stark pine
@strange bronze how about "show that if a function f(x) has a horizontal asymptote, the derivative must approach 0"
this isn't true, but it looks true. You could prove it for polynomials or something (or give a counterexample), but looking at it, an average cheater would probably go "yeah that seems right" then copy a chegg proof
stark pine
Rational functions*
stark pine
Yeah oops the polynomial proof is a little easy
But that sort of question?
Alternatively, make the question “prove or disprove”
Make the chegg proof a proof of the wrong statement
strange bronze
yeah i've heard that tactic being used before
and it works although, to avoid false positives, you should have an "identifying step" in chegg's false proof
like
maybe your fake solution constructs a special function or whatever
that doesnt actually prove anything
so if a student constructs the same function
you've found a cheater
lots of proofs using, say, MVT end up creating a "dummy function" dependant on what you actually care about
so this isnt exactly an uncommon technique
turbid zenith
So ... worth reading
My number theory final had some of these questions on it, and I really appreciated it
kind salmon
i always appreciate cognitivism over behaviourism...
but who shall read all that text in what time to grant fair grades in the end?
tawdry venture
would this be a good analysis exam question?
Let n ≥ 1 and let f: [a,b] -> R^n be a continuous function which is differentiable on (a,b). Is it true that there exists a point c ∈ (a,b) such that f(b) - f(a) = (b-a)f'(c)? If yes, provide a proof; if not, provide a counterexample. If the answer depends on the value of n, state the values of n for which the statement is true, and prove it for those values while providing counterexamples for the rest.
(answer: ||yes if and only if n = 1||)
strange bronze
i mean, every student should be able to recognize that the n = 1 case is immediate, right?
since its just MVT
so i think it'd be more direct to phrase it as
"We know it's true for n = 1; is it true for any larger n? If so, prove it, and if not, construct counterexamples."
or something of the sort
"construct counterexamples" might not be the best phrasing
grand laurel
but isn't n=1 a nice way to give (little) partial credit
i'd phrase it like "for which n is this true"
is it not obvious that students should prove their claims
tawdry venture
"any unjustified claim will set your score for this question to -1"
grand laurel
end every sentence with "which follows directly from ZFC and the definitions"
tawdry venture
grand laurel
i had students write stuff like "which is true if we accept the axiom of choice" for stuff which does not require choice at all
tawdry venture
grand laurel
peak comedy
winged urchin
I don't know why but that question got me thinking first about the answer, cause at first I didn't see your answer, and then why. It's been a little while since I've been at university and mostly focused on lower level maths I guess. But anyways. Was wondering if this is the right understanding in essence...
So if you imagine the domain as time and the range as some space or whatev. Then for n=1 your space curve always moves in the same direction as the secant line between f(a) and f(b). And if it didn't go at least go as fast as the average velocity (speed really in this case) then it simply wouldn't make it to f(b)
For n=2, your curve necessarily must go in the same direction as the average velocity at least once (might be incorrect to use that term but I hope the idea is clear) but could just go way slower at that part of the curve and speed up elsewhere where it isn't in the same direction
For n=3 and above your curve doesn't even have to go in the same direction as the average velocity at any time
kind salmon
if you ever slow down or accelerate you must meet the overall average speed at some point... if you aim for that
winged urchin
But you can accelerate where your velocity isn't in the same direction as the secant line between f(a) and f(b) to compensate for any slow downs when you are in the same direction
I suppose
It's a little late so sorry if I misunderstood your comment. Anywho. I just wanted to ramble off my musings on it and see if it's the right idea
I'm sure a more... Technical proof was desired for such an exam question in practice. I suppose. Anywho.
kind salmon
the idea works, you just need to llok at the gradient instead of partial derivatives for n > 1
and i appreciate the 3d drawing on that very much 🙂
the function mentioned is diffentiable, your case can only be constructed with a continuous but non-differntiable one
strange bronze
i had students write stuff like "which is true if we accept the axiom of choice" for stuff which does not require choice at all
@grand laurel well they never said "iff"
feral vector
What are some of the thoughts on take home exams? Like for exams, you are given 1 week or 1 day.
winged urchin
Hey @kind salmon. Just wondering. Why wouldn't my 3d picture be differentiable? It might be a drawing issue but I tried to essentially draw a coil that spiraled out and then back in towards the center of the coil. I think I can imagine a tangent vector to the curve that changes smoothly without jerking suddenly to another direction or instantly changing speed. Just curious
For the take home exam question personally I do like them. Of course one could point to the potential for students to work together or get solutions online (cheat) but the former should probably be built into the take home exam to begin with and the latter.. well there's already been recent discussion here on cheating and I don't really want to get into it
Some students get the jitters when they have to do it in a fixed place and for a constrained length of time
And other students might like time to reflect on what they've written and possibly change or completely rethink their thought process for a question
feral vector
Yeah personally I am of the view that take-home exams best prepare one for their mathematical career since they would not only learn to use their own knowledge, but also learn to effectively search through internet and understand arguments coming from different writing styles.
winged urchin
The one cavaet I would say is ideally I'd like students to really give it a fair shake without looking up anything
It might be my subjectivity but I really think that someone working on something needs to spend time using only what's in their head and really build their familiarity with the problem first
Maybe looking up definitions or such to unfold any mathematical lingo to make sure they have the right perspective
But I would worry that a student would immediately start looking for information more close to a solution without understanding I guess...
Sorta similar to students who do something for one problem because they saw it down in another seemingly similar problem but the approach is completely mismatched for their problem at hand
feral vector
Yeah but if we require that the students have to be clear as much as possible then even if they immediately start looking for information, if they are able to write down their solution as to show no ambiguity whatsoever then I guess their own ability passes the mark. Because even on the internet you will not find completely elaborated answers, just people who will give the main idea of the proof or something.
It might be my subjectivity but I really think that someone working on something needs to spend time using only what's in their head and really build their familiarity with the problem first
@winged urchin True but that shouldn't be the goal of an examination, no? I would more say that this kind of thing is more suited for weekly assignments rather than examinations.
winged urchin
Perhaps you are correct. If we understand examinations as a tool to gauge who has grasped the material. I go on a bit of train of thought sometimes and lose the plot
It occurs to me that sometimes for me, final exams (as a student) are a time for final reflection on all the tools I learned and reinforcement of the skills involved
And as I think about it more... I think that effect occurred me for me after take home exams rather than in class exams
turbid zenith
This is why I still want to try the idea of having an in-class portion that's mostly definitions, examples, counterexamples, etc
And then the out-of-class portion is the problems that you have to chew on
By the way, any of y'all have suggestions for how to do informal grouping via Zoom? I've got that interview/guest lesson next week at a college and I want to use student interaction however I can.
But, say, breakout rooms are a bit overkill for a quick Think-Pair-Share.
kind salmon
But you can accelerate where your velocity isn't in the same direction as the secant line between f(a) and f(b) to compensate for any slow downs when you are in the same direction
@winged urchin i meant this idea, if you do a "harsh" turn at some point you could overcome the initial problem - but then the function is not differntiable at that point anymore.
winged urchin
Ah, I think I miscommunicated then. But I think I can ask a question to rectify my potential misunderstanding
So
If Ann's statement is only true for n=1
What do you think of the following
Given the same assumptions in Ann's original question
Is it a guarantee that there exists a C>0 such that
f(b)-f(a)=C(b-a)f'(c)
as long as n = 1 or n = 2?
But once you go n = 3 or above then there may not be a C satisfying that
kind salmon
i see a c in any dimension, as long as we talk about the directional derivative for n > 1
winged urchin
Ah maybe we have a confusion of terms then
In my 3d picture, I'd argue that the space curve never moves in a way that is parallel to the x-axis
yet f(b)-f(a) would be parallel to the x-axis
The tangent vectors always have some y- or z- component to them along the curve
kind salmon
screw me 😄
winged urchin
How appropriate to the picture haha, =p
But at least with n=2 I'm pretty sure you have to go in the same direction as f(b)-f(a) at least once
Ah dimensionality...
ionic dagger
What's the definition of continuous and differentiable?
wispy slate
continous is when the fnction is differentiable at every point on the function and differentiable means when a point has a derivative
strange bronze
???
severe kelp
rent free you need to stop talking about things you dont know and misleading ppl who ask for help in this server lmao
wispy slate
being differentiable at a point and continuous at a point are not equivalent statements
differentiable at a point implies continuous at the point but the converse is not necessarily true
wrong channel anyway
desert hamlet
Weierstrass shook rn
kind salmon
wrong channel. try unoccupied math help
feral vector
Now that we have renamed the foundation of math channel, it's time to rename this channel as well so as to reduce the questions not meant for this channel.
rigid thistle
the name is pretty appropriate, it's just people that don't know how to read
winged urchin
I find typically the only people posting here mistakingly are students looking for help
I'd argue, mostly panicked or stressed students looking for help
turbid zenith
I was actually just thinking that yeah
winged urchin
So if you want to change the name to stop that, make it look scary or unhelpful to a panicked student
turbid zenith
I would suggest #math-pedagogy
winged urchin
yeah that's obtuse and scary enough haha
turbid zenith
But still very helpful to anyone who knows what it's for
weary ferry
changed!
turbid zenith
Thank you!
ionic dagger
woog is the mouse
winged urchin
The mouse with the plan
unreal ledge
The mouse with the house
wispy slate
wearing a blouse
novel tendon
conversing with gauss
wispy slate
talking to klaus
ionic dagger
his Net worth is Gross?
tawdry venture
why exactly did this channel get renamed again
cedar fable
The mouse in the house with two rooms
loud dirge
while studying the work of Weierstrass
sage crag
Milking the cows
modern solstice
What happened to teacher's lounge
flat sonnet
Hey guys—I’m a bit new to this server (long-time lurker), and I’ve just decided to take on an Instructional Teaching Assistant position at my school (I’m a student). The position is for Algebra I, which is a level of math that I haven’t helped teach before, I have helped teach Algebra II though. In my lessons for that class—I tried to truly emphasize the beauty of mathematics, why things work, and how math isn’t just an esoteric field where only elite students can excel. I did this with also teaching them the curricular material. My question is, what are some theorems/proofs/formulas, in Algebra I that I can outline in the same manner that I have done prior? I was initially going to be helping with Trigonometry, but those plans fell through as they did not work with my schedule. Thoughts? I figured that as math educators, you guys would be a great resource for this.
rigid thistle
do you have some exemplary theorems/proofs/formulas? im not sure what you're asking precisely
turbid zenith
@flat sonnet - if your students already know the Pythagorean Theorem, then the Distance "Formula" is just the Pythagorean Theorem again
I like how that shows a connection between thinking of things visually and thinking of them algebraically
As much as possible, you want to draw that connection. That algebra and geometry aren't completely disjoint, but rather they inform each other. The idea that you can use algebra to solve geometry problems, and use geometry to solve algebra problems.
Also, if your curriculum has you covering exponent laws in Algebra I, it's never a bad time to look at why those laws are true. Lots of students mistakenly think that x^a * x^b = x^(a*b) because they're just trying to look for a symbol-pushing rule, but if you get them to think of it as "we have this many x's and then this many more x's all being multiplied together", the correct law feels a lot more natural.
What's more, you can really "milk" that equation, x^a * x^b = x^(a+b), to "discover" the other exponent laws. What is x^0? How can we have no copies of something? Sure we could just come up with a rule, but let's see if we can trick the math into revealing its secrets. Let b = 0. Then x^a * x^0 = x^(a+0) = x^a. So multiplying by x^0 does nothing -- what number is that? 1. So we must have x^0 = 1.
Recommended relevant reading: https://www.amazon.com/Burn-Math-Class-Reinvent-Mathematics/dp/0465053734
frigid crest
Hello. Has anyone had success in teaching combinatorics to children? How would you explain to them how binomial coefficients arise?
turbid zenith
Hello. Has anyone had success in teaching combinatorics to children? How would you explain to them how binomial coefficients arise?
@frigid crest
What are you using them for? Combination problems, or binomial expansion?
frigid crest
@turbid zenith They should be for combinatorics problems.
Due to certain the time constraints, I have had only minimal success on three occasions teaching children under 10 years old how to think about factorials and compute permutations. I would be intrigued to know how well it could be learned given a few days of instruction.
turbid zenith
Hmm. I see. Well I mean like ... the kids would need to know multiplication obviously
But I think personally what I might do to start off with simple concrete problems. How many ways could they pick 3 markers out of 4 to color with, etc. They'd most likely list out all the possibilities.
Then make the numbers bigger so that "oh no, now there's no way we could possibly write it all out"
But get them thinking about the permutations (draw blanks, how many could the first one be, how many could the second one be) and then turn that into combinations (are red-green-blue and green-blue-red really different sets of markers to pick out of the box?)
winged urchin
They are different if they're being inserted sequentially into a three-marker holding / writing device
=p
One of my pet peeves with combinatorics was always the distinction between the use of choose and pick
Typically teachers remark the saying of "If order matters you use pick, ... " etc etc
But I always found it confusing as to what order they were referring to
And found that sometimes when I thought order mattered, it actually ended up being choose but I was just thinking of the wrong order or something...
Given that I misremembered which one it was maybe it's just my memory, but yeah, I always had a pet peeve with that saying to inform you which is for what situation
turbid zenith
Yeah and I've seen deliberate attempts to hide that in some worksheets before
My favorite is how "committee" is like always a cue for a combination as opposed to a permutation
Now how many high schoolers do you think have an intuitive lived understanding of what a damn committee is 😛
frigid crest
The major obstacle for understanding is to get kids to recognize when they are overcounting, how much they are overcounting by, and what to do to compensate for the overcounting. From there, many counting problems involving binomial coefficients and summations of integers can be handled.
frigid crest
@winged urchin The language can greatly depend upon context. In my line of work, I am usually able to get away with using "ways" to mean performing an operation where order matters without ambiguity. To me, "ways" communicates some kind of motion. When I am not concerned with order, I am usually fine using "structures" or "patterns" or "configurations".
winged urchin
The problem with any kind of notation or common understanding of a word is that inevitably someone will use it in a way it wasn't meant to be used either purposefully or accidentally
Like log and ln
Like WHY do universities use both log and ln to represent log base e?
It's so confusing for students
In high school they, logically, use them for different purposes
strange bronze
i'm not sure that's as much of a problem as people claim it is
i've never seen a student get confused after telling them "mathematicians use 'log' for base e, and this course will be following in that same vein"
winged urchin
A bit of a tangent and it's unavoidable of course, we need to talk about things without a dictionary of terms at the start of every paper or book, but it's annoying =p
frigid crest
Every student should have access to a math dictionary.
winged urchin
That's something I have thought of ya
I haven't gotten to it yet, but I think over this summer I'm gonna put together some help sheets for students like that
Sorta... ideal cheat sheet
It's helpful to be able to look over your toolbox of identities and theorems and such
jade warren
@winged urchin we could just stop using logs with any base besides e if that makes you feel better...
winged urchin
That's contrary to my desire in fact. My main grief is that log and ln are both used to represent log base e in most university classes. This combined with the conflicting usage of log in highschool to mean log base 10 is unnecessarily confusing
And it's not a hill I'd die on, surely, it isn't that significant. But ideally, I think notation should be consistent throughout all levels of education and we should avoid duplicative labels, imo
jade warren
I get what you are saying, and I'm not sure why the upper division courses (where students are expected to know that all logs are the same anyway, just use change of base) decided to eliminate the ln notation, other than that they were annoyed with having to type fewer letters...
strange bronze
it's because mathematicians never use ln
jade warren
on the other hand, I have found that lots of students struggle with the ln notation anyway
thinking that the l is an I
and when you write In(x) it's very wrong even though it looks almost identical to ln(x) which is correct
so if we are doing away with symbols, we should elminate ln
make the natural logarithm log(x)
and require any other log base (including base 10) to be displayed
and don't get me started on the computer science people who use lg(x) for log_2(x)
winged urchin
Ahaha. It is a tricky balancing act to get as simple a notation as possible within somewhat disjointed fields =p
strange bronze
again this is honestly something ive never seen confuse students
the only student's ive had confused by this are like
intro CS students who asked "when we talk about O(n log n) what log is it"
which is more misunderstanding big O notation
than misunderstanding logs
[though it includes both i guess]
winged urchin
Yeah I guess in that case if a student doesn't have in their mind the change of base formula then they might not recognize that logs only differ by constant multiples which
Or just not realizing what the definition of big O is, yeah
I think I've seen log confusion more in students who are... not really thinking through their answers/work I suppose
jade warren
It's that last one
And it's because they've been trained that math problems are things you put into a calculator and then the answer pops out
so there is no thinking to do
the projectile launched from a building and impacting the ground question shows this up too
because you get solutions like t=-3, 7
and when you ask them what t=-3 means they have no idea
winged urchin
Yeah I was helping a student today in Calculus and I was trying to guide them through a problem "Find all points on the curve y=3x^2 where the tangent line passes through (2,9)"
Those sorta... indirect questions always give them troubles
I was glad that they first tried to sketch it out and had the idea of finding the tangent line but mistakingly just took (2,9) as the point of tangency
It seems they have a lot of difficulty defining unknowns if they aren't explicitly defined. Like yes we don't know where the point of tangency is, but we can sorta... act like we know by giving it labels and trying to mathematicalize the given information
Idk, I'm not 100% pleased with how I approach helping someone with those kinds of questions. If anyone has a good idea or has had success with a student in that type of question, I'd be interested to hear your take
indigo grotto
I think the issue with those situations is how to actually solve problems
There is a difference between following an algorithm and figuring out how to put the pieces together yourself
tidal whale
and don't get me started on the computer science people who use lg(x) for log_2(x)
I've seen faculty use log for log_2 as well lmao
civic tree
lol
turbid zenith
So it looks like, so far, the name change to #math-pedagogy has fixed our problem 🙂
Also for log₂(x), the abbreviation I like is lb(x)
winged urchin
Yeah that's true deekaan, which is particularly difficult I find in tutoring or teaching students. I think we hope that students have some natural curiosity and that would definitely help, but when trying to help students with questions like this... it's somewhat difficult to know when you're holding their hand too much or leaving them stranded
Hey DM Ashura, you usually have interesting thoughts on teaching students. Do you have any insights into helping students with problems like "Find all points on the curve y=3x^2 where the tangent line passes through (2,9)"?
turbid zenith
Hmm. THat's a really specific kind of problem 😛
Really comes down to getting them to be able to break down the problems
Like knowing that, say, "tangent line" is a cue for using the derivative somehow
winged urchin
Yeah.. yeah... It is a pretty difficult question to think about those sorts of problems
That's one thing that irks me about teaching. Or at least... Worries me that I'm not a good tutor? I guess? We can say general statements about teaching but at the end of the day I feel like in the moment it's mostly intuition and a sort of unthinking reaction to how you see a student approaching a problem. Difficult to really judge the best way to help them or whether you're being effective
Sort of an... Imposter syndrome version for teaching. Which isn't surprising since I felt that way while I was in graduate school too. At the end of the day I just have to trust that because a few students will thank me and say they did well, that I am decent enough ?
But on the other end I'm fairly sure I've always had students that failed ultimately... And just never told me
brazen pendant
Find all points on the curve y=3x^2 where the tangent line passes through (2,9)
you can construct them with just a straightedge btw, which is neat
(though with that particular example it’s rather ugly cause everything is so narrow)
this works with all conic sections, i.e. circles, ellipses, parabolas and hyperbolas.
A is the point through which the tangents should go.
stone tusk
whoa pascal's theorem?
trim violet
the 9-points cubic theorem :o
winged urchin
That's a really neat solution to the problem. I'm super interested in uhh.. I guess constructive geometry like that? Shame it isn't taught in high schools here anymore.
frigid crest
Not much is taught anywhere anymore.
winged urchin
Though I'm afraid it isn't at the heart of what I was musing on. Like... How would you help a student to see a solution for themselves? In a more indirect question such as this?
I think I'm asking more rhetorical questions to be honest... I barely know how to answer it myself but that's why I find it's interesting to think about.
In the moment with a student I tend to just... Go off of... Intuition I guess? But I find it hard to narrow down into words how one helps a student with indirect questions
trim violet
maybe pick a point on the curve and ask them if it's the solution and how they would check that it's a solution ?
jade warren
sometimes it's helpful (when you are not actively tutoring or teaching usually) to step back and think what it would be like to be doing something like the problem for the first time. what things do you or should you already know? how would you decide if those things were helpful? is there a way you can break the problem down into more manageable sub problems? No one eats a large pizza in one bite, but with many small bites it can be eaten.
and then of course you have to think of questions to ask that help the student arrive at those conclusions themselves and hopefully develop the ability to ask themselves questions like that too
the couple of times I've had to teach a basic class on math to people planning to be elementary or middle school teachers, one thing I've done is make them learn to do some modular arithmetic. because that lets them relearn a basic skill they learned long ago now that they have matured enough to consider what it is they are doing to learn something. and (i think) it makes them less likely to just draw the standard addition algorithm on the board and tell the students to do it that way because they said so
ionic dagger
I find that there is a common, valid complaint from 7-12 grade teachers that there is just too much to teach. Every non-teacher has to get their slice of math represented to prepare students for their fields: Statisticians, architects, EECS (matrix/base 2), pure natural science majors, and of course pure math majors. Also calculator and paper and mental math methods are all required in some curricula.
Finding time to train math intuition among other math-related topics is a problem.
jade warren
Indeed. Which is why math intuition needs to be taught starting from as soon as numbers and reasoning in general begin. So that by the time the 7-12 teacher is tasked with getting a little bit of everything into the class the student already has some practice using their intuition.
frigid crest
Teachers waste time trying to get everyone on the same page.
Have you tried tutoring more than 5 people at one time?
In university, professors do not waste time trying to get everyone on the same page.
jade warren
“Here is the page. We will be on it. If you are not, remedy that situation post haste.”
frigid crest
Unfortunately, the 5-12 system is stuck in a stone age with the same educational processes and outcomes repeating for 70+ years.
Until we talk about getting students to perform at about a college level, the only appropriate level, by the time they start middle school, we should expect no change in our lifetimes.
ionic dagger
I'm a tutor and I honestly hate my job. Usually my job is dependent on a bad teacher.
Ramblings on HS teacher facts: There's an article out there somewhere that says the best teachers can teach 1.5 times the material that a satisfactory teacher can teach. Also, teacher's salaries haven't kept up with inflation. Also class sizes have increased (which at least means more grading and likely means better classroom management). It's a mess.
meager bronze
all true
ionic dagger
Sorry I'm missing citations. And I hope "It's a mess." isn't a trademark. 😛
winged urchin
Thanks for all the responses everyone. Was nice reading all the different perspectives.
I'm a tutor as well, as it may have been clear, and while I don't hate it I do find it tiring sometimes. But I suppose all teachers of any form think the students need more of X and less of Y or what not and there's only a finite amount of time to cram stuff in there
About getting everyone on the same page, yeah. The degree of repetition I see, more so in elementary, sometimes raises an eyebrow. Like.. I'll be helping a grade 6 student and think "well this is quiteeee similar to what my other student who is in grade 4 is doing... Hmmmm"
Personally though, Ive always wished to teach a math class that was more like .. English class I suppose. Once I realized that equations are actually just sentences written out in more compact form.. it made understanding it a little easier for me
And Id be overjoyed if a student was able to be literate with math the same way you can be literate with a typical language
It's also been a sick dream of mine to take about a sentence long mathematical statement at a high level, then break it down into English and see just how long it is in words haha
Sometimes I like to think that's part of the reason it all works so well. I'll tell myself that because we can only typically hold 7ish ideas in our head at one time. Being able to write a symbol to represent a whole ton of symbols which also means a whole bunch of words if written out... Somehow turns what would be too many ideas into just a single idea... I suppose. Pretty abstract and loosey goosey but eh. :-P
Maybe there's a shred of truth to it
frigid crest
The current math curriculum in America looks like this. 1st grade: arithmetic, 2nd grade: arithmetic, 3rd grade: arithmetic, 4th grade: arithmetic, 5th grade: arithmetic, 6th grade: pre-algebra, 7th grade: pre-algebra, 8th grade: Algebra 1, 9th grade: Algebra 2, 10th grade: Geometry and Trigonometry, 11th grade: Pre-Calculus, 12th grade: Calculus 1
frigid crest
That curriculum is considered by some to be generous.
winged urchin
Yeah haha. Sometimes I'm tutoring calculus students in University and they barely have a good arithmetic grasp
I find it's interesting how once you move past arithmetic or algebra and move into the next stuff, what you learned gets dumbed down
frigid crest
It's mostly arithmetic all the way through and they still need to use calculators.
winged urchin
You do 3 or 4 digit multiplication in elementary but then you almost never have to do it to that degree later
You learn factoring by grouping or factoring by completing the square, but then almost always your quadratics just use the simplest factoring method of just what two numbers multiply to c and add to b
I realize there are reasons why this is, the division algorithm is completely nonsensical with small numbers as you just don't require many steps
But... It's interesting
frigid crest
Similar calculations recur in finite fields and numerical analysis.
Yes, students are not really taught to think algorithmically.
The early grades really should be redesigned to computation and number sense.
quasi musk
Can anyone talk about what it's like to do a master's in math education?
I'm interested in getting one, but I'm uncertain
I have a bs/Ms in pure math
I've worked as a tutor for five years now, probably going to continue with teaching at a CC soon
brazen pendant
the concept of a math education masters is odd to me but I suppose it’s probably very comparable to what I’m planning to do after my master’s (getting the teaching diploma that allows me to teach math at high schools - prerequisite is a master’s in math, statistics or another closely related subject)
which entails basically a bunch of pedagogy classes, some “practice” courses and an internship at a school
frigid crest
A master's degree in education should allow you to teach at community colleges.
turbid zenith
I'm doing kind of the opposite
I have a Master of Arts in Teaching Mathematics, but now I'm going for my MS in Mathematics
I don't know how an MAT compares but I enjoyed it
It had a lot of stuff about how to do educational research.
tidal whale
Is it obligatory to have a teaching degree or diploma to act as a prof or teacher there?
meager bronze
If you want to teach at the collegiate level then yes
quasi musk
@turbid zenith what was the course work like? What was the most impactful experience you had? Do you feel that doing an MAT math gave you a better grasp on how people learn math?
turbid zenith
Hmmm. I feel like it did somewhat. Honestly the thing that helped most was the experience.
Coursework I had some things on pedagogy in general, some things on math pedagogy,, one course on math history & technology... a bunch of stuff about research methods ... and then practicum stuff
The practicum was the best part because I actually got feedback on my teaching and learned to be deliberate about it
ionic dagger
"If you want to teach at the collegiate level then yes" -- unless you are erdos
then people beg you to please talk at their college.
(well not anymore. even zombie erdos wouldn't want to be on US campuses as he might get Corona)
jade warren
You do not need any sort of Education degree to get a job teaching in Uni or community college in US, unless you want to teach education courses. You need a Masters or Higher degree and 18 graduate semester hours in the area you will be teaching.
brazen pendant
ideas what I could do in my last tutorial class of the semester? usually I invite my students for coffee and just chat a bit with them but, well, kinda hard to do that over zoom
maybe I’ll just do a quiz
unreal ledge
Kahoot is fun @brazen pendant
unreal ledge
Haha Kahoot always wins
turbid zenith
Twitter
(Feel free to discuss here but please also consider sharing your ideas on Twitter, or spreading the post further so that the discussion can gain traction)
brazen pendant
not sure why assignments have to look any different than they did before
we carried on a semester with the usual assignments and the only difference was that students had to scan their things and hand them in digitally, and we had to correct them on our pcs (or print and rescan)
really, nothing’s changed at my uni
turbid zenith
Do you know about Photomath, @brazen pendant
brazen pendant
never heard of it
next relic
Personally, I think you should keep the courses the way they are. If you give quizzes, continue giving quizzes. Assessments shouldn't be easier than usual, though I reckon you should allow a bit more time (let's say, 1h instead of 50 mins) for students to upload their work and other technical difficulties.
I'd go for a mix of both critical thinking and regurgitating procedures, because both are important. Critical thinking questions can be considered as the "extension" section of the worksheet, which I think should worth 15-20% or as extra credits, depends on your institution's marking policy.
For quizzes, I think they should be short. 2-3 questions, in order of difficulty. They should be fairly standard, for instance, a question from past exams. Not too challenging, not too easy. Students need to think hard to get 100%.
turbid zenith
@brazen pendant You take a picture of an equation or expression, and it gies you the answers to whatever you might need to do with it AND shows all the steps.
A lot of students unfortunately use it to cheat on homework and even on exams.
next relic
One of the methods I can suggest is using an unknown in your answer. For example, find the value(s) of 'm' such that the equation has two distinct solutions.
@turbid zenith
turbid zenith
Yeah I like that. 🙂
Like ... suppose f(x) = x² + 6x + k.
a) Find a value of k for which f(x) has two real zeros.
b) Find a value of k for which f(x) has one real zero.
c) Find a value of k for which f(x) has no real zeros.
Something like that perhaps?
rigid thistle
i think that's more difficult to chug into a calculator but i suppose that means cheating is reduced
turbid zenith
Yeah. That's the idea. 🙂
tidal whale
What about having problems be about scenarios and the students have to figure out the model?
brazen pendant
in my opinion, the existence of photomath is irrelevant. if students feel like cheating themselves out of learning, then that’s their privilege
the cheaters probably won’t pass anyway (if they do then there is the problem you have to fix)
to disincentivize cheating you just have to make sure it’s not a winning strategy
rigid thistle
if cheaters don't pass why do they cheat lol? clearly they'll have an upper hand just by googling what they're not supposed to in an online setting
grand laurel
in my opinion, the existence of photomath is irrelevant. if students feel like cheating themselves out of learning, then that’s their privilege
i agree for university classes, but those seeme like algebra/precalc questions? (not sure at what age they are taken)
i would have totally cheated in highschool because i was a dumb teenager
strange bronze
the cheaters probably won’t pass anyway
im unsure this is true
i could see a borderline B+/A- student thinking
"everyone else is cheating, so i might as well cheat to try and stay competitive"
or whatever
and yeah, this isnt excusable but like
you can see why they have these sentiments
brazen pendant
I suppose for context I should ask whether the stuff being cheated at here (a) is mandatory and (b) influences your grade negatively if you don’t do it correctly
cause both of those factors greatly incentivize cheating
strange bronze
well yeah, kids wouldnt cheat otherwise
brazen pendant
no, they still would. someone who studied with me during my first semester cheated a lot cause she felt like even if she couldn’t solve the homework herself, writing it down correctly once would help her anyway
it didn’t
and she got into a habit of not even trying properly anymore
and then naturally she failed the exams
and changed studies from math to I think economics?
which, last I checked, she’s enjoying and doing well at
but I haven’t really had contact with her other than meeting her in corridors sometimes
but yea it strikes me that graded homework is the issue here
frigid zephyr
We have a saying at my (weird) school (where everything is based on peer-learning, long-time projects, and there are no teachers; only students selected on their work diligence, competence, curiosity, and pedagogy): copying isn't cheating; cheating is simply not being able to reproduce something again when you're on your own.
Granted, it does help that every time you get a failing grade, you're encouraged to re-submit the project with your fixes, to learn from past mistakes and refine your understanding.
(Even if you get a passing grade and it's not to your liking, you can work on it some more and re-send it)
lethal leaf
I suppose for context I should ask whether the stuff being cheated at here (a) is mandatory and (b) influences your grade negatively if you don’t do it correctly
@brazen pendant
You need these things though
My physics teacher is trying to get rid of grades
So it's 60% participation, 20% on a midterm and 20% on the final
There is no homework catagory
So what happens is that people don't do the homework because it's not mandatory because your grade isn't affected
And so we fall behind on material cause the learning people are supposed to do at home he has to teach in school
If there was a grade for completion of homework
It would be done by students
brazen pendant
you don’t need those things if you’re okay with making people fail courses
that’s a cultural difference though
lethal leaf
I guess
brazen pendant
my math degree has a roughly 50% fail rate in the first year
brazen pendant
high school was different of course
but high school here is very different from e.g. american hs as well
cause it’s already not considered anything needed
lethal leaf
So everyone gets a 100% in that participation so you automatically can't fail
It's literally impossible
brazen pendant
high school here is for smart kids, so to speak
lethal leaf
Yea I'm in America
But I'm talking about our advanced physics classes
Honors and AP
Not the lower level intro physics classes
brazen pendant
i say smart, really what I mean is people who aspire to get a job that can’t be gotten with a vocational education
lethal leaf
Ye
brazen pendant
but that ends up being sth liek the top 20% only
lethal leaf
I think in HS grades serve a great purpose of being a motivator
brazen pendant
I agree that basing your grades fully on exams is bad, or at the very least based on high stress exams and standardized test, that sorta thing
lethal leaf
Well lemme give you an exam of two classes I took in HS
brazen pendant
but I’ve always been of the opinion that lazyness is a choice
brazen pendant
and people should be allowed that choice and learn from it
lethal leaf
I already told you the grade distribution of our physics class
In calc it's 70% on in class exams, 5% homework, 5% quizzes, 20% final
And then when the end of the year exams come around for college credit
Our pass rate for calc BC is very high
And our pass rate for physics is very low
brazen pendant
(for reference, at my high school grades were ~95% in class exams, 5% convincing the teacher to give you a better grade participation. no finals, no graded homework)
lethal leaf
And I think that stems from the fact that the kids in the calc classes are actually doing the work
The homework isn't graded for accuracy here, it's graded for completion
Just attempting it
But it's like
The grades make the kids do the work
And in math and science you need to put in the work to do the problems
brazen pendant
here not doing homework earned you either: nothing, a stern look from the teacher, or detention if they felt like it
(most teachers never used option c)
(but they could)
lethal leaf
I agree laziness is a choice but I feel like some places make laziness not a big enough choice
Like you shouldn't be able to do 0 homework and fail both the big exams
And then still pass the class
brazen pendant
for sure
next relic
If it is graded for completion, would the kids just make something up and hand it in?
brazen pendant
it’s probably graded for “did a solid attempt”
lethal leaf
And yet that's how our physics at our school is and it reflects in the college credit exam results
Yea as long as it looks like you tried somewhat
You're good
brazen pendant
like, it doesn’t have to be correct but you have to have tried something reasonable
the prof I’m tutoring under does sth like that rn
lethal leaf
Or even just saying "I tried it but I don't know how to do it, can you help"
Admitting that gets you the points cause it shows you want to learn the material
brazen pendant
homework is optional but you can get a small bonus if you “reasonably attempt” at least 40% of the homework and hand it in
lethal leaf
But yea in HS 90% (or more) of kids will not learn for the sake of learning
They want the grades
You can't motivate them with the pursuit of knowledge
Or "you'll need this for your job"
Or even "you'll need this to pass this next class"
brazen pendant
mhm, you need to somehow manage to set the game up in such a way that learning is the winning strategy
and that that is actually clear
lethal leaf
Which sucks but that's how it is here
In college the motivation comes with the big price tag and the promise of a valuable degree
brazen pendant
that’s only cause you live in a culture where for some reason you need a degree in philosophy before being allowed to work as a carpenter
even if you really like carpentry and not philosophy
next relic
One solution would be giving weekly quizzes.
You feel bad if you get a failing grade every single time, thus you study.
lethal leaf
Well no trade school is big here
next relic
Probably .5% each week, really low stake.
brazen pendant
I could see sth like doing quizzes that are easy if you did the homework
and where failing a few doesn’t really matter
lethal leaf
You can not go to college and instead go to a trade school if you want
brazen pendant
but you can get a nice cushion if you do them consistently
next relic
I'd say 5% of the course, max.
next relic
Just, you know, some motivation to keep studying.
brazen pendant
like say on your homework you practiced taking derivatives of polynomials and then the quiz is just another one of those
next relic
Make them a bit harder though. :p
brazen pendant
yea yea
lethal leaf
The kids I'm talking about in these classes I'm taking that have these grading structures want to go into medicine or law or engineering or business/finance/econ
brazen pendant
what you don’t wanna do is have it end up failing people who are just a bit slow to start up
like a 5 minute quiz is high stress for some
next relic
I'd do 15.
brazen pendant
because they just need longer than that to even get into the right mindset
lethal leaf
In my calc classes we had 20-25 minutes for a 1-2 page quiz
1 page front and back
Was like 6-7 problems, if they were multipart it was just 2-3 parts but most were just 1 part
next relic
High school or college?
lethal leaf
This is in my calc 3 class
next relic
Ah.
lethal leaf
I took it in HS but it was a college class
next relic
Well, you have more instruction time in high school.
brazen pendant
in my analysis I–II class at uni we had a 10 minute quiz at the beginning of each tutorial, just one or two questions, and if you got enough points you got a bonus to the grade
(you could however still ace it if you didn’t do so)
next relic
So you can afford to give longer quizzes.
brazen pendant
I generally prefer giving bonuses over just splitting the grade between different things
next relic
I'd stick to 10-15.
brazen pendant
like, don’t make it
grade = 60% exams + 40% other
make it
grade = max(100% exams, 60% exams + 40% other)
or whatever
next relic
10 homework, 50 for 3 tests, 40 final.
Or 20 class mark (out of 60 now out of 20) and 80 final.
brazen pendant
for my analysis class there were three parts that could influence the grade:
•the final exam, weighted as 0.8–1
•oral presentations of homework, weighted as 0.15
•quizzes, weighted as 0.05
the latter two were only taken into account if that increased your grade
so like it was
max{
1 exam,
0.85 exam + 0.15 presentations,
0.95 exam + 0.05 quizzes,
0.8 exam + 0.15 presentations + 0.05 quizzes
}
lethal leaf
Huh that's interesting I've never seen that
brazen pendant
I believe
next relic
Hmm interesting.
lethal leaf
Certain catagories impacting grades only if it helps
brazen pendant
the formula may have been different but it was sth like that
next relic
I'm in Canada so perhaps our grading scheme is stricter?
brazen pendant
that’s generally how it goes at my uni
the “base grade” is 100% from the final exam but the professor may give ways to improve the grade further, but not ways to worsen it
with things like graded homework, presentations etc
a common thing is to just give a flat bonus to the grade (think sth like increasing the grade by 5%, capped at 100 ofc) if people hand in enough homework
I think the bonus can’t account for more than a 5 or 10% grade increase relative to the final’s grade but it’s something
next relic
We only have extra credits on tests/exams and that's pretty much it.
Some profs do it, some don't. Usually don't.
Class average is 70.
Around that area.
brazen pendant
also note that a lot of these systems are fairly newly implemented and still experimental
because they changed regulations about 5 years back or so
before that there was a school-wide rule that in order to be admitted to an exam you had to have worked on at least half the homework given in that class
next relic
That's cruel.
brazen pendant
where “worked on” is somewhat up to interpretation
main effect: students are less stressed but over-all pass rates have gone down
though I’m not sure to what extent dropout rates were affected, which are the actually relevant number
since you can’t compare average grades between the two systems
next relic
Hmm yeah our passing grade is 60.
Yeah you also need to know the dropout rates and the passing grade to be able to compare the grades.
brazen pendant
grades here don’t correspond directly (not even necessarily linearly) on points on the exam
as in, getting 60% of the points on an exam doesn’t mean a 60% grade
usually they make it a piecewise affine function
next relic
You're grading on a curve?
brazen pendant
no
it’s not curved relative to the other students
but like what they do is
our grades are given as numbers from 1 to 6, with 1 = worst, 4 = pass and 6 = perfect. in other words, 60% = pass
but then if an exam has, say, 100 points, they will put it like
1 = 0 points
4 = 40 points
6 = 90 points
and affinely in-between
next relic
Ah I see.
brazen pendant
and those three values can be shifted around to account for difficulty and test-making philosophy of the teacher
next relic
So you're grading against a set of standards, kinda, correct?
brazen pendant
but in general as long as you’re not passing, one point will have a larger effect than once you’ve already passed
So you're grading against a set of standards, kinda, correct?
dunno if that’s a good description, no
different profs have very different philosophies
and they have some amount of freedom
next relic
If a student demonstrates that they've mastered objectives 1, 2, 3 and 4, for instance, they should pass. You look for evidence of objectives 1-4 and hence calculate the minimum mark the students need to get to considered "achieved" the four objectives?
brazen pendant
e.g. my phsics II prof gave us sht like 10 exercises, with the expectation that the best students could solve about 8 in that time
meanwhile my analysis prof expected the best students to get pretty much everything on the exam right
next relic
I'd join the first group. ;p
I'd give long exams (would because I'm still a student, though I'm writing IB mock exams and stuff.)
brazen pendant
for the most parts the profs just agreed to disagree on what is considered a good exam
next relic
Yeah it's highly controversial.
brazen pendant
for the most part I agree with the analysis prof’s philosophy, though in practice his exam was also the worst I’ve written so far
it was a four hour exam split into three equally weighted parts: calculations, proofs and theory from class
calculations were things like “find this limit” or “solve this integral” where usually you just had to figure out the right trick (e.g. using the divergence theorem)
proofs were, well, little proofs of statements not covered in class
those first two parts were basically akin to homework problems, though generally easier
theory from class was repeating definitions and proofs given in class. 10 days before the exam we were given a list of 40 possible topics that could show up (this was considered enough time to learn for it if you had gaps left but not enough time to only start learning once you had the list)
I aced the last part, completely failed at the proofs and did okay on the calculations
end result was a 4.75 (75%), which then got raised to a 5 (80%) with the other bonuses I had gotten throughout the year
brazen pendant
it stands as my worst grade but it’s objectively considered a “good” grade
I think in total I had sth like 35/60 points at the exam
a 6 needed somewhere between 50 and 55
the worst exam written in recent years must’ve been last year’s complex analysis exam tho
it was bad enough that they’re now implementing a rule that another professor has to check each exam. the exam had something like 80 points… and you passed it with 12
that was not intentional, mind, they adjusted the grading drastically in order to have a somewhat reasonable amount of people passing
they only do that in exceptional cases
anyway I should finish preparing tomorrow’s tutorial
strange bronze
why do people continue to think this is funny
holy shit the dude's history is just full of this stuff
he hasn't figured out that the joke's never been funny
meager bronze
oof
strange bronze
imagine trying to force the same lame joke for like 7 months
this post made by the Ban Namington gang
frigid crest
I think there should be a national move to make math courses pass/fail, where pass is consistent with traditional A grades.
And I don't see much point in testing in courses since most of work life is not a academic test environment.
Tests are more practical for acquiring certification for employment.
Ideally, course grades should consist mostly of homework exercises where critical thinking and creativity is sought after.
civic tree
that's noble and all but like
you cannot expect students not to cheat on these hw assignments or whatever
or collude
or whatever might be the right word
indigo grotto
collusion 🤔
civic tree
lol
indigo grotto
I like to think of it as collobaration, you know the leaders of tomorrow learning to work together
civic tree
yeah
i personally prefer letting homework be free for all
like
work with whoever u want
the more ppl learn the better
but then test day is the day to assess individual understanding
bruh
indigo grotto
yeah
civic tree
that pounce tho @serene bloom
serene bloom
indigo grotto
i think at the end of the day the cheaters will find a way to get through life
civic tree
yea
indigo grotto
has to be something you have to decide how you want to go about it
no way to force it, because there will always be something you can't anticipate
civic tree
the idea that course grades depend more on the hw and less on the tests is how cheaters have a field day because you're gonna have ppl that just copypasta things or whatever
since that's how they're just gonna like
ace the class
meager bronze
or you could just write good homework problems that make it harder to cheat
civic tree
oh
indigo grotto
or just accept that people cheat
i mean it's annoying especially if you don't but c'est la vie
meager bronze
I also ask my students occasionally to write reflections on the problems they did, and you can kind of tell by reading those how much work the student actually did
civic tree
oh thats neat
to see how many ppl actually got their hands dirty and failed at proving something a few times or sth
before actually getting sth
ig
meager bronze
there's also room for open-ended problems, too. like in class we were talking about solids of revolution and stuff and I said "go read about gabriel's horn on wikipedia and write a couple paragraphs about whatever part of it you find interesting, either the math or the history or whatever you want"
civic tree
oh
civic tree
ah oke
meager bronze
but I think these ideas are applicable in any math class
civic tree
right
meager bronze
just like "go read about this topic and write about what you found interesting"
I had a riemannian geometry class once and there was no final exam, but there was a final project which was "go find a connection between riemannian geometry and whatever field of math you find interesting and write 15 pages about it"
meager bronze
I wrote about like, moduli spaces of genus g curves from the perspective of geometry and algebra
it was a really great project
indigo grotto
that sounds nice
but in school that would have totally fucked me
because I couldn't write anything
just straight up couldn't create shit
too embarrassed
meager bronze
I think you would have made it work haha
the point is supposed to be that it's hard
like, sure doing proofs on homework can be hard
but you have to learn to write at some point
indigo grotto
yeah I get that and now I would be enjoying it, but at like 16
that would have been an automatic fail
meager bronze
I mean this was a course suitable for late undergrads/early grads
most 16 year olds aren't taking courses in riemannian geometry haha
indigo grotto
bro there are some crazy 16 year olds on the streets these days
meager bronze
I would imagine that they'd probably be able to handle this kind of project then
indigo grotto
true
winged urchin
If we're talking about reflecting on your process to solve a problem, what would you think of a student who said something like "I tried this thing, it didn't work. I tried this thing, it didn't work. But then I tried this! And this worked! Done"
Maybe a bit more done up than that but in essence.
meager bronze
I would think that's great. I might ask them to also see if they could describe what didn't work about the first attempt, like where they got stuck
(when I've done this exercise before I've given my students more guidelines than just "reflect")
but like, that's basically what I would be going for. documenting your process
I try to model that for my students, too. like I'll tell them "this example was hard to come up with, here's what I tried first and why it didn't work"
or like "I think this example is really cool because of these reasons"
I guess I also like to see it when my students include their own feelings in the reflections
like soemtimes they'll say "I thought these problems were interesting" or "I thought they were kind of boring"
which actually I appreciate the second one more haha
I think being able to articulate what catches your interest and what doesn't is good, and I'm happy when my students feel like they can open up and say that
I also model this in class, to an extent. sometimes I just say like "look, this particular topic is a little technical, and hard to see the motivation for, but just stick with it and we'll get to see the payoff soon"
indigo grotto
I'd be impressed with the show of frustration tolerance
you know the ability to deal with the failure and keep on moving
meager bronze
yeah
that's a great point
one issue that I have with math education that I don't really now how to fix is that like, we present math as this nice little box with a bow on top and it's all perfect and it all works out
indigo grotto
thats such a huge factor in trying to achieve anything and so many people that fail at different points in life, just can't handle the hurdles
😦
meager bronze
but like, it took centuries to get that, and many failed attempts and incomplete theories
it's okay to fail and try again, but we present math as like a perfect things that you either get right or wrong
indigo grotto
ya I think the exploration point is important
showing it's not something that is, because it is but something that can be explored, be played with
you can make it your own in a way
meager bronze
yeah
one of my favorite projects that I've given as a teacher was in a calculus class a couple years ago, we had been playing with graphs of implicit functions in class when we were doing implicit differentiation
and I was showing them using the online graphinc calculator Desmos what families of curves look like
(basically include a parameter in the coefficients, and Desmos lets you change the parameter continuously and watch the curve move around in real time)
and they thought that was cool, so I gave them a project which was basically "go and play with Desmos"
they all had to find their own family of curves (and I gave them some suggestions for how to get started)
and then I asked them to answer some specified computational questions like find an integer point on one of your curves and compute the slope of the tangent line at that point and stuff like that
but then I asked them to write qualitatively about why they chose their particular family, like why they found it interesting
and it was really cool to read their responses
and a lot of them even tried asking their own mathematical questions
for example, one person's family of curves had 2 components for some values of the parameter, but then as they changed the parameter one of the components shrunk and vanished
(so think like y^2 = x^3 - x + a and vary a)
and they calculated the precise value of a where the component vanished
which was so amazing, like this student asked a question that we had never talked before in class
and then figured out how to answer it
indigo grotto
that is truly amazing
meager bronze
everyone kinda did their own thing and it was really really fun
indigo grotto
and the project sounds great true
I like the introspection part, showing an interest in whats happening inside their heads
you know treat them like actual people and show them that what happens inside of them actually matters
indigo grotto
no worries 🙂
winged urchin
That's lovely Buncho. Heartwarming
Yeah whenever I tutor students and they ask me something not directly related to the question it's so nice
Then I get to go on a bit about matrixes, vectors, complex numbers, even derivatives at times
stark pine
@meager bronze my algebra class in my first semester at university was structured like that. Assignment questions were half coq proofs, and half deriving problems from computations. For example, we were asked to expand 23/42 as a continued fraction (numbers may vary), and then the question ended with (paraphrased) "can you generalize this process? any theorems about it? conjectures? state as many conjectures as you can. Prove as many of these as you can"
and that was how every question went. We'd be given an interesting computation, and then we'd be asked to take that computation, and turn it into a generalized pattern
stark pine
I think the very first one was to find a bunch of elements in the ring of the integers mod p. Could we find i? the square root of 2? -1? a half? a third?
It was the entire class, and we got up to a bunch of highly nontrivial statements, and it was entirely self driven
meager bronze
oh that's really cool!
stark pine
similarly, for the coq proofs, the first assignment had two statements we couldn't prove off the first axioms we were given, then the last question was roughly "Can you prove all the questions on the assignment with the axioms given? is the list complete? if so, explain why, if not, create new axioms and justify their inclusion"
and that was a rule for the rest of the course
if you could justify a statement as an axiom - even a problem on the assignment - you would get full points for it
apparently though, the assignment placed a ridiculous amount of strain on graders. There were usually only 2-3 questions, but grades were stretched to the breaking point. It was like a 50 person class? it started with 80, but about 30 dropped
meager bronze
oh hmm
yeah that's definitely somethign to be careful with
these kinds of problems can take more work to grade
especially in large classes
stark pine
I just thought you may find that interesting, given what you said above
frigid crest
I took a course in stochastic processes that was disguised as a combinatorics course in which the professor came up with problems that neither he nor the students had answers to.
next relic
one issue that I have with math education that I don't really now how to fix is that like, we present math as this nice little box with a bow on top and it's all perfect and it all works out
@meager bronze one of the solutions I have is low barrier high ceiling problems. Despite most of these problems are designed for grade 10 (gcse), the same principle applies.
https://colleenyoung.files.wordpress.com/2015/11/edexcel-gcse-coursework-tasks-and-projects2540-new-v2.pdf
frigid crest
So, it was practically impossible to look up solutions.
next relic
I mean, then the problem of grading shows up.
How are you going to mark the work?
How can you ensure that the grades given are consistent and justified?
next relic
Does that mean students can google a similar problem?
If the problem proposed is too difficult (or perhaps impossible, because the prof hasn't solved it), is it really good for learning?
frigid crest
There were no graders since homework was not required.
next relic
Oh that's another story.
But then if it is not graded why would students bother looking up solutions?
frigid crest
There were tests.
next relic
So the prof had answers to test questions, yes?
frigid crest
No.
next relic
I still don't get it.
You can't look for the exact question online, but you can look for similar questions.
frigid crest
Of course.
next relic
Isn't it the way all tests and exams are supposed to be?
frigid crest
Yes, ideally.
Of course, homework has always had the problem of distributing answers for everyone to find.
And with internet technology, that is to be expected.
Still, homework gives you more control over the depth of questions you can ask.
halcyon light
im not studying to be a maths teacher (studying to be a science teacher), but im tutoring yr 7-10 maths and reading this channel is really fascinating even if it's not directly related to my degree/goal!
inspires me to be a better tutor even though im not a huge fan of maths
stark pine
not a huge fan of maths
HERESY
next relic
@halcyon light uwu I'm working on y11-12 maths.
Do you have any tips on teaching trigonometry?
I can share with you some algebra stuff I just made.
I'm a newbie, so please feel free to butcher my resource. ;p
halcyon light
im also a newbie so i havent actually taught trig yet. i think this is v good for piquing interest!
https://www.youtube.com/watch?v=0jltioeaEyY&feature=youtu.be
next relic
I find trigonometry hard to teach. Once it clicks, the beautiful world of trig opens to you.
The thing is, we suffer from the curse of knowledge as we can't imagine how the lesson(s) would look like to a novice learner.
winged urchin
Personally whenever trig comes up in my tutoring I find myself always refreshing their memory on the unit circle
A couple points I tend to hit on:
next relic
Sorry for interrupting. What if this is the first time they see trig?
winged urchin
No worries
next relic
Thanks for the advice though!
winged urchin
I suppose they're just solving triangles then?
next relic
I'll be teaching trig circles soon, but right now I'm making lessons on trig from scratch.
From the very beginning SOHCAHTOA to trig equations.
winged urchin
Right
Admittedly while I have taught students at that level, most of my students are near university or in university so my advice might be off
I might be tempted to even tell them about circles anyway
next relic
To be honest, you need to know trig circle to get out of the right-angled triangle area.
winged urchin
It's just the most useful way I know to interpret the trig functions and their values
And you can draw triangles in the circle if you must
next relic
In your opinion, what is the hardest part in trig?
winged urchin
It's a pretty broad topic, of course
Off the top of my head
Proofs trip students up
next relic
Trig identities?
next relic
Thank you!
winged urchin
On a more general level I'd say any lesson you want to give them, show it in multiple ways
stark pine
I'd honestly say that the biggest pitfall I've found when tutoring trig is that they look at everything as a series of formulas. What I mean by that is that a lot of students can tell you that sin(theta) is the y-coordinate on the unit circle, but many can't tell you why that's the case, how it relates to the other definition, or even why you need both. Many look at the quadrant sign rules as a magical abstraction instead of a very clear consequence of the unit circle definition.
winged urchin
Rarely do I just tell them what I want them to grasp and leave it at that. I'll say that, draw a diagram to represent it, show them a couple values to reinforce other facts they may know
stark pine
this leads to a lot of the misunderstandings, like applying sin directly in a non-right triangle, or thinking something is the wrong sign or being unable to set up word problems
and I honestly feel that trig identities make it worse. Students learn a couple of rote tricks (the basic identities), and just apply them in sequence until it works, which makes the trig functions lose a lot of their geometric connections imo
next relic
I think trigonometric equations give trigonometric identities a purpose.
Otherwise it's just a matter of manipulating one hand side to get the other, right?
One of the reasons I can think of that perhaps justifies why students are bad at trig identities: their algebra foundation isn't in place.
stark pine
Right, which is why I prefer when students find trig identities in natural contexts, ie when doing another problem, not when doing them just for the sake of getting better at algebra
I will say though, trig identities are excellent prep for trig subs in calc
winged urchin
Nicholas is right, for sure. I'll add that there's an inflexibility and lack of extension to their knowledge at times.
Like you say that they might understand that sin is the y-coordinate. But if I were going over that, I'd also make the connection between say... "sin(theta) = some number" and "y = same number". Then reinforce that idea by sliding the horizontal line up and down to show why sin(theta) is restricted to between -1 and 1. When they should expect two solutions and where they can find them. Etc...
Like I wouldn't just say the rules... if we have theta as one solution, then we can get the other solution as pi - theta, or whatever rule they have. I try to get them to realize on the diagram how it's clear to get the second solution
next relic
I'll probably expose them to both methods - the unit circle and the sinusoidal function.
Let them choose the one that clicks for them.
winged urchin
Right! Definitely. And reinforce, hey the function is zero here and one here. Why is that? Look at the unit circle?
Hey tan gets super large at pi/2 and -pi/2, why is that? Oh look, if we draw the ray for those... how would you describe it's slope?
I have a really poor memory myself, but I find since I've learned so many facts that support and interact with each other. Even if I forget one thing, I can rediscover it with other facts I do confidently remember
And I take that idea into my teaching. Ideally I'll draw a web of facts that connect and let them check the things they are less confident with with the things they are more confident with
next relic
Hmm, interesting. Thanks for sharing. :D
halcyon light
honestly lowkey questioning whether i should continue tutoring mathematics. i feel like i dont do the subject justice and teach it in a lot of the ways that have been criticised here
winged urchin
Eh, you'll never get away from mistakes and those sort
next relic
^
strange bronze
a lot of pedagogy strategies are built with the goal of making classroom management feasible
winged urchin
I've been tutoring for over 10 years now and Im still unhappy or doubtful with my process at times
strange bronze
one-on-one tutoring is a different affair
next relic
The one tricky question that I want to ask. Should I talk about the proofs for compound angle formulae?
strange bronze
yeah, similar principles translate
but you have the opportunity to personalize and adapt a lot more in a one-on-one setting
next relic
You might risk the students paying attention to the small details, or they are intimidated by the proofs.
halcyon light
im doing one on one teaching and im still questioning whether im doing it right to this day 😂
winged urchin
You probably aren't doing it right if right to you is perfect =p
next relic
^
winged urchin
And no one is doing it right then
halcyon light
yeah true
winged urchin
Just do your best and if your students do better, if they can answer the questions, and if they perform better on tests. Take that as a better indicator
next relic
Have you heard of the discussion between performance vs learning?
winged urchin
And also remember that sometimes... some students just won't click with you.
halcyon light
i struggle on how to prepare my students for more problem solving focused q's in exams that arent' as easy as plugging in a formula
hmm i think i've heard about it but havent really gone deep into it
next relic
Just a blind guess - are you teaching GCSE Maths?
halcyon light
it's like the equivalent of gcse maths
winged urchin
I actually have to go for a bit but I'll be back later if ya'll are still chatting. 👍
halcyon light
i should be workign on an assignment rather than talking
next relic
I'm sorry but I should be asleep by then. ;p
halcyon light
is the performance vs learning discussion about how a student's performance =/= their understanding?
just guessing from the name
next relic
Performance refers to the data gathered right after instruction. Let's say, exit tickets, or even unit test.
Learning focuses on the long-term effects.
halcyon light
ohh so like short-term learning vs long-term understanding
next relic
If a student is able to solve a trig equation now, would they be able to solve it a few months down the line?
Yeah, I'd say so.
Test results aren't really reliable when determining whether the student has gotten the concepts or not, but the tricky part is, how else are you gonna assess the students?
Sorry I'm going a bit off track here.
next relic
Ideally you should wait 3 weeks before testing your students, but if you're tutoring, you can't afford such time right?
I'm looking for an alternative, if possible.
halcyon light
yeaaa i getchu
ive been meaning to include some questions abt previous topics in the hw
next relic
I'm also thinking about that.
How much weight do you give on previous topics questions?
halcyon light
what do you mean exactly?
i tend to focus on supporting their understanding of the current topic
by exposing them to many types if q's
ah man i really gotta go and postpone this convo for ~2 weeks. got a lotta assignments due and exams to study for
next relic
Good luck with your studies!
halcyon light
thanks! gl teaching trig
jade warren
To those discussing teaching trig functions: why only explore the right triangle, unit circle, and sinusoidal function definitions? There’s also the general angle definitions.
next relic
I assumed that the students are familiar with the basic angle facts, angles on parallel lines, naming angles and triangles, classifying angles and triangles.
I'd briefly go over them when we start right-angled triangle trig.
About the other types of angles, honestly, if they're not really important I'd consider skipping them.
It'd be a shame if a student does not name angles properly by the end of the trig units, right?
@ Malix
frigid crest
Are curious students not allowed to know about alternatives to mainstream mathematics?
strange bronze
that's a different question
wildberger's criticisms of "mainstream mathematics" are, in general, unfounded or alarmist
and i don't see the advantages of his "alternatives"
frigid crest
To me, they are not unfounded, but they may be alarmist.
He is certainly making hopeful/wishful claims.
stark pine
I mean a quick google just tells me he's a garden variety ultrafinitist. What would you say the advantage of listening to him is?
frigid crest
I do not subscribe to ultrafinitism.
frigid crest
Wildberger motivates intuition for much of mathematics by focusing on finite approaches that are virtually never discussed in primary and secondary education.
lament wraith
Can you give an example?
frigid crest
To begin with, the fundamental underpinnings of most of discrete math outside of elementary arithmetic is almost completely disregarded in the modern education system.
The basics are not discussed until students reach their 2nd or 3rd year of college.
lament wraith
Sure I agree, but this is a well known issue and I don't see what it has to do with Wildberger at all?
frigid crest
This is evidenced by things like students encountering the delta-epsilon definition of derivatives in high school and being shown rules for differentiation rules without ever having used discrete methods to derive or to discover those rules.
Generally, students lack experience to understand the former and they are almost completely ignorant of the latter.
strange bronze
what "discrete methods" do you have in mind
frigid crest
Basically, discrete calculus.
strange bronze
that clears nothing up
in the sense that discrete calculus, at least as im familiar with it
is studying a fundamentally different thing
maybe you're using the term in another sense idk
like obviously there's relations but im not sure of the point of presenting this "alternative"
just to be contrarian?
frigid crest
The incremental approach to calculus motivates intuition for the continuous approach.
There is even a famous book that takes this point of view.
Wildberger has attempted to extend this point of view to trigonometric identities.
strange bronze
wait, kline does discrete calculus?
i thought it was a fairly conventional, though application-focused, treatment
i'm also not sure of the relation to what wildberger's doing, considering kline's approach is certainly not finitist
just looking at the amazon preview, he invokes limits a fair few times
maybe i'm just confused what you're trying to argue for.
lament wraith
Yeah, scrolling through the book, there's nothing close to what I'd call discrete calculus at least
After talking about some basic function things, he immediately talks about the notion of limits
unreal ledge
I don't disagree that Wildberger has a great amount of teaching experience and has good points of view. But the thing he's "known for" is saying the reals just don't work and shouldn't be used by anyone ever. That's ehh
frigid crest
Wildberger is an advocate of introducing propositional logic by using a Boolean algebra almost exclusively.
I would say that a lot of math can be done without appealing to irrational numbers, but many such problems won't be interesting to most mathematicians.
But those problems should be engaging enough to non-mathematicians.
rigid thistle
wdym without worrying about irrational numbers
does that mean many quadratic functions with irrational roots are non sense?
winged urchin
can't describe natural growth exactly without irrational numbers
frigid crest
Instead, you use extensions.
From that point of view, things like natural growth are amenable to approximation.
Expressing things could become quite inconvenient though.
rigid thistle
but like, if you're trying to approximate an object, say X, then does X need to exist? otherwise, how do you know how good of an approximation you're getting?
frigid crest
It's hard to properly say that X is a thing.
You can say that you are doing something that looks like it is giving X.
You can get into things like using a finite sum of binomial coefficients to express the exact terms of the Fibonacci sequence and finite alternating series to express types of exponential growth.
But this sort of thing can get computationally problematic.
quasi musk
Another point Wildberger routinely makes is that just because you write it down doesn't mean it exists
I personally don't really see any advantages to the system that he develops, sure it might be more "intuitive" in some sense of the word, but I don't think this will help people use mathematical tools to explore/build new ideas
Philosophically it might be nice because what do you mean when you say the "Real numbers" are complete, when almost everything we model is discrete. I think this tosses out the point of calculus, saying "You're close enough under an error tolerance for all practical purposes"
There are lots of valid criticisms and issues that Wildberger has with the modern formulation of mathematics, it's just that his system seems more archaic, less flexible, less useful, and possibly slightly more intuitive
halcyon light
to my understanding pedagogy is defined as how you teach smth
when you help people here, you have a certain pedagogy you use. e.g. are you trying to just give them the ans? or are you helping them discover the ans themselves
(feel free to correct an edu-student-who-isnt-doing-well's take on pedagogy)
turbid zenith
To be fair, helping people in this server IS a good example of pedagogy
Some people are more effective than others
And it would be totally fair as a discussion point in here
stark pine
Although saying stuff like [person x] is more effective than [person y] seems rather problematic (if both are members of the server)
turbid zenith
I mean speak for yourself, I have seen some terrible pedagogy from members before but that was a long time ago and it was much more prevalent on the IRC server I came from before this
I have no problem calling someone out if they answer a high school algebra question with graduate algebra terminology
For the most part though, the vast majority of people in this server do a great job!
next relic
In my opinion, as long as you don't give them full solutions but dropping hints one at a time until they can figure it out themselves, you're doing a good job.
turbid zenith
Yeah absolutely
next relic
Obvs some people are better at teaching than others, but that conclusion is really subjective.
turbid zenith
Just giving them the answer doesn't really help
next relic
Some students prefer to have someone to walk them through the problem. Others only need some pointers on how to start.
turbid zenith
I think people here generally do a great job with "okay what have you done", "what do you know about this", "what if you try that", etc
next relic
I'd argue that just giving the answer doesn't help at all.
A topic that many students find difficult is optimisation.
And it's usually the algebra and geometry that throw them off, not the calculus part.
halcyon light
what level of maths is optimisation? i havent heard of it before
next relic
It would be much more useful if we have a lesson (or even half a lesson) to refresh their memory on calculating area of a compound shape or volume of a prism before starting the lesson on optimisation.
turbid zenith
I think you first see it in precalc or calculus
next relic
It's usually calculus 1.
turbid zenith
You can usually solve optimization problems graphically in precalc and then analytically in calc
halcyon light
ah okay, i never really got past 1st year maths and did terribly in 2nd year maths
jade warren
Optimization is the “fancy” term for finding local or global minima and/or maxima
Huh. Didn’t really get away from the fancy terms with that....
next relic
Optimisation gives the process of finding extrema a purpose. It's really just a fancy term. ;p
winged urchin
I often tell my students to look out for words like least, most, greatest, smallest, etc... to identify optimization questions
But yeah, it normally does involve area to some degree. And squares are often the solution haha =p
Optimization actually offers a small door to discussing the interesting complexity of the reals too
When you talk about open intervals and absolute extrema =p
"Where would you claim the absolute maxima? Oh.. at the boundary? But that's not in the interval, right? Oh.. very close to that? But what number specifically? Oh well what if I take your answer and find the midpoint between it and the boundary? Etc etc..."
winged urchin
Sometimes I wonder about the whole 'not giving the student's the answer' pedagogy if you will =p
And I say this fully noting that I'm going on a train of thought and might not be fully fleshed out
But sometimes I kind of feel the need to show them a substantial step, typically algrebraic or some interpretation or defining of terms in some way
And I think it comes from a couple sources...
Though that last one is more tied with my abhorrence to confrontation
I know some tutors who will call students out on perceived laziness or thought-to-be ridiculous gaps in knowledge, and I just don't see the point
I don't think being adversarial in a teaching role is helpful, though perhaps it's just how I perceive it
But anyways.. of course I don't give answer to any graded work. But there are times when I will give the full solution and walk them through it. And I don't think they gain nothing from that
In particular, in reviewing midterms/exams or when doing practice midterms/exams I find myself really holding their hand through a question more sometimes
And I question if that is of no help to them. Because I can do that, and get through a whole practice midterm in a 2-hour session or I can really try to get them to go through the questions and maybeeee get through 3 long questions
indigo grotto
its not about being adverbial its about letting people walk the walk
getting an explanation is nice but working things out for yourself is a huge part of learning how to solve problems
meager bronze
but I think that sometimes people take that way too far
like, if students just don't have a certain skill
they aren't going to magically develop it
and there are lots of things that are skills that we don't normally think of as skills
like just "exploring" a problem
the ability to work something out on your own and to be resourceful is a skill
there should be a balance
indigo grotto
sure I can totally agree with that
but sometimes you also have to have the patients to let people figure things out for themselves
meager bronze
(patience?)
indigo grotto
it many cases just telling the person the answer is just a form of laziness
meager bronze
I'm never suggesting just giving the answer
indigo grotto
you know because you don't want to sit and wait for them to get to the goal
winged urchin
I'd just add that the adversarial part I'm talking about is like this...
Sometimes when I take the approach of let them walk the walk, sometimes I find they just get irritated and I feel like they are just lost without any help. That's where I'd rather come in and show them a bit than have them see me as someone perhaps.. not putting in the work they paid for...
meager bronze
it's not about you, it's about the student
indigo grotto
rolf and yeah i meant patience
meager bronze
I think gemini is absolutely right
if the students just don't have the background skills
indigo grotto
yeah but thats about learning frustration tolerance as well
meager bronze
then they just can't do it
indigo grotto
you know experiencing frustration and not giving up
there needs to be space for that
meager bronze
like, imagine throwing a 10 year old into a calculus class and giving them an integral to solve
and being like "look you gotta figure it out for yourself"
indigo grotto
I know its hard and you can't let them stay with that too long
but its something that needs to be experienced
so it can be overcome
okay well that would sick
I dont think anyone in their right mind would debate that
meager bronze
like, for another analogy, if you think of "where the students are" to "where you want them to be" as a gap that needs to be jumped
if it's a small gap you should help them jump it on their own
winged urchin
I suppose I should clarify that... when I say "giving the answer" I don't mean... like... "Oh the slope will come out to 2/3" or whatever.
The 'answer' to me is a very detailed, step-by-step solution (often more than would be expected in a test question to really show the links between steps and where things are coming from) along with motivating statements/diagrams and speaking all the way through to add little bits and look for them 'getting it'
meager bronze
but if it's a huge gap, jsut saying "sorry you gotta learn to do this" is really just setting them up to fail
indigo grotto
yeah but you can turn that into smaller manageable steps
meager bronze
you, as the teacher who has experience in both math and teaching, should evaluate for each student how much assistance they need
indigo grotto
like instead of having to jump the whole gap they just have to do manageable jumps
but they DO have to jump
meager bronze
of course lmao
what are you saying
that is what I have been saying this whole time
sometmies you need to lay out the steps for them
or some of the steps
indigo grotto
I agree with you guys, its just I dunno
indigo grotto
Im trying to make the point that there is a really fine line here
meager bronze
I don't think the line is that fine
it's "fine" in the sense that it's easy to just give everyone all the answers
which would definitely be crossing the line
but like, it's not the case that there is a unique string of words that you should say to perfectly aide the student
indigo grotto
well no every person is different
winged urchin
I do think that "giving the solution" is sort of a dirty phrase... or seen that way sometimes
meager bronze
I try to separate the words "solution" and "answer"
indigo grotto
because in a lot of cases its a shortcut for (instead?) actually teaching something
winged urchin
Yeah... referring to my description of what "giving an answer" looks like... perhaps there is some difference in what is assumed in "giving the answer" =p
indigo grotto
yeah I mean after you explained it
its clear you mean something different then just handing out answer sheets
winged urchin
Yeah. I would add to that as we are finite beings in a finite world, we ultimately run into balancing issues of time/effort/etc...
Like... Yeah, if the students could fully explore the problem, play with it, etc etc...
There is an ideal way to learn, in a way, but sometimes you teach students who just don't have the time/interest/whatever to go down that route
meager bronze
or the background necessary to go down that route
indigo grotto
yeah and then they get lost
winged urchin
Or that, yeah
meager bronze
but that's jsut not true though
indigo grotto
its basically like an entry requirement if you're unlucky
meager bronze
just because you explain something to the students
doesn't mean they're "getting lost"
indigo grotto
no I mean if there isn't the time and they don't have the requirements
obtuse widget
yea, that's the big problem which I encounter while tutoring. I have no idea how to balance teaching material vs teaching how to solve tasks for exam, because a lot of time I feel that I'm paid to prepare for exam
and teaching actual material is obviously much slower
meager bronze
haha thanks for posting here because I think your name is the perfect example of what I see sometimes in calculus. I've had students who actually thought that a^b + a^c = a^(b+c) and similar algebraic "rules". And just telling them "sorry, that's wrong, now go figure out why" is just not useful at all
obtuse widget
my nick came after in like a week I've seen this problem like 3 times
on this server
meager bronze
students without much experience tend not to think of statements as things that have reasons, they just think of them as something that's true or false. I think one of the things math teachers/tutors can try to do is to "pull back the curtain" a little bit
and I feel like you (deekaan) are suggesting that it's better to let the students try to pull back the curtain on their own to figure out what's going on
but the issue is that lots of students don't even know there's a curtain
or they know there's a curtain but they don't know how to pull it back because they've never seen it done before
tall citrus
stone kite
stray furnace
limpid dirge