but that is another problem is it not
#math-pedagogy
civic tree
when there is a difference across students in understanding
but the grade numbers are relatively close together
faint yarrow
it is a lot easier to grade a class if you've taught it a lot before.
stark pine
@faint yarrow how are you using "above and beyond"?
faint yarrow
well, back in the olden days when I taught algebra I'd say 70 was the cutoff for an A, and a lot of students would bunch up around there. And then the cutoff for A+ was a 97
unreal ledge
An A represents being a certain standard deviation above the mean, in terms of tests and assignment performance. This interpretation is forced when a professor "curves your grade"
So if we expect a student to perform to the mean, then an A means, in the most literal sense, "exceeding expectations" haha
But so does a B+
turbid zenith
...grading on a curve should be illegal but that's a different story lol
stark pine
@turbid zenith does "grading on a curve" mean "boost everyone's mark by x%" or "give a certain number of people each grade"
strange bronze
the latter roughly
it means "adjust grade cutoffs to fit some grade distribution"
turbid zenith
The latter, yeah.
tidal whale
would that be every system of grading relying on relativity to everyone?
turbid zenith
Yeah.
In a nutshell, I don't think grades in a course should be norm-referenced, but criterion-referenced. Your grade should represent what you have demonstrated that you know, not how well you competed with your classmates.
tidal whale
ah, that's understandable
strange bronze
my undergrad's math department disallowed curving but had some "clustered cutoff" system, the idea being that the prof would look trhough the grade distribution and try and find "natural cutoff points"
like if the grades look like:
98%, 97%, 97%, 96%, 92%
then there's a very clear cutoff between 92 and 96
so it would make sense to divide between A+ and A there
the system still feels kinda competition-heavy for my tastes, i'd rather have students collaborate
but i prefer it to just a curve
tidal whale
We have this system simply titled Relative Grading where the average of the middle 50% of students is used to generate a grading table, with grades at roughly 5% apart
it's either that or a fixed grade table splitting 50+ into the grades
MCA as in the avg of the middle 50% of students
stark pine
@turbid zenith How do you feel about the "the average was low, apply some function to everyone's grade to fix this"? for us, a percentage shows up on the transcript, so you need to do something
ie if there's a 30% average
what would you suggest doing to the grades? the system here is usually either
- add a grade to everyone
- apply some (slightly more complex) function
One of my classes had the "curve" as
38.2296+1.01992x−0.00322957x^2
meager bronze
is it standard in canada for the explicit percentage to appear on a transcript?
stark pine
yeah
meager bronze
oh interesting
stark pine
actually, I'm not even sure if anything else exists
meager bronze
at most places in the US, it's just a letter grade
stark pine
Up until grade 6 we have letter grades
then grade 7 and onwards, report cards and then transcripts show percentages
meager bronze
so like, when I teach calculus, my grade distribution ends up being like, in 5 point increments
A+ = 95-100
A = 90-95
A- = 85-90
etc
but I don't have to go and "curve" individual grades
tidal drum
My schools never had an A+
meager bronze
my undergrad did but my grad school didn't
I don't have to curve individual numbers because in some sense they don't really matter
I could just decide "60 and above is an A"
stark pine
Since our final grades show a percentage, you sort of need to curve the percentages, especially if it's a sufficiently hard class, because otherwise everyone fails
stark pine
yeah no, here if a 60 shows up on your transcript, it's a 60
meager bronze
yeah, the numbers just don't mean anything for us
the only thing which appears is the letter
and I decide how to translate numbers to letters
stark pine
here the call is really how to translate numbers to other numbers ;P
the upside is that no class is allowed to lower the raw percentage you get
ie if you get a 90 on everything
the teacher can't curve you down to a 75
you get your 90
meager bronze
yeah
I would need to think a while to really be happy with a "pure numbers" system like that... like, to me, the value of that kind of system is that "numbers don't lie" -- the number should represent the percentage of the material that you mastered
but then as soon as there are curves, that goes away
and as soon as people realize "oh there are curves so I can just give tests with an average of 30%" then the numbers just kind of lose their meaning
like, at that point, it's just "we have numbers because we have to" but the numbers don't mean anything
like, you and I talked before about why there are cases where giving a test with a 30% average could make sense, but now I'm wondering if the test should just be graded differently
"only count the best 4 problems" or something
stark pine
well that's effectively what the curve does
but the idea of a curve like that is that you get fewer marginal returns
meager bronze
I guess
stark pine
which makes sense
meager bronze
I just want grades to mean something
and "38.2296+1.01992x−0.00322957x^2" doesn't really mean anything to me
stark pine
That amounted to about "count the best 15 problems on the exam, unless you got over a 90 in which case count the best 20"
meager bronze
I just wish that were the policy then
and that that was set out ahead of time so everyone knew how they would be evaluated
as opposed to retrofitting
to get numbers that look nice
like, the menu prices at a restaurant are displayed when you go in. it's not like everyone eats dinner and then the manager comes out and says "oh so uhh you were all richer than I thought you would be so I'm gonna raise the prices"
stark pine
hmmmm
meager bronze
because at that point, it becomes impossible to compare prices at different restaurants
I want my grades to be evaluating students, not ranking them. I understand that grades get used for ranking, but I still think they should mean something else
stark pine
I guess I'd argue this was a class with like 40 students out of 3000, and this isn't standard for the normal level. This was an exceptional course, and grades honestly didn't mean much; the idea was just to get most people an A, and fail nobody who didn't deserve it
meager bronze
yeah, I understand that this was a special case
and there is value in being able to teach a class where you don't even have to really give numerical grades
like, "as long as you contribute and make progress, you'll get an A"
I think that system can work really well in a class like this one
but at that point, why even have numbers at all? Why not just give every student with an A a 95 at the end?
or -- and this kind of gets into stuff we were talking about the other day -- why not have a class discussion halfway through the term about like, "what do we think a 90 should mean in a class like this? what should a 95 mean? a 100?"
agree on some standards
and then at the end, let students evaluate their own progress
turbid zenith
If everybody's grades are like a 30% that could mean a number of things
stark pine
The rest is very explicit. For example, my combinatorics course, taught by the same prof, has an explicit policy
"Grade is 33% midterm 1, 33% midterm 2, 33% assignments, 33% final, and 1% mock exam. Count the best 2 of the first 3 components. Only your best 8 assignments count"
turbid zenith
Either everybody just did REALLY terribly
turbid zenith
Or I did my job poorly as a teacher
meager bronze
ashura, there is some background here about this test
turbid zenith
But my solution would be to build revisions into the grading system so people can bring their grades up 😛
stark pine
yes there is some BIG background about this test
meager bronze
this was a special class and the exam was designed in a special way to encourage this
basically "everyone picks the one or two topics out of 10 they want to study, but the exam has all 10 topics on it"
turbid zenith
Interesting 😮
stark pine
- this class was entirely inquiry based. Everything we knew about the course had been independently proven. Everyone walked in with different knowledge, and ability to solve different problems
- The exam was so absurdly long that not even the professor could finish it
- roughly half the points were bonus points in case you literally didn't study
meager bronze
I think something like that could be really effective, but then I feel like trying to grade that on a numerical scale just isn't as effective anymore
stark pine
so ie if on your assignments you never figured out how to approximate real numbers with continued fractions, you just wouldn't do that problem on the exam. To make up the points, you might do a random bonus question, or do one of the things you did derive how to solve
meager bronze
actually I am a little distracted too, I need to finish writing a homework assignment for my class and then send some emails
I should step away for a bit!
stray furnace
guys what's 2+2
turbid zenith
Why do you ask @stray furnace
stark pine
@meager bronze I think I can properly answer your question. Most courses have no curve, and instead have a grading policy like "best 8 out of 10 assignments". In those cases, a percentage is (intended) to be a "you understand this percentage of the material", which is imo more reliable and generally better suited to things than letter grades. I would still prefer a breakdown by component on transcript but w/e
As a consequence of this, the policy at every school is that transcripts have numbers. Courses that don't fundamentally work on that "you're supposed to finish the class understanding 100% of the material model" suffer, so they make arbitrary numbers that end up just as bad as number grades, but overall the system works better
since most of the time, it means shit
wispy slate
@stray furnace 6?
meager bronze
@stark pine I see -- thanks!
stark pine
as bad as *LETTER grades sorry
frigid crest
dusky elbow
Factoring is better for ppl learning basic algebra
@wispy slate
Is it, though? If they always factor, then "you only have to cancel one thing" is reinforced in their mind, and I worry they might forget when that's true
wispy slate
Imo yes it is, in this way they see easier where the whole denominator is being multiplied by that, and hence they can cancel
Idk if im explaining
dusky elbow
What you said makes sense. Thanks
turbid zenith
Cancelling is just ... a struggle
It's a constant battle to stop students from thinking they can do it all the time anywhere
hard prism
A lot of times not canceling solves the priblem too
remote pumice
What's up teachers! I would love some feedback on this one if any of you guys have nothing else to do 😊 https://youtu.be/WVJBKT3jn10
ionic dagger
lol 🙂 I'm being purposefullly logically silly. Just because I have nothing to do doesn't mean other people don't want me to do stuff. A bit pretentious, no? 😛
civic tree
? is this in reference to something
obsidian oasis
Do teachers mind if students study ahead of the curriculum?
grand laurel
well, you dont want bored students in your class, because they will distract others
hollow heath
unless you are in high school or younger, I dont see how that is the case
Id actually encourage people to study ahead
thats a very healthy practice
grand laurel
obviously in uni nobody will care
in highschool or lower as a teacher i would just give interested students additional material to study
ofc you cant force students to not study ahead and if it helps with academic success, do it
obsidian oasis
in highschool or lower as a teacher i would just give interested students additional material to study
@grand laurel So would it be recommended that I ask my teacher for additional material?
hollow heath
if they like their teaching, theyll probably enjoy you asking
showing interest is giod
grand laurel
if its a good teacher, yes
when i was in school, some teachers approached me with additional material to study on my own and it was nice
stark pine
@obsidian oasis not a teacher, standard disclaimer. I did exactly that in high school!! Several of my best teachers were accomodating, and the materials they gave me eventually got me to major in math. It definitely can't hurt, but it can help
if you want resources, people on this server can also link plenty :)
next relic
Some do mind, but if their exercises aren't engaging enough, do your own stuff!
Legit how I saved myself from boredom in high school calculus.
Busy doing stuff from ocw or other subjects.
midnight haven
hello everyone
I am currently tutoring at college level, and it's a very cool kid that was diagnosed within the autistic spectrum
he's an undergrad - first year - and, among others, I've been helping him with a very fundamental math class that deals with propositional logic, set theory, functions, etc.
and it's something a little heavy on proofs, which is something he doesn't quite grasp yet.
so I'm looking for any resource that might help us drive him into the right direction
tidal drum
If he struggles with the concept or how to do a proof in general, some basic number theory stuff might be good
midnight haven
i've gone through a few
but he still struggles a bit
something he's clearly not comfortable with is the idea of not being able to prove by example
i.e., prove a general proposition on all the reals with an example of one real
of course, it's quite obvious
and I tell him very often that would be quite impractical
tidal drum
I mean maybe try grilling in some "proofs" of things that are wrong via "proof by example"
midnight haven
of course
tidal drum
Like if you take the statment for all n, n + 1 = 5
this is true for n = 4
but clearly false for a lot more
midnight haven
yeah, that's a way
I don't know -- of course he knows you cannot prove by example
tidal drum
Others might have thoughts, but I'm not sure how you can get someone to realize that a statement for all can't be proven by showing it for some
midnight haven
on the other hand he really is very uncomfortable with abstract proofs
well that n + 1 = 5 example is actually pretty good
very simple and quite obvious
so thank you for that
tidal drum
struggling with abstraction is ¯_(ツ)_/¯
I feel like that's something everyone goes through
tidal drum
but I think trying to foster a skeptic mindset is best
midnight haven
some of these concepts he really has to battle himself
tidal drum
I, and I'm sure most everyone else, looks at our proofs and actively tries to find flaws
midnight haven
no one other than him can drill it into him
tidal drum
so trying to get him to adopt a mindset of "let's try to prove myself wrong"
Because then you're actively looking for counterexample which shoot holes in bad proofs which implicitly rely on some assumption
this at least should end proof by example
From my memory about what classes like that really use in the proofs it's about properties of reals, naturals, rationals, etc.
Going to the more abstract setting is kind of hard because instead of 4, 17, pi, etc. you suddenly have to just let x be an arbitrary real number or something, and I think it's hard to sort of think about what properties of 4, 17, pi are inherent to being a real number, and which are due to it being that exact number
seeing a lot of counterexamples hopefully can get him to start understanding stuff that's true for all real numbers or something
stark pine
@midnight haven have you asked him what he doesn't understand specifically?
"proofs" is very broad
(I'm not a teacher but I'm also a tutor)
midnight haven
yes @stark pine -- he really has a hard time comprehending why he cannot prove by example
which is very odd, I suppose...but it's a struggle for him
stark pine
@midnight haven sorry haha I got that at 3:53 am so I couldn’t respond right away. The n + 1 = 5 example is obvious. Another one I really like is circle subdivision by chords
If you draw the chords, you get the nice pattern of 1, 2, 4, 8, 16 areas
Lovely right?
Then it breaks at 31
Then it stays broken for a bit
Then it goes back to 256 at n = 10 iirc
Then it is never a power of 2 again
That ones a little more abstract imo.for the circle, you could make the claim “adding a point and drawing all the lines doubles the number of areas”, which looks true
stark pine
More generally, the thing that made “for all” click for me was that it means we don’t need to look for an example like this
If we’ve shown for all, we don’t need to check if it’s accurate to a trillion decimal places
We just know
And if we can’t show, then we know to look for a single counterexample, because that’s enough to disprove the statement.
Certainty is just so valuable, because you know you can apply it, you don't just think so
wispy slate
Any general tips for creating lessons plans that anyone has found helpful?
I'm trying to use beamer for mine 
next relic
I recommend thinking about a sequence of lessons rather than a one-off.
What do you expect your students to achieve by the end of unit?
What are the steps you need to do to get there?
If you’re tutoring uni students, this doesn’t apply though. They are prepared for more unstructured lessons, though it might be useful when you’re introducing them to a new topic.
I think it’s better than planning lesson by lesson rigidly because you have more flexibility on when you can give students the content.
vestal quiver
steal material from your colleagues
next relic
^ this also works.
Use it as a starting point, or if you’re happy with it, as the teaching material.
wispy slate
Thanks!
next relic
Which course(s) are you teaching if you don’t mind sharing?
wispy slate
Nothing yet, a friend who is already doing professional tutoring is helping me start out by connecting me to some of his students and clients
It will probably be a lot of high school topics but I would advertise myself and accept offers to hold courses at the undergrad level too
it might not even come to fruition but I want to be ready anyway because this will be something I will try to do likely over winter break as well as next summer
and the whole year after that potentially
wispy slate
@wispy slate honestly, it depends heavily on how many students there are. something that works for 2-10 will not work for 200-500 at the same time.
i feel like this is something that is often not taken into account. that is, classroom sizes
graceful cloak
https://www.pnas.org/content/pnas/116/39/19251.full.pdf
abstract: "We compared students’ self-reported perception of learning with their actual learning under controlled conditions in large enrollment introductory college physics courses taught using 1) active instruction (following best practices in the discipline) and 2) passive instruction (lectures by experienced and highly rated instructors). Both groups received identical class content and handouts, students were randomly assigned, and the instructor made no effort to persuade students of the benefit of either method. Students in active classrooms learned more (as would be expected based on prior research), but their perception of learning, while positive, was lower than that of their peers in passive environments. This suggests that attempts to evaluate instruction based on students’ perceptions of learning could inadvertently promote inferior (passive) pedagogical methods. For instance, a superstar lecturer could create such a positive feeling of learning that students would choose those lectures over active learning. Most importantly, these results suggest that when students experience the increased cognitive effort associated with active learning, they initially take that effort to signify poorer learning. That disconnect may have a detrimental effect on students’ motivation, engagement, and ability to self-regulate their own learning. Although students can, on their own, discover the increased value of being actively engaged during a semester-long course, their learning may be impaired during the initial part of the course. We discuss strategies that instructors can use, early in the semester, to improve students’ response to being actively engaged in the classroom'
faint yarrow
I like to make the homework questions for the course first, before the course starts. Then that basically writes the lesson plans for me.
next relic
^ I think it's a good idea to think about how you can assess performance and learning before planning the lessons.
rotund sandal
What's the right way to approach someone who seriously cannot do very basic math at an advanced age? How do you teach very basic math?
frosty flame
I wonder what inspired this question
rotund sandal
Haha yeah
next relic
The first thing you would do is to identify what the student can and cannot do, then set/teach them appropriate content on cimt. I think these resources are pretty good and demanding.
@rotund sandal
rapid kettle
What's the right way to approach someone who seriously cannot do very basic math at an advanced age? How do you teach very basic math?
ask poly
next relic
Can they perform four operations with natural numbers? integers? rational numbers and decimals? fractions? Can they order decimals in increasing order? Can they order fractions in increasing order?
frosty flame
Any recommendations specifically on working with negative numbers?
They could do 15/5 but not -15/5, as an example
rotund sandal
Yeah it's mainly introducing very simple concepts in an intuitive way quickly, not a long term teaching plan
rapid kettle
Any recommendations specifically on working with negative numbers?
@frosty flame -a = (-1)*a
frosty flame
Thanks. Now I know.
rapid kettle
well i mean that show that they can split it
frosty flame
Yeah, but you tell them that once and show them, and in the next question they forget it again.
rapid kettle
well it is always up to student
i mean one can not get anything disregarding how you present
frosty flame
This is good advice, that I usually try to follow. Thanks.
faint yarrow
non-examples are way underrated
hollow heath
true
wispy slate
@faint yarrow I've heard this as a much more general complaint about math research. That you only ever publish theorems... which tells you which ideas worked out, but leaves little evidence in way of what ideas don't work out.
next relic
if anyone wants to suffer with me, let's read Theory of Instruction by Engelmann and Carnine
rapid kettle
@next relic what about is it
next relic
probably one of the most effective instructional method out there. I can link you some papers to see its effectiveness. Sadly reading line 1 of page 1 already made me yawn, and it's really really hard to get through the book
rapid kettle
sure dm me with that links
wispy slate
@next relic can you send me some links
next relic
frigid crest
turbid zenith
I had an interesting idea today for how I'm gonna do my College Algebra class!
Normally the sequence is Fundamentals → Functions (including Linear) → Quadratics and Polynomial Functions → Exponential and Logarithmic Functions
I'm going to swap Exp/Log with Quad/Poly.
The ideas being:
-
Exp/log have a lot more immediate applications (Interest! Earthquakes! Disease! Population! Sound intensity! Radioactivity! Cooling!) than quad/poly (Um... gravity? And some random curve that happens to model this phenomenon on some tiny interval. And ... uh ... gravity?)
-
Big theme of the course will be "how do you solve equations, overall? I want them to see that there are really two techniques: (A) "Undoing" things, and (B) "splitting" things. This way we can talk about "undoing" exponentials not long at all after covering inverse functions, and then we can spend the second part of the course more getting into the splitting part with quadratics and polynomials (and then even combine the two techniques toward the end).
wise onyx
Quadratics model celestial mechanics
wise onyx
Rather conics
turbid zenith
😂
Okay yeah conics are awesome! But we don't get to see them til next semester's course.
wise onyx
I would have loved someone to point out that polynomials are the simplest algebraic 'things' we can construct out of variables, addition and multiplication
turbid zenith
Yeah I'm definitely planning on doing that when I get to that point
How they're just slightly more general extensions of what you learn to do with whole numbers in elementary school
So that there's like a running thread throughout the math you learn
wise onyx
I dont know if you wanna introduce them to various polynomial rings, like Z[x] vs Q[x]
turbid zenith
Not by name obviously
THis is college algebra, not abstract algebra
But if I can get them to understand "polynomials are actually a lot like whole numbers, just in base x", I've done my job
wise onyx
Like I didnt understand why we as mathematicians collectively cared about polynomials until I got to Galois theory. Before then it just seems like yeah arbitrary functions for no real purpose
Anyway
You can talk about computer graphics and polynomials, ex Splines
strange bronze
polynomials are actually a lot like whole numbers, just in base x
are you going to cover polynomial long division? that helps a lot of the relations "click" for a lot of students
since regular long division is just polynomial long division with x = 10
turbid zenith
Absolutely
Though I'm thinking I might do it a little differently than the textbooks usually do.
Using the area model of multiplication to motivate polynomial division, and turn it into just a problem solving thing!
strange bronze
count the "gee"s
next relic
@turbid zenith what do you think about this sequence?
Fundamentals --> Functions --> Quadratics --> Logs and Exponential --> Poly?
You kinda need quadratics to solve logarithmic equations and inequalities right?
turbid zenith
Not really
next relic
Otherwise you lose a bit of rigour imo.
turbid zenith
I mean if you want to do things like multiple logs, then yes, but I plan to bring that kind of thing back at the end to tie things together
What do you mean by "rigor" in particular here?
next relic
As in you cover examples/problems that stretch students.
Instead of log(x) + log(x+3) = 1, you do something more involved?
I feel like it's a good opportunity to tie quadratics to logs.
Then you can certainly recap it at the end of course.
turbid zenith
Sure, and I plan on doing that still, but at the end
But I kind of raise an eyebrow at the word "rigor" used in math education
Because that usually just ends up meaning "Ummmm make it hard"
next relic
As a non-mathematician, I use the word in a much looser sense I guess?
turbid zenith
I'm not even talking about mathematical rigor
next relic
Oh, I've just written a unit on quadratics. Do you want to take a look at it? I'd love to get your feedback.
turbid zenith
Sure!
turbid zenith
Here's a mini-assignment I just made for my College Algebra class. This is something to be done at home before class.
Just a little bit of snark. XD
turbid zenith
Hi @wispy slate !
burnt vesselBOT
Result:
8.2462112512353
also, prolly better do this before introducing midpoints
like i mean it would be logical development
turbid zenith
Nah. They've actually seen midpoints before if they're taking this class.
This is partially a review.
But the point is to send the message of "hey, in your past math classes, you probably learned this stuff, but probably not in a way that got you to think about what it means"
rapid kettle
ah ok
nice
on 9 i would say: becoz i am not able to draw even in R^1, not speaking about R^10~ 
turbid zenith
Which is why I always get students in my tutoring who when I ask how to find a midpoint they're like "Oh crap, uhhh, there's a formula but I forgot it ... (x1-x2)/2, (y1-y2)/2 or something like that?"
If they thought of it as "Average the x's, average the y's" they'd never forget it.
rapid kettle
yes, considering that they know how to calculate average, but if it is a review this is nice practice
turbid zenith
Really this entire class is review to be honest
If a student is taking "college algebra", chances are they took all of these things in high school but didn't do well
rapid kettle
but maybe would make sense to add some execrices connected to real world since not all are interested in math for the sake of math
turbid zenith
Oh sure we're gonna be doing mathematical modeling pretty heavily in class
I'm specifically telling them up front that on day 1 there's two big reasons we do math:
(1) Model and explain real-world things.
(2) Explain why the rest of math works.
And that some things you learn because of #1, and other things you learn because of #2. Not EVERYTHING has to have a real-world application.
clever jetty
Why not explain with vectors? The gap between the 2 vectors (BA) is (x2-x1,y2-y1). So the gap between midpoint and A is half of that.
turbid zenith
Vectors are gone into in the next class actually!
rapid kettle
tbh i am not very fond of geometric interpretation of vector, for me interpretation of them as just arrays of data is more clear
(because i firstly start to work with vectors actively in programming)
oh ye i recalled one use of vector for real world
document distancing
turbid zenith
So-called "formula" = we've decided to write out a formula for this as if it's one of the long list of formulas you need to remember to do well in math
Like "Midpoint equals x one plus x two over two comma y one plus y two over two"
Instead of, you know, "find the average of the x's and y's and that'll be halfway between because that's kind of exactly what an average does"
wispy slate
oh lol
turbid zenith
The activity I posted? @supple topaz
supple topaz
yes
turbid zenith
Woohoo
Hopefully that means it'll have the intended effect
How long do you think it would take you to complete that?
supple topaz
umm
for me as a student
i would love to have as much as i can
just to emphasize the thinking part haha
cuz for me this would be a new waay for learning math for me
so not worrying that time would hold me thinking and discovering this on my own
would be cool
thats just my bad opinion tho 😄
@turbid zenith
next relic
iirc he assigned it before the class (hence pre-class), so students can spend as much time as they want.
turbid zenith
I haven't actually assigned it yet! I have two weeks still before classes start.
turbid zenith
Anyone have recommendations for a good document camera?
faint yarrow
IPEVO VZ-X is the one I use
winged urchin
Honestly I'd recommend trying a tablet. Ive used both a document camera and a tablet and have found tablets do be a little better.
Your hand doesn't block anything and I found the tablet was less fuzzy/laggy. You do have to get used to not using a normal pen but it's mostly a non-issue
faint yarrow
I have used both a tablet and a document camera and could not more loathe the tablet option
strange bronze
imagine not just screensharing a latex editor and typing as you go
i've legit considered doing this
or rather, i've considered a "hybrid" approach
where i keep a tablet pen handy to circle things or draw arrows or whatever
turbid zenith
@strange bronze ngl I've actually considered that
One of these days I'll invent a good environment to do that live
wise vine
Ok so I need to record my TA lessons. I have a board at home, but I'm concerned about how should I record. I mean, I tried with my phone and a 17sec trial video is 34 Mo... How to record a video with decent quality (at least enough so people can see), without it taking so much space? Please ping me if you answer 🙂
stark pine
@wise vine do you have a tablet/laptop? What several of my professors did was just screencapture and use some writing/drawing app on the tablet. They never seemed to have issues with filesize.
If that doesn't work and you're recording with your phone, have you considered just changing the resolution? I know iphones can do it, and that usually reduces file size significantly
wise vine
I have a laptop, but nothing to write on. I am considering getting a tablet indeed @stark pine
stark pine
@wise vine in case of emergency you could then handwrite, scan, and sort of put it in a powerpoint/image app and just slowly scroll through it as you teach?
I had a prof who did that at the start of the term, until he figured something better out
wise vine
I do still have 12 days, so I think I'll be able to find a solution 🙂 A drawing tablet seems a good idea
turbid zenith
Hey so good luck to those of y'all who are teaching this semester. 🙂
faint yarrow
it's an adventure for sure 😛
wispy slate
@faint yarrow yo I liked your video (didnt finish it though) but I feel like you should add some music to them
faint yarrow
Thanks! maybe with time. As of the second video we have intro/outro music!
wispy slate
Just saying - 3blue1browns videos would be way worse without the music playing in the background (in my opinion), dpn't know why, maybe because witohut it there are moments of awkward silence for few seconds sometimes
faint yarrow
yeah, it is definitely something I would love to be able to include
turbid zenith
I imagine it's to give you time to think
Ruminate, etc
https://twitter.com/solidangles/status/1302637101784150016
Cross-posting from my Twitter. Would love y'all's thoughts.
Twitter
wispy slate
desmos, really? Like if you could somehow make a Mathematica class that would be wayyy better. It's not free though, but maybe there is a chance you can figure that out, there was this calculus/analysis course in my uni that used mathematica and every student got it for free for the course or sth
meager bronze
I think Desmos would be so much better for a college algebra class
mathematica is so much more complicated and you have to jump through so many hoops just to get it installed and started up
desmos is just "open the website and start typing it in"
turbid zenith
Basically exactly what @meager bronze said.
I'm trying to get the barrier to entry to be as low as possible
stark pine
@turbid zenith Setting up a unit circle with all the trig ratios can be really enlightening. Walking your students through the rough process of
- Creating the unit circle
- Creating the triangle within the unit circle (this can actually require some thought - given an angle, how can you tell desmos to draw the triangle with that angle?)
- Labelling the point where the triangle intersects the circle.
It may give some insight to trig!
Something I personally found really enlightening was also just straight up graphing the general form of functions with the coefficients as sliders, and messing with the sliders, to see what each parameter did
if you graph ax^2 + bx + c, you can mess with a, b and c and get a surprising amount of information about what each parameter does. It can be interesting to compare that to graphing a(x - p)(x - q) and messing with p and q
turbid zenith
They're not going to see trig until the next year
But I like that idea of labeling things
And using the various functionality mindfully, "How can we get this to show what we want?"
I'm actually working on a crazy idea now actually
. . . having them look at elliptic curves, lol
Nothing too intense with them, but just like ... get into groups of 3, look at the graph of y² = x³ + ax + b. Make as many observations (using mathematical vocabulary whenever possible) about the shape of the curve as you change the values of a and b.
acoustic ridge
Anyone here reading Francis Su's Book Mathematics for Human Flourishing?
on page 26.
I was tutoring a kid about taylor series and he got really excited when i showed him how sine and cosine got better approximated when we kept adding terms.
I was showing him on Desmos the graph of the polynomial expansion vs the actual graph of the trig functions, and he started all kinds of questions.
it was kinda thrilling.
this particular day sticks out in my memory because it was one of the first times he didn't as a "what is this used for" question, but rather asked about the math behind expansion.
"This looks like magic, how did anyone come up with this? "
"Can you do it for other functions?"
"How good does this get?"
stark pine
"how good does this get" is such an insightful question
wise vine
Just finished my first TA lesson with my drawing tablet (Wacom), using Whiteboard app on Windows
That was... painful, for my hand, I guess
Should get used to it soon
turbid zenith
Oops
turbid zenith
wispy slate
dang I remember making stuff like this up on my own a long time ago to experiment with, was a lot of fun for me
hope they enjoy
turbid zenith
😄
tawdry venture
has anyone else had trouble with bridging the gap in the student's mind between the notation a ≤ x ≤ b and "x is between a and b"
faint yarrow
Hmm that's fascinating. I definitely find they struggle with the |x-a| < R thing that comes up in Calc 2, and in general I'm often surprised by students not finding intervals as intuitive.
strange bronze
honestly i feel like the main way to make this really "set in" is lots of examples
like for a lot of students, this notational symbol soup is overwhelming
they havent developed the "eye" that lets them "chunk together" parts of equations or "block off" stuff and process it one-by-one
if that makes sense
and that mental parsing can only be developed with practice
quasi musk
Yeah I think you're right on the notation thing
They just haven't processed/put it together all the way
kind salmon
i encounter that problem very often aswell, no matter if in university or school. in most cases i can solve it very easy by drawing the numbers on a line (or circle and ellipse for higher grades). i think many people just miss this simple idea.
fathom vessel
In the calculus class I'm teaching right now, I've been trying to write (and pronounce) "a-r < x < a+r", "|x-a| < r", "d(x,a) < r", and "x ∈ B(r,a)" together, when such an expression comes up. Hopefully seeing and hearing those expressions together will drive their equivalence home.
meager bronze
does anyone have any motivation for why students should care about FTC problems where you have to differentiate an integral with functions in the bounds
like, find $\frac{d}{dx} \int_x^{x^2} sin(t) dt$
burnt vesselBOT
Buncho Bananas:
meager bronze
of course plain FTC is important
but these kinds of problems always just felt so contrived to me. I don't know how to explain why this is an important conceptual point
I guess it's a proxy for testing how well students understand the idea of defining an "accumulation function" in the first place? since in order to solve the above problem you have to recognize that as a composition of functions. but I just feel like there are better ways to reinforce that point
(please @ me if you have thoughts!)
polar flint
@meager bronze I'm sorry for the vague answer, but in solving elementary ODEs in beginning ODE courses
in which you don't just "multiply by dx", and you actually use the change of variables theorem explicitly, things like this appear iirc.
I think the example in physics in which people derive the kinetic energy / work theorem uses stuff like this... I guess... not sure though
meager bronze
hmm, thanks, I'll see if I can find anything on that
austere inlet
was also going to name ODEs, for instance if $a:\mathbb R \to \mathbb R$ is continuous then the solution to the IVP $$x'=a(t)x, ~~~~ x(t_0)=x_0$$ is $x(t)=\exp \left(\int_{t_0}^t a(t),\mathrm dt\right)x_0$, but then again that's a very simple application and you aren't really taking a composition of functions in the bounds
burnt vesselBOT
bastian.uwu:
austere inlet
but maybe something along those lines
faint yarrow
@meager bronze I was in grad school when I first saw one of these, and was like "oh taht is cool" so I of course did it in my calc classes for years afterwards. It took a bit for me to realize "because I was in grad school before I learned this" was a very bad reason to do something in intro calculus.
or, more generally, "This subtle point took me years to understand, so therefore I will spend an hour talking about it" is a very appealing notion, but usually wrong
(For what it's worth, I don't think that fundamental theorem of calculus belongs in a calculus course at all. It's an analysis fact, not a calculus one.)
polar flint
@meager bronze If I recall correctly, this is something done in ODE courses:
In a physical spring-mass system, maybe $ma = Kx$ ($x$ being 1D position as a
function of time). $a = x''$. For any time $t \in \bR$, we have
$mx''(t) = Kx(t)$ and then $mx'(t) x''(t) = K x(t) x'(t)$. Which gives,
since that is true for all $t \in \bR$,
$\int_0^t mx'(s) x''(s)ds = \int_0^t K x(s) x'(s)ds$, using
the change of variables theorem, we have,
$\int_{x'(t)}^{x'(t)} mu du= \int_{x(0)}^{x(t)} K u du$. FTC does the
rest of the job.
burnt vesselBOT
phao:
meager bronze
ah interesting
polar flint
Not so sure if that helps you, but examples like this and similar are actually present in ODE courses. I think in advanced ODE courses, "tricks" like these are even more common. There is also Analytical Mechanics, which I image there are more of these things.
meager bronze
also @faint yarrow I think I would disagree that FTC doesn't belong in calculus, but I think I would agree that there doesn't seem to be any good reason to include these kinds of problems
phao your example makes sense -- thanks!
faint yarrow
My list of things to trhow out of calculus is long
meager bronze
I haven't ever actually seen this come up before
lol I also want to throw some things out of calculus courses but FTC is not one of them
faint yarrow
well there is a good FTC and a bad FTC
and talkign about the bad one distracts from the good one
and sure, the bad one is easier to prove and implies the good one, but I don't think proving the good FTC is the goal of a calculus course.
meager bronze
what? the "first FTC" is such a nice theorem and the picture is really all you need
like, the fact that the derivative of an area accumulation function is the function you started with is such a nice geometric fact
faint yarrow
oh, it's a beautiful theorem! but calculus is, to me, not a collection of beautiful theorems.
meager bronze
and you can justify it pretty easily without going into a technical proof
then why don't you want to show it to your students?
austere inlet
(For what it's worth, I don't think that fundamental theorem of calculus belongs in a calculus course at all. It's an analysis fact, not a calculus one.)
@faint yarrow I believe it fully belongs in a calculus course since it's easier to calculate primitives with it. I think students should think of derivation and integration as dual operations rather than try to memorize "integral tables" and "derivative tables" separately. I agree that the proof belongs in an intro analysis course (as is the case for most calc proofs)
polar flint
@meager bronze That kind of example I saw in a somewhat more advanced ODE course.
Which wasn't focused on following those traditional computational rules.
It was a motivational thing before the actual theory in the course.
meager bronze
oh I see
faint yarrow
haha I'm very big on teaching students "math is cool" in calc, but I think that FTC about accumulation functions is the kind of math that people with PhDs in math think is cool, not the kind that calc students do.
polar flint
@meager bronze It has to do with the math that appears in classical mechanics studied from a more mathematical point of view. It's sick interesting!!! hehe
meager bronze
I'm not saying I teach my students as if they're all going to go to grad school in math
faint yarrow
but the in real analysis, the audience is right, and eveyrthing makes sense, and you can give the fake proof and the real proof and help all the theory behind calculus click together.
meager bronze
but I think that FTC is definitely up there in terms of like, historical human mathematical achievement
faint yarrow
but I think a lot of people, and I'm not attributing this to you, have the premise that "even if it makes no sense, it is good to see it earlier, because it will make it make more sense when they see it later," and I have not seen evidence of this being actually true.
meager bronze
I just don't believe your claim that FTC makes no sense
also, "nothing makes sense until it does"
faint yarrow
but I think that understanding why the FTC is cool requires a very nuanced understanding of the difference between a definition and a theorem, which is one of the things I find students most lacking in.
(this is not the students' fault, to be crystal clear. A definition is a very mathematiciany way to think about the universe)
meager bronze
agreed, but I would also say that this is a point where you can sort of advertise the difference to students
I don't think that the idea of "this is an unexpected connection and that is interesting" is completely lost to students
faint yarrow
but if I were to explain it to an analysis student I would say "the point of this is that the relationship between integrals and derivatives is a theorem not a definition"
meager bronze
so you have to explain it a different way to a calculus student, sure
but I think you can ask calculus students what the connection is between slopes and areas before you talk about FTC, then draw some pictures and have them compute a small example like \int_0^x t dt
give them several functions and ask them to sketch out the "area accumulation functions" and look for patterns
like, that can be a great classroom experience, and students have an opportunity to feel like they figured it out themselves
faint yarrow
I should say, I think talking about this is a perfectly reasonable thing to do in a calc class. I think ideally calc syllabi have some flex slots where faculty can choose what they like to aid the story they want to tell.
meager bronze
you don't have to go into the difference between a definition and a theorem or talk about a technical proof (or even tell them what a "proof" is).
I agree with that statement as well. I don't think that calculus classes need to be 100% uniform
okay I should get back to actually writing this lecture
which I paused to do to ask what people thought about FTC
I'm teaching from powerpoitns for the first time
since I'm making asynchronous lecture videos
powerpoitns take more time than I thought... but I've found so far that they're actually pretty good teaching devices
bye all for now
faint yarrow
bye!
meager bronze
and thanks for the suggestions!
polar flint
@meager bronze Care to elaborate on powerpoints being good teaching deviced?
devices* as a student I've always disliked PP lectures.
I've never taught anyone... so I don't really know
meager bronze
some powerpoints are bad
writing a paragraph of a text on a slide and reading it aloud is a terrible way to teach
my slides have generally been like, "key point(s) written on the left, and I work out an example or two on the right as I talk about them"
for exmple here was a slide from my antiderivatives class http://prntscr.com/uka0ex
on the right, I wrote on the PPT and solved the problems on the slide
as I talked about them
students told me they really liked how it was organized
if you just use a PPT as a script for what to say, that's bad
polar flint
Ok... I see.
@meager bronze Bt.w.. in that example I gate, I just realized... I forgot the conclusion.
After using the change of variables, we had
$\int_{x'(t)}^{x'(t)} mu du= \int_{x(0)}^{x(t)} K u du$.
Applying FTC, we then conclude $\frac 1 2 (mx'(t)^2 - mx'(t)^2) =
\int_{x(0)}^{x(t)} K u du$, using more traditional physics notation:
$\frac {mv^2} 2 - \frac {mv_0^2} 2 = W$.
burnt vesselBOT
phao:
polar flint
And thus $W = \Delta K$
burnt vesselBOT
phao:
polar flint
In that integral $\int_{x(0)}^{x(t)} K u du$, $u$ "ranges over positions" so we're talking about $Ku$ as in the spring force here. That $W$ is the work done by that force.
burnt vesselBOT
phao:
polar flint
So this is a particular case of the work/kinetic energy theorem. I don't remember off of the top of my head its generalization, but I remember it not being complicated.
And also being based on extremely similar ideas.
next relic
To me (a non maths major), FTC part 1 is kinda formulaic, and I don't think I'm the only person finds it that way. Maybe my prof didn't do a good job at explaining it, so it ended up a somewhat pointless exercise of plug and chug.
next relic
Differentiating an integral-ish.
We call it FTC part 1 for some reason.
Part 2 is the age old F(b) - F(a).
strange bronze
ah sure, i see what you mean
next relic
I honestly don't remember if my prof actually explained why it works.
Coming from the other end of spectrum, I can understand why some people don't really want to know why it works. Honestly, I don't think it should be evaluated on exams for non-maths majors, but that doesn't mean profs shouldn't teach it.
strange bronze
in the sense of "How do you apply FTC here" or "Why does FTC itself work"?
next relic
Why does FTC itself work.
strange bronze
ah sure
i mean the formal proof is a bit involved
but its certainly possible to justify intuitively
next relic
My very informal understanding is basically adding infinitely many rectangles.
strange bronze
well, that's how a definite integral works
FTC says that the derivative of this gives the function itself
intuitively, the idea is that the "area" of the rectangles you add increases as you "go along" (move right) on the function
and the amount it changes directly depends on how "tall" the rectangles are, i.e. the y value of the function
so the rate of change of this area is the function itself
hence, the derivative of the definite integral is the function itself.
obviously this is far from a formal proof but thats a "good enough" justification for most students, i'd feel
next relic
Yeah, I think for the majority of students that'd be a "good enough" explanation, though personally I'd like a more formal proof. :p
strange bronze
well you can consult http://www.math.ubc.ca/~feldman/m105/intFundThm.pdf if youre interested in that
it leaves out ssome of the nitty-gritty but
if youre familiar with epsilon-deltas (or just the extreme value theorem + squeezing actually)
you can fill in the gaps
a proof that fills in the gaps is available here http://mathonline.wikidot.com/the-fundamental-theorem-of-calculus-part-1
next relic
Oh I see. Thanks!
lucid monolith
I heard an interesting explanation before. You can view integration as multiplying a changing quantity (multiplying a function by dx). A derivative is the opposite, it’s like dividing a changing quantity (dividing a function by dx). It then makes sense that the derivative of an integral is that original function, since division and multiplication are inverses
strange bronze
hmmm
next relic
Just going back to a point I made earlier though, unfortunately I don't think many science students are really interested in the proof because they just wanna take calc 2 to be allowed to take their required courses.
strange bronze
that plays well with the notation but im worried that encourages bad habit interpretations
at least in the multivariable case
the intuition i gave generalizes quite well to the multivariable case making appropriate simplifications
lucid monolith
Yeah, I agree that considering d/dx as literally a fraction isn’t useful, but true notion that a derivative is a generalization of division can be pretty helpful at times. Of course, it’s not the only explanation, and there are some time when it’s more or less helpful than others. Still, I think it’s at least useful to know more than one interpretation of different theorems in calculus
It’s very often true that the analogous process for division with changing quantities is derivation. For example, in the case of speed=distance/time, the analogous for changing quantities in speed distance and time is speed=d(distance)/dt
strange bronze
yeah sure, that makes sense
perhaps its best to say that both ideas are connected with the idea of "change in one thing per change in other thing"
and division is just a particularly well-behaved case of this
lucid monolith
Yeah, I’d agree
The other idea is useful often times, but I wanted to add this since it’s a way that isn’t common to think about integral and derivatives
And it’s useful to have at times
pearl onyx
I'm no teacher, Im just wondering what you guys think. When should students be comfortable with reading and understanding a formal definition? For example, when I tutor people in algebra 2 and define an inverse function as "a function g such that g(f(x)) = x for all x" they don't really get it and prefer to think of it as something else, like the the original function reflected over y = x, or that the x and y coordinates are switched in each function.
I guess I'm asking when it's appropriate to introduce them to formal definitions. Another example would be defining a polynomial as $a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0$.
burnt vesselBOT
abs_0:
wispy slate
I think between last year high school and 1 year university is when it happened in my country (roughly 18 yo) but it depends
Can't give you my opinion because I'm not a teacher, this is my experience and I believe earlier than that very few students are confident enough in using math to actually step up and really get why it is important to know the theory and definitions very well. Most people younger than 18 would just memorize definitions and use mental examples to get "what it actually means"
pearl onyx
I agree a lot with the confidence part, a lot of my classmates are kind of afraid of the technical language when it has words like "such that", "there exists", "for all", etc. The intuitive definition just kind of works for the exam and they don't have to deal with edge cases or anything
turbid zenith
Formal definitions are nice but should not be the first thing students learn.
Get them to understand it intuitively first, then let the formal definition further strengthen that intuition.
An inverse function is a function that "undoes" the original function.
Those mental examples (like the inverse of x+3 is x-3, etc) are what helps students to nail down the concept and are therefore important.
I'd argue though that this needs to be continued much further than high school. I'm not a fan of "definition theorem proof lather rinse repeat".
round robin
personally i prefer reading a concise book with just like defn theorem examples while playing around with examples to gain some intuition as i feel books that try to build intuition tend to have the feeling like 'idk why im reading this i want to learn like something' but in lectures ig it's different there are already books with the definitions and theorems
pearl onyx
I find a balance between both of yours I think. I like when they derive a concept out of “need,” because then I understand the definition and have the intuition. One example is https://youtu.be/tt2DGYOi3hc and Part 2
wispy slate
As a tutor with no control over curriculum and often no control over the expectations students have of the time you will take to teach them, how do you combat "math hate"?
Fundamentally changing students' mindsets is hard even when you have full control of their curriculum and their deadlines, but it feels nearly impossible when you don't
Like I would love to take what the student is learning and reframe it in terms of a very exciting and human problem where they can play with the concepts and feel empowered
But instead there's an expectation from students that I need to be using my time to make them chug through as much memorized content as possible, giving some judgement to where I can give tangential explanations that clarify what some concepts really are
round robin
honestly for high school students (at least here) i find most of them only care about getting good grades instead of actually learning the subject, so perhaps if you have control over the structure of the tests it could be better, like make it less on 'how much practice have i done in the past month' and more like fun kinda, still hard tho:/
i think for some subjects my sch is trying to do more of like 'choose your own adventure' kinda style where you get the freedom to choose from a list/area to learn from yourself and show what you've learnt
austere inlet
I agree, it's way easier if you have total control over the grading and testing. But even then there's little hope in combating "math hate" when your trench is the one class you're teaching or tutoring, since education in general turns out to be a chore of sorts where you're supposed to grind for grades (which are about the only quantitative measure of success and thus important universally for life prospectives).
Best you can hope is not changing the mindset of a whole class but that of a few students. A good chunk of that happens in non-formal interaction, in the form of small math-talk right before/after the sessions or in office time, or in the hallways, etc. Clearly this happens a lot less or not at all due to the pandemic.
i think for some subjects my sch is trying to do more of like 'choose your own adventure' kinda style where you get the freedom to choose from a list/area to learn from yourself and show what you've learnt
@round robin that sounds fun, when I did probability & stats in undergraduate our prof (a really old guy in his 70s) did something like this but a ton more chaotic lol, still we had a good time and even learnt some stuff about Markov chains which wasn't part of the syllabus
round robin
for my music class our test was literally a quiz show
was fun
also less work if you get the students to teach each other the topics jus give them the resources to learn
austere inlet
just give them the resources to learn
@round robin I think it's that + \epsilon, you still need to somehow convince them to teach each other. Last year some prof. at my dept. tried this approach to teach junior algebra (groups, rings, modules) and the students hated it b/c all they did was read the course notes aloud while using the board pretty much, so it was perceived as lazy on the prof's behalf
round robin
oh my class was basically like the students present in class about some era in history and that is the stuff that is graded but ig yea can be perceived as lazy
austere inlet
maybe it's more fun if it's open-ended, like solving a problem or something if it's maths
a presentation about history is usually more open-ended than say presenting a chapter off a math book
next relic
It's a tricky question. By the time they get to you they've had several years of plugging and chugging or even worse, a lack of appreciation for maths because no one teaches you why things work the way they do. You may wanna try giving them some sort of investigations when you think they're ready for it. On your practice tests you might wanna go beyond and above the expectations of the states, because let's be real, the expectations are pretty low.
After a bunch of drills questions you can give them harder ones to force them to think rather than to plug and chug, or you can use a sequence of variation theory questions and ask them what's changed and what hasn't. Might be a step to force them to reflect a bit on what they're doing.
spark flare
I think visual examples of what can be done with the tools they have can be a great way to grab their interest. Cool geometric constructions (at hand or even with a software) might impress them, especially if it's something they can do themselves
I still remember how my hs teacher introduced trigonometry with a geogebra animation of a line that rolled up on itself infinitely to create a circle
frigid crest
@wispy slate Students are human beings, not robots. Give them problems to solve. Let them choose a path of discovery. Along the way, give them advice from your experience. And make sure to give them opportunities to present their findings to you and to their peers.
wispy slate
Thank you all for the advice, but I tutor students who are already enrolled in classes and come to me for help outside those classes. I can have no curriculum of my own or new problems to give, and as much time I do have to sink into making cool animations and the like for students, only rarely will a student coming to me have the time to do study anything besides their professors' homeworks and exam study sheets.
With this little control, what can I employ to help people change how they view math?
strange bronze
unfortunately your options are a bit limited here, and i'd temper asking too much of yourself
don't expect to convince students of some mathematical euphoria or w/e
you'll just let both of you down
you might have to appeal to cynicism a bit
like "I know this is doesn't seem the most useful, but you're being tested on it - and trust me, if you understand this stuff, 'the basics' that you actually need will get a LOT easier and faster"
it feels kinda defeatist/like you're "losing ground", but the reality is that you cant be expected as a tutor to fix institutional and cultural problems with mathematics education
you should, of course, still try and engage students as much as you can
anyway, as for more concrete tactics
one thing you can try and do is engage the student's interests directly
this isn't always possible - i mean, if the student is deadset on becoming a barber or whatever, you're unlikely to convince them of some divine importance of mathematics
but maybe you know they're into sports, so you could try and find a way to draw connections to sports statistics (even if a bit contrived)
or you know they're interested in some scientific field, so you can say "if you make the math easy now, in the future you won't have to worry about getting tripped up by math in your bio/engineering/whatever courses"
"and can just focus on the content itself"
still, dont expect there to be some easy, unambiguous solution
that would be great, but it doesnt exist
wispy slate
Thanks that was very insightful
lucid monolith
Does anyone know of a nice proof for the area of a polygon given by this formula? I'm trying to help a student in geometry understand why this works.
I'm aware of how to prove this using Green theorem, but of course, thats not very helpful to a someonw without multivariable background. If there is not a simple proof (which would be my guess, since proving this for polygons proves a pretty significant subset of greens theorem), is there good intuition I can give that wouldn't require understanding advanced concepts?
wispy slate
drop perpendiculars down from 1,2,3,4 to make 1',2',3',4'
area = [1 2 3 4] = [3' 4' 4 3] + [4' 1' 1 4] - [2' 1' 1 2] - [3' 2' 2 3]
is this nice?
requires knowing how to express the area of a trapezium
lucid monolith
Yeah, that’s is very nice. This would work for all polygons right?
wispy slate
ignoring concave ones what can fail here?
lucid monolith
I guess self intersecting polygons, but original formula doesn’t work for those anyway
Also, I think I’m a little bit confused. Where do you drop the perpendicular from? It seems like the location along the line would effect whether or not you split the shape into trapeziums
wispy slate
i had the x axis from the picture in mind @lucid monolith but the choice of that line is arbitrary anyway
graceful cloak
Thank you all for the advice, but I tutor students who are already enrolled in classes and come to me for help outside those classes. I can have no curriculum of my own or new problems to give, and as much time I do have to sink into making cool animations and the like for students, only rarely will a student coming to me have the time to do study anything besides their professors' homeworks and exam study sheets.
With this little control, what can I employ to help people change how they view math?
@wispy slate If I ever TA calc in person again I might go the "Wow this is Amazing! How cool is this???" Strategy after every example problem.
Because really its amazing stuff, even if its dry for "grown ups." You can actually calculate these things! It took people thousands of years to figure out how to do that! Wow! Amazing!
Wow integration by parts is so useful! Wow u-substitution is so useful! Yessss look at the chain rule coming through for us! etc.
But I guess you have to actually believe it for it to not feel fake. But it shouldn't be that hard to convince yourself that the material is incredibly cool.
lucid monolith
My calc teacher did that. It felt weird for me (I already knew most of the calc before I took the class), but other students enjoyed the more impassioned teaching
quasi musk
Anyone know any good sources on Math Tutoring practices?
I'm doing a Math Ed. course on it (not really by choice) and this was a last minute thing
I have a meeting on Friday and have no clue
austere inlet
never read anything but try https://matheducators.stackexchange.com, perhaps the tag [reference-request] if you specifically want a book rec
next relic
Avoid reinventing the wheel whenever possible.
If you're writing something very similar to, let's say, Stewart, then there's no point doing it.
You're way better off thinking about how you're going to use the textbook to improve your teaching.
quasi musk
Right - I completely understand that point however there seems to be a lack of decent math tutor resources that is geared toward a kind of 'best practices'
And I need to show my professor that I am doing something for the course credit
next relic
You might wanna look at students' misconceptions on some topics of a course you're teaching/you'll teach.
You can also try (given the constraints in lecture time, this is VERY hard to achieve) atomisation, ie breaking a complex topics down to subtopics and teaching each subtopic independently before bringing everything altogether.
Also given the class size you're teaching (I'm assuming you're teaching at university level) it's hard to know if your students are prepared to move on, but there are ways to get around it, such as clickers.
Not a perfect solution, but if, let's say, less than half the class gets the question correctly, it definitely benefits from some clarification.
next relic
Then I think you have enough time to try it out.
civic tree
hmm
wispy slate
Do u guys pronounce it peda go jee(soft g) or peda gah gee(hard g)?
spark flare
Pedo Goji
frosty flame
Do u guys pronounce it peda go jee(soft g) or peda gah gee(hard g)?
The first
light pond
second one
loud dirge
jif
gusty coral
Are there any good websites/articles/other resources that you guys would recommend to read about general pedagogical techniques and practices for teaching advanced math?
turbid zenith
wise onyx
Lmaoo
raven folio
Hello! Does anybody know some good (non-trivial, challenging) problems or examples using the Cantor's intersection theorem for calculus-1 students?
strange bronze
I'd assume they're not comfortable with proofs?
raven folio
Well, it depends on the complexness of the proof. For example, the proof of the Cantor's intersection theorem itself wasn't really hard for them, but sometimes they have problems, yes
burnt vesselBOT
notwhale:
tidal drum
Why would a calculus student learn cantor’s intersection theorem? Don’t you need to define compactness to use that which falls far outside of a typical calculus student’s toolbox
raven folio
Yeah, just for the reals. Still, somewhat interesting could be done. For example, originally I wanted to ask some pair of students to prepare a talk on the completeness of real numbers, in particular, about proving that in axiomatic approach to real numbers we can assume the Cantor's intersection theorem (for segments) and Archimedean property instead of Dedekind's completeness. For additional points. But then I disappointed in this idea cause I couldn't come up with some beautiful proofs.
I thought that it would be great if they learned something in pairs or groups and learned to give a talk properly. And of course, I wanted to give them something that is kind of advanced, but without need in learning some new concepts
tidal drum
this seems so unnecessary for calc students
unless this is like some honors calc class for math ppl
My thoughts on teaching calculus is that it's gonna be taught primarily to engineers who straight up don't need, don't care, and don't benefit from this type of thang
quasi musk
If you're going to do proofs in Calculus (which you should), just keep down to epsilon delta, and maybe the proof of FTC
Where you assume continuity
faint yarrow
My proofs in calculus are things like proving the integral of sec(x)
where they can simultaneously say "I understand every step of that" and "I would never have thought of that in a million years." But proofs are like art, once one person makes them, we have them forever.
that's my calc pitch
faint yarrow
I am also fond of proving the error bound for lefthand Riemann sums of monotone functions, but I don't call it a proof. But Calc II ramps up the amount of proofs I expose the students to, though again, I usually avoid the word "rpoof"
turbid zenith
@quasi musk By when you say "If you're going to do proofs in Calculus (which you should)", who is the "you" you're referring to? The instructor or the students? Or both?
wise onyx
you should only present the most streamlined basic proofs so as not to 1. scare them away, 2. confuse them further, 3. mislead them to thinking this is what they will be tested on
turbid zenith
I'm not so sure that the whole "proving the things work" thing is really what students need in introductory calculus
I think there are better places to work proof in, and proof that more students would be able to wrap their heads around
AP Calculus does well it in, say, explaining why a given function must hit a particular value, using the IVT.
Like how do we know that there exists a value of x such that x^2 = 2?
Well, f(x) = x^2 is a continuous function, and f(1) = 1 while f(2) = 4, so by the IVT there should be a value x in (1,2) for which f(x) = 2.
faint yarrow
I think the idea of proving things is so foreign and so different from how every other discipline thinks about things that it is hard to meaningfully communicate it except in passing. Even science, which some would call our sister discipline, everything is just "This is how it works, get used to it"
So when we ask "Why is the area of a circle pi*r^2" the sort of answer is very different than "Why is the sky blue?" and it takes a lot of soak time for the different types of "Why" to get internalized.
The hardest things to teach are the things we take for granted.
quasi musk
I learned calculus by proofs at community college, Soo it's definitely doable. Lots of students sign up for this classes, you have to be careful that you set up models for students to follow
Like every student has to prove on an exam that if the eigenvalues are distinct then the eigenvectors are linearly independent on the exam
raven folio
Well, I am not sure that what I am teaching really matches Calculus in American uni-s. It is called here "Introduction to Mathematical Analysis" and pretty much all proofs are done. It is a course for freshmans and basically problem solving part more or less is the same as Calculus. There are about 120 students that listen one lecturer who starts from some axiomatic way to introduce real numbers and proves every theorem that will be used in the course. Some lecturers don't like axiomatic approach to real numbers and they start as they have integers, then using construction of equivalence relation and equivalence classes first introduce rational numbers, then sequences of rational numbers and then real numbers as classes of equivalence of fundamental rational sequences. And these 120 freshmans are divided to 6-7 groups and every group has its own instructor (I am one of them) who teaches them to solve problems and understand the material given by lecturer. At the winter session it is expected that every student will be examined both for problem solving and theorem proving. Is it how Calculus done too? It is just strange to me that you say that you need or don't need to do proofs
The program for the first semester consists of real numbers, sequences, the convergence of sequences, the Cauchy criterion, the Bolzano-Weierstrass theorem, topology on R; functions of one variable, limit of a function (both in the sense of Cauchy and Heine), continuos functions and their properties (intermediate value and image of a compact is a compact), derivatives, mean value theorems, the Macloren and the Taylor formulas, L'Hopital's rule; uniform contiuity; curves on plane, smooth curves, Frenet-Serret formulas
Sometimes indefinite integrals included too, but not always
quasi musk
That's usually akin to an Honors calculus course in the US notwhale
raven folio
Well it kinda make a sense
Because about 70% of students here don't like the course and think proofs are useless
But it is usually after winter exam)
After some random dude from chair of mathematics shows the student that he knows nothing and his reply was terrible
But failure rate is not really high
About 10% fails the course, and about 25-30% gets less than B-
austere inlet
Over here (Chile, heavily inspired by European academic culture) that's how most "calculus" courses are, even for engineering/physics majors.
@ applications of Cantor's intersection theorem, I think the only one that makes sense at that level would be proving that every bounded sequence (w.l.o.g. any sequence in a closed, bounded interval) has a convergent subsequence. This can be done by a "lion chasing" argument which your students might come up with more or less naturally if they're reasonably smart and maybe with some guidance.
@ the question of "calculus or analysis first" here's some interesting arguments https://matheducators.stackexchange.com/questions/10620/why-would-you-teach-calculus-before-teaching-real-analysis/10621#10621
raven folio
Well, the Bolzano Weierstrass theorem was proven to students by their lecturer, but maybe using the monotone convergence theorem not Cantor's intersection theorem. But students already gave their talks last Monday. If you are interested I gave 4 topics to 4 pair of students, 2 out of them consisted of one strong student and one not so strong, and 2 other pairs consisted of normal students (in terms of their problem solving abilities shown on lessons).
The topics were
- Cardinality of a set. To prove that rational numbers are countable and the power set of natural numbers is equal to cardinality of real numbers (this was given to 2 normal students, they did well)
burnt vesselBOT
notwhale:
raven folio
- To prove the Cauchy criterion for real numbers if assume that real numbers are classes of equivalence of rational fundamental sequences. (this one was to the pair consisting of one strong and one not so strong student, and the requirement was that the latter will give a talk; the strong student had to make a written report before the talk and both of them had to study the topic together and explain it to each other; report was alright, but the presenter mixed everything and therefore it was kinda very bad; I still praised both of them and give them some points)
and the last topic was to prove that Archimedian property and the Cantor's intersection theorem (A+C) is equivalent Cauchy completeness or Dedekind's completeness or the least-upper-bound property (the task was to prove one of these equivalences, not all and it was necessary that A+C was there; it was too hard for students and they only proved how to get A+C; I was kinda disappointed in this idea and therefore I praised both of the students and give them some points)
We will have same event next month but on new material (functions, mean value theorems, derivatives and so on)
So if you have some ideas what could be given for topics I will be very happy
But it shouldn't be something standard what will most probably proved by their lecturer
I already have in mind one topic
that is Darboux's theorem that states intermediate value property for derivative of some differentiable function. If function is continuously differentiable then it is obvious that its derivative has intermediate value property, but in general not every differentiable function is continuously differentiable. This might be the second question of this topic
I didn't know about it when I was freshman and it was kinda fun when my examinator formulated it and asked to prove it. That exam was very fun
So, all of this is done to prepare them to their exams, usually they have very little experience in giving talks and proving something and really has no knowledge outside their curriculum (and most of the examinators are not really interested in listening to standard curriculum)
spark flare
The program for the first semester consists of real numbers, sequences, the convergence of sequences, the Cauchy criterion, the Bolzano-Weierstrass theorem, topology on R; functions of one variable, limit of a function (both in the sense of Cauchy and Heine), continuos functions and their properties (intermediate value and image of a compact is a compact), derivatives, mean value theorems, the Macloren and the Taylor formulas, L'Hopital's rule; uniform contiuity; curves on plane, smooth curves, Frenet-Serret formulas
@raven folio This seems like a very ambitious program for a single class in one semester, how many hours does this represent?
Hmm wait, maybe not that ambitious 
AGhhh idk
raven folio
it is 2 lectures and 2 seminars a week, so it is 4x85 minutes = 5 hours and 40 minutes every week
and there is about 14-15 weeks
some lecturers are insane and finish the program 2-3 weeks before
and start teaching the next semester program
spark flare
Yeah looks a bit tight imo, tho to be completely fair I'm unsure about it
Do you usually have time to see it all?
raven folio
wym by see it all? you mean attend the lectures and seminars?
spark flare
I mean do you have time to finish the program without going super fast
raven folio
well, there are 2 points of view, when I was a student the first two semesters was hard and you really learn the course and understand everything when you prepare for the exam, right before the exam after you study the subject for 3 days in a row: sleep 5-6 hours, and study the rest of the time (even when eating or doing some other activities). But after the first year you either start to keep up and want more material or just completely ignore the lectures and try to learn the course before the exams
and as an instructor it is quite hard to keep the pace and make everything clear
especially when you want the students to solve the problems during the seminars
of course you can solve them all by yourself
but it is not the best for students cause then they don't learn how to solve the problems
wispy slate
Yo question to all the teachers (especially ones doing lectures) about the online classes. In my uni it's been advised to the profs that they record the lectures so the students can watch them whenever they want, any times they want. There's been quite a big discussion about it, as supposedly many (some say even majority) of profs having lectures, are against recording the lectures and uploading them online. What's your opinion about it? Are you pro or against recording classes? My take is: if they're not recording, someone else will.
raven folio
Godel, why they are against the recordings? (I am totally clueless for both pros and cons)
spark flare
Why are they against it? Recording them is better since it allows students to go back to the lecture and rewatch a part they didn't understand
wispy slate
For example: they might say something that shouldn't be said, or make a mistake, or something interrupting in the background and they dont want those recordings to be online.
spark flare
That's quite weird
wispy slate
Some say that they make students 'lazier', not come to consultting hours, missing live lectures so they can't ask questions live
spark flare
They just have to make lectures mandatory by keeping a register then?
spark flare
Also students seem a bit afraid to interrupt the lecture on live from what I've seen, so they wouldn't ask the questions even if they attended
But it's gotnothing to do with it being recorded
wispy slate
yeah taht's true, although it very often is the case in normal lectures as well.
raven folio
In my uni there were similar story, but not the same. The lectures were recorded way before all corona stuff, just so that students can study the courses by rewatching lectures and most of the courses didn't really have a book to stick to but rather multiple books. The lectures were recorded and when corona started uni decided to cancel all lectures, cause students had recordings of previous years. But uni decided to reduce the payment for lecturers "because lectures are cancelled" and that what pissed everyone
spark flare
Just that students aren't very comfortable with the online format
yeah taht's true, although it very often is the case in normal lectures as well.
Yeah but I think it's exacerbated by the online format, even students that would ask them do it less often
wispy slate
I don't agree. I think there is roughly the same amount of questions
spark flare
Hmm idk, maybe it's just the case in my uni then 
raven folio
I guess the right thing would be just to give some additional payment to the lecturers who are willing to record the lectures, and uni should hold the rights for the recording. That way if something unnecessary was on the lecture it could be edited out and the recordings should be uploaded to a uni's private website. All recordings then could be used for next generations as additional material
wispy slate
Of course, but that requires giving out money...
raven folio
then if I were the lecturer I would record lectures myself and told it to students, so that they wouldn't record. I would edit the lectures and upload them to YouTube but with the "access by link" only. Of course it would require some time, but I guess everything could be done within 2-3 weeks.
every video could be hided at any moment because the channel is mine and also there is a possibility to hold the rights for the recordings
also doing recordings is actually useful for lecturers
cause they can use recording in the future
spark flare
Tbh I'm confused, at my uni we use Teams and only the lecturer (the one hosting the channel) can record it. Then the recording remains on the conversation of the class
Students can download it ofc
Unis can't use recordings for future classes tho that way
Which is good
(prevents the uni from using recordings and paying teachers less, tho tbh teachers are paid by the government here, not the uni so it's not even an issue)
viscid oasis
What is the easiest way to get animation of transformation of grid lines(of R^2 or Complex Plane) under a function?
wispy slate
not the answer you want but a really hard but awesome way is probably manim
hard as in time commitment lmao
Easiest might be lists in desmos and finding a single constant to homotope things around with
lethal leaf
Oh uhhhhh I swear I did something like this
in Mathmatica for calc 3
this is from my school's provided notes
@viscid oasis does this work? is this what you're looking for?
viscid oasis
not the answer you want but a really hard but awesome way is probably manim
@wispy slate Yes manim is beautiful, but i don't have time to learn it currently so.
does this work? is this what you're looking for?
@lethal leaf Sorry I am noob currently. Will try mathematica for sure.
raven folio
@lethal leaf are these notes available for everyone? If that's the case I would appreciate if you shared the link
lethal leaf
@viscid oasis well I was more asking if that's what you were asking to animate. I hope that helped
@raven folio I'll see if I can download the notes somehow
the issue is that I can't do any more cell evaluations in the software cause the access code expired
so I have the code but can't evaluate any of it (yay for college software)
raven folio
@lethal leaf if it is not too much trouble
lethal leaf
it's not that it's too much trouble
it's just that sadly I can't run any code anymore
cause the software gets locked a certain amount of time after the class ended
which is dumb
but it means I can't run the code and then get the resulting graph/output
spark flare
When you're a grad student but the tutoring company wants to have you pass a hs test to see if you master the domain 
(I have a 2 hours test on Friday)
wispy slate
"alright, now show us you can use FOIL"
spark flare
wtf is foil
spark flare
I've not done maths advanced enough
wispy slate
this is phd level
clearly you haven't mastered the domain
ok i shouldn't shitpost too hard in the adv channels
spark flare
Tbh they'll prob ask me to explain to them something like that as if they were a student
I had a similar question two years ago when I applied to another tutoring company (however they wanted tutors with a bs to teach high schooler so they didn't keep me: "oh wow you did great, so btw you're a masters student right ?
-Nope
-Well too bad")
lethal leaf
Tutoring companies are a scam imo. When I did tutoring I made more money just by advertising on FB and stuff
spark flare
Hmm, the problem is that I've been on a few tutoring websites for more than 1 year, maybe 2, but haven't gotten much answers yet
spark flare
Eh their test is trivial
I made only one mistake because I misclicked. Expecting the same kind of thing this time too
However I hope they'll let me tutor college students as well
lethal leaf
Hmm, the problem is that I've been on a few tutoring websites for more than 1 year, maybe 2, but haven't gotten much answers yet
@spark flare
wdym tutoring websites
And haven't gotten answers
Like you haven't gotten customers after advertising yourself on a tutoring website?
spark flare
Yep
Those that are on the top pages are tutors with lors of tutoring hours already
So it's hard to be even seen
lethal leaf
Yea that's not a good way to advertise IMO
I got hella customers by advertising on neighborhood FB pages around me
What I did was this
My background (HS senior, accepted into XYZ college for major)
Subjects I'd teach
Contact info
Price
Giant flex list of grades and test scores cause parents love that shit
It feels scummy listing all the math classes I did well in + all my standardized test scores but it fucking works
Also FB is best cause that's where the parents are
And it's parents who are looking and paying for tutoring, not students most of the time (we're talking MS-HS students
If you can tutor for SAT/ACT and AP Tests that's the top thing you should advertise
spark flare
hmm
I'll look into it
Gotta find a fb group where I can advertize myself tho
Maybe just my uni group 
quasi musk
@spark flare have you tried Wyzant?
It can be a big help in getting lots of students quickly
quasi musk
Oh I see. That is a different scenario
Do parents get in a frenzy to get tutors when the BAC is coming up?
spark flare
Probably
quasi musk
I know in the US anytime the SAT/ACT or AP tests come up, there's a lot more tutoring options
quasi musk
Right
lethal leaf
I know in the US anytime the SAT/ACT or AP tests come up, there's a lot more tutoring options
@quasi musk
AP season is crazy for tutoring. It's nice cause calc AB/BC are some of the most common AP tests so customers aplenty
quasi musk
Yeah it is, it's like 3 months of unlimited Tutoring hours
turbid zenith
Can confirm
wispy slate
@wispy slate 
stark pine
People who hire TA's - when you interview, what qualities do you look for? I'm interviewing for a 40 hour/week CS TA type job, and I'm wondering what aspects of my personality I should highlight/emphasize
(idk if this is an appropriate channel but I feel like here is where the teachers are)
faint yarrow
Being pleasant to talk to, caring about teaching, being enthusiastic. I flat out expect that most TAs are going to start out being... not good. I cringe thinking about my first semester TAing. But doing it is how you become good. So I'm looking for someone who has the drive to do that. And particularly someone who views it as an opportunity, and not a job. @stark pine
(also for the love of god, show up to the meeting on time, don't reschedule, respond to emails promptly, use sentences, don't call the Mrs., etc etc.)
stark pine
@faint yarrow Thank you so much! I really do want to be a TA, I want to teach, and yeah, I'm assuming that I won't be good right away lol
faint yarrow
also, I don't know if you get a say, but TAing for bad professors is much more rewarding than TAing for good ones.
quasi musk
LOL
austere inlet
also, I don't know if you get a say, but TAing for bad professors is much more rewarding than TAing for good ones.
can confirm
stark pine
I don't think TA's are assigned to individual professors
because all professors have the same assignments/lecture modules/etc., and your tutorial section isn't based on your professor
austere inlet
eh depends on the institution, in mine everyone does each course their own way with their own material and TAs are personally selected by them (though there's an application process)
stark pine
sorry I specifically meant here, for this department
meager bronze
@faint yarrow when I was a TA, students complained to me about the bad prof. Now that I'm a prof, students complain to me about the bad TAs
hahaha
faint yarrow
That's the life @meager bronze
wispy slate
you both have the rainbow background so i thought buncho was replying to themselves for a moment 
meager bronze
hahahaha
tidal drum
I thought zeta was talking to himself
turbid zenith
Let the un-FOIL-ing begin.
(We're introducing polynomials in College Algebra today, and my goal is to get them to see how the area model can be used to explain basically all of polynomial multiplication, factorization, and division.)
And I'm going to try to make the "polynomials are a whoooole lot like integers" thing explicit.
faint yarrow
That seems ambitious!
tall hedge
faint yarrow
haha what percentage of math dissertations do we think contain no errors. I bet it is less than 20%.
clever jetty
How significant is that error?
next relic
@turbid zenith An alternative way that I've heard about teaching polynomial multiplication is by breaking the brackets and treating as the sum of the products of single bracket multiplication. Let's say,
(x-3)(2x-1) = x(2x-1) - 3(2x-1) = 2x^2 - x - 6x + 3 = 2x^2 - 7x + 3.
How do you find this method? Is it easier for students to understand?
turbid zenith
I mean it's okay, and it's the same as what we're doing
But I would still rather do it using this area model because I'm trying to get my students away from thinking that the distributive property has to do with parenthesis/brackets but rather breaking down a problem
winged urchin
I'm a little at odds with the area method personally
I mean, it really is the distributive property that lets you decompose factored terms
Is it not?
And personally, in tutoring elementary kids, whenever they have questions about the area squares for multiplication they get lost. When they already understand how to multiply. It just seems like some unnecessary view of multiplication that confuses what they already understand
I am interested in any counter arguments though. Of course I want to be the best educator I can be
turbid zenith
Yeah it is just the distributive property. The area model is just a way of visualizing it.
I think the big counter argument comes from what it means to "understand" multiplication. Does it mean they can do it without making any mistakes?
winged urchin
Right, there are different levels of understanding.
I'd say making no mistakes is not necessarily the goal. I often tell my students that they will always make mistakes but they should endeavour to catch their mistakes
Because if we solely want to make no mistakes then you use calculators
(Of course even there you have potential user error)
austere inlet
I'm not entirely sure if it's different levels in this case, just different ways to understand. I know some people like the geometric way of understanding e.g. (a+b)^2 or (a+b)^3 and myself I always tend to understand things better with some visualisation. Though in the case of multiplication it's almost useless to understand the geometry and not be able to do the algebraic computations quickly
winged urchin
Right, visualization is very useful
And to be fair, I do something akin to the area method when I help students
But that's mainly with double digit times single digit numbers
Like, if I have 37x6 I'll hold my left hand out and say here is 30 (really just think 3 and add a 0 later)
And then hold my right hand out and say here is 7
Then I ask, 3 times 6... with an extra zero. And then 7 times 6... then put them together
Which is in essence, the area method
But I never decompose it like that for larger numbers
If they have to do 364x934 then I'll generally tell them to either use the
364
x934
Or just use a calculator. However, I try to get them to think about whether their answer makes sense
Often I think the 'number sense' there comes from getting them to think of close numbers that are much easier to calculate. Like 900*300
If they can do that and get 270000 then they can look at their answer and sometimes tell when it's absolutely wrong.
Such as if they forget the extra 0s. And get something far too small to be near the right answer
Hmmm.. I guess though if we're talking about polynomials... you don't have the tabular method of multiplication. For instance, doing
984
x329
Although you very well could have that...
Hmmm.. perhaps for polynomials the area method makes more sense if we want to be accurate...
austere inlet
you could do a tabular method for polys. as well; numbers are just polynomials over 10 in some sense
984=9 * 10^2 + 8 * 10 + 4
winged urchin
Ya, you could I suppose ya
You know, in talking this over.. I'm a little more partial to the area method for polynomial multiplication but only as a starting point to get them to understand which things need to multiply which
winged urchin
Though I would question whether you need that much space to accurately do it..
I mean... is it so much worse for their understanding to think of (3x+6y+9)(4x-9y-12)
And just like... mentally draw boxes around each term and just say that every term has to multiply with every other term in the other bracket
austere inlet
yeah and the point of the tabular method is that you can line up the 1s, the 10s, the 100s, etc. with each other. Polynomials like (3x+6y+9) don't go too well with that reasoning so it's probably better to just distribute
winged urchin
Eh you can still use a tabular method even then no?
For instance....
It's not as cleannnn since you don't just add down columns necessarily and is much more like distribution
But again, at the heart of it, it'll all come down to distribution. I suppose we're seeking the most effective way of getting a student to 'get' expanding
Though I doubt there is one supreme way to expand over all others. In fact I believe that showing students multiple ways of approaching a particular problem helps them more than sticking to one single method
winged urchin
Oddly enough, I had a session with a student recently involving polynomial multiplication. A couple things I want to leave here that I noticed.
They were able to check their answer by me asking questions like... "What would give you a x^2 (or xy oy y^2 or y, etc...) term?" They could then do the multiplication and sometimes addition relatively quickly in their head to verify their answer.
I was able to hit on a personal pet peeve involving the use of the word "simplify". Their book gave something like (3x-5y+3)(2x-3y+5) and then part a instructs the student to expand it and simplify. Part b then asks the student to evaluate the 'simplified' expression at x = -2 and y = 1. I usually like to talk about how in higher level mathematics you have a purpose for why you're doing something, in that you don't just expand or factor because you can and this is a prime example. In this case if our goal is to evaluate at (-2,1) then, as the student found as I made them calculate it both ways, the factored form is simpler.
next relic
Yeah, makes perfect sense. For higher order polynomials more often than not we want to factorise, not expand.
wispy slate
Wow great timing, I was supposed to prepare similiar example for tutoring. I like it and I don't think its too long, 15 mins sounds perfect. You wouldn't mind me using your video?
wispy slate
Just to steal the problem or send the video to the kid I tutor
turbid zenith
That's fine. 🙂
turbid zenith
next relic
Oh wow, fascinating!
stark pine
@turbid zenith When you get answers to this question, I'd be curious about the results. Are you expecting anything insightful/mostly correct, or do you think that most students will just sort of guess/bullshit?
turbid zenith
Mostly it's just to get students to reflect.
I'm sure some will guess, but it'll make them think a bit regardless.
And that way in class they'll be primed to be introduced to complex numbers.
novel kraken
this is fun, you could also use this way of thinking but take a left turn at the rationals and think about how to get the p-adics
although from this perspective, you wouldn't be lining any numbers up on a number line at all, even up to the point of making rationals, because it's the absolute value you put on the rationals that gives it the number line structure for the completion to get reals or gives you the fractally "number tree" structure to get a p-adic field
austere inlet
how can i have 2+2i apples bruh
you won't, but one of the smart kids will.
clever jetty
You can say 2+2i represents 2 apples of kind 1 and 2 apples of kind 2
wispy slate
a debt of 4 apples
stark pine
I think the much more helpful way to think of complex multiplication is rotation
winged urchin
Lol, ya for sure. I think in fact that rotation idea is why it couldn't be used to describe two different kinds of apples
clever jetty
I mean,you could see i not as the imaginary number but just as a formal symbol
stark pine
"This is an apple and this is a twisted apple. 2i * 2i is because if you rotate the apples then you now have an apple debt"
that doesn't quite work
winged urchin
Yeah haha
round robin
people are generally comfortable with multiplying polynomials so why not intro complex numbers as the same thing with x^2=-1
turbid zenith
Random thought. Have any of y'all had math profs who give PowerPoint/Beamer notes and color code theorems, main ideas, etc?
wise onyx
I've been to a talk with that
turbid zenith
Do you remember how they used their colors off the top of your head?
I'm making a template for making new videos to teach with
At the very least I imagine it would be good to have one thing for definitions, one thing for theorems/lemmas/corollaries, one thing for key ideas, one thing for examples.
winged urchin
Oh ya, books do that too
And some books even have all the theorems or definitions or whatevers collected at the end of each chapter
I've often mulled over the idea of either giving or encouraging them to make a toolbox of theorems and definitions or graph shapes
And that could be a neat way to do it. Just tell them everyone that was in red were theorems you should remember. Everything that was in blue are important graphs you should remember the shapes of. etc, etc...
next relic
It might distract the learners and split their attention though.
strange bronze
i colour definitions blue, major theorems red, lemmas/corollaries/minor propositions orange-yellow, examples green, side remarks grey
in my lecture notes at least
vagrant meadow
when making written resources for students, how do you know how many steps to show?
i tried making a short write up for a linear algebra class and sent it to a few of the students for feedback to see where it lost them. it was usually like equation 1 or 2 out of 14, which were just the basic results/theorems from previous sections/weeks the main topic was based on. but then the more explanation and steps i add, the more bloated and clunky it becomes. i cant find a good middle ground and it makes me feel out of touch with the students 😦
austere inlet
Oh I struggled with this a bit when distributing exercise solutions as a TA. I think the optimum is showing the steps of the solution while omitting small details such as the ones involving in, say, solving for a specific term in an equation. These are things that the student should be able to fill in.
however if they're getting lost in
the basic results/theorems from previous sections/weeks
perhaps it's a good idea to write a short reminder, or reference the specific result if you're writing a big document with course notes. It's good to emphasize the logical cadence of the course as a whole.
faint yarrow
How much to skip really depends on the level of the students. But generally skipping less is usually a good idea. It is easy for a student to get totally lost because they don't see where a term came from
And students in calculus classes don't usually have the skill of ignoring small details
quasi musk
Even when the students are strong/confident I like to show lots of steps
Because it's easy to get lulled into a false sense of security
Or when the students take the test they might say but I showed as much work as you did
vagrant meadow
Right those are good points.
But it just feels weird. this single equation caused confusion for some students (v1 and v2 are eigenvectors of A)
https://puu.sh/GKuMm/a8b54ed235.png
And then immediately going from that to this also caused confusion
https://puu.sh/GKuOG/dea9060db7.png
This is in the context of diagonalization and at this point in the class eigenvectors arent new
And it seems like the confusion comes less from the eigenvectors themselves and more from just a lack of understanding of basic properties of matrix multiplication
Which is alarming considering how far into the semester we are and the fact that these students are no dummies. They worked at the same tutoring center as me until recently.
So I'm not sure what to do exactly with such a lack of foundational knowledge this far in i guess
novel kraken
if they're having trouble with (1) I don't think you should blame yourself, that just seems lazy on their part, simply distributing and thinking for a second would make it clear.
strange bronze
i kinda feel like "core" issue here might not be the step-skipping
but rather, students not understanding the fundamental importance of definitions
since these identities should be very clear looking at the definitions, but many students - especially in these sorts of intro courses - think of mathematics as applying chains of "rules" or "algorithms" or "formulae"
and this sort of reasoning-from-definitions is new to them conceptually
@vagrant meadow pinging since it's been a few days since you posted that
vagrant meadow
okay that does make me feel a bit better
yeah there does seem to be a big lack of understanding for a lot of the important definitions. when prompted on what the rank of a matrix tells us, one student i tutor pretty regularly responded "the number of leading 1s". technically correct, but thats just a way to compute the rank, not what it means fundamentally. definitely a case of thinking of things algorithmically and not understanding them conceptually :/
winged urchin
Curious, what would you say is an answer that shows better understanding?
The number of linearly independent columns? The dimension of the column or row space?
winged urchin
I would perhaps suggest that our target for their understanding in this way is not just one statement. It's more how it connects with other concepts in linear algebra and how to use that for clever solutions to problems
But then I might put forward that this is less a product of how you choose to present the material and more a product of how much time the student themselves spends on the subject
clever jetty
In that case,The number of nonzero rows in the rref
winged urchin
Ah but that is equivalent to the statement nix gave actually when you think of it
clever jetty
Yes
winged urchin
Again I think any one statement is not sufficient to show the kind of understanding nix is talking about imo
clever jetty
True,It depends on what you want to use linear independence for
winged urchin
Knowing rank + nullity = number of columns of the matrix. That the nullity tells you about the dimension of the space of solutions one might have to a given system. So forth and so forth
The ideal understanding, I would suggest, is a sort of mingled combination of all these sorts of statements
It's a web of familiar ideas that all kind of talk about the same concepts in different perspectives in order to give students the ability to twist and turn a given problem in different directions and configurations until they find one that works