Google Colab

[wispy slate]

yeah, I do too, but out of 30 students I'm guaranteed to catch one without it

[idle estuary]

Another option is for the school to host a Jupyter instance.. but.. idk how much trouble that would be

[wispy slate]

Yeah, that'd be out of my time-budget ;-). But I wanted a jupyter-hub from day1; It's the future. Unfortunately IT-management is overworked (1/5 of a guy for 600 students+staff) and has no experience with server management. Well he does with things like "Office365" and all kinds of proprietary stuff I don't really want…

[idle estuary]

If the school has openings.. recommend a friend there, then collaborate with them to build out the infra and capabilities you want

[wispy slate]

We don't get any budget to hire our own IT. It's done by the district, so it's a political thing. From what I've heard, they want to double down in their upcoming open bidding for school-IT, but they really don't see that for a serious calculation we don't need them to merely double down, but more like an increase by a factor of ~20.

And obviously no one does finance any open dev-work. Instead schools throughout the city spend probably north of 8 figures each year on licences for products a team of 5 could do in half a year.

[idle estuary]

Sounds familiar. It is what it is

[wispy slate]

yeah… not my money being wasted after all...

[wispy slate]

Can I get simpy to give a tree of unreduced expressions like 6*3/2 ? When I try this, I just get the node 9.

[idle estuary]

Yes, the reason for that is that the Python interpreter does the arithmetic which is not useful in this case.
In order to avoid that, you can write a symbolic expression with only symbols, and then tell SymPy to replace those symbols with constants but prevent evaluation and any symplification so your original expression remains unaltered.
Then, you can serialize it into DOT and compile it to SVG and print it as output in the Jupyter document.
Here is a notebook usable with Google Colab that does just this:

[idle estuary]

If you like Sage, you might also like their CoCalc. They're building nice things, and they have presentations and events where they show off new functionality. They have a youtube channel where they've some recordings of those online presentations https://www.youtube.com/channel/UCle9EN6pjJrcirOBqffamgA

[idle estuary]

@wispy slate UPDATE: it's possible to do it even easier than that through parse_expr , so no more substitutions needed and the result is the same as the one in the picture above. The only caveat is that the spacing in the expression string seems to be required to get what you want (so spaces before/after operators and parentheses).

from sympy import *
from graphviz import Source
x,y,z = symbols('x y z')

with evaluate(False):
  expr = parse_expr(r'6 * (3 / 2)')
  display(expr)
  src = Source(dotprint(expr))
  src.render('output.gv', view=True)
  display(src)

[wispy slate]

Yeah I'm aware of cocalc. It's really neat.

[near cloak]

I'm making a lesson plan for 6th "11-12 y/o" grade and would like to find real world examples for the classification of numbers (rational numbers, integers, whole numbers) but couldn't really think of and real world applications. not sure if this is the place to ask but thought it was worth a shot.
what I'm looking for is a closing, I'm in a education class and I'm in need of a "eye-opener" or something to "help students remember my lesson"

[cosmic ibex]

A practical application of the "rational/irrational" distinction is going to be very hard to find.

Integer / non-integer is easier, if you don't insist on having a situation where looking at a number and recognizing it as an integer or not is the crucial skill.

For example: suppose we need to move 5 tons of stuff, and the truck we have available can fit 1.5 tons, and it takes 3 hours to drive where the stuff is needed and back again. How long will it take? Naively one might punch (5 ÷ 1.5) × 3 into a calculator and get a too tight schedule. To get a realistic plan we need to round the number of trips up to the nearest integer before we multiply by the duration of each trip.

So the point of having and learning a concept of "integer" is to make it possible to speak about that mistake and how to correct it.

I don't think that ought to be a "closing", though -- such motivation deserves to come as early as you can fit it.

[near cloak]

Thanks, that helps alot.

[tawny slate]

rational/irrational distinctions are everywhere, but I'd have to hunt for one that 6th graders can understand and care about

[long pelican]

One piece of caution, don't look too hard for a real life application to the point that you're looking at things adults do but kids don't do. For example, as a 6th grader I can assure you taxes were more abstract than a good puzzle

[woeful tusk]

Is this channel only for undergraduate education or is k-12 education (specifically elementary/middle allowed)?

[cosmic ibex]

All levels of math education.

[winged urchin]

"Hey I'm a math tutor tutoring a PhD student right now involving their project. How tf do I do this?"

I'm taking a bit of a shot at the previous comment but I would be hard pressed to 'tutor' someone at a graduate level

[cosmic ibex]

The channel is not restricted to tutors, though.

It does seem somewhat unlikely that any professors would want to use as public a forum as this one to discuss individual advising of doctoral students. But talking about how to run a graduate course would be perfectly on topic.

[winged urchin]

Fair points 👍

[ember beacon]

https://www.youtube.com/watch?v=ent5g6_gnik

[south raptor]

This is literally some dude saying his opinions with no sources

like the only thing he has linked is his blog and the content I saw there makes me want to gouge my eyes out. Why would you post this clown?

Like this guy thinks science is the reason so many ppl died from smoking rather than the reason ppl stopped smoking. Just insane.

[ember beacon]

cuz I'm too lazy to look through his blog

[south raptor]

well fortunately for you I wasted my time doing it for you

[ember beacon]

tysm ❤️

just bc it has no sources, does that necessarily mean it's wrong?
maybe he made some shitty blog posts on other topics, but I find that this video captures every issue I have with math education, and I'm now doing so much better ever since I put the video's ideas into practice (I came up with similar ideas independently)
sure, I never looked at the literature or anything,

[ember beacon]

but the good studies usually have sample sizes so large, that what works the best on average for those people, will likely backfire on me (I know this from experience)

I guess I'm just an outlier

[south raptor]

I didn't watch the video because I only have 24 hours in a day and based on the information I have I decided this video was not worth my time. If he does say anything of value then there's probably someone else out there who is saying the same things but isn't as shitty

If those ideas have helped you personally then that's great and you should keep doing that

[ember beacon]

2x (his speech is still perfectly understandable), it's only 5mins

[south raptor]

that's 5 mins I could be doing something else

like sure I'm spending time here anyway but still

[ember beacon]

science isn't what caused those people to die from smoking due to inaction (in that screenshot)
the word he would be looking for is scientism

and scientism (and the shit that parents and teachers used it to justify) is what fucked me over educationally and took away so many years of my life I'll never get back

[ember beacon]

I'm going to sign off for today
so if anyone is capable of suspending disbelief for 5 mins, maybe you'll try it, and find something that helps you so much that you can finally stop banging your head against the wall.

[tawny slate]

What the hell is scientism

[wide latch]

watching this video rn and i really hate this too

he advocates for both putting the meat at the beginning of your lecture as well as reviewing prerequisite content in the beginning of that lecture

those are just straight up contradictory

in general youtube analytics are not good evidence for how a lecture should be ran because youtube as a platform is fundamentally different to the goal of learning things at a deep level

he also advocates for rambling? wtf

[tepid smelt]

I personally don't think a heavy lecture style works well for k-12 at least. I like the approach AOPS uses which is instruction is driven through problems. I find lessons run so much smoother if we are working on problems for the majority of the time and those problems reveal the topic over time. I find good problems at the appropriate level for the students hook them more than trying to explain why this topic is important or potential uses of it in application.

The biggest struggle I face is kids are not at grade level and it's hard to differentiate when the class has students at very different levels.

In this channel I have enjoyed most reading some of the problems people have written up. Icy for example has some really good problems they have shared.

[tidal mica]

do yall take points off for not having any english words in a proof, to the point that it's hard to follow

grading analysis hw rn

ive been pretty lenient so far but

prof has said "the focus of the class isn't proofwriting" so i could let it slide but i feel like by the time someone is in analysis they should write more mature proofs than thiss

[winged urchin]

I'd just call that sloppy overall. The student does seem to grasp the core idea behind the proof though which is nice at least

[tidal mica]

yeeah i didnt take points off just made a note

[austere inlet]

I totally would, it's a pain to read

however you should've made it clear at some point that you'll dock off points for that, and explain that if you ever write a paper or class notes or whatever you'll be expected to write something legible

[austere inlet]

it's HW, not even a timed exam or anything. They certainly have time to write correctly

[tidal mica]

it was a late submission too

[tepid smelt]

I had a professor mark me up for similar issues and I am grateful they did. I think it's good to learn to write clear proofs and just knowing the right approach is not enough. In general it just builds sloppy work and you would not write a paper that way or lecture that way.

I think being a little lenient on exams is ok due to the time constraints but on HW you should practice writing clear proofs.

[distant fractal]

seeing as how they concluded the opposite of the claim...

[tidal mica]

ik

[distant fractal]

its also like... not very precise

I can pick an a much smaller than the lower bound and everything up to the last line is true

If this is an analysis class they need to figure out that this proof writing is pathetic

id take off at least a little

[tidal mica]

i agree

i feel bad doing that retroactively now tho

[distant fractal]

for doing what retroactively?

[winged urchin]

Talking about bad student answers, ahah, what about badly designed exam questions? I was just helping someone with a first year calculus course and these are two snippets of apparently an actual factual test that was given to students in previous years.... This makes me more depressed than the sloppy student answer ahah, but really they're probably related problems (bad student habits turning into badly designed questions if that student becomes a teacher)

2. is either a mean trick question or I maybe made a mistake I suppose. One can always be wrong

3. I'm pretty sure is mostly okay but one could beg the question about the limit at x=0 (yes I know that typically we imply a one-sided limit at the boundary of a domain. Honestly 3. is not so bad tbf

1. Is just whack. I haven't looked at it too much but it kinda seemed like a weird combo of IVT with the continuity stuff

EXCEPT we'd require absolutely for a<1 else the function isn't even defined for some x's, let alone continuous.

[quasi musk]

There's a difference between "I'm not going to nit pick you on every little detail" and "I'm not going to care if you write things clearly"

The way I say it is in college classes, every answer requires a complete sentence

[long pelican]

First year calculus course exam problems have proofs with nested quantifiers (looking at #3 here)?... Extraordinarily high expectations this professor has of 18 year old math students fresh out of high school

[trim violet]

did they just say that the limit of bn could be any lower bound of the sequence

[cosmic ibex]

I don't see any (1) there, but I see two different (3) ...

All three look to me like the teacher was trying really hard to avoid "crank the handle on a cookbook procedure from class" questions, while still testing that the students know the procedures. That's a laudable goal in itself, but perhaps executed with insufficient respect for the extra cognitive load that results when values that are usually givens in the procedures suddenly become unknown parameters in the exam questions. That needs practice questions of the same general kind during the course and pretty soon it all begins to eat into the time there is for understanding the core content.

[tidal mica]

i like this, thank you moonbears

[winged urchin]

Oh shoot that is another 3 ahah. The second 3 was kinda cut off and I didn't look very closely so I thought it was a 1.

The difficulty of these questions is one thing for sure. But my main gripe was that they're broken questions.

The one where they find k is broken because you go... Okay I'll plug it in and oh shoot the limit looks like 5k+2 divided by 0. That'll be something infinite or DNE unless 5k+2=0 so we do that and get k=-2/5. Except! That value of k doesn't even make the limit equal 4 but we've run out of flexibility in the question and it's impossible

Hmmm.. actually the question for finding a is not broken like I was worrying about. There are two intersections of sqrt(pix) and cos(2pi*x) within x from 0 to 1 so we can choose those two values for a and have more than one value of a that makes it continuous. I suppose. Fair play question but still a tricky one if students haven't seen similarly difficult questions

Oh but, and you'll love this Icy I'm sure, another question from this test was a true/false question stating...

True/False? If f(x)=5 then f(a+h)=5

🤨

[long pelican]

🤨

Implicit quantification might be a useful keyword here (search it!)

[winged urchin]

Oh is that basically the idea that we know they are implying the x to be any element of the domain of f and also for a+h to be a specific element in the domain?

Like an... namely... implied quantification

Oh I see it's a little more formally defined than that ahah

[long pelican]

Yes, I have a hunch students are led to believe that "let f(x) = ..." is defining something wishy-washy with a function f and a variable x, and hence questions like the above confuse them whether they should be true or false, but it's unconfusing if you properly read it as a case of universal quantification: "for all real numbers x let f(x) be ..."

[turbid zenith]

PSA: Making videos for class is a lot of work. If you're gonna flip, be ready to commit. x.x

[turbid zenith]

https://www.youtube.com/playlist?list=PLZzHxk_TPOSv_pPkrhVfKC8ACpW82tXeS

This semester is going really interestingly in calc.

[long pelican]

Have you given exams yet? Any surprises?

[ripe mantle]

:)

[light garnet]

Yes. This is not the type of thing you want to do without expecting a ton of work on your end.

Not only did I edit my 3 hours of lectures each week down to 45 minutes, but I also had to make them ADA complaint via captions

I quit teaching after doing 3 semesters of that

(because, unfortunately, data science tech seems to change every year)

[turbid zenith]

Yup lol

I was ahead for a while but I've been slowly catching up to myself

I'm not gonna do it for Calc II/III/IV, just for Calc I, because I'm doing a drastically different order than any textbook

[winged urchin]

You can't just leave us on a hook like that

What order are you teaching it in?

[brazen bluff]

So, I'm tutoring this undergrad in Calculus, and he seems to lack a lot of the foundations expected from someone in a University Calculus class. They're now starting limits. How can I go about
1.) Getting him caught up with the foundations
2.) Prepare him for quizzes on content they're covering

Basically, how do I play catch-up

Because at this point, I feel like I'm trying harder than the student. At one point, I was the one telling him where his class resources were, and what they were covering next (outlined on his syllabus)

[light garnet]

I highly recommend getting a copy of Precalculus Mathematics in a Nutshell and getting through that quickly

[brazen bluff]

I appreciate the resource, thank you

Math Pedagogy is a whole 'nother can of worms it seems like

[turbid zenith]

Oh sorry I fell asleep x.x

Didn’t mean to leave people hanging. I’m doing derivatives first, then integrals (that’s where we’re at right now), and at the end we’re doing limits

[turbid zenith]

Also for each topic, we’re starting with applications and an idea of what we want to know, using those to motivate computations, then getting graphical intuition from those, and building up to having algebraic rules and definitions.

As opposed to the textbooks who do the exact opposite of that, where applications (“word problems”) are an afterthought shoved at the end of the chapter if anyone gets around to them.

[winged urchin]

Ooo that's neat. So of course without limits you can't derive the derivative formulas so I guess you're just giving them the formulas and perhaps motivating that some handwavey way?

Do you then explain where derivatives came from in a sense once you get to limits? Or is the limit stuff more for the series/sequences material?

Ooo word problems sound good

I am curious how you might motivate the power rule with a word problem

[turbid zenith]

Judicious hand waving is a skill! :3

But we’ll bring in limits toward the end to kinda fill in the gaps once they know what’s being talked about

As for power rule that’s one of the few things I didn’t explicitly do with an application … but once we did the product rule I have them a clue on how you could derive the power rule with it

[winged urchin]

So for product rule, a word problem thing... You need to motivate how looking at the change of a product somehow intuitively is the same as the change in one part of the product multiplied by the other and vice versa..

[turbid zenith]

Actually what I did was more use the geometry to explain why it works

The whole "tweak two sides of a rectangle, and the smallest piece is negligible" thing

But the motivating example I used was revenue as price times demand

And showing why taking the product of the derivatives gives entirely the wrong units

[turbid zenith]

https://youtu.be/zVw-OoegUuw

[winged urchin]

I appreciate the effort you take in dressing up for the videos ahah. I go on cam for tutoring but I do not dress up quite yet

I loveeee talking about units and I also hate writing them in equations at the same time ahah

I always tell my students you can actually check whether you have the right answer by thinking of the units involved and making sure LHS units are the same as RHS units

Lovely

My go-to for showing that (fg)' is not f'g' personally is typically to just consider (x^2)' vs. (x*x)' assuming they know the power rule

But this is a nice real world based version

Great video. If I had anything to say I think the music may be a smidgen too loud in some parts but I'm really stretching with that comment. Love it

[deep kindle]

Has anyone here ever taken a particular interest in how different students best learn math as differentiated by personality type?

[unborn osprey]

no, but i ask my students to share their zodiac signs on an anonymous survey at the beginning of each semester.

[cinder rover]

@deep kindle Do you have any experience with this?

[hollow kernel]

I'm curious - what's your pedagogical reasoning behind taking this nonstandard order of topics?

[restive apex]

I don't know about personality type, but I definitely approach things differently with different students based on previous experience with them.

[turbid zenith]

Students often struggle with limits right at the beginning of the semester, and I don't think they really help students understand the calculus. They do a whole bunch of algebraic hoops and don't see much purpose to them.

So I want students to get used to what derivatives and integrals are first, and then later on we can formalize it and I can frame it in terms of "this is what mathematicians do — take intuitive ideas, and figure out how to make them more precise"

It's largely inspired by a book called Discovering Geometry: An Investigative Approach by Michael Serra, where he doesn't do any proofs for most of the book and leaves everything as a conjecture, but at the very end introduces the idea of proof and then proves everything all at once at the end

So the students aren't juggling trying to learn geometry with simultaneously learning proofs (and having no idea why they're doing so)

I'm of the opinion that mathematical foundation ≠ pedagogical foundation

[cosmic ibex]

Hmm, does that mean you don't even present the concept of limit to start with, or you tell what a limit is but hold off on how to compute them?

[turbid zenith]

I've actually mentioned a few times "we'll make this precise soon, when we talk about limits"

I've been phrasing things in terms of infinitesimals, but in a handwavy way

Just enough to have the intuition of "zoom in infinitely far until the curve looks straight" or "cut the shape into infinitely many infinitely-thin rectangles"

What we're doing involves limits though, just not by name. Like for derivatives, I showed them how to compute the slope over smaller and smaller intervals, which is a limiting process, but instead of saying "okay take the limit" I'm saying "okay imagine we took two points that are infinitely close together"

[cosmic ibex]

Ah, okay. For a moment it sounded like it was "these symbolic rules define how to compute a magic number that we're calling 'slope' for no particular reason".

[turbid zenith]

Hell no. I'm still giving them a reason to believe it's a "slope at a single point / of the tangent line" and a way to visualize it. 😛

[lost pebble]

that seems really interesting and the kind of spirit I want to incorporate. Can I copy your order and approach?

and if I may, a question about that. How do you deal with having to bring in formality "suddenly" at the end. In other words, how do you manage gradually bringing in the algebraic rules, edge case highlighting and proof-making into derivatives and integrals?

[turbid zenith]

Im still figuring that out but I’ll let you know how it ends up going

[winged urchin]

Did you see the video he posted above of one of his video lectures? Pretty great imo. I enjoyed watching it as an educator

[turbid zenith]

https://www.youtube.com/watch?v=Ra_gPCId2-k Just finished this one!

[earnest trail]

calculus 1???

[turbid zenith]

Yup

It's not a usual Calc I topic but I think it'll make implicit differentiation and related rates easier to understand and easier to calculate

Since my usual way of doing imp diff is to take the total differential of both sides and divide by dx

$$\begin{align*}
x^2 + y^2 &= 1\
\mathrm d(x^2 + y^2) &= \mathrm d(1)\
\mathrm d(x^2) + \mathrm d(y^2) &= \mathrm d(1)\
2x,\mathrm dx+2y,\mathrm dy &= 0\
2x+2y\dfrac{\mathrm dy}{\mathrm dx} &= 0\
\dfrac{\mathrm dy}{\mathrm dx} &= -\dfrac xy
\end{align*}$$

[burnt vessel]

DMAshura
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[earnest trail]

oh I see

that's cool

[long pelican]

Total differentials seems scary to teach because you have to tell the students not to think too hard about what type of objects dx and dy are (cotangent vectors on the real line…), so it’s hard to avoid reducing it to procedures

[earnest trail]

cotangent vectors? what are those?

[long pelican]

They are vectors in the dual space of tangent vectors, so they are basically linear functionals from the space of tangent vectors to the field of scalars

[turbid zenith]

I mean small/infinitesimal changes works fine for me ;P

[long pelican]

But what type of object is dx? We have numbers (1, 2, 3, 1/2, etc), we have functions, we have variables (which are really just numbers in a logical context). Clearly dx isn't a number, nor is it a function. (Well, it is a function if you think of it as a linear functional on the tangent space but that's not for calculus students)

What are the rules for what you can do with dx? Can you add dx with a number? Can you multiply dx with a number? Can you plug dx into a function? Can you divide a number by dx? Can you say dx - dx = 0? What about dy? dy/dx means something, but does dy - dx? How about dy * dx?

All of those can be answered ad-hoc-ly of course but what theory binds them together?

The fact that I can't point to a unified treatment of differentials that answers all those questions and is developmentally appropriate for calculus students why I'm sort of uncomfortable doing too much with dx in calculus (even though I have to...)

Actually I take back dy/dx meaning something, because dy/dx is defined as an atomic piece of notation, and we always say that dy/dx is not actually a fraction

[deep kindle]

What are your guys' thoughts on calling shorthand evaluation methods "tools" instead of "rules"?

I had a teacher in high school that felt very strongly about trying to make math feel less like memorizing strict rules and instead conceptualizing tools for solving problems

So e.g. she would call things the "product tool", "chain tool", so on

[deep kindle]

It belongs to the world of forms I guess lol

[turbid zenith]

One way I've seen dy, dx, etc handled is that they represent movements along a particular tangent line to a curve and thus just represent real numbers, where dy is defined as f'(x) dx, so dy/dx really is a fraction

I still encourage thinking of them as very small and potentially infinitesimal

[deep kindle]

Yeah I was gonna say, if they'r ejust an infinitesimal change in some number I would think they'd fall under the real numbers

[turbid zenith]

If they’re infinitesimal they’re not really real

[earnest trail]

yeah it's like how infinity is not real

[deep kindle]

Could an infinitesimal be represented in $\aleph$ form or is that just for infinities?

[burnt vessel]

lexitorius

[earnest trail]

no

aleph is the size of the natural numbers

doesn't have anything to do with infinitesimals afaik

[turbid zenith]

I'm fond of the surreal infinitesimal ε = 1/ω

[cosmic ibex]

The cardinality of $\bN$ is $\aleph_0$. The letter $\aleph$ \emph{without} a subscript is rarely assigned any meaning, but a few authors use it to refer to the cardinality of $\bR$.

[burnt vessel]

Troposphere

[cosmic ibex]

(whoops, thought I was in #foundations)

[turbid zenith]

I don't think I've ever seen ℵ used that way 😮

[deep kindle]

I've seen it used $\aleph_1$ for the cardinality of $\bR$

[burnt vessel]

lexitorius

[turbid zenith]

The way I've seen it, ℵ₁ is the "next" cardinal number after ℵ₀. Whether that equals the cardinality of ℝ or not is precisely the Continuum Hypothesis.

[deep kindle]

Anyone have thoughts on this? #math-pedagogy message

[turbid zenith]

I think it's cute! I might have to start using that sometime soon

I've never been a fan of calling things "rules" myself

[deep kindle]

I've tried to start doing it myself but in fear of being made fun of I usually compromise with "method"

Like last night in one of the help channels I called the chain rule the chain method for differentiation

[turbid zenith]

Method, technique, property... still trying to come up with something I like

But the issue is, those seem to be the "standard names" for the things

[deep kindle]

Touche

[cosmic ibex]

It's quite rare, and the sources that use it that way are generally not particularly advanced. I've seen it claimed that the notation goes back to Cantor, but haven't found a primary source to confirm that.

[turbid zenith]

Also ... man, calculus quiz grading is rough ...

[deep kindle]

Sounds fun

[turbid zenith]

Here's the quiz I'm grading. It's done on students' own time rather than in class.

[cosmic ibex]

What on earth would be the use of integrating a stock's price over time?

[turbid zenith]

Initial price + integral of rate of change = final price?

Is that not a thing?

[cosmic ibex]

Ah, no the function is a rate.

[deep kindle]

total money spent on stocks at time t ig

[cosmic ibex]

I didn't read closely enough.

[deep kindle]

I assume at least

I mean the integral of a distance vs. time graph is total distance at time t

So I just swapped the words tehre

I guess it would likely have applications but I know nothing about stocks

[turbid zenith]

I'm trying to deliberately have a big variety of applications

All the problems I usually find for integrals usually either involve distance/velocity/etc or water in a tank

[deep kindle]

Which is a good thing, students often feel more confident in mathematics when they're given problems that have clear real world implications

[turbid zenith]

But ... okay on the last problem, with the linear approximation? SO MANY STUDENTS keep plugging the function into the linear approximation

Like you're finding a linear approximation L(x) of tan(x) near x=5π/4

So that you can approximate tan(4) by finding L(4)

But ... lots of students are finding L(tan 4)

[deep kindle]

What was the linear approximation lecture like?

I assume there were graphical displays of what the linear approximation looks like and achieves

Maybe if there weren't it wasn't very clear what it's doing in the first place

[turbid zenith]

https://www.youtube.com/watch?v=ARp7VkLppdI

[deep kindle]

But I'm just extrapolating from zero information

[turbid zenith]

(I'm calling (a,f(a)) the "anchor point", for where the tangent line is attached to the curve)

[deep kindle]

Valid

[turbid zenith]

But yeah it's clear they're just symbol-shuffling :/

[deep kindle]

Yeah I think the video a pretty clear about computing L(x) and then evaluating L(4.1)

Which would imply evaluating L(5pi/4) for the other problem

Maybe the fact that tangent is there at all threw them off?

Loving the trifold tie knot

I wore my tie like that to my high school graduation

[turbid zenith]

They did the same on the previous assessment too

Approximating ln(x) close to x = e³, and using that to estimate ln(20)

Instead of finding L(20), they found L(ln 20)

[turbid zenith]

Thanks! 😄 I always wear it like this

[deep kindle]

Have they seen more examples than the one in the vid?

Maybe seeing a walkthrough of problems involving trig, exponentials, logs, stuff like that might help

Repetition always helps

[turbid zenith]

Well we did one more in class, and when I posted the last quiz I posted a detailed explanation :/

That lays it out in this exact same step by step fashion

[deep kindle]

Hm I'm not as sure then

[turbid zenith]

You'd think that students would be able to, at the very least, look at one and pattern-match to solve the other

"This goes here, that goes there"

[deep kindle]

That's how I'm getting through calc III lol

My prof never really explained parametrizations so I brute forced it into my brain by looking at other problems

[long pelican]

Cheers, that's the same problem I'm trying to solve! Giving an exam tomorrow and I'm hoping (fingers crossed) that most students have an idea what's happening!

(Calc II, looking like much stronger students than last year)

[deep kindle]

Do you guys administer gateway exams for your calc classes?

I had a gateway option when I took calc II and I really liked the idea of it

I also tutored a high school student who said he had a similar exam in his precalc class, so I've been thinking about whether it would be a good idea for high school level maths

[turbid zenith]

I haven't though I've considered it

https://www.youtube.com/watch?v=9Ac-RII6Y0A BTW, unlisted right now, but here's my take on implicit differentiation... featuring Bitcoin!

[deep kindle]

Dude your mic is awesome

Makes me think of my whole friend group on discord where we all have expensive headsets that sound awful and there's this one guy with a $13 amazon mic that sounds incredibel

[turbid zenith]

It’s a Blue Yeti

[deep kindle]

That's an approach to implicit diff that my calc I prof did not take

I like it

I haven't worked with partial differentials (I think that's what it was iirc)

It was introduced to me as a sort of chain rule applied to y since it's a function of x

Just like how sqrt(x) is a function of x

Tbf I like your way more

[turbid zenith]

It also makes related rates dead easy

Just divide by dt instead of dx

[deep kindle]

Do you have a video on related rates?

[turbid zenith]

That’s my next video I’m gonna make lol

Which should be done later this week

[humble raptor]

omg its [person i know!!!!]

[deep kindle]

,w implicit derivative of y^2 = x^3 - 4x + 1

[burnt vessel]

Results provided by WolframAlpha

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

Requested by lexitorius#4749

[deep kindle]

Hell yeah I got that right

I like your way more

Actually I guess they're not much different I just never happened to be told to write dx's

We didn't abuse the leibniz notation lol

[turbid zenith]

It’s the typical “don’t treat dy and dx separately” thing ;P

[deep kindle]

Like I would differentiate the x^3 as just 3x^2 and not 3x^2 dx

[turbid zenith]

Despite then doing so when you do separation of variables

[deep kindle]

And then differentiate y terms as 2y dy/dx instead of 2y dy

[turbid zenith]

Which usually results in “uhhhhhh it’s just a notation”

[deep kindle]

Yeah I was always told to treat it as one symbol

It makes me want to abuse more notations

I think implicit diff is my favorite differentiation method, it's just fun

I doodle in class by doing random implicit problems

Isn't one of the millennium prize problems about elliptic curves?

[turbid zenith]

Me:

[turbid zenith]

Yup. BSD conjecture.

[deep kindle]

I wish my HS teachers had gone more into modular arithmetic

I've played with it on my own and love it, especially that "mind trick" where you can calculate the day of the week of any date

Ooh I love the little "check your understanding" problem at the end

Giving them a bonus problem to think about

[winged urchin]

I've always found it funny and a little misleading that the 'implicit' in implicit differentiation actually really has little to do with the differentiation itself

It's just that it's differentiation applied to an implicit equation as opposed to an explicit one

Same sort of deal with logarithmic differentiation, really ahah

[deep kindle]

@turbid zenith nailed your "check your understanding" problem

[tawdry venture]

i am once again seething at the use of "whole number" to mean naturals with zero that is apparently common in america

[turbid zenith]

Never understood that

Like give it a name sure but “whole numbers” is a weird choice

[tawdry venture]

so many languages' word for "integer" translates literally as "whole number" too... it just sticks out like a sore thumb

[cosmic ibex]

Including "numerus integer" in Latin. 😆

[deep kindle]

I like integer over whole number

I also strongly prefer complex over "imaginary"

I think calling any numbers imaginary is dumb; if anything variables are imaginary because they're arbitrary values

[restive apex]

I don't think it really is that common. I was taught that definition in grade school, then it was promptly discarded and I never saw it ever again.

At least, not until joining this server.

[deep kindle]

I grew up calling them positive integers

[winged urchin]

I see whole numbers used in elementary/highschool here in Vancouver BC

[long pelican]

After the exam today I feel like Americans are uniquely handicapped in basic probability compared to, like,... Chinese students and Russian students

[pale wasp]

A lot of American high school education is not very good

[winged urchin]

What kind of questions did you have Icy?

[long pelican]

Interested in the exam, eh?

[winged urchin]

I personally know I always had trouble understanding when I should use Choose versus Pick, at least when I was a student

[long pelican]

Oh it was just one question

that had dismal results

[winged urchin]

I always resorted to alternative ways of getting the probability rather than choose or pick

[long pelican]

And it was meant to be a freebie question

to set up the equation for the second part, which involves calculus

The freebie question didn't even need any calculus

[pale wasp]

Probability class?

[winged urchin]

You're really selling it ahah

[long pelican]

Calc 2

But probability was a big part of it

2nd time a freebie question unexpectedly got low scores lol

[winged urchin]

Oh so they needed to think of areas, huh ya

[pale wasp]

Random variables might be a bit advanced for calc 2

[long pelican]

They've had a lot of practice with random variables in problem sets

[pale wasp]

Standard us college calc 2?

[winged urchin]

If I wanted to emphasize the area understanding the goto I guess would be calculating the area of a circle via random points in a square

[long pelican]

Ya, but I have basically complete freedom as to what the syllabus is

which is nice

I don't think anyone in their right mind enjoys the standard syllabus

[pale wasp]

Right but it's calc 2

[long pelican]

They're still doing fine overall

This question was just the surprising bit

They did very well in the proof question in problem 5

[winged urchin]

I assume at least some of the probability homework involved using the area perspective? I'd agree if that sorta.. approach wasn't done it might be a stretch

But ya