math-pedagogy

[winged urchin]

The mixed fractions are written most like they might be said however

Saying five halves is a bit awkward

Saying 2 point 5 is also a bit awkward. Although in this particular case you'd probably look at 2.5 and just say 2 and a half anyways

But 2½ literally reads as 2 and one half

Not to betray my own beliefs since I do think that it's silly to take points off if you don't have the answer in the prescribed manner (unless it's specifically that manner being taught. Like if they are teaching mixed fractions. Though there should still be a request in the question to display the answer as a mixed fraction rather than just an expectation)

My previous point is more apparent I think if you have more awkward decimal expansions.

Like if you have 34/7.

Saying thirty four sevenths is silly.

Saying four point eight, five, seven... blah blah blah is silly.

But saying four and six sevenths is, I'd argue, the most clear

[earnest trail]

ya

[earnest trail]

how can I explain the concept of flipping an inequality symbol when multiplying by a negative

for example:

-x<5 implies x>-5

[long pelican]

Show one is true iff the other is true

If necessary, explain the concept of true and “if”

[near oriole]

concrete examples? (Like -1 > -2 -> 1 < 2)

a few would be enough to convince it should be true

Perhaps a graph of y = -x can help too...
This is a special case of applying a monotonic strictly decreasing function to both sides of the inequality and needing to flip the inequality sign. The proof of this can be shown visually.

[long pelican]

Actually I amend my answer to:

Show the statement that one is true iff the other is true and make sure they understand what that statement says, proof comes after that

[earnest trail]

I wouldn't go through a proof

but ye

[shadow basalt]

If I have more money than you, I am wealthier

If I have more debt, I am poorer

(thinking of -money = debt)

[earnest trail]

oh okay that makes sensd

I'll definitely use that

[long pelican]

The correct answer in general is to find out what they’re missing then come up with the explanation to fill it in

My answer guesses that they’re likely missing the meaning of how the procedure relates to logic, and the other answers guess that they’re missing intuition or why the theorem is true

But every student misses something different

[earnest trail]

tbh idek how to prove it

[long pelican]

Start with -1 < 0 maybe

[burnt vessel]

the G

[quasi musk]

Today I was tutoring a student and the fire alarm went off at the college. While we were evacuating I had a whiteboard, marker, and eraser in hand

And continued the session during the evacuation

[earnest trail]

lmfao

[quasi musk]

The english tutors were like "And this is why I'm not in math"

[tawdry venture]

ok so like

can someone explain to me what 'grading on a curve' is, and why are students whose raw(?) scores are high 'curve ruiners'?

[real mauve]

you apply some (possibly nonlinear) function to the grades. how this is done depends on your aim

i think in the states it is common to try to force grades to approximately follow a normal distribution, for instance

and so grades that are naturally high affect the function that is applied

(though not necessarily)

in germany it is common to simply rescale the grades

take one of the highest grades and make that the new 100%

so if a handful of people get a good grade, the scaling is very small

alternatively, a fixed offset can be used, or something like a sigmoid function

at any rate, how high the grades naturally are affects to varying degrees how much the grades are modified

[tribal delta]

It just means fitting to a normal distribution no matter what distribution the raw scores follow. It tends to push top grades boundaries up, possibly all the way to 100%. So a class with 39 100 raw scorers and 1 98 raw translates to 39 As and 1 C.

[real mauve]

i think icy and i agree in that that is the worst kind of curving

[tawdry venture]

so curving like that is basically making tests competitive?

[shadow basalt]

Yes. I think people often refer to forced-normal-distribution curved classes as "strictly competitive"

I think it is generally more common to use a curve to improve students grades, at least in my experience. So a curve in that sense would be more similar to the german one edd mentioned, where the real goal is to adjust if the exam was harder than expected

Curve ruiners would then be people who scored highly on an exam where others scored poorly, which often causes teachers to curve less for the other students

[pastel horizon]

You could just convert a raw mark into a z score and grade based on that?

[cosmic ibex]

That's essentially what the normal-fitting curve does.

[quasi musk]

It depends on how the instructor curves; most instructors who have a basic understanding of statistics know how to curve so that everybody improves

It was frequent in my HS that the curve was based on the highest scoring student, and that would set the "100%"

And everyone else was graded based on that, so the higher the top student's score was, the lower other peoples were

[tepid smelt]

At the high school level I don't curve but do grade partial work generously especially to lower skill students(IEP or students repeating the class). That and I allow retakes for students who fail so if someone wants to pass they can eventually. It has helped pass a few who wouldn't. The bell curve is awful I am glad I never had a class like that. Some of my classes had a curve in that they lowered the % for passing so like a 60% was still a C bit you needed a 90% still for an A. I have seen many schools make it easy to pass probably to juke the stats.

[flat cargo]

I'm writing an introduction to generalized least squares and I'm wondering if anyone knows an intuitive proof that for basis $e_1,\dots,e_n$ the coefficients $a_j = \langle v, e_j\rangle$ give the smallest L2 error $|\Sigma a_je_j - v|^2$. all the proofs I came up with do some series manipulation, cancel the orthogonal terms then low and behold we find the minimum (either by rearranging into a quadratic or taking the derivative)

[burnt vessel]

uli

[flat cargo]

with tensor notation and abbreviating the inner product as vector multiplication it's not that bad

but ideally there'd be a rigorous and intuitive proof

oh there's also an inequality argument which is a bit better, maybe ill use that. @ me if anyone knows something better though

[earnest trail]

is this the right channel for that?

[vapid herald]

Hey can I share a book here (PDF)? I'm curious what you guys would think

[shadow basalt]

If the pdf is legally available for free

[quasi musk]

Looks like I'm gonna be TA-ing a Spivak based Calculus course & an honors linear algebra/differential equations course that goes over the basics of Topology/Dual spaces

[earnest trail]

interesting

can I take your courses

[quasi musk]

no I don't teach

I just provide study group stuff

I'm only teaching middle school & high school

[light pond]

Hey has anyone tutored over Zoom before

i suspect the kid doesnt have a ipad

so I’m just wondering what the best way for me to see their work is

I have an ipad that I can screenshare with to teach stuff

But should I just get them to like. tell me what to write? that seems wrong

[severe kelp]

thats kinda what i do

like, i give em a problem

and then ask em to work it out

[light pond]

On paper?

[severe kelp]

and usually ask em to reason outloud

[light pond]

Right

[severe kelp]

so i can write down what they r thinking

[grim spindle]

I had a student whatsapp me their solution then put it up on the screen or put a camera on their paper

And yea reason out loud

[light pond]

I was thinking that shin but doesnt the text get flipped

so its hard to read

lol

ya I could get them to text it

hm the jds way seems nice too

[grim spindle]

They could log in from their phone

[light pond]

Maybe for longer problems it is nicer to do the text thing

[severe kelp]

yeah i usually dont do terribly long problems

[light pond]

I think yes I will do the speak outloud prt

its all like

algebra 1

[grim spindle]

Depending on the class it's important to sometimes let them write without interruption then critique them once they're done

[light pond]

or whatever

[grim spindle]

Because writing properly is important

[light pond]

3x+1=5 stuff

[severe kelp]

oh lol pain

[grim spindle]

Oh ok

[light pond]

yeah

[severe kelp]

im sorry jesse :(

[light pond]

LMAO

it wont b that bad

[grim spindle]

Then yea this isn't relevant really

[severe kelp]

thats what u think lol

[light pond]

i cant confidently teach anything harder

[severe kelp]

like idk how one would even explain like

things in alg1

[light pond]

slowly and multiple times

[severe kelp]

its just "figure it out" ykwim

[light pond]

im gonna watch some like khan academy

and get the vibes

[grim spindle]

Khan academy at that level is just working out solutions to the same problem with different numbers out loud

It's different when you're tutoring vs teaching

[light pond]

hmm

what do u recommend

i wish i could see them work a problem live

maybe ill use an online whiteboard

bad idea

[severe kelp]

thats probably a good idea

[grim spindle]

I've never tutored at that level but I think it's usually best to let the students to problems and correct them as they go along

[severe kelp]

inb4 technical errors take up half the time

[grim spindle]

See what misconceptions they have

And try to address those

[light pond]

yea thats what im worried abt jds

[severe kelp]

yeah and let them go down the wrong paths

[grim spindle]

If they don't have a tablet online white board will be painful

[light pond]

yeah thats good shin

[severe kelp]

i think its v tempting to push them really hard in the right direction

[light pond]

hm yeah ok ill just let them tell me ahat to write maybe

[severe kelp]

but its not good pedagogy usually

[grim spindle]

This is very important

Don't immediately correct them

[light pond]

oki

[severe kelp]

if my student is like "does this u-sub work?" im like "lets try it"

[light pond]

[severe kelp]

even if its wrong

[light pond]

right

okay i can do that

thank GOD i dont need to tutor someone doing. usubs

lol

[severe kelp]

yeah it takes a while for most students to get the right intuition ig

[winged urchin]

This is normally what I do too.

Sometimes I'll kind of ask them to 'guide' me too

Like I'll write down the problem and ask them what they want to do first

It can actually be nice because if they answer more informally like "I would remove the 1" or something like that then I can put it back on them and say something like "No you need to tell me what operation youre doing on both sides. You can add, subtract, multiply, and divide by the same thing on both sides or you can multiply and divide top and bottom of a fraction by the same thing. Now what would you like to do?" Just to get them to think more formally about the problem.

Then also Ill sometimes tell them what the result of their choice is and ask if it's what they wanted.

Kinda nice

For their work I can usually ask for photos of the problems or just screenshare if it's online

[tepid smelt]

Is this at RSM? What age group for that level? Also my new pet peeve is students complaining when given problems that don't exactly match hw/classwork. Its like they refuse to even try it thinking they can't do it. It goes back to what Icy says though in that students really lack conceptual understanding

[quasi musk]

No, that's not RSM. That's at my community college

[tepid smelt]

Ah cool I know one community college in costa mesa who uses spivak. They even had a class I think that was on tensors and calculus on manifolds. I remember it having lots of classes not offered at pretty much any other community college. In fact many don't even offer linear algebra sadly.

Sounds like a really fun class. What textbook do they use for honors linear algebra?

[quasi musk]

It's the same one

[quasi musk]

I went through this very same program, starting 9 years ago in 2013. Transferred out after completing the Calculus on Manifolds class

So we used to use the Multivariable Mathematics by Williamson & Trotter second edition

But we're now moving over to Linear Algebra Done Wrong, and in search of a good differential equations textbook

If you're in the Costa Mesa area I can tell you it's a pretty good class/college, especially for math

Also a remarkably strong Physics, Chemistry, and CS department

[vapid herald]

Any one have experience with this book, teaching a non mathematicians from scratch?

[tepid smelt]

I always wanted to go back and learn differential equations properly. That was one of my least favorite classes. https://web.williams.edu/Mathematics/lg5/Rota.pdf Giana carlo rota talks about several of the things I hated about my experience with the class. I think that is a good read as what to avoid as your helping your students out.

[tepid smelt]

I don't really know how your expected to learn much from that book as it has no exercises. Really if your goal is to learn math your should get some good textbooks that you can find suggestions for in the book-recommendations.

[cosmic ibex]

I don't know the subject, but boy that is a very angry paper.

[tribal delta]

Why the attack on variation on parameters though...

[quasi musk]

Ok so in my research we actually had integrating factors

in a hardcore PDE/harmonic analysis/Probability paper

[quasi musk]

I think that text lays it on a little too thick

[wise onyx]

Whats the point of lectures?

If people can just read the books on their own and learn by themselves?

I have an idea

You know they say if you wanna learn a new word, you don't just read/ memorize the definition, but you also hear it used in practice and use it yourself in your own speech

I think the function of lectures is to ease the transition between the reading phase of learning and the doing exercises phase

Its to set the student in an environment where they not only read the material, but also hear it used so its a bit easier to use it themselves

What do you think of this idea?

#math-pedagogy

[quasi musk]

Because people can't do this

[wise onyx]

Good point

[shadow basalt]

There is intuition not in the books that a lecture can provide, including, but not limited to, more examples, historical motivation, and even just things like "you can think of X as Y which is not entirely true but can help"

Books leave out a lot

Lecturers also help you separate important parts of a book from things that aren't so important

and they choose good exercises

and they set a good pace

[earnest trail]

do books have things that lectures don't?

[long pelican]

Time 😎

[earnest trail]

true

[cosmic ibex]

And the ability to go back at will.

[tepid smelt]

Really I find lectures often leave out a lot and books are much more comprehensive. I find good lectures do motivate the material well and highlight important ideas and overall condense the material well. I still can't learn without reading the chapters myself. Yet by reading combined with good lectures I can really pinpoint key concepts. It also helps when professors have a good eye for problem sets. A book can be really intimidating with the amount of problems and sometimes difficulty. Good professors have a really good way of picking out what problems to focus on.

Really though my professors were most helpful in office hours breaking down key misconceptions I had. It saves countless hours going to an expert to break down your misconceptions rather then struggling alone.

Yet bad lectures are also common and I found I learned just as much reading and doing pronlem sets and using study groups or online resources to clarify things for me rather then sitting through a lecture going through the exact thing in the book

[shadow basalt]

Sorry I think you misunderstood me

lectures need to be supplemented with reading

I am just saying that they provide different information

It is better to have both than just one or the other

Books leave out a lot, lectures do and should leave out more, but they leave out different things and create a com,plete picture together

[tepid smelt]

At the k- 12 level kids brains are not developed enough and they need a lot more structure as they can't yet learn alone. I think slowly the goal is to develop them into independent thinkers but its a difficult challenge on how to do it

[tepid smelt]

Thats a good point. Good professors have a clear picture of the material not only in that textbook but how it connects to the content at large. I will say I have often had bad lectures and I wonder if those can actually do more harm. Lecturing well is an art and takes a lot of effort. Even simple concepts at the secondary level are tough to present well let alone more abstract ideas.

I mean even books can vary a ton in quality despite presenting the same thing.

I think a lot of professors really don't care about teaching well and view it as a required chore to do what they really enjoy which is research.

[shadow basalt]

this is true

also like

fairly understandable

[wise onyx]

You can care little and teach very well, you can care a lot a teach poorly. I don't think how much the teacher cares matters all that much, as long as they care the minimum ammount to do their job

[shadow basalt]

this seems like

incredibly bad logic lol

just because the relationship isn't perfect does not mean people who care more don't tend to be better teachers lol

[earnest trail]

exactly

[earnest trail]

this doesn't seem to be the case so much in most classes, even in high school

it seems like even in your senior year, information is spoon fed to you

of course it depends on the teacher but I've really only had a couple of teachers in high school that really want you to do critical thinking on your own with as little guidance as possible

I'm sure when I get to college it will be a huge shock

[tepid smelt]

From a teachers perspective its easier to spoon feed. It requires less thought and helps with classroom management. High school kids don't respond well when you don't spoon feed them and they get angry and act out because they are not used to thinking. I would have a much easier time if I did cookie cutter no thinking lessons. Becoming an independent thinker is hard and most kids are conditioned to not think for years. Its one reason why so many drop out of stem/college because the school system has failed them in many regards.

The privledge kids can get by because they have a ton of resources like private tutoring and educated parents so most of their learning is not happening in the school to begin with.

[long pelican]

Thoughts on using wedge product to introduce determinants for a theory-based linear algebra class?

[quasi musk]

It's a good idea

That's what we did at my little community college ~ you can do it if you have an engaged group of students

if it's a disengaged group of students it's most likely to flounder

[tall beacon]

Does anyone have any ideas/resources or thoughts on teaching the concept of ratios to 6th graders? A lot of them seem to struggle with this and I was wondering if anyone had any suggestions on how to teach this particular subject.

[tepid smelt]

What have you tried so far? Younger kids respond well to visuals and manipulatives

[quasi musk]

I usually default to coins

A quarter is 1/4th a dollar (four quarters make a dollar)

Have them add up the money, then show that fractions work

[near oriole]

This thought has plagued for a long while. Certainly, when the lecturer motivates and explains well, it's great, but if they're just reading their notes or reading+writing proofs word for word, I just cannot find it in me to follow...

[grim spindle]

To me be the biggest purpose of lectures is being able to ask questions in the moment, stop the lecturer to clarify things that aren't in the book, etc.

[earnest trail]

you can stop the lecturer during class?

I always got the impression that it would be disrespectful in front of like 200 other students

[real mauve]

this varies culturally

but i would agree it's part of what makes lectures good

[grim spindle]

I mean, if you're at the point where you are mathematically mature enough to read books to learn, your lectures don't have 200 people

Well that's a bit reductive but tbh even in big lectures, I don't think it's disrespectful. Ideally the prof should stop for questions often but if theh don't then there's a reason you're sitting there listening to them in real time instead of a recording

And that is to ask questions

[near oriole]

no one does this here unless theres an error
most that happens is asking at the end

That would be great if it was encouraged

[earnest trail]

so lower level classes are typically in lecture halls while higher level classes are typically in classrooms?

[near oriole]

So classroom -> lecture -> classroom?
jks aside, the transition to lectures gets a lot of ppl

[earnest trail]

oh rip

[cosmic ibex]

Some of the instruction I see people describe in this server sounds like the point of lectures is to pretend that neither the printing press nor the photocopier has been invented, so students sit down to hand-copy a textbook for themselves from the professor's dictation ...

[real mauve]

in some countries it kinda does work like that

[pastel horizon]

Yes and no. Sure you could make hand outs with useful information but then why bother having a lecturer? The point is they share their expert input and advice

A combination of both is optimal

The best ones I've seen are simply just printing out the slides and having a section for note taking next to them

[earnest trail]

to me it feels like, what's the point of going to college if I can learn everything I need online? for the hands-on experience? there are internships and opportunities for that kind of stuff outside of college. with all the technology and stuff it feels like college is kinda useless

im sure my opinion here is uneducated and under-researched but would someone be willing to have a discussion about this?

[real mauve]

i would say having a fixed schedule and study plan, professional supervision, colleagues/classmates and additional resources is the point

along with supervised research & feedback

you don't get any of those if you do it on your own. nevertheless, going through the system is then supposed to enable you to study by yourself afterwards

[earnest trail]

okay that makes sense, thanks

[pastel horizon]

It's like a tree. Without a strong trunk you won't be able to "branch out"

Uni helps develop the trunk

[earnest trail]

true

[quasi musk]

It's very common in southern california to ask questions in lecture for math classes

However as a student at UCLA I rose my hand and asked a question

Lecturer looked like he was going to kill me

[earnest trail]

o

damn

[quasi musk]

But that was just one lecturer

Most instructors/profs were fine with questions

Even Terry was ok answering questions. It was quite amusing to watch him get stumped on technical details

[earnest trail]

wait... he was your prof?

[quasi musk]

At one point, I did take a class from him

I had a lot of profs at LA

My main ones were Elman & Garnett

Garnett probably had the most influence on me and the way I view mathematics

[earnest trail]

damn what class

[earnest trail]

oh that's interesting that you had ones that repeated a lot

[quasi musk]

246C, Spring 2018

Topics in Complex Analysis

[earnest trail]

I see

that's cool af

[dreamy yew]

But of a random question I guess, did you ever have Balmer?

[earnest trail]

the way geometry proofs are taught in high school is horrendous

it's completely based on memorization and students don't actually get taught the logic and reasoning behind everything

either teach them properly or don't teach them at all

thanks for attending my ted talk

[wispy slate]

tbh that's with every proof in high school

[earnest trail]

what other proofs are there?

trig proofs?

they're more straightforward for students to understand

[wispy slate]

pretty much

true

[earnest trail]

just manipulate trig identities and boom

[wispy slate]

I agree with the geometry part

[wispy slate]

true true

I'm in I guess the 1st year of Hs

[earnest trail]

oh nice

[wispy slate]

I panic in math exams

eventhough I know everything

I write the correct answer, cancel it out and then wrong answer and I swear to god that it's genuinely one of the most frustrating things

[earnest trail]

that's why I don't go back to check my work

I trust that I get it right the first time and leave it at that

[wispy slate]

yeah but there's a sense of paranoia for me

[earnest trail]

just ignore it

don't erase the answer even if you think you have to

[wispy slate]

I'm trying to get into that habit because I would've gotten a 100% in these midterms but I got a 90% due to my paranoia

[earnest trail]

damn

[wispy slate]

I was 1 of 3 students who got one of the harder question right but I got the most basic question wrong

sad

[earnest trail]

bruh

rip

[brazen bluff]

I feel like this is true for a lot of math topics in hs (at least when I was still in hs)

[earnest trail]

yeah true

[brazen bluff]

Do you know a reason why a lot of hs math is taught this way?

[long pelican]

Many reasons, but the overarching reason may just be that the written goals of the math curriculum are basically "students should know such and such" and the enjoyable aspects of doing math don't have any part of the goal. Plus the teachers have no idea or support for teaching math in a better way

Standardized testing also pressures teachers more towards cramming information and the practicing of procedures

[brazen bluff]

Hmm, that makes sense

[narrow nest]

How would you explain to a child that there exist numbers that are more than 1 and less than 2

It's actually a surprisingly difficult concept for a young mind to grasp

[real mauve]

how young are we talking?

i think the standard way is to introduce that with fractions, which are taught at different ages depending on your country's grade school curriculum

how abstractly you can introduce this depends on which operations they're already familiar with

so, as concretely as "one pizza and a couple extra slices", this can be done very early

if you want to introduce rational numbers, the "how" would include all of the arithmetic up to that level

[narrow nest]

The context is when I was trying to get him to measure the length of a line segment with a ruler

And I'd say that he's around 5 years old or so

[real mauve]

maybe with a prop like a big ruler and a small ruler

and trying to fit some rectangular shape into a slot

then with the big ruler the piece either doesn't fit at all, or is very loose

but if you then use the smaller ruler, etc.

(i'm making this up off the top of my head, i don't know whether there is a very good method to introduce this to someone so young other than so concretely)

[tawdry venture]

how would y'all explain the multiplication principle in combinatorics to kids who may not have heard of it before

im having some difficulties coming up with a good explanation

[real mauve]

draw a tree?

[tawdry venture]

a tree with 64 branches

or i guess an order 9 tree for the other problem my kids are solving rn

[real mauve]

you can do the tree diagram for a couple of small scale examples while linking it with the multiplication principle

convince them of their equivalence, and then for the large scale case only do the multiplication, and verbally mention it represents the huge underlying decision tree

might be a good way to introduce the idea that objects may have equivalent representations, each with strengths and shortcomings

trade in the "visual" component for more representation power

[cosmic ibex]

A table might be better than a tree -- if you can actually see all the combinations in a rectangular arrangement, that ought to connect to intuition about multiplication.

[wispy slate]

I know it sounds a bit idiotic but what about using time

between 1 AM/PM and 2AM/PM

and yes it would realistically only go to 1.6

[near oriole]

even more confusing surely. decimal vs base 60

unless u do half past, etc

[cosmic ibex]

It feels important to keep in mind that it's ultimately a choice/definition/convention to call those things between 1 and 2 "numbers" in the first place. So it's not a matter of convincing the child of a necessary truth about the world, but just to explain how we use the words.

[quasi musk]

No, and thank the lord for that

He frequently would make fun of students in front of everyone

[deep knot]

Hi all, I have been faced with a dilemma recently. Now that in my faculty that we are reintroducing exams, I feel like they are detrimental to how I've structured my class recently. Before the pandemic, I was a firm believer that exams were crucial in testing my students' abilities.

During Covid, instead of exams, we had some evaluations and I allowed anyone who wanted to to make up the mark they received by taking an oral examination with me. I have already talked with some people, and it turns out once I have marked an exam or midterm, that mark is final. They said that I could run either all oral, or all written which leads to my problem.

It takes me 30 minutes to give an oral exam, and I have near 200 students. Is my only option to go back to written exams even though I feel that they don't actually represent a student's ability?

[long pelican]

Written exams can represent a student’s ability if you have correct expectations for what a good student will score and what a poor student will score. A moderately difficult exam where you expect a good student to solve 60% of the problems in the allotted time would be an example. Another thing you can do is write a completely trivial exam that you would think no one would fail on, but this kind of exam does a good job at catching people who are faking their way through the course. Bottom line is the utility of written exams is up to how you design them and interpret their results

(Side note: even in the moderately difficult exam example I gave, you will probably have to write questions that are trivial to you)

It is true that written exams are nearly not as noise-resistant (i.e. stress or random factors) as oral exams but that's a tradeoff for saving time

[dreamy yew]

Idk, I quite like him as a prof. I certainly haven’t seen him do this, he does have a sense of humor and likes to joke around in class sometimes though.

For context I took 115AH with him last quarter and 212A with him this quarter, and he was great in both.

[deep knot]

yea thats what I found myself doing, I would have very easy written exams but on my oral tests there would be some sort of insightful analysis that the students would need to do.

but i often ended up giving two grades, one off percent completion and one off my judgement

and i let the students choose which one they want me to give

[quasi musk]

Great! It's just a few times he picked on some students. I had 115AH and B from Elman, and the 110H sequence from Elman

Elman was great for 115, but his 110H series I dunno what happened

Everything that made him a great teacher in 115 was gone

[dreamy yew]

Yeah I’ve heard others complain about his 110H series as well. I’m substituting 210 for my 110 reqs, so I don’t have to deal with the series, but I have heard it is rather… mediocre

I am quite a fan of Merkurjev for 210 though, he is a great lecturer/teacher

[quasi musk]

I've heard great things about the Merk

I thought 110H would go well because honestly 115AH & B by Elman were some of my favorite classes

[tepid smelt]

I have found a mix of projects and traditional exams do give me a good idea of a students ability. Writing good exams and projects is hard though.

I have opted to do orals for retakes for the fee that take advantage of it.

I am curious why you think traditional exams don't reflect understanding? I realize its not as good as orals but they can definitely give me good data on what they know.

[dreamy yew]

I haven't had a bad experience with a professor yet tbh. I certainly am a fan of Merk, Balmer and Sarkar.

[deep knot]

I feel like theres always students who can game the system. On a written exam I can't put much actual thinking on the exam. It becomes a test of triviality

maybe this is me not being able to make a good exam

I would be able to put stuff like this on the oral exam

but i know if i put this on my written exam it wouldnt end well

[earnest trail]

@deep knot use $\log$ and $\exp$!!!!

[burnt vessel]

guh mode

[deep knot]

yea thx

those are part of my notes for the oral exam i gave last year

but like i cant give that on a written exam right?

[clever jetty]

I have a question with pedagogy in general. So let's say I do something really obvious. Would a prof in general require me to prove the obvious thing?

For example take the obvious fact

If f(x) is a polynomial,f(x) is in O(x^deg(f))

I can understand people here would just agree with that, because it's really obvious

Not sure how actual profs or TAs would treat that

[tawdry venture]

probably varies case by case

if you're just learning about big O notation this would probably require some proof

if you're past that point idt your prof will require you to prove it in detail

[pearl onyx]

How do you teach factoring quadratics to kids in Algebra I?

I was trying to go the route of writing $$(x-r)(x-s) = x^2 -(r+s) + rs,$$ to show that you seek two numbers multiplying to the third coefficient and adding to the negative of the second coefficient

[burnt vessel]

abs_0

[pearl onyx]

The issue with this presentation is that students in Algebra I struggle with all the variables

They don’t understand that r and s are “fixed” but x can “vary”

And so I just don’t know what to say instead

Anyone have any suggestions for this?

[severe wing]

Could try having them multiply a bunch of (x+1)(x+2)s first, and then ask them how you might go in reverse.

Like.

(x+2)(x+3)
(x-2)(x-3)
(x-1)(x-6)```

"See how the last number is always six?"

[pearl onyx]

Ah that’s interesting

[severe wing]

Then from there something-something make them realize that you have to get all factors of the last number.

Then try various combinations of those factors until you can find the middle one.

Have a good answer for "But what if x^2 +8x +6? This one doesn't work with any factors of six!" in your back pocket.

Which I would answer "We get to that later in the semester", but whatever's your style.

They won't be able to do problems of this type if they can't factor integers.

So.

Should probably make sure that their last educator didn't shit the bed on that.

First.

[pearl onyx]

Ah ok yeah those are good suggestions

[tawdry venture]

this saturday i was explaining the solution of a problem to my students. i wrote down a thing and made a pause to say "Does everyone understand what i've just written?" one kid raises his hand to say no. fair enough i think, i'm gonna re-explain. which i do. except that the kid is literally not looking at nor listening to me, instead looking at something on his deskmate's phone

is this just him being rude to me or what

like
if you're going to ask to re-explain, you might want to listen to said re-explanation...

[south raptor]

maybe he doesn't listen with his "eyes" if that makes sense

like even though he was looking at something else he was still listening

or maybe he just got distracted even though he intended to listen

I wouldn't take it too personally

[quasi musk]

A lot of kids have short attention spans, a good way to get them motivated is to just throw a problem at them

I find when I lecture, most of the class falls asleep

Only my college students are attentive, but 5-12th grade? They can't follow an actual lecture

[tawdry venture]

this was during the part of the class where i lecture them on the solutions to the current pset

[quasi musk]

yeah ~ it might be best to throw them in the deep end

Let them work on problems, then lecture as they are stuck

Cuz when I teach something they'll be like "Absolute value is easy"

and not pay attention

But then we do things like |x-3| < 1

and they all lose their minds

And wonder why they can't do the problems

So I always tell my students "Don't ever tell me something is easy, because it can be complicated"

[earnest trail]

yeah by the time he finished asking the question he probably forgot what was going on lmao

I would personally "call him out" in a nice way if I were you to make sure he's paying attention

[quasi musk]

That's something you can do too

Do it all the time

[molten urchin]

Can anyone point me to pedagogical resources for elementary school mathematics? I'm using the term resources in the broadest sense, so anything from actual student/instructor resources to blogs, research articles, books on pedagogy/comparative studies/alternate curricula works.

[shadow basalt]

khan academy?

https://www.jstor.org/stable/3072368?seq=1

[narrow nest]

Is there a specific name for this process:
(x+3)/(x-2) =
(x-2+3+2)/(x-2) =
(x-2+5)/(x-2) =
1 + 5/(x-2)

People don't seem to have much trouble with this when it comes to actual numbers (7/5 = 1 + 2/5) but when you add variables into the mix you lose them

[winged urchin]

I don't know if there's a formal name... From third to fourth line I would describe as breaking the fraction and from first to second line I would describe as adding zero in a clever way.

I also suppose I might emphasize how in the process we have converted the expression into a form where we can sketch the graph via transformations using 1/x as the base function

[near oriole]

Long division

[winged urchin]

Oh... oh heck you're right eh ahahah. Yeah that's it

[cosmic ibex]

Also "polynomial division" for greater specificity.

Or "polynomial long division" if you want to hammer it all the way in.

[narrow nest]

But that usually refers to the specific algorithm

Of setting up like an actual long division problem

[cosmic ibex]

Well, what you wrote is a fair rendering of one run of that algorithm ...

[narrow nest]

And that's much more tedious

[winged urchin]

It's also showing a kind of disconnect between algorithms and relations I guess

You could come up with the equation (x+3)/(x-2) = 1 + 5/(x-2) without polynomial long division as you shown

Or infact I'm sure someone could guess or try other methods to show exactly that

[near oriole]

The specific algorithm is not tedious full stop

it is tedious in certain cases

the case above is not one of them.

And also, you might find it tedious depending on how you execute it (check synthetic)

All in all, its just euclidean algorithm

[earnest trail]

I hate how glued students are to specific variables - like some students find it difficult to use variables other than x and y in graphs, and functions other than f(x), etc

[wet slate]

i mean yeah x and y are kinda ingrained

idk abt f(x) tho

[earnest trail]

less so but still

[wet slate]

we were forced to do it with f(x), g(x), h(x), and sometimes t(x)

anything else is still weird

[earnest trail]

and people use y and f(x) interchangeably

which sucks

[wet slate]

the only things i ever see replaced for x are t and sometimes s

[earnest trail]

because y is not always equal to f(x)

[wet slate]

i never see anything but y

[earnest trail]

my calc teacher always uses y and f(x) interchangeably and it pisses me off

oh and with integration by substitution

some of my classmates can not conceive using variables other than u

like wut

[wet slate]

lmaooo whenever we need to do multiple subs we use w

and v after

and we've never needed to do more

also saying "double u-sub" is funny

[earnest trail]

I cant tell you how many times I've heard "wait we can do this even though it's not [] variable?"

like what

we haven't done any double u subs

im in ap calc ab

[wet slate]

hm

really?

my friends in bc have done some double u-subs

[earnest trail]

thats bc

im in ab

[wet slate]

idt any triples yet but we only needed to do that like twice

is ab like significantly different?

[earnest trail]

not really

it's just slightly lower level

[near oriole]

yh sure, but u would confuse the hell out of most people if you start writing x(f)

i think not to stick with just one letter for everything, but having some regularity is necessary
generally ---
f, g, h functions
x, y, z variables
a, b, c constants
m, n integers
etc.

[wet slate]

j(i)

[wet slate]

fwiw i also see k as a constant a lot

[earnest trail]

yeah

[long pelican]

I know someone whose preferred explanation of why $\int_1^4 f(x),dx=\int_1^4 f(u),du$ is "u-substitution"

[burnt vessel]

Icy001

[long pelican]

You might not believe this I think that in the minds of inexperienced math people, the whole "what can you do with these symbols" is pure guesswork and inconsistent and varies from teacher to teacher

[winged urchin]

I was wondering when you'd show up in the this convo Icy haha. Immediately thought of you when I was lurking

[long pelican]

I'm here now hehe

I haven't thought very much about calculus students this semester because my challenge right now is to make the theory of dual spaces, linear functionals, determinants, and transposes understandable to (strong) freshmen

[winged urchin]

Also if you really want more freedom in variable use then questions need to be more rigorously laid out which could lead to further confusion but honestly it's probably better to get them use to that kind of notation

Like if I say y = a*x + 1 and ask for y' then the question is ill-defined

[narrow nest]

normies can't abstract

[molten urchin]

Ahahhahahaha, thank you. I was looking into Elementary Mathematics from an Advanced Standpoint by Klein but this seems to go a step ahead. There also seems to be Enzyklopädie der Elementarmathematik by Weber-Wellstein but I've failed to find a translated version.

[south raptor]

I feel this so much. Often when teaching algebra my strategy is to let the student pick a number to plug into the variable then once they do whatever manipulation the problem is asking I point out how there's a parallel between what they just did with concrete numbers and what you're supposed with the variables.

[tepid smelt]

Yes beast academy by AOPS is the best I have found outside of RSM or math circles. It is the curriculum I use with my daughter and it is just as good as there high school books. I am also really impressed with the math puzzles they have in the curriculum though sadly only available for 2nd and 3rd grade level so far.

It does have content that are not typically taught at a lower level such as probability at the junior/high school level for 4th graders but the sequence they use builds to those topics nicely. I can't stress enough how beneficial its been for my daughter who is bored to death with the standard curriculum even khan academy was much too simple.

[tepid smelt]

I had one of my students got made at me because he got a B on a test I gave in our geometry unit which stressed some proofs that he struggled with.

He mentioned my test is too hard because he is taking calculus and those tests are easier. I mentioned something similar in that anything can be made easier or harder.

Geometry in particular is a interesting topic because it can have a lot of material approachable to an elementary school kid yet also offer challenges for the strongest student well versed in Olympiad ideas.

[molten urchin]

Thank you! This is the kind of thing I was looking for. May I ask how you go about teaching your daughter? Do you have any specific methods (if any) to make them spend more time thinking about a problem or try to read/learn some of the material on their own?

[little spindle]

unsw

[quasi musk]

There are things I wish RSM would include earlier

I wish we'd start with matrices in 8th grade

We don't have a fully fleshed out Calculus curriculum yet, so when students finish their "second year" of pre-calc & trig

They're considered done

One of my local branches is offering a Calculus course and a Linear Algebra course

[quaint grotto]

hello i need a program like larp to use on

its better contain a simple algorithm and a flowchart

[tepid smelt]

When we started in the second grade I would read the comics with her and watch the videos then guide her through the problems. Now though in 4th grade she does it all on her own. I make sure she takes notes from the video and does the problem sets in the book so she can reference it.

Now she typically watches the short 5ish min lecture by Richard Rusczky copies the examples down and she only comes to me if she gets a problem wrong and I will give hints on what to try next. Some of the trophy problems(challenge problems) can be quite difficult so I give a bit more hints on how to approach it. Then I will just talk to her during dinner on what she did. We spend only about 30 min up to 1 hour max as I don't want her to burnout. I think it just takes some time to get comfortable trying different things out but its a good idea to keep a notebook to reference strategies they have done before as it builds nicely. Also talking to them about it to show your interested and care and bot making it a extra chore to do.

[solar nest]

What is your opinion on “A Mathematician’s Lament” (AKA Lockhart’s Lament)?

[molten urchin]

Thanks!

[earnest trail]

ugh my calc teacher was going through a differential equation, smth like f'(x)=-f(x) and she asks "what's another way to write f'(x)?" to which someone responds "dy/dx!" and she says "exactly" and I'm sitting here like that wasn't defined in the problem bruh

I hate how she reinforces the idea that y is always equal to f(x) because that is absolutely not true

it's an issue with teachers and students in general tho

why do we say y is just another way to say f(x)? who thought that was a good thing to teach?

[long pelican]

What happened to guh mode's color :O

[tepid smelt]

This is stolen from a post I saw online but I agree with a lot of it "25 pages without any realistic suggestions for improving things? And blaming math teachers for not understanding math sure isn't what we need...

Math Ed programs are fully pushing the model of open-ended problem solving to new teachers. Without a curriculum to support that, and without a department-wide commitment to that model, it's incredibly challenging to implement. There is never going to be a problem that is interesting to EVERY student. Students come in at such wildly different levels that they likely don't really have the background knowledge to engage in certain problems-- but you still have to find a way to teach them that topic. If you're the only teacher at the school doing things this way you're going to get a ton of pushback from students and parents, and after you struggle through the year they'll just go to a different teacher's class and back to the way they've learned before. It's not a mystery why this happens.

It sucks but until students are held accountable on all levels to actually learn content to pass a class, and we can stop worrying about standardized testing and grading so much, this problem isn't going away."

[inland wren]

i honestly think the main issue with math education is just that most students dont care. they don't want to learn, they don't want to think they just wanna do the bare minimum to pass the class. even if the "bare minimum" is actually harder than understanding the material.
i mean i did this with english and social science in school too, i kinda regret it but back then i don't think you could have convinced me to actually put any effort into understanding those subjects

[south raptor]

ok but there are reasons that student's lose motivation that are not completely in their control

personally I've noticed much lower motivation in middle school students compared to elementary school students which I think is caused by environmental factors

[dense edge]

I am writing a book with formulas and theorems/propositions in diverse mathematical fields for my personal usage. I am writing the trigonometry (circular and hyperbolic) chapter. How can I introduce trigonometric functions ? With their geometric definition inside a triangle or with their definition as functions ?

[earnest trail]

yah I lost it bc of inactivity and I'm gonna continue to be pretty inactive these next 2 months with preparing for college & exams n shit

[long pelican]

Local optimization indeed! Using the fact that the human brain is a good local optimizer, perhaps we can change the landscape somehow so that there is a direction away from this local maximum leading to wanting to actually understand

For example, most students have never been exposed to good (and understandable) math in their life and don’t know anyone within a 10 mile radius who can, and it never crossed their mind you can find such a thing on the internet

[long pelican]

I must have thought you were still active because you gave a talk recently

[quaint grotto]

guys anyone know a program like larp that have a simple algorithm and a flowchart ?

[earnest trail]

I lost active role like a month ago

[quasi musk]

I think a lot of it is overly idealistic, and over estimates what one teacher can do

I made one of my students cry this week because he was asking too many questions while I was trying to get through the examples

He'd repeatedly say "But I don't understand!", yet I have 10+ other students to teach too. Within my lesson plan, I give example problems fully solved

Give practice problems for students

Go around and help each student as they need it

But this student kept interrupting me when I was giving solutions to practice problems or helping other students

I had to basically say stop it, shut-up, take notes, grow-up

I don't like being like that, and I didn't want to make him cry. But he came back a better student after that incident

[woeful wren]

It's one of those things you can generally take as standard from how we graph problems; it might not be a pleasant assumption, but especially in lower level math, y coordinates and the function giving us those coordinates tend to be treated interchangeably.

Kind of feels reminiscent of how we shouldn't write polynomials as, say, Σ a(n)x^n (when n=0 is a valid index) because that technically allows for a 0^0 situation. But by assuming 0^0 = 1 for this particular instance we compactify notation. (And with y = f(x), we draw on intuitions from graphing.)

Granted I've run into this exact issue in teaching Calculus 1 and it always irks me, especially when the textbook tries to involve both y's and f's in the same problem. (Our section on differentials for instance, IIRC.) Usually at that point I just forego the book's nonsense and stick to what actually matters or what is really, properly meant.

[tepid smelt]

Being firm is good teacher pedagogy you can't let one kid hurt the learning of others. I would talk to them one on one to reiterate your not mad at them but that they can't distract other students.

I hate being the mean teacher at times but its necessary especially for kids who likely don't get that at home.

I remember my mentor teacher describing s power battle they had with a kid where they ended up getting kicked out of class daily. Until one day he lost it and berrated the kid the entire way to office. From that moment they were one of his best students because it showed he was serious and cared and would not put up with nonsense.

[cosmic ibex]

(There's nothing wrong with a 0^0 situation, by the way).

[earnest trail]

I hate 0^0 situations

they're scary

[wispy slate]

Is anyone good with Quantum Computing? Would you change anything in this plan?

[dawn walrus]

Looks like a fine plan to me, but this probably doesn't belong in #math-pedagogy

[strange bronze]

also you should know that that kind of doxxes you

like someone can work out your university from the number codes, and some of those classes are probably only gonna have like 10 people

just so you're aware

[real mauve]

i'll go ahead and remove the image for now. you can re-post it if you don't mind sharing the personal info, but we discourage it

@wispy slate

[wispy slate]

I'm not enrolled anywhere

but I'll respect the rules of the channel

[real mauve]

it's more discord TOS than rules of the channel. you can share the diagram if you obscure the class codes

or if you really don't care, just go ahead and post it again. this was just so that you were made fully aware that if those are real class codes, people might be able to figure out personal info about you

[tawdry venture]

is it good pedagogical practice to ask students to confirm whether im understanding something they say correctly and then showing them hands-on that what they're saying is not true?

or am i actually fucking these kids up for life like that

something like

yesterday, one of my 5th graders was showing me his solution to a problem that involved a chess bishop, and at some point he made a claim that sounded like "after an even number of moves the bishop will end up on a red cell"

i thought that sounded strange given that the cells he colored as red didnt seem to have that property

so i asked "wait, so you're saying that after an even number of moves, the bishop will ALWAYS end up on a red cell?"

and he said yes, and i took his diagram and drew a 2-move path that didnt go to a red cell

how pedagogically unsound am i being in this situation

[real mauve]

that sounds good. you can only reply to what you understood, so better make sure you understood what they meant. you've probably had the experience where you or a classmate asked a question and the teacher/lecturer/prof answers something completely unrelated, then the person that asked the question gives up and just says "yeah, we can move on"

it can also be helpful for them to acquire the ability to verbalize thoughts more clearly

[winged urchin]

I think any potential 'damage' would you due to an appearance that they are dumb for making that mistake, just guessing off my instinct

So I would take whatever opportunities I could do to minimize that

I try to emphasize that sometimes math is hard and you don't just have some algorithm to get to the end that you can memorize. Sometimes you gotta stumble around and try things and we can't be afraid to make silly mistakes

We just have to do our best to critically evaluate our own solution to find those silly mistakes

And thats not an easy habit to pick up either sometimes

[tawdry venture]

verbalizing thoughts more clearly is something i see as a goal for my teaching

[tawdry venture]

also like, if i ask "do i understand you correctly" and they start trying to tell me something else and/or acting dodgy is it ok for me to say "please answer the question i asked you. do i understand you correctly that <statement>, yes or no"

[molten urchin]

I tend to hear them out first, then point out the difference between their initial claim and the one they just explained, and then ask for a clarification.

Stepping in midway might be fine especially if I'm in a classroom environment, but I'd try my best not to come across as intimidating in doing so.

[real mauve]

yeah, manan's way is a good approach. explaining clearly why the two (or more) things are not the same is good

saying "please answer the question" doesn't add much, because presumably they are already trying to

[long pelican]

“Please answer the question” is also reminiscent of a police interaction or something and is generally unpleasant

[tawdry venture]

@long pelican honestly you could have been more direct by telling me "you sound like a fucking cop, cut that shit out" w/o beating around the bush

but also that point might be a cultural difference since i personally do not perceive the russian "Отвечай, пожалуйста, на поставленный вопрос" as being something from a cop interrogating a suspect. it's formal and firm but not cop-like.

[cosmic ibex]

My experience from trying to help people here is that a significant number of people will assume a clarifying question was meant to be rhetorical, and instead respond to whichever point they believe I was making by asking it.

[earnest trail]

that's a problem through text primarily when it's harder to tell the tone

I would say (through text and in-person) "so you're saying [.....]?"

that way there's no confusion

[long pelican]

Отвечай, пожалуйста, на поставленный вопрос could be such a long sentence that it loses its confrontational nature

@tawdry venture (just a guess)

[tawdry venture]

not really, it's just five words

[long pelican]

But the words are so long

I'm a Russian noob

[earnest trail]

lol

[long pelican]

practically a wall of cyrillic

[earnest trail]

each of those words is probably one syllable

[tawdry venture]

no

[long pelican]

That's French

[tawdry venture]

the syllable counts of these words, in order, are 3, 4, 1, 4, 2

however people do not always enunciate every syllable with absolute clarity and unstressed syllables tend to be pronounced in a very sloppy manner if at all

i just recorded myself pronouncing that sentence in the same tone i would say it to one of my students and it ended up being the same ish length as "Please answer the question I ask you"

[long pelican]

Interesting

[lofty garden]

Hey all. So today, monday, I'm doing a section of the chapter I'm in (4.2 out of 4.4) and today is fine. Tomorrow though, my first block class won't be in class. First block, no one will be there tomorrow. It's a holiday thing. And I'm having trouble thinking what I should do because I have the same class for 4th block and I wanna keep them aligned.

I was thinking of making an activity for them but I wasn't sure.

[earnest trail]

this is a common thing that teachers have to deal with, most of my teachers in these situations have assigned activities or extra review assignments so that the time is spent but not wasted

at my school toward the end of the year, state exams cause a lot of rearrangements with the schedule so a lot of teachers son't worry about keeping the schedules aligned because the extra time will 'cancel out' and correct itself later

[shadow basalt]

do kahoot

the children love kahoot

[quasi musk]

Looks like the Curriculum department at RSM is putting together a Calculus course

[earnest trail]

oooh nice

[pastel horizon]

Basically anything online

[earnest trail]

eh

not if it's a lot of online work

[pastel horizon]

I've heard a teacher even having success moving a worksheet online and printing out a QR code

Obviously that's not gonna work forever but it's an interesting gimmick once in a while

[earnest trail]

ah

that's interesting

[pastel horizon]

Really it's just the novelty of kids whipping out their phones and scanning a QR code that makes it interesting

[shadow basalt]

My students are “children” ie 18-20 year olds

So QR codes probably aren’t enough novelty

[pastel horizon]

Maybe but the other benefit is not having to print stuff and saving paper

Your QR could just be on a slide

[tawdry venture]

does anyone know of a name for these diagrams/visual mnemonics/whatever that ppl who are unskilled in algebra use to solve basic distance/time problems?

[earnest trail]

they're [...] triangles where you just insert the name

in this case velocity

or you could have force or whatever

I don't think there's a general name for it

at least not that I've heard of

formula triangles maybe?

[pastel horizon]

Formula triangle is exactly what the general name is

Btw I hate them, I don't think they help reinforce the concept of rearranging equations

[earnest trail]

yeah they're stupid

just teach them how to manipulate algebraic equations

[quasi musk]

I call it the holy trinity

Speed, Velocity, Time

I mean distance, rate, time

d=rt. Dirt is holy

[earnest trail]

the holy trinity lmao

triangle of love

[winged urchin]

Am I being crazy.. is this question language just terrible?

By the answers it's clear the book wants the student to just insert k terms between the given terms such that they all follow an arithmetic progression

These are in no way arithmetic means right? I'm just making sure I'm not seeing some deeper connection for the student ahah

Even the idea of there being multiple arithmetic means is ludicrous right?

Gosh sometimes these badly worded questions make me question my own understanding and it rattles me

[tawdry venture]

tbh yes

@winged urchin "insert k arithmetic means between a and b" seems to have the meaning of "find the arithmetic sequence of length k+2 starting with a and ending with b"

[winged urchin]

But that's just terribly worded then right, you'd agree?

[tawdry venture]

yeah it's. old fashioned wording i think

[winged urchin]

Okay thank you ahah. Like I said, I was worried there was something I was missing but ya. Anywho.. =p

[earnest trail]

I had a student struggling with equations such as:
solve for k, 10k+3m=15

(solving for k in terms of m)

I forgot the exact questions but this is what they looked like

when I asked for what his first step, he always gave some random answer like "uhh divide by 10?"

idk how to explain it to him, can anyone help?

[long pelican]

You can read a lot into the "uhh divide by 10?" response, basically he's struggling to see the big picture

[earnest trail]

exactly

[long pelican]

There's also something else -- the task "divide by 10" all by itself is ambiguous and makes no sense whatever way it's interpreted

Divide the whole equation by 10? Why would you do that
Divide just the 10k term by 10? That doesn't preserve the equation

[earnest trail]

I would love to talk about inverses of operations but I feel like that is overly complicated and would confuse him

[tawdry venture]

Divide the whole equation by 10? Why would you do that
i mean why not tho

[long pelican]

Well actually dividing the whole equation by 10 is something

Dang

[tawdry venture]

you can do that and then subtract 3m/10

[earnest trail]

true

[tawdry venture]

from both sides obv

[earnest trail]

okay bad example then

[tawdry venture]

so its a viable strategy

[earnest trail]

but there were other examples where that wouldn't make too much sense

I forget exactly

[long pelican]

There's a third thing I see in that type of response

[tawdry venture]

i think maybe try to take him through some equations starting from the simplest ones and working your way up

[long pelican]

They ask that question and stop instead of continuing, meaning that they aren't sure whether it's legal maybe, or they don't want to go down a path they aren't sure is the "right" path

[earnest trail]

true

like waiting for my validation lol

[tawdry venture]

get him accustomed to the teapot principle

or like. idk what its called

[earnest trail]

wot

[tawdry venture]

does the phrase 'teapot principle' ring any bells to yall or is it getting lost in translation

[earnest trail]

lemme look that up

[long pelican]

Russell's teapot?

It reminds me of that but I can't see how that's related

[tawdry venture]

no not that

[earnest trail]

idk

[tawdry venture]

ok so theres this joke

a mathematician and a physicist are faced with the same problem: you are in a kitchen with a working sink and stove and you have an empty teapot. your goal is to boil some water
both the mathematician and the physicist fill the teapot with water, turn on the stove and put the teapot on the stove

[long pelican]

oh yeah I know that

[tawdry venture]

but then they're faced with a new problem: same kitchen, but now you have a full teapot. the mathematician dumps the water out of the teapot and says "we have now reduced the problem to one we know how to solve"

[earnest trail]

oh i see

[tawdry venture]

@earnest trail is it linear equations your student is struggling with?

or like, what kind(s) of equations generally

[earnest trail]

mainly algebraic manipulations

in general

but yes they're linear

[tawdry venture]

right, linear

but in two or more variables, i'm assuming

[earnest trail]

yeah

[tawdry venture]

how about single-variable ones

is he able to solve these

[earnest trail]

I'm not sure

next time I get this student I will try that with him

[tawdry venture]

in roughly increasing order of complexity from a pedagogical standpoint, there are several different types of linear equations:

• x + a = b
• ax = b
• ax + b = c
• a(x+b) = c
• ax + b = cx + d
• other, messier equations

for these purposes, addends are assumed nonzero and coefficients are assumed not equal to 1

[long pelican]

I feel like "reducing something to a previously known problem" is a mathematical practice math teachers neglect to teach often

[tawdry venture]

yes

it's a very good and necessary mathematical practice

[long pelican]

In the shoes of a math student it feels more like Unit 1 is a collection of these methods, Unit 2 is completely new methods, Unit 3 is new methods, and you just have to remember each set of methods for each unit

[south raptor]

Hey I had a very similar situation the other day. In my case the student did know how to solve for equations with one variable so what I had them do was plug in a number for m, like 1, and then solve for k. Then I had them plug in a different number for m like lets say 2 and solve for k again. Then I would point out that there's really no difference in how you would solve it even if you don't plug in anything for m since you can just treat 3m like a number.

I made a point of stressing that variables are basically just numbers you don't know the value of so you can treat them like numbers in cases like this

For students that are still learning to solve for one variable one strategy I've tried is to tell them they have to apply PEMDAS but in reverse

basically I'll explain that the reason the variable isn't by itself on one side of the equal sign is because it is having a bunch of operations done to it (multiplication addition, etc). So to get it by itself you have to undo the operations in reverse order.

[tawdry venture]

one strategy I've tried is to tell them they have to apply PEMDAS but in reverse
that honestly sounds pretty limiting to me

and doesnt scale well at all when you have anything more complicated than ax + b = c

• it leaves students with the impression that there is one and only one correct way to solve any given equation and everything else is just wrong by fiat

[south raptor]

I think I see what you mean. It's a strategy that might work in the short term but in the long run would make the student take a rigid approach to solving equations

[tawdry venture]

yes

overly rigid.

i don't know of a good metaphor to go with here.

[cosmic ibex]

That's the eternal problem with methods, isn't it? You want the student to gain sufficient experience with the method that they're able to follow it, but you also don't want them to get the impression that the method iss the only valid way to deal with the problem in general.

[wise onyx]

Yes

[earnest trail]

branching off of that, do you think it's a good or a bad idea for teachers to make you use a certain method on an assignment/test? I can see both sides: forcing students to conform to one method makes it seem like there is one way to do the problem, but you want students to have experience with different methods

I remember this type of stuff with solving systems of linear equations (graphing, elimination, substitution) and solving quadratic equations (quadratic formula, completing the square, factoring)

[cosmic ibex]

I'm not sure there's much of an alternative to requiring the use of a method in at least some homework. Some damage control might be possible, though. Perhaps have a problem that's obviously of the same type but simple enough to just wing it, and require that it be solved in any way other than the method?

[tepid smelt]

I try to set up problems that are solved easier with one method. In your example of systems this is easy to do. I still had some that would always go with one method and it generally made it harder but I wouldn't take points away if they got it correct. I think its important to show plenty of methods and how they are useful in certain scenarios.

In classwork/homework I think you should force them to practice other methods. On a test though thats where its up to them to recognize that some techniques are more appropriate to use.

[earnest trail]

makes sense, I agree with these ideas

[quasi musk]

When you're teaching derivatives, you want to know that they understand the limit definition, a geometric interpretation, and other things

You don't want students to just go and use the power rule

[earnest trail]

that's true

[quasi musk]

If a problem says "solve", "compute" "simplify"

I think it's anything goes

If it says, use integration by parts to show that....

Then you best use IBP

[earnest trail]

yeah

[woeful wren]

God I remember even when I'd explicitly state (when teaching Calc 1) to use the limit definition on an exam, they'd still just use the power rule or something else.

Like. It's nice you can do that but you're missing the point of the question

[long pelican]

This reminds me of an article I read by a grad student writing about teaching calculus and his interaction with this one insufferable student who argued relentlessly that an exam question that asks to use the limit definition of derivative is unfair because he knew the power rule

[woeful wren]

"okay fine, calculate the derivative of e^x"

[pastel horizon]

A "show that" will usually get them to answer in the way you're looking for

[austere inlet]

honestly if they're asked to use the limit defn. for something that can be easily computed with previous results (such as the power rule), I think they have a point

the whole point of showing computational lemmas like the "rules" for the derivative is to ease these sort of things

if I wanted to make sure they understand the definition I'd ask for the derivative of something that can't be just expressed as a composition of fns. at that point

like f(x)=e^{-x^2} for x≥0 and f(x)=0 for x<0

[austere inlet]

and asking them to suddenly forget these life-savers on an exam can come across as arbitrary

[woeful wren]

They're not being asked to forget them though. The "shortcuts" can help validate their answer and usually there are other problems on the same exam which are not restricted to a certain method

Sure, you can write up a function they can't yet differentiate....but then you run a risk of making it quite difficult to actually do via limits

Something basic like x², sure, they can get via power rule. But more fundamentally it's also simple via limits. They're not being tested on a complicated function, because we'll eventually develop methods (e.g. chain rule) to not use limits for those more complicated functions

It tests their knowledge of where derivatives come from, in other words, without bogging it down in difficult computations. I don't think it's unreasonable for them to do it in that light

[cosmic ibex]

Perherhaps the objection can be neutralized by saying "prove that symbolic differentiation gives the right result for such-and-such function. You may either prove the rules for all functions from first principles or just evaluate the derivative in both ways and see that they agree".

[stark pine]

Do you have a link?

[long pelican]

No 😦

[stoic python]

Not really pedagogy, but at the end of the last semester I taught, I did an informal review session and my class basically complained about how shitty the other profs compared to me and how everyone is tenured and the reviews don't matter. I explained to them in detail that the reviews at our institution actually make an enormous difference.

I got my teaching review summary today from faculty and they sent me a summary. At the top of the summary they said this is collected from n = 1 samples.

I don't care because I already quit but they didn't know that. I can't help but roll my eyes that these students complain about bad instructors and then don't fill out the one form that can influence anything.

[neon radish]

How the heck do I teach my student with special abilities how to do elementary row operations. His number sense and basic skills are phenomenal, but his ability to see a whole row of numbers get added or subtracted by a multiple of another row just isn’t connecting.

On top of that, teaching how to get a matrix in rref is even more difficult

can’t find a good resource that teaches it in a way that resonates with him so i figured i’d ask y’all

[real mauve]

i think a couple of things that might help are first motivating this type of operation using linear systems of equations and adding and subtracting equations from one another to isolate variables

then look at matrix multiplication from the standpoint of linear combinations and see that matrix multiplication acts on columns (or alternatively rows) of a matrix independently

and building the elementary row operations as products with special matrices. using the previous results, these are the same as taking linear combinations of rows, which is reminiscent of the first thing done (adding equations to one another)

maybe?

[neon radish]

he’s been doing systems of linear equations but i suppose that hasn’t stuck enough with him yet. might just need to have him run through problems more to get the hang of how it works before re-explaining how to use a matrix to solve.

Thanks.

[earnest trail]

what level are your students?

[long pelican]

Tensors are so scary when I proposed them as a topic to teach my linear algebra students they were too scared to say yes

[earnest trail]

you let them decide if they want to learn certain things?

[quasi musk]

You can say they were tense

[long pelican]

Yep my syllabus is very flexible

[earnest trail]

oh damn that's cool

I'm sure not many profs are like that tho

[tepid smelt]

Are you using a textbook as reference or just noted you have created? Linear algebra was the class that convinced me to study math so I think thats a class you should explore a lot of different things.

[long pelican]

There's course notes written by another professor in the department and also Axler, and I wrote 7 pages of latex'd notes for subjects not in either reference so far

[tawdry venture]

is it just me or do schools in the US ban students from writing things in pen

[real mauve]

in my experience, american teachers very strongly encourage students to use pencil for maths

and one can get points shaved off for scratching stuff out instead of erasing. so for practical purposes, may as well be the case.

[tawdry venture]

oh, so one scratched out thing = zero on entire assignment

got it

[woeful wren]

In grade school at least, yes, violently.

thank god no one at my college gives a shit about that, I prefer pen - easier to read

[grim spindle]

That's so weird

Here you are strictly not allowed pencils from secondary school onward for exams and are encouraged to use pen

[woeful wren]

I don't understand the hate boner for pens either... BS about standardized tests maybe?

Then again can't those scanners read pens now?

[molten urchin]

Pens are mandatory here beyond primary school, and especially so on standardised test answer scripts that undergo OCR scans.

[pastel horizon]

Pens are used because it shows in the scan more easily. Scripts are electronically marked

[earnest trail]

I forget if pens are required for the AP free response portion but I know they're required for the Cambridge AICE program and it fucking sucks

I hated writing my aice math exam in pen

[severe kelp]

I have not used pen and paper in 2 to3 ish years

[earnest trail]

what have you used, latex? lol

[severe kelp]

And ipad

[earnest trail]

oh okay

[proper trout]

in my experience pens are also used so 1) it's a permanent record and students can't erase after the fact and claim it was graded wrong and 2) so teachers/graders can see the thought process and any rejected ideas

[proper trout]

pen iirc

[woeful wren]

It's funny, I'm similar

Like I rarely actually work my own homeworks in pen and paper, or an equivalent (MS Paint, a whiteboard), unless I'm just fooling around or experimenting

Usually I just end up typing up the relevant chain of equations in LaTeX, render every few minutes to check, take the obvious next steps, and repeat, if that somehow makes sense?

It's actually nice for messy equations because (a) it looks nicer and (b) when I want to show my work on a sequence of steps, I just copy and paste the previous line and modify according to the step I want next

honestly it feels like it might be more of a headache than it's worth though? no clue

[near oriole]

Algorithmic approach to equation manipulation... like what a symbolic calculator does. Suddenly all those disgusting equations are relatively simple to handle... they were disguesting because the copying out on each line was a pain

[tribal delta]

Hello, anyone here does grading for very applied statistics/data sciencey context?

I'd like to know of people's thoughts with respect to grading on model performance. Specifically, the students could have improved on models used by looking through their data in detail, which apparently only a minority did. (I'm seeing 0 right now but I have not gone though all the students' work yet)

Specifically, there was a column which can conditionally perfectly predict a target. If the column is 0, then the target is 0. Noticing this, one can create models which ignore these rows and come up with models of higher quality than they could otherwise. A nested model predicts 0 when the target is 0, and goes as per normal otherwise.

[real mauve]

was this someone explicitly covered in class? was a rubric given ahead of time?

and more importantly, what were the goals you had in mind when you made the assignment

i wouldn't dock points for this, especially if no one does it, but i would comment on it afterwards. if few people get it right, you could either penalize very slightly (e.g. 99 vs 100) or give some extra credit

but again it would depend on the goals of the evaluation and the course

[tribal delta]

nope not really

[tribal delta]

on this, I don't really know. I didn't make the assignment

[tribal delta]

I suppose I am personally trying to make them see the data more carefully than to blackbox the process.

Honestly I am not sure how much of that is formalised in the learning objectives, but I come from a very applied context, and this is certainly not a 'stats' course

I think the applied context is really important, and students should walk away knowing that mechanization of statistics is not the right way to go

[tawdry venture]

why do american schools teach writing multiplication with an overload of parentheses like (1)(2)(3)(4)(5)?

[earnest trail]

idk bruh my math teacher does it and it bothers the fuck out of me

I only use parentheses when I'm multiplying by a negative since 2*-4 looks weird and potentially confusing

but yeah I have no idea where the parentheses thing originates from

I do remember in 8th grade that it confused students when being introduced to functions

[pearl gate]

in general "x" is avoided as much as possible

American mathematics education is a bit of a joke anyways

in my opinion

[tawdry venture]

ok but why not \cdot

[pearl gate]

not sure, but all of these are taught to students

i dont think the notation used is too important

[tawdry venture]

i think it is

if only for reasons of aesthetic concern

[cosmic ibex]

I suppose there's a perception that multiplication by juxtaposition is a different "symbol" than \cdot or \times, rather than an abbreviated way of writing either or both of them. (For sure you cannot omit \cdot or \times when they stand for dot product of vectors or cartesian product of sets, so it's definitely not an innate property of these symbols that they can be omitted).

[pearl gate]

i mean those who move on to study math in the future usually adapt to conventions used in higher level textbooks or whatever their professors use... im pretty sure the issue u mentioned only exists in high schools (i am in high school though, so i wouldnt know)

[tawdry venture]

and the existence of this issue in high schools is precisely what im unhappy about

[pearl gate]

why is it an issue?

[tawdry venture]

it is aesthetically displeasing to me personally

[pearl gate]

🤷‍♂️ many others seem to think its fine, and usually society goes by the majority

what do people in other countries use?

[tawdry venture]

well over here it's \cdot

[solemn forge]

I need to improve at stepping away from a topic or problem when my capacity for it runs out

The times when I’m forced to, once I return in a day or so, it’s almost crystal clear (or at least where I was going wrong priorly is)

I just need to trust the gut feeling that it’s time to get up

Anyone else have experience with this?

[pearl gate]

yes, it helps to study different subjects in mathematics at once

for me right now that is probability and differential forms simultaneously

if i face a problem somewhere, i take a break or switch to the other subject once in a while

[solemn forge]

Ah

Efficient

I was thinking a new activity altogether

[pearl gate]

well personally i like my activities at least loosely related to mathematics

sometimes i do just put down math and watch a show or go for a jog

then i come back to the problem

really helps for me

[solemn forge]

Yeah I’m going to try rotating subjects

[pearl gate]

i also recommend programming

for me i like to make things

i generally encounter more difficult problems in mathematics than programming

although computer science has plenty of difficult problems

just that programming is relatively straightforward in comparison

[solemn forge]

Yeah I think your brain needs some time for the new ideas to marinate

[pearl gate]

i also like to write about mathematics, so sometimes ill just go and write about something new

[solemn forge]

Something less arduous mentally seems effective

[pearl gate]

it strenghtens my understanding of the subject, and its something else to do

[pearl gate]

yes exactly it needs to settle

[solemn forge]

I do that quite often too, but for my own purposes

Tricky concepts to me I’ll write out in words (no symbols)

Sometimes you realize you don’t really fully understand what you thought you did; or can’t express it

[pearl gate]

i mean more of i like to write blog post type things about math and put them on a personal site

[earnest trail]

I see mostly \cdot but occasionally (like my teacher) I see overwhelming amounts of parentheses

[pearl gate]

its fun, it strengthens my knowlege, and it buids a portfolio

[pearl gate]

yup

ive seen it even with variables

[earnest trail]

that's stupid

[pearl gate]

like (x)(y)(z)(t) instead of just xyzt

pretty stupid yeah

[earnest trail]

bruh moment

[tawdry venture]

it is stupid yes

excessively so when it's with variables

[abstract grove]

I've never seen that

that would be upsetting

you usually see a lot of parentheses in problems like simplify (2xy)(3x^2y)(xyz)

[pearl gate]

parenthases are important for readability and operator precedence

using them for variables or (excessively) numbers is just dumb

[shadow basalt]

It makes your terms easily distinguishable and keeps them separate

if you're doing a bunch of messy algebra or arithmetic it can be super helpful

the \cdots just add clutter than can be mistake, say, for a negative sign or something

I still do it to this day if i am being extra careful for some reason

[patent island]

Round Brackets
Square Brackets
Set Brackets
Asterisk
Concatenation
\cdot
\times
All of the above
etc.

These all have their uses, but, what matters most is clarity. I use and encourage the use of brackets; to first be cautious before assuming that my students are “grown up” enough to not use them.

[earnest trail]

I just use parentheses with the negatives

1*2*3*(-4)*5*6

[earnest trail]

that's a good way of doing it imo

[patent island]

[(16)!]

or

(16)!

or

16! …?

(2•n)!, (2n)!, 2•n!, 2•(n!), (2n!), or 2n!?

[abstract grove]

16!

(2n)! or 2n! are fine for those two non-equal things

(2n!) could sometimes be useful in the middle of an expression

(2•n)! .... no

[elfin karma]

Greetings, people. I'm new on the server. 🙂 I'm working for a tutoring agency teaching german, english and maths, and I'm re-learning a lot of mathematics so I can teach older students with confidence. After the corona lockdown, most of my students basically dropped up to two whole grades (1-6 system here in Germany, 1 being the best), and I've been struggling to pick up the slack that the schoolteachers left.

I'd like to ask if you have any experience teaching students about third-grade functions and beyond. While real-life examples abound for linear and quadratic functions, beyond that I found it hard to find examples for text exercises that the students can relate to, conceptually

[earnest trail]

what exactly is a third-grade function? @elfin karma

[south raptor]

I'm guessing they mean cubic polynomials?

[cosmic ibex]

Yeah: the German for "degree" is Grad and traditionally goes with the ordinal form of numbers.

[quasi musk]

Taylor series approximation to third order?

[elfin karma]

What Troposphere said. I'm not a native english speaker, and some of the terms might not translate well. Cubic polynomials is what I'm getting at.

[tribal delta]

Lol it is actually quite difficult it seems, hope this math.se helps
https://math.stackexchange.com/questions/825699

I think the cubic bezier curves examples takes itself too far, but the cubic drag equation is a possibility. Though I'm not too sure what is the regime for which drag is taken as a cubic

Elliptic curves are more math-oriented, that could be helpful if you see anyone really interested in math (or cryptography)

[cosmic ibex]

Are the students here expected just to know what a cubic equation is, or do they need to be motivated to drill a procedure for solving them? I hope it's not the latter; the available methods are so complex (!) that they're not really worth spending the effort on for practical purposes.

And if it's just to be aware what the concept is, then I think it would be fair to admit that they don't arise that often in practice.

[earnest trail]

yeah solving cubics are hard unless they have nice features

like if they're depressed or whatever

[cosmic ibex]

An arguably-practical example could be something engineering about communicating vessels where there's a conical funnel leading into a reservoir and also a vertical standing pipe of constant cross section parallel to the funnel. If you overfill the system with so-and-so much additional volume, how far up into the funnel and standing pipe will the fluid reach?
(But this is more suited as an example of how it's possible for problems to yield a cubic equation, and not as an example that solving them is important).

[elfin karma]

So, the students are obviously able to grasp what a cubic equation is, but since I usually work with students with little interest in maths (bad grades and equally bad teachers having driven their enthusiasm into the proverbial ground), I try to include real-life applications in all my math lessons to avoid the inevitable barrage of "why do I need to learn this, what do I even need this for"

I mean, linear functions are pretty much everywhere, and for quadratics you can always go the route of using them to model trajectories, but it's hard to explain how higher functions find usage in real life

[real mauve]

tropo's suggestion of volumes is good

you could discuss buoyancy of boats, for example

but although this and splines are good motivations, at some point this falls short. you don't always find "real world" applications so readily (splines are already a stretch)

[cosmic ibex]

I suppose I'm asking what those students are required to learn about cubic equations. The interesting facts about cubics I can come up with offhand all seem to be interesting only if you're already enthusiastic about mathematics for curiosity's sake.

It makes sense to say something like

If we have any complicated equation of one variable that can be phrased only using the four elementary arithmetic operations, we can always rewrite it as a polynomial equation, and then we can sort them roughly into difficulties by looking at the degree. Degree 1 is easy; degree 2 can be handled if you remember either the trick of completing the square, or the ready-made formula. Degree 3 and 4 have formulas, but they're so complicated that nobody uses them in practice. So there we have to either get lucky or rely on systematic trial-and-error. (And by the way, for degree 5 and above it has been proved that there cannot be anything analogous to the quadratic formula).
And I'm not sure if it makes sense to insist that students around grade 8 (which you mentioned in another channel) should really know more than that. However, to make that story relevant, what we need is not problems that give cubics specifically, but just some equations of any degree >2.

[south raptor]

I usually work with students with little interest in maths (bad grades and equally bad teachers having driven their enthusiasm into the proverbial ground). I try to include real-life applications in all my math lessons to avoid the inevitable barrage of "why do I need to learn this, what do I even need this for".

I really question why you feel the need to do this. The reality is this type of math is irrelevant to these students lives and any attempt to pretend it is will come off as disingenuous in my opinion.

The flip side is that students are perfectly willing to learn things that don't have direct real life application if they enjoy it. I think students caring about real life applications is a symptom of an underlying lack of motivation rather than the fundamental reason why they don't want to learn math.

I feel no other subject gets scrutinized over its potential applications the way math does.

[hexed sand]

Hi all. I'm currently teaching an AP calculus BC class (my first one), and we finished the material at the end of March. I've decided to work through practice FRQ section with the class, and on average in our 90 minute period most students are scoring in the 50% range. So here are my questions:

1. I have heard that a 65% or higher is supposed to be a 5 on the exam. I currently have 5 or 6 of the 20 students consistently in that range. If anyone has taken the exam, what are some tips for the remainder of the class. How do I get the rest of the class into the AP 5 level for the FRQ.

2. The multiple choice questions are worse for my class, they take far too long to do them. Throughout the course I have put both multiple choice questions and FRQ questions on my tests. I have noticed that the multiple choice questions on those are fine, but when there is so many multiple choice questions, the students get flustered so far I have given the following proposals, I want your opinions on them:

i) For multiple choice questions, label them. I recommended three labels, ones that you can do (leave blank and answer), ones you have a suspicions on but can't do (leave a star and skip, when going back prioritize these), and those which you cannot think of how to solve (I said to put a star). Does this work in general? (i don't want to be giving bad advice).

ii) For the multiple choice section, make sure that you put your first throughts down no matter what you do. I often find that stuents are better than they give themselves credit for, their first guess is often correct and they over-complicate from there.

I know my class are all capable on doing well, I just don't know how to support them through this part of the course. Any tips?

[shadow basalt]

Hm

i) seems fine I do something similar

Or I used to

ii) just make sure there’s no guessing penalty

I don’t think there is anymore

But I forget

I think for multiple choice the main thing is just a lot of practice

For FRQ it depends what they are doing wrong

If they show conceptual weaknesses that’s one thing

If they don’t know how to approach the FRQ that’s another

Fwiw there’s also an AP server that you might get some advice form

[hexed sand]

oh ill go to that

a lot of the time, they aren't able to think of the approach

[earnest trail]

if you taught them properly and with past exam questions, I dont see why they are struggling

do they have the proper intuition?

[shadow basalt]

Hm. The best advice I can give is to have them do more practice questions and look for like, concrete gaps in their understanding

Make sure to go over some with them in detail and stuff too

Maybe let them like, redo the problems for extra credit?

(or just make them do it)

[earnest trail]

yeah maybe take a day for you to go through some problems and explain your thought process throughout

that way they can see exactly how you think about the problem and apply it to themselves

[hexed sand]

yea that sounds good

it's usually in series that the problems arise so ill do that

[earnest trail]

yeah thats something I've wished some of my teachers have done lol

[proper trout]

Is the only reasonable option, for freshmen who refuse to even attempt to learn the material and instead expect to be pushed through the class like in highschool, failing them?

I don't do grading for the most part so this is mostly just something i've personally been considering

[austere inlet]

sounds like the obvious option to me, the whole point of giving a student a passing grade is certifying that they learnt something. Someone who doesn't understand the material and makes zero effort to learn should fail

any other option is half-assed and likely the result of pressure from administrators or even the students themselves

[earnest trail]

talk to the student, perhaps there is an issue in their lives or something

how are they doing in other classes?

[halcyon light]

for questions like these, how do i explain that you can't take the average of the two given speeds and have that as your answer?

i understand it myself but idk how to put it into words

[shadow basalt]

I would show them a bunch of different graphs that satisfy those assumptions but have super different averages

You could also ask them things like

If I drive at 75 miles an hour for 2 hours and then 40 miles per hour for 10 minutes

Will I get to my destination faster than the other way around?

[cosmic ibex]

Hmm, drive for 10 hours at 1 km/h, and then for one second at 99 km/h. How far do you get, and does that correspond to an average speed of 50 km/h?

[winged urchin]

I'd wag my finger back and forth pretty fast and ask about how many times they think I wagged it. Then I'd wag my finger real slow and ask again. Writing these answers down.

Then I'd proceed to wag my finger real slow for a length of time and speed up to real fast for like a couple seconds at the end.

Then I'd calculate the average in the way they wanted to and hopefully I controlled my wagging well enough for their answer to be absurd lol

Ohhh wait even better is to just not wag my finger at all for the slow period

Or to just use zero speed for y'all more sensible options ahah

[cosmic ibex]

Hmm, actually all of these answers might not work if the real problem is that the person you're talking to has no intuition about what "average speed" ought to mean to begin with. If you only know "average" as the name for taking an unweighted arithmetic mean and perhaps have never thought about what that operation achieves, then it's natural you you would consider the durations to be irrelevant.

[winged urchin]

Yes that's true too. It'd be good too to just ask the student questions about why they think what they do or what they think average speed means and correct the misconceptions they likely hold

[shadow basalt]

It might also be a case of like

Deep down they know that taking the basic average doesn’t make sense

But they don’t have any better ideas

So they just sort of hope it works

Like not understanding the right answer can cause them to default to a bad answer

[abstract grove]

when I show that kind of problem, I start off making clear that average speed means total distance traveled divided by total time of the trip

[south moat]

Hello all, I was in the #discussion tab and they said I should come here. I mainly teach Geometry and I use a lot of PDFs that were created by my books manufacturers. Because it is Geometry, there are a lot of shapes and graphs. I need to convert them from PDF to WORD and everything I use just makes it look bad. Do any of you have a tool that does a good job?

[tribal delta]

Draw/vectorize them if possible? Make code that draws them?

[south moat]

At the moment, just trying to find out ways to straight convert. Save the most amount of time possible. It takes me a lot more time than I'd like to admit to make testing materials than I'd like to admit.

[real mauve]

oh hey

https://www.adobe.com/de/acrobat/online/pdf-to-word.html

this should have an equivalent in english

to my surprise it's free and worked pretty ok with a random paper i shoved in

have you given that a shot?

[south moat]

Yeah, Adobe did a really... Bad job...

[real mauve]

it does move stuff around a lot, but

[south moat]

But, I could maybe copy paste sections from the Adobe editing tool

[strange glade]

Hi are u using manim?

[real mauve]

i'm not sure i've heard anyone around here using that. that's the 3b1b animation library, right?

if you have questions about it, i surmise you'd have better luck looking in the python server

[strange glade]

yeah

[strange glade]

I'm thinking about make vid about geometry

and, maybe I'll need help

with constructions

[real mauve]

you can get help with the math here, e.g. in #computing-software . regarding the programming itself, you'd be better served looking in a programming server, though.

[austere inlet]

manim community probably has their own server too

[real mauve]

that's a good point

3b1b too, for that matter

those would be good places to look

[vast zenith]

Is there mathematical notation for something like r = r + 1? In programming this means "change the value of r to r + 1", how should I say this in maths?

[real mauve]

you could use a different variable, define an arithmetic sequence, or use your favorite notation for successor if working with natural numbers

but i would say it's bad practice to write something like r = r + 1 in a mathematical context

maybe use r := r + 1 at least to distinguish it

[vast zenith]

thanks, think I'll stick to using sequences - I'm doing a maths/programming project & trying to give an outline of the algorithm used in the write-up

[earnest trail]

or maybe $r \mapsto r+1$

[burnt vessel]

gmod

[earnest trail]

hmm maybe start with some motivation like "how would you model ocean currents or weather patterns" and show some slope fields

then from there you can start teaching how to actually solve them

[austere inlet]

:= means definition in mathematics and it can be misleading as well, it's important to use symbols for their precise purposes when teaching mathematics, or in this case a mathematically-minded programming project. I agree that the best approach might be to define a sequence r_{n+1} := r_n+1 instead; it can lend itself to emphasize the iteration behind the algorithm

[halcyon light]

thank you all for your answers to the avg spd q! super useful answers @shadow basalt @cosmic ibex @winged urchin

sorry, forgot to thank y'all

[winged urchin]

Awe you're too kind =p

[zinc wigeon]

Is calculus with infinitesimals easier for students to understand?

[earnest trail]

easier than what? @zinc wigeon

[zinc wigeon]

Than standard calculus

[earnest trail]

what's the difference

[zinc wigeon]

Standard calculus doesn’t use infinitesimals

[earnest trail]

isn't dx basically an infinitesimal

[zinc wigeon]

For example, the derivative dy/dx isn’t a ratio of infinitesimals but the limit of a ratio

[earnest trail]

differential but still

[zinc wigeon]

Well, in standard calculus it’s a symbol devoided of independent meaning

afaik

[earnest trail]

oh I see

[cosmic ibex]

I think it depends on whether "calculus with infinitesimals" means "calculus with infinitesimal instead of the rigorous limit definitions", or "calculus with the standard definitions and also some handwaving about infinitesimals".

[zinc wigeon]

I am curious about both

[cosmic ibex]

The latter is to a certain extent unavoidable, since many commonly used notations are based on an infinitesimal-based intuition, and they lose their mnemonic value unless one knows how that intuition was supposed to work. Saying "these notations don't mean anything; they're just random splotches of ink that we ended up with by historical accident" may suffice for the formal reasoning, but seems to do students a disservice. There's the risk, however, that students will not have the sophistication to deal with a distinction between "what the definitions really are" and "a useful and suggestive pretense that makes it easier to remember what is what, but is not safe to use for proving things". Which is probably why many teachers take the easy way out with the its-just-ink viewpoint.

[cosmic ibex]

As for calculus with infinitesimals instead of limits, I'm not sure it can be done well that way at all. It's the way the historical development went, of course, but there's a reason limits took over.
It seems reasonably possible to get through first derivatives that way. But even then, for the simple case of differentiating a polynomial, you need to pull a rule out of a hat that says we ignore any infinitesimal terms that remain in dy/dx after we've simplified. This can be justified with appropriate handwaving, as long as we're just interested in the slope. However it also naturally leads to f'(x) = f'(x + an infinitesimal), which wreaks havoc when we try to claim that a second derivative is the same operation done twice.
The ancients (in the 1700s) were satisfied with just deriving an expression for f'(x) using the "ignore remaining infinitesimals" rule, and then forgetting where it came from before they differentiated it again (or else consider d²y/dx² to be a separate thing with its own "ignore the remaining infinitesimals" rule that's not a direct consequence of the same rule for dy/dx). But that fits badly with the modern concept of a function where a function doesn't need to have an arithmetic expression defining it in the first place.

Non-standard analysis attempts to make all this work nevertheless, with a formal grounding. While it has been seriously proposed for introductory teaching, I'm quite skeptical how well it can work there. It solves the second-derivative problem by judiciously moving back and forth parallel "standard" and "non-standard" universes -- differentiating a function initially produces a function defined only on standard reals, and the hairy model-theoretic framework then gives a canonical way to fill in the gaps so we get a derivative on non-standard reals too. But in order to avoid everything devolving to free-wheeling handwaving, one needs to teach the two-level universe explicitly, with guidelines for when to pass between one and the other, and so forth. I have trouble believing that is easier to understand for students than limit-based calculus. (It's possible I just haven't seen it done in the right way).

[verbal sparrow]

Can you give an example of f'(x) = f'(x + an infinitesimal)?

[cosmic ibex]

Suppose f(x)=x², and o is an infinitesimal.
To compute f'(2) we calculate (f(2+o)-f(2))/o = (4+4o+o²-4)/o = (4o+o²)/o = 4+o and discard the remaining infinitesimal, so f'(2)=4.
To compute f'(2+3o) we'd calculate (f(2+4o)-f(2+3o))/o = (4+16o+16o²-4-12o-9o²)/o = (4o + 4o²)/o = 4 + 4o and discard the remaining infinitesimal, so f'(2+3o)=4.

[verbal sparrow]

Hmm... in my mind infinitesimals satisfy o² = 0. So (4o+o²)/o = 4o/o = 4 and that problem goes away.

[cosmic ibex]

That would make them zero divisors, and then dividing by them in the differential quotient looks rather dodgy.

But even so, wouldn't o²=0 still lead to both f'(2)=4 and f'(2+o)=4 at the end of the day? How can we then say that f''(2) is nonzero?

[verbal sparrow]

You do have a point about f''(2). Hmm...

[cosmic ibex]

There's also the way Cauchy took in Cours d'Analyse. He had something that looked like infinitesimals in his calculations, but they officially didn't denote particular infinitely small numbers, but instead variables that were supposed to go towards zero in an implicit limit operation. I'm not sure exactly how he dealt with higher derivatives, but it might have been possible for him to get through with saying each derivative needs its own separate infinitesimal as dx.
[And there's at least a plausible argument that this strategy really gives us all the didactic trouble that an openly limit-based approach does, and more so because the limit-taking is now invisible].

[verbal sparrow]

Hmm... I am wondering if we need to discard the infinitesimal. Why not keep it around and when you need a "real" answer, just ignore the infinitesimals.

[proper dragon]

Hi everyone I'm an economist interested in doing some research in the maths of string theory, the most advanced course I took was Optimization and differential equations.
I would like someone here to guide me , thanks

[cosmic ibex]

This is not a channel to ask for math help.

[proper dragon]

So what's the most adequate channel for my question?

[cosmic ibex]

#math-discussion possibly.