Is anyone available to help
#math-pedagogy
raw breach
tawdry venture
toxic spire
ew S
winged urchin
Hey! Any teachers or tutors who have an opinion.
I'm tutoring someone in first year Calculus and they recently did quite poorly on their midterm. However when I looked at the test it was mostly silly mistakes. In two cases a silly mistake made the question impossible for them and they ended up wasting a lot of time and getting no where.
The two silly mistakes were saying (x^3)' = 3x
And
x^7f(x) with f(x) = -sin(x^4) but they plugged f in as...
x^7-sin(x^4)
And quickly mistook it for subtraction after that point.
They were wondering what they can do to improve their grade but I hesitate to just tell them to do more practice or something like that.
I feel like they're overwhelming themselves with practice and stressing to the point of being very myopic while in a test and miss the silly mistakes they make.
I do tell them to try to check their work as they go and think about if their answer makes sense (if they get an answer).
What would you suggest or say to a student in this situation or what are your opinions otherwise?
Thanks!
novel kraken
Sounds like they're rushing too much and when they do practice, they should spend their time striving to get the answer right on the first try on their homework. Often times students at this level will have to redo a problem like 2 or 3 times because of tiny errors like this. If you think they're stressing out you can try suggesting they focus on keeping their breathing at an even pace or just generally look for things like this to help them calm down and slow down
some specific things you can do for this student is start by asking them how much time they had left when they got to the end. If they had a fair bit of time, you can tell them not to use that time to go back, but instead tell them they should be going through the test slower so that they are avoiding these mistakes
you can also bargain with them over the tradeoff between getting fewer problems and say that their low test score might be better if more slow, quality effort on problems is spent rather than just rushing through all of them and just doing poorly on all problems
obtuse widget
I think suggesting them to verify each step before proceeding further would be good solution
At least that's how I solved my problem of tiny mistakes
Maybe propose doing some mental calculations instead of the topic itself?
winged urchin
Thanks for the suggestions guys! Im definitely gonna take some of these and mull it over. Oh those silly mistakes... š«
daring thunder
Can an instructor please give me advice on how to approach a instructor?
My problem is with my instructor is he becomes irritated when I ask for help. Also the things I have learned in his class are not 100% true. For example, Iām in another class that correlates with his. When I say something to the other instructor he always asks me where I learned it. I tell him and I also pull up the slides from the instructors class that I am having a hard time with. Then this instructor will say āDonāt tell him Iām saying this, but heās wrongā. Then this instructor will teach me the correct way. I know itās correct because what heās saying matches with whatās actually happening. I feel sometimes as if he takes me as a threat. I study my ass off and know my stuff! I go above and beyond! Im at my wits with him. He took up my exam today without even saying the exam was timed! I would have taken a different approach of doing the exam if known it was timed. Also itās not stated in his syllabus. All my instructors have it posted or tell you. Most allow us to move to another room to finish. Iām not that student just complaining. I have a 3.93 GPA. (B+ in the instructors class I am having trouble with). I am extremely disciplined about schooling. I am a Junior. How do I approach this instructor about this? I donāt want to make an issue, but I want it to be resolved. Itās affecting my learning. Heās the only instructor for these classes. I donāt think it matters but he has a Masters and the other has a PhD. Thank you! Please feel free to criticize me!
unreal ledge
Where are you at where your exams aren't timed? That's very not standard
daring thunder
Iām at a private 4 year university. I have never had a timed exam. Unless it was open book or online(online class). Usually the syllabus has everything about how the course is structured and what the instructor expects of you. Iām not a mind reader. I read every single syllabus.
unreal ledge
I think you've got a legitimate case with the timed exam. Your school should have a student services in your department
Get a few students to go with you, and plead your case. They'll tell you the correct way to file
Getting people to go with you is important, I wouldn't walk in and start this alone
Or, maybe there's someone you can just email, you can maybe justify a retake. Ultimately, what's your goal?
lethal leaf
Yea just a teacher being straight up wrong is worth complaining to some department head
or student services
winged urchin
Do you have an example of the kind of errors he makes that the other instructor corrected?
Sometimes people just have weird ways of approaching problems and others might think it's the 'wrong' approach because they don't recognize it and perhaps don't consider it deeply enough
Though TBH it sounds like you have a good idea of the situation. I'd wager he is incorrect but I'm interested
daring thunder
@unreal ledge My ultimate goal is for him to help me when I ask without an āattitude of your stupidā. I can tell, because his face turns red when he starts getting impatient with me (because I ask questions that he doesnāt have an answer too on his slides). His tone and body gestures change. Makes me uncomfortable. I donāt ask him anything anymore, because of how he acts towards me. I forgot to add that I go through his slides like a hawk. If I donāt understand something I try to find external help. Well, I found 6 errors and I tried to show him; he instantly got defensive. Although, swiftly followed up with that I was correct in a not so friendly tone and then he fixed the mistakes. His slides are 6 years old. Just goes to show how many students really care in his class.
@winged urchin Itās a computer science instructor. The slides are his and he makes sure to let you know they are his slides. Anyways, one example is him saying in his slides that ASCII characters are 2 bytes. Although they arenāt. They have a variable size. The PhD cleared this up after he called me out when I said an ASCII was 2 bytes. I came back with his slides as proof that I was taught this in data structures. It frustrated me. So then this last week I was trying to figure out how to pack two 32 bit integers into a single long(64bits). I was taught an integer was 4 bytes or 32 bits. Although, this is not true. An int can have a number with max size of up to 32 bits. It doesnāt mean that itās 32 bits. Your compiler will bring it down to a char(1 byte) 0-255 if itās between 0-255 (Pack it into the smallest data type)or(Pack it with other small integers and mask them out when needed). Which then causes a problem when you are trying to do what I was doing. Although, I had no clue. So I asked the PhD for help and he cleared it up for me and he even asked me where I learned that a int is 32bits. He shook his head and said no you need to use a int32_t which forces the compiler to use an int32. See I was taught an int was 32 bits. Itās not true. It frustrated me and then the timed exam thing today. Really pushed me over the edge.
winged urchin
Hmmm that's interesting. Im somewhat familiar with computer stuff but not really involved deeply at the moment.
To me that sounds like the messy details that sometimes get excluded in lectures. The easiest example I can think of (in math) is the square root of x^2. Often you'll see this just reduced to x without any mention of the subtly involved. But of course the square root of x^2 is |x|. This is a sorta 'messy' detail that gets brushed over sometimes in calculus classes I think
Or even algebra classes for that matter... Often leading students to really not knowing where the +/- comes when solving say x^2 = 4.
But I do think that there is a trade off between accuracy and ease of understanding
Ideally we would present 100% correct ideas with the utmost clarity but I think the messy details get in the way of the understanding depending on what level you're at.
Or at least thats an argument for not going into the really subtle points of a question sometimes...
Not saying it's necessarily right but what I've noticed.
It sounds, to me, like you're a student who is above the level of the other students and so you're starting to see more of the subtley and asking questions about what you're learning. I think that's great and I encourage it deeply
I find it unfortunate the instructor was so defensive and unsupportive of your curiousity though
obtuse girder
wispy slate
That's easy. In order to ensure they cannot cheat, their Internet and electricity is cut off before the online test starts.
@daring thunder by default int is 32 bits but stdint.h is machine specific and makes sure you get the right size
in general when you care about sizes use the types in stdint
c compilers dont do a whole lot to simplify types afaik, use chars manually for 1 byte numbers
wispy sparrow
Which channel is best for Lie theory?
wispy slate
Prolly abstract algebra, no?
lethal leaf
Tutors, how do you expand your client base
I've been advertising in some local FB groups
but I get people contacting me
and they're all like "I'll schedule tutoring soon"
andf then they never respond
and then I'll try to follow up and they don't respond to that
and then after first semester basically all of my clients stopped scheduling tutoring
idk if that's normal and I'll just get a bunch of requests in the 2nd half of the semester
but I'm trying to earn more cash and it's hard when my ads go nowhere
violet isle
@lethal leaf It can be tough to get private tutoring work depending on where you live. I would recommend putting up fliers at your local universities; like in the library or something. Also you could try craigslist. I personally had several successful tutoring jobs from craigslist; just make sure to meet in a public location like the library. There are also private tutoring companies you could look at working for, they won't pay as much as just working independently, but its something.
lethal leaf
In the span since I posted my message above I got 2 more clients
but I'm still in HS so idk how kindly they'd take to a High Schooler advertising tutoring in a college
and I offer to meet in my house, their house, or library
their choice
I make that very clear in the ad
violet isle
That's great. I just assumed you were a university student. So yeah, it must be more challenging to get tutoring positions if you are only in high school.
boreal yacht
I did this for a while in high school. I found that a lot of people just want one-time help, which makes things kinda hard, but i did eventually get a few clients who needed me consistently.
lethal leaf
Yea
I have like 3 people who just want occasional help
Which is nice until I get to finals and them I'm like "fuuck"
Cause scheduling is hell
brazen pendant
my high school had a program for people in the last two years to tutor people in the lower classes, but I only ever got one job out of that
(you basically just told them that youād be willing to tutor these subjects, showed them that you had decent grades and then theyād put your name and contact info on a list and put that list on a billboard at the beginning of the semester)
and if someone struggled the teachers would recommend them to get help from someone on that list
inner gazelle
ask ur hs teachers that ur homies with if they had any requests for private tutoring (math will tend to get some reqs)
if ur confident in ap subjects or sat or w/e you alr have experience with
And you did well (eg: 5s or 1500+/1600 for sat)
U can flex that to show u know ur stuff
Dont come off too braggy tho LOL
u prob will have best chance tutoring middle school/elementsry kids
wispy slate
Never brag. In the end you look like an idiot
inner gazelle
u "humble brag" (eg: flexing that u go to good college instead of noname one) to get more attention
winged urchin
You can also 'brag' in a sense by showing them how you might solve a problem. Even if it's not something you'd expect them to think of, I think it's good to expose some students to slightly higher level or more obscure tricks.
inner gazelle
if the obscure trick isnt an easier way to solve the problem
Youre just being a smartass lol
ppl want tutors usually bc they dont understand "normal" explanations and need a simpler method/something they can relate to
winged urchin
That's true. Of course 'easier' is subjective here. Most clever tricks that I'd consider to fall under that definition do make the problem easier. But seeing the trick or knowing the background for it is the difficult paty
Part*
It is only for some students, I agree. Most of my students are on the lower end grade-wise and so I'll keep things simpler and more focused on the core of the idea
But sometimes you get students who are on the higher end and whose parents, or themselves, simply want greater understanding and a broader perspective on math and so for them it's a good opportunity to show them the cool things math can do
inner gazelle
easier isnt really subjective... if your "trick" involves knowledge beyond what they're expected to know you're legit overcomplicating things by introducing something completely new instead of showing what possible approaches they can take using what the average student in tht subject knows
in that case if they are on the higher end you can show it to them privately or say that the method ur about to show is "optional"
most students wont give a shit unless theyre actually passionate about math
They just want to know what they needa do to get the grade
strange bronze
pfft
"just use the fact that the degree of an irreducible polynomial in R[x] is 1 or 2"
doesnt obscure it at all
š
but i mean, if the "trick" is just an alternate way to frame the problem
and the student is interested in learning it
i dont see the issue
assuming you've already spent plenty of time covering the "essentials" and the student feels confident and wants to learn more
like, in an ideal world, the end goal of education for students should be learning, not grades; sadly society's overemphasis on grades as a determiner of merit makes this unrealistic
so these factors have to be balanced
inner gazelle
duh "facts" like that clarify the problem
Like for a counting problem
You wouldnt show a kid how to solve a prob using generating funcitons
Unless ur stroking ur own ego
If kid didnt ask for it
U have to show them how to solve the problem using what they alr know
winged urchin
Additionally what I say are tricks they might know also fall into the category of things they should know but have forgotten. Like how the multiplicity of a factor in a polynomial determines the shape of the graph near the root of that factor.
Or how one can get choose values from Pascal's triangle for relatively low values of n
strange bronze
i'd absolutely teach a student, say, stars and bars
if it would be useful for a combinatorics problem
even if they'd be expected to give a more "just jam combinations" approach
so i guess the question there becomes, "where do you draw the line"
inner gazelle
"Should know but have forgotten" aka average student who gives two shits about math wont know it
LOL
strange bronze
but i think a decently talented individual
should be able to listen to their student
to know where to draw the line
"talented" is the wrong word
"experienced"
(as an aside, "where do you draw the line" is unintentionally funny given the stars-and-bars context)
like if you show an alternate method to a student, and it doesnt help it "click" with them
you know to deemphasize that
now and for the future
winged urchin
Right that's true too. It's much more fluid and adaptable in practice.
Also if I only went off of what the students haven't forgotten I could barely introduce fractions. Some students legit forget how to add fractions. Or at least are very unsure even if they do the right things
It is definitely about listening to the student and their responses to your explanations. Especially in such a global world where kids may have come from completely different school systems
storm tangle
I tutor sometimes, and you can quite immediately see if the student is just trying his/her best to pass the course, or if the student wants to truly excel, in which case you should, and it will be welcomed, give the more obscure, neat and "cool" techniques.
The former student will not even be interested in it, in 99% of the cases while the latter student will thirst for that kind of knowledge.
GeminiDream really hit the nail there; when you tend to help a student with a question in, for example, Calculus you rarely need to stop and reiterate or explain a Calculus topic but instead you notice the student gets stuck at the actual algebra, which could just be adding or multiplying fractions. When helping a student, it usually involves having to quickly go through a concept that should have been taught 3-4 years earlier in their math education.
winged urchin
One of the hardest things I've noticed as far as algebra goes is they just don't seem to play with expressions. It's tied in with their knowledge of algebra rules as well since if they feel uncomfortable switching the order of multiplication or in combining/breaking fractions then of course they can't manipulate the expression
And then it's like... how do you get them to the answer if they can't do the searching? If the answer involves some trick like adding 0 in a clever way to cancel a denominator and get left with an arctan integral you shouldn't really start with "Oh hey, the way to get there is..." in my opinion but then all you can say is things like... "Well, what do you see in the expression? Does it look like anything you've seen before? Maybe sorta kinda?" or "Just try a couple things?"
But I'd say, unfortunately, more often than not I have to eventually just show them the trick
I've been trying to instill a feeling of flexibility into math when I help people now. I don't want them to see it as this problems needs to be done exactly like this... but rather that there are many ways one could get to the answer, the student needs to explore how they move the puzzle, how they can draw the 'picture' so to speak
But there are rules of course. Even in art, if I ask you to draw an owl there's a lot of ways you can do that but eventually if you go too far off track you aren't drawing an owl anymore
Sorry for the wall of text haha. Went on a bit of a train of thought there. Interested if anyone has insights though!
turbid zenith
Feel like this should be shared here too.
https://twitter.com/solidangles/status/1241193403368591361
Educators, thank you for everything you do.
gentle ibex
I'm tutoring my cousin for her math work (she can't go to school, they are closed) but i think i'm doing it in the wrong way. How do you teach math to "non-math" people? She seems to don't understand me even if I use a simple language
boreal yacht
well, thats the whole challenge of teaching. You can't expect someone to just understand what you are saying super easily. You have to be clever about the way you explain things, and when an explanation doesn't seem to work for them, you tirelessly look for different ways of helping them make sense of things.
gentle ibex
yeah but i can't figure out a way to motivate her, she doesn't like math so she isn't listening me.
grand laurel
my take is that you can't force people to understand something
a teacher can showcase different ways to try and understand something, but ultimately understanding is something the student has to work for themselves
so you should try to motivate your students
so that they want to understand
brazen pendant
yea. if someone doesnāt actually want to learn then you can come up with explanations as clever and clear as you want, theyāre just not gonna be putting in the effort to make them click. so you have to first motivate them on why they should care. and that can be an impossible challenge
since the schools are closed anyway you could probably spend some time just trying to engage her through interesting puzzles rather than stuff that is strictly related to what sheās supposed to be doing at school
wispy slate
Does anyone here use (or used in past) some kind of apps to make correcting homework/giving feedback easier? My prof is getting a shit ton of phone pics of the homeworks/latex pdfs and its apparently quite time consuming to correct them. Any tips how he could improve? Could be tablet apps, pc, or maybe just a better system of sending in homeworks idk?
wispy slate
Notability is pretty good
Though this is under the assumption that he has an ipad and apple pencil
Yes, so far Ive heard about Notability and GoodNotes
Like would you be able to load in my pdf latex file in there and write on it then send it to me back corrected with those apps?
Yes
At least with notability that works
You can import pdf files and then write on it and then export those to pdf again
Cool & The Gang then
grand laurel
how do i make my uni sponsor me an ipad for TAing duties in corona times
gentle ibex
I think that depends where are you from and your University
grand laurel
i don't really think my uni will do this or even consider it
feral vector
Yes, so far Ive heard about Notability and GoodNotes
@wispy slate Tried both of them. Ended up with using Flexcil. I use PDFexpert for pdf reading because it can directly access my drive folder, Flexcil for writing and notes, because you can view the book at the same time, Goodnotes if the pdf loads as an image, and notability for audio recording
If the files are LaTeX, I would recommend PDFexpert as it has built-in annotations and doesn't cost at all.
civic tree
yo so like my school moved to online learning
and it's pretty sad when my math teacher is using the built in whiteboard on teams
cus like it goes in and out for him and for us
very slow
anyone know any good substitutes?
manic totem
How about Zoom?
civic tree
our district is forcing us to use teams as the main program to get everyone together
but he can screenshare
ideally if it's like some kind of website he can just
screen capture and send us
would be great
i was actually thinking to use MS paint but idk
maybe there are better lol
manic totem
screenshare would depend on the network speed. I am not sure if it's the network problem such that the screensharing is going in and out or the system itself is not that's strong to support the system that you are using
mixed use of MS word and MS paint would be good as teacher can later on send the teaching files to students
civic tree
thing is my teacher himself is getting kicked from the whiteboard ms teams provides
so i was thinking if he screenshared even if he got kicked he could just continue
and whenever his internet got back to shape in the seconds that passed
it would just like be a quicc update for us kids
instead of him having to wait for his own laptop to return the whiteboard to him so he can continue
but yeah i agree
warm lintel
brazen pendant
I use OneNote for screensharing handwriting
works well enough
better than paint for sure
manic totem
š
lethal leaf
What I'm doing for online tutoring is this
I'm getting a Wacom drawing tablet
And using that for writing
grand laurel
i am thinking of doing the same @lethal leaf
well, not necessarily wacom
you already have any specific model in mind?
lethal leaf
Whatever the cheapest no-screen Wacom tablet is available
Wacom is like industry standard for artists
So it'll be more than good enough for my math scribbling
grand laurel
yeah, i was thinking of getting a huion
it seems to be pretty decent, even for the cheaper models
lethal leaf
It's gonna be a no-screen one cause the ones with screens are pricy
grand laurel
like cheapest one is $40
lethal leaf
So it'll basically just be a super sensitive stylus and track pad
grand laurel
yes, i already have a wacom writing thing integrated in my e-reader, just want the same for my computer
lethal leaf
I think that combined with some software like desmos
lethal leaf
Actually yea what good visualization software is out there other than Desmos
For tutoring
I have mathmatica for 3D shit which is cool
strange bronze
google images
brazen pendant
I have a drawing tablet (about same price range) but it kinda doesnāt work with a lot of software and Iām considering getting a wacom one just cause I heard theyāre better supported
lethal leaf
yea that's another advantage of Wacom
Oh I should llok into Geogebra
that looks pretty powerful
wispy slate
Hi
wispy slate
@wispy slate hi
I got a doubt in power series ..in which channel I should ask that
hoary shale
Anyone willing to translate Grothendieck's "teaching statement" here: http://matematicas.unex.es/~navarro/res/guiseprogramme.pdf?
brazen pendant
deepl.com is very good at translating texts
Iād run it through that and then if thereās any parts that donāt make sense ask a native to clarify
(you could ask on /r/translator for someone to verify deeplās translation)
vestal quiver
mhmm i've had a quick look at it and I must say, it's a quite long piece of text and Grothendieck is flexing an enormous prose skill in there
it'll probably be very difficult to produce a satisfying translation
TIL a new word
on the second sentence ;_;
trim violet
in an alternate universe, grothendieck became a poet instead
brazen pendant
like grassmann who was disappointed that people ignored him and instead became a linguist?
after, you know, essentially founding linalg
hoary shale
long?? Not compared to the many thousand-page manuscripts he wrote all through his life up to his death lol
But thank you very much
I am just beginning to learn French myself. With the goal of reading and maybe translating La Clef des Songes some day.
wispy slate
http://www.landsburg.com/grothendieck/EsquisseEng.pdf this @hoary shale?
hoary shale
That's something different. btw if anyone has read the Esquisse (in your link) and understands the portion on regular polyhedra (or any portion of it, for that matter) I would love to talk about it.
brazen pendant
grading can be frustrating sometimes. first this student got a difficult question right that so far everyone else had gotten wrong. next thing he uses the claim that (x+y=0) ā (x=0 and y=0)
(implicitly ofc, he didnāt write that)
tawdry venture
with x and y not being known nonnegative?
brazen pendant
the actual claim they made was āu+v is orthogonal to all wāW, so u and v are tooā
brazen pendant
$V$ finite dimensional, $W$ a subspace, show that $(W^\perp)^\perp = W$
burnt vesselBOT
Sascha Baer:
brazen pendant
everyone beofre that tried a āproofā that never used the finite dim. hypothesis
and itās wrong in the infinite dim. case
wispy slate
I mean that happens to me all the time, I do the hard question then I get really big brain moment and just do something really stupid
round robin
^^^
rigid thistle
i relate to the second part exclusively
lethal leaf
I mean it depends on what they did wrong
I got my 2nd calc 3 midterm back and I got a problem completely wrong in that I dropped a negative sign very early in the problem
but I did literally everything else right
and interpreted my answer correctly as well so the grader gave me full points
that's probably justified to give me the points (but if they didn't give me the points I'd understand)
also you'd be amazed how far people can get with the "I don't understand what this is but I know how to do the problem"
which is where alot of that stuff stems
winged urchin
Oh that's fortunate that your sign mistake still led you to a similarly difficult problem. I've had students make simple mistakes but then end up with either a far easier problem (thus it cannot be worth the same since they inherently can't show the same techniques on the easier problem that is required for the actual problem) or an inherently impossible problem that they just pluck at and waste time trying to solve
But totally, if you can't answer the question specifically, say how you'd do the problem and that does go a long way as you say
Additionally, at least for me, if you notice your solution must be wrong AND state why it's wrong I'll typically be more lenient than if you said nothing and tried to pass it off like it was correct
strange bronze
meager bronze
as a teacher, I can say that the best advice I can give you is to READ THE GODDAMN INSTRUCTIONS
winged urchin
Getting snippy with us certainly won't help you get help
Try the question channels if you are trying to get help with a question or try the topic specific channels if you have a more general interest or curiousity
winged urchin
Just trying to cut you down I bet. Smh
winged urchin
Too teacher bad English not student an?
rigid thistle
we should just boot this annoying ass already
lethal leaf
Or just don't respond
drowsy crest
Ok
ebon grotto
Q: how are online courses going? I don't know how to gauge the level of engagement my students have been giving me so far, or if there are obvious things I should be doing
strange bronze
gauging engagement is probably the biggest challenge of online lectures, yeah
my recommendation would be to try asking questions with an answer that should be "obvious"
every few minutes
and see how enthusiastically students respond
ebon grotto
heh .. well ... I'm not doing live (online) classrooms
ebon grotto
yeah. there's too many issues in my class with availability etc
strange bronze
totally understandable, and there isnt really a great catch-all solution to that
ebon grotto
I've been posting lecture videos, and giving simple simple quizzes after checking that they watched and got a little out of them..
separate from the quizzes that I had before and continue to give which delve further into doing the material
strange bronze
yeah, i mean part of the problem is
its really tempting to give students more and more work right now
to compensate for missed in-class stuff
but obviously students are going through stressful, busy times
now really isnt the moment to pile on workload
ebon grotto
yep. the total of the video lecture + simple quiz is shorter than the original class was, right now
fiery cove
So ya Teachers? That haves discord?
rigid thistle
evidently yeah
lethal leaf
I feel bad, my calc 3 teacher also teaches calc 1
And he said he held a calc 1 lecture where ZERO PEOPLE showed up :(
nova seal
publius i cant imagine a teacher having yours pfp
if by evidently yeah you meant youre a teacher too xd
rigid thistle
yeah lol i'm not
nova seal
oof
rigid thistle
you can see logic is not my strong suit
meager bronze
what kinds of profile pictures should teachers have?
grand laurel
anime
earnest sandal
big facts
hidden jasper
i need jesus
spice nexus
hi
loud dirge
here's an example of a profile pic a teacher should have
barren sinew
I have an idea to combat covid 19.We must create stealth technology at a large scale and then use the stealth technology to hide everyone from the virus and the virus cant infect what it cant see.
ionic dagger
wouldn't it be easier if we blinded all the virii?
feral vector
or switched all the lights off
marsh thistle
just cover your eyes
if you can't see it, then it can't see you
winged urchin
Wow so much misinformation. Somebody's gonna think the virus can't get them if they cover their eyes!
oh my lord!
winged urchin
Sorry I don't have Twitter but it is an interesting thread of conversation
Depends on what your definition of a trick is. Adding 0 or multiplying by 1 in clever ways is a trick sure, but it's necessary whether the student realizes they are using it or not.
Though the kind of trick where it's like... "Consider this polynomial (or function) that is seemingly disconnected entirely from the question. Oh look if I do this to the polynomial it turns out to solve our problem! Tah dah!"
That is the worst kind of trick
But I think the essence of why it's bad is because its unmotivated
An unmotivated trick is bad.
If you motivate why we would use a particular trick then I think it's great
Finding out how to motivate a trick properly is difficult though
One example that springs to mind is with
$\int \frac{x^2}{x^2+1} dx$
burnt vesselBOT
mkovacic:
winged urchin
If a student can't use trig sub for this but knows the derivative of arctan(x) then all they can do is add and subtract 1 to the top
When students come with a problem like this I have to get them to think about how they could come to thinking of that trick.
I'll try to get them to realize that the problem here involves not really being able to deal with the x^2 in the numerator. And what could let us get rid of that? Well if we had a +1 in the numerator then it would cancel and we would be golden! But we can't quite do that... To balance the expression we'd need to subtract 1 as well... But hey! Follow that through!
If anyone has any better ways to motivate that sort of use of a trick there though I'm certainly open.
And perhaps an alternative is to say well they should really learn trig sub first then before having to deal with that.
It's interesting at least
Sometimes though I wonder if part of the problem is students not knowing or not being confident in manipulating or exploring how they could change around an expression.
Like... Sometimes you just try to twist and rearrange the mathematical puzzle and something clicks and all the pieces align. I question whether all 'tricks' can be reasonably motivated
wispy slate
I have a question who knows how to solve this
grand laurel
tawdry venture
winged urchin
I hope that little disruption doesn't stall the conversation @turbid zenith started
If anyone has comments or anything to add to that conversation I'd love to hear it
tawdry venture
well i can say for sure that some tricks are more resilient to motivation than others but i'm not sure if there exist any which defy motivation utterly and completely
and sometimes what little motivation i can find is kinda shoddy
usually i try to make a point of distinguishing scratchwork from proof work when providing said motivation
ig it's more of a continuum rather than a division
a kind of trickiness scale if you will
with formal techniques like collection of like terms at the bottom
and shit that defies all prior intuition flying near the top
leaps of faith ig you could call them
winged urchin
Sometimes I like to kinda visualize playing with equations or expressions like a map or set of directions if you will
In that way I can see the 'collect terms' kind of idea as locations that should be basically adjacent to the current expression
They take but a tiny little bit of manipulation. Really a simplification of what you already have
And the 'leaps of faith' to be far off locations from where we currently are
Like, if we accept that this location exists and it somehow helps us solve the problem and hey look, if we walk north then east then ... etc etc.. we get that seemingly disconnected expression or idea or equation
In that way I totally get where you come from there
And honestly I feel it's an often overlooked part of proofs and textbooks
They'll be like.. consider this.. and then the results unfold naturally from that
But why did we consider that?
What led us to considering that?
Is something I feel textbooks don't often delve into
You could perhaps say that a lot of explanations in textbooks definitely address the "How" question
But not the "Why" question
turbid zenith
Just got back to reading this
Yeah I have to agree it depends on what you call "tricks"
turbid zenith
For me it kind of depends
There are some "tricks" that are just clever manipulations like you talked about (adding 0, multiplying 1)
There are some that are quick ways of doing something other than going the long way ā like multiplying a number by 5 easily by just multiplying by 10 and halving it
And then there are some that are just blindly followed procedures with a cute mnemonic that replaces the understanding, and that's the kind where I have a problem
FOIL is my go-to for that ire š
remote inlet
is anyone good with differential equations?!?
meager bronze
this channel is not for you
remote inlet
do you know where the right channel i can go to
turbid zenith
Anyway, sort of on the topic of "tricks" still...
https://twitter.com/solidangles/status/1247215157450809347
Have fun with this. š
tidal whale
Yo, DM, if you hated FOIL then I have an eastern mnemonic for the signs of trig functions
"Add Sugar to Coffee"
All +ve in first quadrant
Sin +ve in second quadrant
Tan +ve in third quadrant
Cos +ve in fourth quadrant
Oh yeah, SOH CAH TOA got replaced by this:
"Some People Have Curly Brown Hair Through Proper Brushing"
strange bronze
i dont think memorization mnemonics are necessarily a bad thing
FOIL is different because it isnt meant to help students memorize
it's meant to sidestep actually teaching students mathematics
because i guess the term "distributive" is scary or something?
winged urchin
Oh ya Carl I've seen things like that. And everytime anyone tries to use something like that I try to explain how if they just understand how sin, cos, and tan related to the unit circle then not only would they understand the signs but also be able to reason about when they might be larger or smaller or undefined
turbid zenith
I do hate ASTC, though I first saw it as "All Students Take Calculus"
But again that sidesteps understanding
If you know that sin = y, cos = x, and tan = slope, you can reason it out in like 2 sec
strange bronze
part of the blame lies less on students/teachers and more on the cultural conception that
being fast at math = being good at math
you see this everywhere from standardized tests with tight time limits (hello, subject gre) to the constant media popularization of "human calculators"
"man from singapore can multiply twenty-digit numbers together in his head in 10 seconds!!! a true math genius!??!?!?"
as well as just a general idea that
if you have to think about something, it means you dont understand it yet
i'm not really sure how to fix this cultural problem, admittedly, but i do think teachers can try and emphasize how unimportant speed is - at least when standardized tests arent looming over students
turbid zenith
Yeah. Being fast is just one way to be good.
But it's not everything, not by a longshot.
strange bronze
anyway, this is why "tricks" often end up being a substitute for learning in many students' minds
like
"why do i need to understand this process when i can memorize x procedure and do it in half the time?"
these are the same students who complain about "word problems", and who feel like their foundations are weak when they go into later years
[its worth noting that im not equating "understanding" here with "rigorous proof"; i just want everything a student does to be at least somewhat motivated. nothing should seem arbitrary]
turbid zenith
Yeah I'm with you absolutely
I've had students literally be resistant to learning why something works
stone tusk
Yay for magic
lavish crest
so, which way is better?
rigid thistle
i think it is implied that the square method is superior
turbid zenith
It is at least as an introductory model so that students build conceptual understanding.
Nobody's saying the "old-school" way should go away. Just that starting off with the other one has a lot of advantages.
brazen pendant
I must say I find the "old-school" method very intuitive. it was taught to me and explained via (examples showcasing) distributivity
our notation was slightly different but the idea should be the same
the 0 obviously comes from multiplying by 90
of course in this contrived example the righthand side approach is more efficient
as the numbers are very close to 100, so it's more sensible to do (100-3)(100-2) instead of (90+7)(90+8)
I was always rather slow at arithmetic, and part of it was definitely because I much preferred the algorithmic "just get it over with" approach to thibking about how to tackle a problem smartly - because arithmetic is so fucking boring, the "clever" approaches are still not at all engaging to think about, so why bother
on a related note, in Algebra, the fastest way to solve a system of equations often involved adding equations to one another, but at that age no one could explain to me why that magic trick should work, so I always did everything using only substitutions
because those at least were very obviously valid operations and not just tricks from out of a wizard's hat
(I donāt have a point to make here, itās just some rambling)
turbid zenith
Watch out for "obviously" though.
It's obvious to you either because you see it or because you were taught it well.
Lots of teachers, unfortunately, teach that you put the zero there "because that's the next step, get back to work".
brazen pendant
in that particular case, substitution was obvious to me because āI have the same side on either side of the equation so obviously I can substitute one expression with the otherā
it was the only operation among the bag of tricks that I was convinced was true
because I could see for myself that replacing a thing with itself should not change the outcome
turbid zenith
Yeah that makes sense. š
brazen pendant
whereas adding two equationsā¦ well, how was I supposed to be certain that the solution I get is still even related to the original one?
turbid zenith
And yeah, the example was contrived, but I feel like the example I was combatting was equally contrived to have tiny digits and therefore require no carrying.
brazen pendant
to be quite honest, I would still have to think about why it works if you asked me
I can see that the solutions of the original equations would still be solutions of the sum, but itās far less obvious that it also works the other way round
in the case of linear equations you can make sense of it with some linear algebra trickery but idk, still feels odd to me. Iām not really comfortable with doing things like gauss elimination to solve a system of equations. still not convinced it works, deep down
and Iāve read the proofs
and Iām a linalg tutor (though I never tutored linalg I, where this stuff is covered)
(only linalg II, which is about eigenstuff and some other shenanigans)
turbid zenith
I just think of it as "if two things are equal, and two more things are equal, then adding them they should still be equal"
brazen pendant
yea thatās the direction that is clear
what is less obvious is āI have a solution to the summed equation and from this I can actually recover the solution to the original thingā
turbid zenith
Like if I have two identical cups of water, and two identical packets of lemonade powder, then when I mix them together I should have two identically-tasting glasses of lemonade š
Hehe
brazen pendant
ultimately I donāt think I could ever teach elementary math
(nor do I have any intention to)
the geometry class we had some time during primary school was fun (it involved a lot of constructing shapes given some informations, very much puzzle solving)
but the rest was just
mindless
I fail to see the point entirely to this day; I see no purpose in those years spent adding and dividing larger and larger numbers
no one practically does computations involving numbers with more than two nonzero digits in their head
it doesnāt teach anything
turbid zenith
I am 100% with you on that.
We should be teaching students to be sense-makers, not just answer-getters.
brazen pendant
later on, e.g. during calculus (which is mandatory in high schools here) you at least get to solve interesting optimization puzzles and what not
but Iām pretty sure you could cut four years of primary education and the students wouldnāt miss out on anything
but that is not feasable in a society where in most families, both parents have to work - or often donāt even live together
turbid zenith
I agree with a whole bunch of it.
Maybe all, though it's been quite a while since I read it.
I really do hate two-column proofs with every fiber of my being.
lavish crest
what are two-column proofs?
lavish crest
ah those
yeah those suck
how would anyone who's not a mathematician remember all those theorems?
never mind high schoolers?
winged urchin
That's part of the issue at hand I think
I've only skimmed through it right now, will read it fully later, I've gotta go soon
I think there are some good points but I can't help but come up with objections to some
Or at least... thoughts concerning it
Like the apparent objection to notation
Or at least.. unambiguous notation
I've often seen students mistake functions for multiplication incorrectly because it looks the same as multiplication if you aren't confidently aware of the context
And I've thought that making notation as unambiguous as we reasonably can is something good
But I also have, in helping students, tried to reduce complicated statements into simplistic ones with a (I'd argue) necessary loss in exactness
Like simply stating the intermediate value theorem as the idea that if a curve goes from one height to another height then it must go through all the other heights in between those
Of course because I'm avoiding mathematical notation and lengthy statements that could be interpretated incorrectly
It's more helpful in that case though to be able to draw the two points and emphasize what I mean by trying to draw different curves from one to the other
It feels like some of this document is arguing for more intuitively simple ideas at the potential cost of not being 'exactly' correct or leaving out edge cases where things aren't as clear
But again, I gotta run and will read it in full and digest it more then
strange bronze
how would anyone who's not a mathematician remember all those theorems?
i can confidently state that less than 2% of mathematicians remember those theorems
no one cares about euclidean geometry
and if people do, its usually for stuff like logic (having a consistent, complete system with so much structure makes for pretty convenient examples) moreso than actual geometric statements
meager bronze
wait, there are theorems in euclidean geometry?
strange bronze
zoph mentioned that we apparently have an algorithm to determine the truth value of every statement in euclidean geometry
i knew one existed since consistent and complete, but he claims weve actually constructed one
meager bronze
oh interesting
I feel like I might have heard something along those lines before but it's really muddy
strange bronze
anyway i dont think anyone cares about euclidean geometry for any other reason than logic
i mean ok, part of the reason for that is
every interesting statement you could make about euclidean geometry easily generalizes
but the point stands, mathematicians dont just remember laundry lists of theorems
honestly i dont think high schools should be teaching euclidean geometry at all besides trig + basic perimeter/area/SA/volume/etc calculations
like even if you believe that it "improves critical thinking" somehow, which is a very suspicious statement
so does like, every field of math
teach combinatorics instead
meager bronze
I agree
strange bronze
most proof questions in high school geometry look like "this is obviously true, now prove it through some dumb formalism that no mathematician would ever actually use"
but you dont really get the same thing in combinatorics
so it's more approachable to studnets IMO since you're not making mountains out of molehills
as well as the fact that it's like, you know, actually useful
especially with CS enrollment on the rise
sorry, that was a total side-tangent from the prior conversation
but high school geometry fucking sucks and should be nuked from orbit
or at least from curriculums
wispy slate
i agree
boreal spire
strange bronze
yeah vector stuff is totally reasonable
im more referring to like, competition math-esque problems
except competition math does it better since
a) the problems are usually actually somewhat interesting [and often generalized]
b) they dont expect some shitty two-column hyper-formalist fill-in-the-blank template, but rather, an actual mathematical argument
lavish crest
suspicions that the people deciding which math course goes and doesn't aren't even teachers nor mathematicians. is this true?
meager bronze
yes
feral vector
most proof questions in high school geometry look like "this is obviously true, now prove it through some dumb formalism that no mathematician would ever actually use"
No wonder a lot of highschool students raise the concern where exactly am I gonna use all of this (like pythagoras). Formalism without motivation is just time-waste.
wispy slate
agreed
ionic dagger
Each state government has some say in the qualifications required of teachers, the content of every class, and even the graduation requirements (5ish years ago Texas changed to requiring only Algebra I and Geometry and a "advanced class" in math).
honest sequoia
What dumb formalisms that are used in euclidean geometry are you talking about?
Srs question
I thought learning proofs in geometry was one of the most enlightening things
But im a noob so, im curious
@strange bronze
stone tusk
this abomination:
https://www.sparknotes.com/math/geometry3/geometricproofs/section1/
@honest sequoia
wispy slate
lmao
ionic dagger
I disagree @stone tusk
stone tusk
I think it should be optional to use it
like people who can write good prose can write good prose
ionic dagger
It's certainly too "command command command" to allow for creativity. Yet, I think if you put "One tactic that might help is ..." then it would just be a list of methods that might help.
stone tusk
definitely not targeted at the microbiology teacher who taught the researchers that are trying to research out this epidemic
ionic dagger
I dunno. How else would you recommend building up to a somewhat easy endpoint like Pythagorean Theorem?
wispy slate
why would any microbiology teacher be teaching two column proofs 
ionic dagger
Two-column proofs are the "let's write sentences with simple subjects and simple verbs" of the math world.
stone tusk
@wispy slate well they wouldn't be teaching math and hence are they teaching useless rubbish?
wispy slate
what
ionic dagger
They're a necessity at the beginning.
wispy slate
why?
ionic dagger
You have to start somewhere. And if a hot-shot student can do amazing things, his homework will be a piece of cake and we ALL know that success is often just showing up or doing other mundane tasks.
I calculated I could have gotten 2 PhDs waiting in line for lunch back in college. š
God, the annoying list of things you need after you get done with classes but before you can officially graduate is somewhat long in most colleges.
brazen pendant
lemme tell you one thing: I have never encountered two-column proofs in my life
they are not part of the curriculum I went through
they are in no way a necessity
the high school curriculum I went through prepared me well for a math degree that jumped straight into proofs. I for my part took extra math classes (we have a flexible slot where we can choose one subject to deepen our understanding in and I chose physics&math) and we did a little bit of proofs, I think literally just induction. friend of mine didnāt even do that and she did just fine too (way better than me in fact, and Iām a good student according to my grades)
I would hate to read two-column proofs, they are the opposite of how I like proofs to be written
grand laurel
the only thing you need to start writing "real" proofs is a teacher that tells you when you fuck up
two column proofs and even "intro proof" classes are kinda unnecessary
brazen pendant
yea we didnāt have intro proof classes
the first two weeks the linalg and analysis prof worked together to give an overview of predicate logic and naive set theory (introducing notions such as what functions and relations are)
and then we just started on the topics
grand laurel
i kinda see the idea of two column proofs in that it forces you to think about why something is true and state the reason explicitly
but i would much rather see that written down as an actual sentence
or maybe annotate your "=" signs
brazen pendant
yea exactly
feral knoll
I did two column proofs in 8th grade, they almost made me want to quit on math because honestly they feel terrible to do, and no one could explain why they were useful. (nearly all of us in the advanced math sequence thought they were a torture implement of the teacher)
tawdry venture
two col proofs are garbage
brazen pendant
I wonder whether Iād have enjoyed them. I would probably have been good at them
and I tended to enjoy things Iām good at
but like
Iām glad I didnāt have to go through that anyway
tawdry venture
they make proofs seem like a series of magical incantations, with the names of theorems invoked as if they were fucking magic spells or something
grand laurel
this is my head canon too
brazen pendant
āand by the inverse function theorem waves wand there exists an open neighborhood U of p such that f|U is a diffeomorphismā
tawdry venture
oh yeah. arcane incantations
brazen pendant
I guess the actual arcane incantation is āby the axiom of choiceā
you know something stupid is about to happen when you read that
tawdry venture
what i'm saying is that two col proofs make students bad at reasoning
bc it allows them to circumvent explaining themselves clearly
that's just my onion
feral knoll
My personal experience is that they're garbage because they don't actually teach the student how to defend/explain their steps/logic. 2 years later, we went from 2 col to standard mathematical proofs, and I don't believe I carried over anything from the 2 col proof to standard math proofs other than knowing to invoke theorems when making explanations, which the teacher covered in lecture anyway.
feral vector
More importantly, they are garbage because the students aren't taught what exactly are they trying to accomplish. Rather they're instead taught in a pattern-finding manner, that is, "If you find a question that is similar to this, just rewrite this". (From my experience)
I had zero idea I was "proving" something
I guess the actual arcane incantation is āby the axiom of choiceā
Also "by quantum physics"
ionic dagger
You have to understand that most people aren't going to be math majors nor need extensive math training. In the US the highest math that any state requires is Algebra II (induction is taught the year after in Precalculus). It's the only training in "logical" reasoning they might get in their lifetime.
@feral vector The fact that you didn't know that you were proving something is the fault of the teacher. The method is a practice in low-level, but sound logic.
feral vector
Yeah, but it fails even at that. There's no sense being conveyed that the students are being taught how to logically reason. It seems to me more an exercise about how you can just copy paste a "correct answer" in different contexts (considering how almost all the exercises have the exact same form). If you really wanna teach logical reasoning, maybe something that's less geometrical would be better for that purpose.
ionic dagger
I'd be up for that. Instead of Geometry ā¦ Proofs in Geometry first semester and Proofs in Everything Else second semester.
feral vector
It's also debatable whether logical reasoning only comes under the domain of math. Many non-stem majors give a course on argumentation which begins with an intro on classical logic to teach how to logically reason in one's arguments.
ionic dagger
In the end proofs, at least in many cases, decide what we can know/do from a few axioms.
And I'd also be okay with two-column proofs for half of Geometry (be the Geometry class one or two semesters) and then half free-form.
I think my bias comes from never seeing a Geometry class without two-column proofs.
feral vector
I guess one point in favour of two-col is that it continually acts as a reminder for justification. A student might go about writing x+5 = 3 and then in the next line x =2. The unfilled right column would serve as a reminder that the student have to justify and thus force a habit of reminding oneself to justify their assertions
ionic dagger
On a similar note, I have had more than one calculus student who just didn't get logic. They had these terrible, new-fangled things called "need statements." Basically if you satisfy the "needed" conditions, you can invoke the rest of theorem. This is at the calculus level. It made me want to cry.
feral vector
How's that different from saying you have to satisfy the hypotheses to invoke the theorem?
ionic dagger
It's not. It's just more formulaic. Kinda like two-column proofs ā¦ but 2-3 more years into math.
By that point, you should really know how simple explanations work.
It's out there. Google "need statements calculus"
honest sequoia
Ah
I see, i never learned 2 column proofs. I was a really bad student as kid. I self taught basic math in my 20s and learned how to do proofs with predicate calculus, set theory, etc.
I still suck at, as i went engineering but id like to think i am somewhat mathematically literate. I know how to navigate proofs by negation, etc.
ionic dagger
um, why? Are you interested in reading the proofs of others?
Or interested in becoming the next Beal?
honest sequoia
Why what? I got sick of lugging boxes in a warehouse as a 20 yo and decided to go back to school. I was struggling with basic math classes in community college and felt like im smarter than this. So i started reading textbooks and now im now an engineer.
rigid thistle
then they would probably be earning less
brazen pendant
Proofs in Geometry first semester and Proofs in Everything Else second semester.
that would mean you couldnāt start actually doing things until your third semester! thatās madness!
provided you can assume the students have encountered calculus before, you can drop them right into real analysis
and linearl algebra doesnāt have any real prereqs anyway so you can start on that right away too
grand laurel
you can also do real analysis fine without doing calculus before
brazen pendant
sure, you just need to spend more time explaining what the hell youāre doing
meager bronze
Do whatever you want
I mean it -- if you want to do math research for a career, go for it and be a professor
if you want to teach math for a career, you can also do that at either the university level or pre-university
if you don't want to do those things (which many people who get a phd decide not to) then don't
PhDs attract people who really like learning math (because that's all you know how to do as an undergraduate) and that's different from really liking research
So maybe you decide that research isn't the career for you, or maybe you decide that it is
Note that my advice would be exactly the same if you were 25 instead of 20
you have plenty of time, you're only in your second year
I'm assuming you're from europe given that you write "maths" instead of "math" -- phds are normally 4 years, right?
assuming you got a masters beforehand
or are you getting a phd in the US
ok, so you're more an equivalent of a 3rd or 4th year phd student in the US, since in the US you go straight from bachelors to PhD
so you are a little closer to finishing than I thought at first
so I am/was the same way as you. I also decided to pursue teaching a little more than research
there are academic jobs where you can focus on teaching at the university level
though they aren't as common as the pure research positions
it's rare but not unheard of
there are a handful of highschool students in this discord server
who are learning university-level or higher math
these people are like, 15-17 and learning algebraic number theory
which is like traditionally "junior undergrad to 1st year grad student" level
like elementary number theory or algebraic number theory
so like number fields and factorization of primes and class groups and stuff
ok just checking
yeah so there are other people around here who are doing about the same
idk if our terminology differs because of the ocean between us haha
also yes it is cool stuff
it's what I do lol
class groups are my jam
also, about starting a phd so early, in the US I think that would be much more uncommon. instead of speeding through university, you just keep learning more advanced stuff while in university
so at 20 you would still be in your 3rd year of university, but you would be doing graduate-level research
instead of graduating early or whatever
there are plenty of undergraduate or masters theses that are of a higher quality than phd dissertations haha
I don't mean to say you weren't special, it certainly is impressive for anyone to be learning that advanced math at that age
I wasn't learning algebraic number theory at 15
but at the end of the day, having a head start doesn't always mean that you'll stay ahead. so don't get complacent haha
I only say that because I've seen several people who were always the best at math in their school, then all of a sudden they go to get a phd and they're not the best anymore
and they dont' know how to handle it
yeah like I said it is special to be learning that level of math at that age
but again I just want to caution you about holding on to that for too long. your mathematical ability and talent isn't defined by what you were doing a decade ago, it's defined by what you're doing now
I'm graduating with my phd in two months
and I have an academic job lined up for the autumn
I'm 27, which is standard in the US
:P
graduate high school at 18, then 4 years of college (to 21 or 22) and then 5-6 years of phd
keep in mind our phd also contains a masters
so it's kind of like "1-2 year masters, 4 years phd"
that's not a thing in the US
you just get a phd
undergraduate degrees typically come with some kind of honors
cum laude, magna cum laude, or summa cum laude
or separately you can just have "with honors"
sounds like the latin honors we giev
but those don't exist at the graduate level
yeah, here you would just get a masters degree
how do they even decide
in the US, you don't even take courses during a phd
you don't get any grades
it's literally just "do research and write a dissertation"
(at least post-masters that's how it is)
but who decides lol
also I'm reading online about this
and it says that only undergraduate degrees and combined bachelors-masters degrees are given honours
also
from wikipedia
"In England, Northern Ireland and Wales, almost all bachelor's degrees are awarded as honours degrees; in contrast, honours degrees are rarely awarded in the United States."
it's just a different system
it seems like to get an honours degree you just need to take more classes
whereas in the US, there aren't multiple levels
it's just like "here are the requirements to get a bachelor's degree"
in any case, I don't think phds can come with 1st/2nd/3rd honours
it's fine, it doesn't really matter to me
I do algebraic number theory
my research involves trying to understand the structure of class groups via galois cohomology
so basically I take arithmetic questions about class groups, translate them into cohomological questions, "do cohomology", and then translate back
what about you?
I agree :P
(which is why I do it haha)
what kind of research do you do
ah ok
cool
there really isn't any mathematical physics at my institution
so I don't know very many people in that field
haha
alright, I need to go work on my thesis
deadlines are coming up...
:x
haha good luck to you too
queen acorn
I don't know if this is the best place to ask this... Would anyone be willing to video chat with me and go over a math equation. It's a simple equation, just have some questions on notation.
rigid thistle
wrong channel
queen acorn
alright thanks
tawdry venture
how often do y'all get students who go from 5x = 0 to x = -5 or something to this effect and how does one effectively deal with this error
feral vector
I used to teach my students error-checking methods like after solving any identity, plug in some numbers and see if the answer matches.
Now after they've realised the errors they made I would usually either try to explain why does the actual result makes sense by building a real-world example or just ask them to do more practice.
winged urchin
I think error-checking is okay. It's a great way to just demonstrate that it is wrong. I'll often say something like "Alright so you claim that x=-5 is the solution. It should satisfy our original equation. If I plug it in, 5 times -5 is what...?" and they answer -25 (hopefully) and realize it's wrong
I suppose it depends on the level of the student too. If it were an elementary student I would address it differently than a first year calculus student.
I might also talk about what they actually did to the equation. I'll describe that there are only so many things we can do. we can add to both sides, subtract from both sides, multiply both sides, etc...
Then I can hopefully lead them to realizing that to get -5 they needed to subtract by 5 on both sides. Then ask them to consider the left side then... we would have 5x-5
And hopefully they see that that doesn't become x
Lastly (sorry :p) I might just talk briefly about why they might be making that mistake
Because I believe part of the problem is the casual way we talk about equations sometimes. "Alright class, here we just move the 5 over to the other side..."
And just let the student know that that is informal language and there are different ways to "move a 5 over"
obsidian oasis
@kind salmon
kind salmon
goodday. can you help me find an exercise (for 8th grade)?
i need a real life example of any circle, where you know the surface area but not the radius. (i only find ideas where you know/measure the radius first.)
@tawdry venture this is a systematic problem, i wouldn't even call that a real error. your students still don't see the difference between subtraction and division. i would do both cases so they see it more clearly and also it helps to write the multiplication sign, like 5*x = 0 vs. 5+x = 0
(rather search for the pattern in the error, than for the error only. if you got the time in class at least.)
strange bronze
i could see a problem like, say
"you have x amount of [paint or material or whatever]; what's the radius of the largest possible circle you could create out of this?"
obviously rephrase as appropriate, but something in that vein
maybe something like, you have enough fabric to make cover 10 cm^2 (or whatever) and make a circular badge with this fabric, how big is the badge's radius
etc
another situation might be, say, you have a perfectly round island and want to know how far it is from one edge to the centre
since the land area of islands/countries/whatever is usually fairly easy to look up, but radii arent
obviously this is still somewhat contrived since perfectly round islands/countries/etc dont really exist
but some places get fairly close, like sierra leone or swaziland
obsidian oasis
wow teachers can get creative
kind salmon
ty, that's very helpful. i did introduce density already so the paint will work i hope
ebon grotto
a 3D example would be measuring the surface area of the earth to determine its radius (since we can't drill in, but we can measure the outside)
but I really like the clothing patch idea. like ... we have a circle-ish of fabric measuring area A, and we have a rip measuring length L; how big can L be and be guaranteed that we can cover it using patch A ?
Assuming that the rip is a straight line, otherwise, we get near unsolved territory https://en.wikipedia.org/wiki/Moser's_worm_problem . (I personally like that, since I like showing that math is not a "completely known" subject at that level.... too many students grow up thinking it's just rules and procedures, not a pursuit of new knowledge.) @kind salmon
kind salmon
i really like the earth idea but then i have to introduce spheres a year early... hmm... where are tha flat earthers when you need it
ok you got me. the clothing patch is applicable even in quarantine. great one!
ebon grotto
especially in quarantine. can't go buy new clothes at the mall, can ye?
kind salmon
can't go buy new earth either š
btw, i agree about the pursuit but i think you have to be very carfeul how far you go. can get overwhelming fast.
for example i already told that student, density is not as constant as taught in physics so far. meaning what the student just learned well last year was suddenly 'wrong'. understanding why we make this simplification of constant density for now was crucial i'd say.
ebon grotto
I usually prefer things like the worm problem because they understand that rips are not necessarily in a straight line, but maybe don't intuitively (enough) understand that density is not constant
kind salmon
ah right. i just teach 1:1 mostly and the student came up with density, so i can go into detail and make sure 'everyone' understands
turbid zenith
Hello mathematicians
Anyone have any suggestions for how to teach non-Euclidean geometry to bright high school students online in a student-centered way?
My summer program I teach at is likely going to be online this summer
ebon grotto
I don't have anything to contribute. But do you mean ... finite geometries, or hyperbolic/etc geometry?
turbid zenith
I'm also teaching combinatorial game theory and numerical analysis but I already have ideas for those
Hmmm...
ebon grotto
I'd guess Fano plane etc
turbid zenith
We could hit a number of topics. But I definitely plan to do taxicab, spherical, hyperbolic, and projective
ebon grotto
interesting
turbid zenith
These kids soak up whatever you throw at them
But I want to make sure it's engaging
And giving them the opportunities to do stuff rather than just listen to me lecture
trim violet
maybe some interactive app where you have the disk model of hyperbolic geometry and you can drag things on it and you see how everything rotates ?
winged urchin
Yeah an interactive app would be the ideal here I think
Really in any mathematical education I'm a fiend for anything I can manipulate myself, at least when I want to learn
kind salmon
if they pick up new stuff fast just use matlab or whatever. i am overwhelmed how fast my students anticipate any software i use, even if it's years ahead of them
brazen pendant
french railway metric best metric
(itās the one where distances are measured by always going either towards or away from paris aka the origin)
if they pick up new stuff fast just use matlab or whatever
matlabās expensive af, if youāre gonna do some computational stuff Iād much rather show python cause they can actually get that for free if they wanna play aorund with it.
as for actual ideas perhaps talk about geodesics on different surfaces in āĀ³?
not necessarily with all that much math but e.g. Clairautās Relation is fun and you can show some qualitative stuff with it
such as the statement that if you have a geodesic on a torus that is at one point tangential to the ātop circleā then it is confined to the outer half of the torus
kind salmon
matlabās expensive af, if youāre gonna do some computational stuff Iād much rather show python cause they can actually get that for free if they wanna play aorund with it.
@brazen pendant octave is free. you should use what you can work with, simply.
ionic dagger
@turbid zenith Escher has some nice drawings using spherical and hyperbolic.
honest sequoia
Matlab is better for production stuff
Rapid prototyping models. Really, unless you need Simulink and its codegen/testing supports OR some super specialized field where the Matlab toolbox makes it trivial, theres no reason to use Matlab over Python.
kind salmon
@strange bronze @ebon grotto ty both for the radius idea. after all, my student came up with an own: what radius does a ring need to fit my finger?
kind salmon
well it at least involved finding r. it's circumference, true. but i am so happy when they finally do think alone anyway š
kind salmon
'three girls and two boys from you class sit on a bench with 6 seats, waiting for the school bus. how many possible combinations are there?'
'two - all girls to the left and boys to the right or vice versa'
tawdry venture
kind salmon
i was close to link you to this, hope you like it š
jolly pollen
And you have $Var(A)*(n_A-1)=\sum(A-\bar{A})^2$
burnt vesselBOT
Limit City:
proper stratus
I'm applying to a tutor job, what kind of math is covered in grades 3 through 8?
In the USA, that is
I'm really not familiar with how the common core math system works as I never went to school
strange bronze
the curricula should be available online
the rough idea is
everything from an introduction to fractions and geometry (as in like, "identify shapes" or maybe "find the area of a square")
to actual algebraic manipulation and a brief look at very very simple probabilistic concepts
again, read the link for more details
aside - reading that link, i was surprised at this outcome under "grade 8 functions"
Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
like i'm surprised that they define "graph of a function" semi-formally
the rest of the definition is ass but whatevver
its simplified
proper stratus
Thanks namington 
turbid zenith
I think that's a good thing they try to define it that way
A lot of my students come to me thinking the equation of a graph is basically a magical incantation that makes the graph appear but only if you say it right
They don't think of it as a description of the points, or a relationship between the coordinates
meager bronze
have any teachers here taught precalculus before?
I'm teaching precalculus to first year university students in the fall and I'm being asked to choose a textbook
I've only taught calc before, not precalc, so I'm not as familiar with the landscape of precalculus textbooks
any recommendations?
pale vessel
strange bronze
š
topaz sail
im planning on majoring in math...ye not a good one. but what courses do i need to take? I already finished linear algebra
can someone list the entire math courses that I need thx
and in general math not applied
it's*
topaz sail
lol im using it for something else
math is better than psychology major
psychology major is a fcking joke
strange bronze
The degree requirements will vary from institution to institution
topaz sail
but isn't there a general pathway
strange bronze
Look into your school's description and requirements for the major
I mean, sure
Introductory analysis and algebra (groups/rings/fields/modules) into some proper complex analysis, Galois stuff, measure theory, etcetc
With some DEs added on the side
topaz sail
oh ok tyty
strange bronze
But the exact requirements will vary.
For example, American schools tend to have less stringent math requirements than European ones
And even then, Harvard will expect you to take more high-level math than your average state school
Etc
winged urchin
This is sorta in-line with the discussion we had a.. week or so ago about the arcane incantations and opacity of some maths.
I was wondering what you all thought about questions like these
Are they a waste of time? Is it possible to give an intuitive explanation of how to approach these kind of problems or do you just have to resort to saying things like...
"Well here we use the fact that exponents distribute over multiplication and get..."
Anyway, I'm more interested in what you all think about such questions so I'll stay brief and hopefully let a bit of a discussion unfold š
ionic dagger
I really like concepts, but mostly proofs of even the most basic things require induction or the like. Also, unfortunately the most boring math is generally the stuff most students Will Need to know and thus it is emphasized in K-12 math classes.
I mean I'm just a guy talking ... but it would be nice if 2-5% of school time was specifically set aside to let the instructor teach whatever they wanted. This wiggle room might be used to help a class of struggling students or prepare an advanced class for the AMCs.
acoustic ridge
rugged oriole
@turbid zenith you know our HyperRogue? lots of stuff about hyperbolic (or spherical) geometry can be shown there (if something cannot be shown, please tell me;) )
rugged oriole
yes š
lament wraith
wow, thanks for making a super cool game
rugged oriole
thanks!
tawdry venture
yeah it's amazing and also kinda surreal that i'm like. talking to you rn lol
rugged oriole
thanks! I think it is easier to get an answer from an author than it seems, though I guess the most popular ones are swamped with questions and are unable to answer everything
paper gust
any math teachers here who can PM me? I have some questions
lethal leaf
And read #āhow-to-get-help
paper gust
@round robin Discussion for teachers, tutors, TA's, and professors about math teaching techniques.
This is not a channel to ask for math help.
I'm confused what that has to do with me?
stone tusk
@paper gust sorry, I think they have misinterpreted
but I think it shouldn't be a problem asking it here
paper gust
It wasn't about math help, I wanted to ask questions about becoming a teacher. Maybe this isn't the place for that though.
stone tusk
yeah that's fine
but the fact you said "question" and not "questions about becoming a teacher" set off false positives
lethal leaf
^
Typically when people ask something like it it's more along the lines of "can a teacher help me with my precalc homework"
Sorry about that
Yea if it's about that, totally ask here
round robin
else just ask openly idk the need for dms
proper stratus
@meager bronze It might be too high-level but sections of AoPS Precalclus are really useful
They don't require a very extensive background in mathematics
meager bronze
i'll look into it
but I'm not just looking for resources, I need to choose like a textbook for the bookstore to stock
so like if it's too high level then maybe it wouldn't be good
to exclusively teach from
but yeah I'll look into it! Thanks :)
proper stratus
@meager bronze Chapters 1-5 are all on (i assume) typical precalculus curriculum like trigonometry (in both geometric and algebraic applications), function, advanced graphing, parametrization. Chapters 6-8 are all complex mathematics, complex numbers/trig, roots of unity, complex plane. Chapters 9-13 are all intro LA material.
proper stratus
I'm really not familiar with how advanced high school math goes, but if my memory serves me correct, only a fairly introductory-level understanding of geometry and algebra was required
turbid zenith
Anyone here subscribed to the Chronicle of Higher Education?
ionic dagger
@proper stratus that depends on where you live (even in the US the criteria for high school education differ state to state)
In California there is also a list of standards you must / are recommended to achieve to get into a University of California. These standards are higher than those for just a high school diploma.
wispy slate
I will help you solve math problems
I am lockdown and I am not able to go to office please help guys
rigid thistle
wispy slate
"so while i suspected there was some cheating going on, there was no way i could prove it" š¤£ š¤£ š¤£
cerulean barn
So a 6 point average increase is enough to tank as many people as possible?
frigid crest
unreal ledge
Oof, that's an awful strategy. What happens to all the kids who spend forever on the question?
strange bronze
yeah that's my concern
like if it was a legit question and they put a fake answer on Chegg
like an answer that has some obvious, distinct "indicators"
but the question itself is totally normal
i don't think there'd be a problem
frigid crest
In the future, AI software should render cheating virtually impossible.
strange bronze
ehhhhhh
i'm suspicious
like obviously an AI wouldnt be able to help much with cheating on a multiple-choice test
i guess it could analyze things like, if a student took unnaturally long on one specific question, or if they messed up an "easy" question but got a "hard" one right very quickly
but i'd be afraid of false positives in that case
and IMO preventing false positives is more important than catching cheaters
now multiple choice tests are one extreme
frigid crest
Tests will be administered in controlled settings.
strange bronze
the link you gave (which doesnt seem to cite anything?) seems to be referring to specifically "danish" [sic] essays
im not sure it'd be as effective at fighting cheating on a math test
especially since most "cheating" in that context is in the form of
texting someone "hey, what approach did you take to x question"
or quickly googling something and checking stackexchange
or whatever
students are still writing answers in their own words
its just they had help in the ideas
turbid zenith
Small brain: Try to trap students who you think are cheating. š§
Big brain: Write tests with less answer-getting and more introspection/analysis. š§ š
Galaxy brain: Figure out WHY your students might be compelled to cheat and rethink the ubiquity of high-stakes grading. š§ š«
strange bronze
i mean, that's great, but IIRC that specific email was from a 100-level chem prof at a state school
im not sure they have much power to change things on such a broad level
i totally agree that we should advocate for reform
and in any case, the action taken on this specific test was certainly a misfire
but sadly, things arent always so flexible that you can tell profs "just write better tests"
especially since the move to the online format was so sudden
frigid crest
There are fancy high-tech deterrents that are potentially invasive.
turbid zenith
I get that @strange bronze
But this prof took the time to develop a trick question WITH the TA's and do a whole bunch of legwork to entrap the students
I think there are better solutions
strange bronze
oh yeah i agree
turbid zenith
But ever since the lockdown almost the entire conversation online has been "BuT wHaT iF tHeY cHeAt?"
If anything I think continuing to be hardass during this crisis makes people even MORE likely to cheat
turbid zenith
My thought tends to go toward having more open-ended questions but I'm not sure how that could be done at scale
ionic dagger
There's a long line at the unemployment office. There have got to be at least a few qualified graders ...
stark pine
@turbid zenith just curious, what issue do you have with a fake question? I'm a student, and when I read that, my first reaction was "that's a great idea". What issues does it present that I might not consider?
strange bronze
it could end up hurting honest students if they try and spend time on the fake.
especially if the fake question is meant to "look convincing" [and if not, it's not all that effective]
like
imagine an instructor trying this with something like ``Determine whether $\sum_{n=1}^{\infty} \frac{1}{n^3\sin^2(n)}$ converges"
burnt vesselBOT
Namington:
stark pine
oh fuck lol that's evil
strange bronze
this series is literally an open problem, but it "looks like" something that can be solved
that's a particularly extreme example, mind, but i'm still opposed to it on principle
stark pine
I guess what I'd say is that isn't it, to some degree, the students responsibility to choose which questions to do, and which ones not to do? like if you can't solve it, move on to the next one
that specific question though hmmm I see how a question like that could turn into a time sink
turbid zenith
@strange bronze basically explained my issue with it.
I feel terrible for those students whose grade was on the line and then thought they were doing well but then were faced with that question.
Especially if it was partway through the test rather than, say, the last one.
tepid citrus
But would an informed or legitimate student not know that the question has no solution? I am not entirely aware of what the trick question was
strange bronze
i mean thats literally a problem you could give to a calc 2 class
stark pine
if an informed student could determine that, cheaters could determine that. The prof probably made it convincing
strange bronze
and it'd seem fairly standard
yeah kids might realize "wait i have no idea how tf integer values of sin^2 behave" but
they might figure they can bound it or something
and try some stuff with the ratio test or w/e
since it "looks like" it might be able to simplify nicely
especially if you add in trig identities, etcetc
stark pine
I just took calc 2. If I saw that on an exam, I would try it. I would try to come up with a clever comparison or try the intergral test, try trig identities. Yeah, ok, that definitely makes sense
I was imagining the prof put the question as the last question, but if it wasn't, or if any honest students tried it that's pretty shitty
strange bronze
like dont get me wrong
if the test started with something like
"Some of the problems on this test might not be solvable. You only need to attempt x many" or whatevver
and that was factored into designing the test
then maybe that'd be fine?
idk it'd definitely be fairly radical for a math test
but in any case, thats not whats happening here
wispy slate
what if each question on the test was given a certain amount of time
turbid zenith
Okay, imagine this. You have a 79. You really need that B. You've put your everything into studying for this final. Then you're working it, you're doing okay, you think you might be able to pull your grade up. Then you get hit with the "impossible" question, but you don't know it's impossible. You try it, but it's not working out. And you need that grade, so you start racking your brain while the clock ticks away, just so your professor could catch the cheaters.
strange bronze
botnuke im not sure how you enforce that cleanly in an online setting
turbid zenith
Who knows ... you might even turn to the internet out of desperation.
wispy slate
and you couldn't see any others or go back to any
once the time was up
surely with modern technology that's possible
strange bronze
idk i feel like that'd end up being a massive mess
and probably increasing individual student stress
since like, as a student i'd try and knock out the "quick" stuff right away
and spend more time focusing on the trickier stuff
turbid zenith
@stark pine often you have A = [90,100], B = [80,90), C = [70,80), D = [60,70), F = [0,60).
strange bronze
and sure, you could say "well, the prof could adjust time intervals to give the trickier stuff more time" but
stark pine
A = [80, 100]
B = [70, 80)
C = [60, 70)
D = [50, 60)
F = [0, 50)
strange bronze
and in my experience, profs are already prettttty inconsistent with figuring out how much time their tests take
especially proof-based ones
stark pine
my algebra class had an... interesting approach to that problem. The prof made the final 30 questions, and then curved a 30% to an A
like 30 involved, long proof questions
strange bronze
dude i would just blitz through that for part marks
get 3/10 part marks on every question and that's an A
stark pine
getting 3/10 on a question was very difficult
strange bronze
yeah thats fine then
stark pine
it was like "your proof is incomplete, have a pity mark"
or
"your proof is complete, with a minor error, lose a mark"
there wasn't much in between
strange bronze
i mean 30 questions still seems ridiculous but like
10 questions with a 50 curved to an A
is totally fine
and normal in some upper-level courses
my algebra quals were literally
7 questions, had to answer 2 correctly to pass
turbid zenith
Still feels a little ridiculous to me that the scale is so "F-biased"
stark pine
if that exam - the one from that email - had a disclaimer like "no need to solve every question" I feel like a trap question would be a great way to handle cheaters
turbid zenith
That's potentially fine then yeah. Some choice would be great.
strange bronze
yeah as long as students know going in
and ideally if the trap question has some "giveaways" that more keen students could pick up on
that lets them know its unsolvable
like, hmmm
stark pine
posting it in notes but not going over it in class?
cheaters rarely check course notes/resources from the prof
turbid zenith
See I don't want to resort to tricking students
stark pine
ahhhhh that's fair
turbid zenith
I think the most important thing to do in this situation is find out why the students would WANT to cheat
strange bronze
just the act of recognizing it's unsolvable
demonstrates an understanding of course concepts
like say
stark pine
also please note I'm talking from the perspective of a student, not anyone with any pedagogical experience other than a little tutoring, so my perspective is severely limited
turbid zenith
And try to design the course so that students don't feel like they have to cheat
strange bronze
for example, suppose a question is:
stark pine
that's really great of you
strange bronze
Show all functions $f\colon \bR\to \bR$ satisfying $f(x+y) = f(x)+f(y)$ are continuous linear functions
burnt vesselBOT
Namington:
turbid zenith
For example using a grading system that allows students to make revisions if they do something wrong, so it's not like you start with a 100 and watch it slowly slip away
strange bronze
as long as students know beforehand that it MAY be unsolvable
this example isnt the best one since
cauchy's func equation is pretty famous
but something in this vein
i'd have to think about it more
or like, if a student is to respond to that question and say
"I can't prove it for ALL functions f, but i can prove it assuming f is continuous"
they'd get "full marks" for the question
if you get what i mean