math-pedagogy

[turbid zenith]

I don’t like forced real life situations but I also don’t think real life situations are always forced. I’m always on the lookout for real life situations that would actually make sense to use a technique

Because I usually use them to introduce a topic in the first place

[long pelican]

The Folium of Descartes one in the pdf you sent was nice actually, but only if you ask where is the tangent line vertical or horizontal right away

with no guided steps in between

(obviously still guided, just not in the sheet itself)

[turbid zenith]

Well it seems like if I don’t ask guided steps what I get 9 times out of 10 is flailing about or going far into left field

[long pelican]

Yea and that's your opportunity to guide them in how not not flail about when thinking!

[turbid zenith]

I do. But then the next time they still flail.

[long pelican]

Without improvements?

Even a little improvement is welcome

[turbid zenith]

So I guess what I’m trying to figure out is if there’s something I’m NOT doing

[turbid zenith]

I do get a little improvement but only on that specific topic

It doesn’t seem like anything transfers

[long pelican]

Hmm do they not eventually come up with stuff on their own?

[turbid zenith]

Nope :/

“I’m confused. What’s the first step?”

[long pelican]

I guess that the problem was outside their "zone of proximal development" (not sure if I'm misusing psychological terminology here)

[long pelican]

Oh yeah, that's something you have to model not thinking in that way

Because they have had zero experience not thinking in that way

(Ok I guess you already do model that)

[turbid zenith]

I apologize that I’m probably pretty frustrating right now

[long pelican]

I don't feel frustrated at all 😛

How often are you being the one suggesting the correct next step?

vs. them coming up with it on their own

An idea might be to dial down the difficulty (and content-level) of the problems until it's something they actually can come up with viable methods for on their own

Even if it has to be dialed down so much that it just asks them to look at a function in a different way and no longer involves computing a derivative

and then you build up their knowledge from there

[turbid zenith]

It's usually me suggesting it because they just ... stop

Part of it is I don't think the group work is being done as effectively as it could. It usually just turns into students working alone even though they're all at the table. No discussion of ideas, etc.

So I'm looking into ways to make the group interaction better by assigning roles.

[long pelican]

I did one group at a time at the board while the class watched and gave suggestions mainly because I'm deaf and I would not be able to function with multiple people talking at once, but I also incidentally feel like it worked very well

I only had to give a serious input for the hardest problems

(Sometimes I wrote something 'obvious' on the board just to model that it's okay to write something you think is obvious)

It definitely requires the right difficulty level

It might be that with your students, juggling derivatives and a lack of foundations might be too much for them

[turbid zenith]

I've been thinking maybe I need to make the foundation stuff more explicit somehow. Like ... somehow predict what foundational thing they might have forgotten and give them deliberate practice on it beforehand, somehow

I have students who for the LONGEST time couldn't turn, say, (\dfrac{1}{2\sqrt x}) into (\dfrac12 x^{-1/2})

[burnt vessel]

DMAshura

[long pelican]

Missing foundation may be different for each student

Probably not worth it to have whole-class practice on something you prepare beforehand

Best way would be to find out what the missing foundation is with problems that touch a lot of different foundational areas

I think the function graphing one was exactly such a problem

which is why it gave you so much information

There is something you can predict the whole class is missing: namely, good mathematical habits of thinking, and what problem solving is

also to a broader extent, what math is lol

[turbid zenith]

Yep yep

I've tried to address that in the openings to a lot of my videos ... broader messages about what math is about

So I guess if nothing else they'll leave with that, if they remember it

[long pelican]

yeah me too (with lectures), but I gotta say they pick it up so much more efficiently when they are experiencing it themselves

[turbid zenith]

I wish they picked it up when they experienced it themselves in my case

[long pelican]

I think they need to have one satisfying instance of solving a tricky problem completely on their own

just one

"tricky" being relative to their level though

So like a 2 step logical deduction made independently counts

1 step is too similar to "retrieve this information from this box"

[turbid zenith]

Mhm

[winged urchin]

I love when there's been a big discussion in here. And it's from two of my favourite people here! So insightful to read through ^^

[turbid zenith]

❤️

[fathom hemlock]

Hey, hs student here, just got hired as a tutor, wondering what I should do to prepare. I'm tutoring a precalc student and they're studying sinusoidal functions, their graphs, identities, harmonic motion, etc... I've never done any tutoring before further than just helping friends with homework. Any advice?

[cold trench]

I'd recommend making sure you're clear on how to do the likely problems you'll be working through

the first time you tutor you will likely make some mistakes, you may quickly realize that a certain method of teaching or explanation doesn't work, so you also have to be flexible and open to adjusting your methods

also don't simply give answers, instead use socratic questioning in which you ask the student questions - this build their understanding

so questions like "how did you get to that answer?" or "why do you think this is true?" or "what do you think we should do as our next step?" are really good as it forces the student to think

that way, they come up with their own answers through your guidance

[fathom hemlock]

alright, thank you :)

[stable trout]

First year teaching circle theorems and man is at struggle. My students seem to understand the formulas fine and even got them to practice proofing some of them, but when ever they have to apply the theorems they can’t do it. Anyone got some mnemonics or tips on teaching this topic? Majority seem fine with applying one theorem but are lost when it requires more than one and/or application.

[long pelican]

@stable trout This part of the conversation seems extremely relevant to your question!

[stable trout]

This summarized my thoughts better than I could explain them. Sorry I didn't read above before typing lol. Yeah sadly enough the students I have are getting there first experience of braking this mold of compartmentalizing stuff just to remember it before the test then drop it the second the test is over with me. Seems like I am the first teacher they have had that does this. It just feels weird with this topic of circle theorems because of how many of them have hit a wall and are still struggling to get back up from the impact of said wall. The concept seems different than other units we have covered. Why that is, I am not sure.

[long pelican]

Something that could get you started is to think back to whenever you showed an example of you solving a problem... it was more than likely that the students took it as "When you see a problem like this, do this" and took no problem solving lessons out of it

[stable trout]

That's probably it. I just felt so bad on Friday when I taught it because I didn't have the ability nor experience to tell what was going wrong. I am spending tomorrow to redo that lesson with this foresight.

[turbid zenith]

https://www.youtube.com/watch?v=1ly-56jXnrg&list=PLZzHxk_TPOSv_pPkrhVfKC8ACpW82tXeS&index=24 Aaaaand that's the last video!

Trying something with this to make sure students watching it are still engaged even though I'm doing the proof ... because I wouldn't expect students to intuit the right step to do in each case, a lot of them might seem out of nowhere.

But the thing I'm trying is listing a bunch of potential reasons at the beginning, and saying "these will all appear at some point in the proof — can you figure out where?"

[long pelican]

"And that includes you" 😄

and that Lockhart reference too!

speaking of math and music analogies, I was thinking during dinner about an analogy between Chopin's Etudes (which are are both practice as well as beautiful pieces of music in themselves) and math problems, and between regular etudes (which are mainly for practice) and exercises

[turbid zenith]

Interesting analogy

Chopin definitely does seem to straddle that line!

[turbid zenith]

Man I really wanna do more of these videos but there’s no way I’m gonna be able to keep up this two per week pace XD

[cloud zealot]

http://toomandre.com/others-articles/engeduc/Zvon-eng.pdf

https://faculty.utrgv.edu/eleftherios.gkioulekas/OGS/Misc/ARUSSIAN.PDF

pretty neat

[lethal leaf]

Ooooo I'll give these a read instead of working on my final papers

[lethal leaf]

This first one is a great great read so far

[long pelican]

I love these Russians' way of writing, it's so fresh. Also reading these makes me have the thought that a lot of educational fads/research are backwards in thinking, but I'm not bold yet enough to actually claim this yet

[quasi musk]

A lot of educational fads are cyclical

If you look at math education over the past 120 years

It's strange because we are undoubtedly in the golden age of mathematics, more people do math, more people produce math results, more people know math than ever before

[violet sapphire]

true, but mostly because of the increase of population

Many people go to schools or are self-taught, are forced to learn mathematics, weather they like it or not, or they teach themselves because they find a great interest in it, weather it’s on a higher level or a lower one, they are still people learning maths, for one reson or another

[tribal tiger]

Yeah it's pretty crazy how far math education has come. My grandmother and I were talking and she said the highest she had to take was arithmetic and my grandfather was considered an advanced student at the time because he had taken trig and geometry (this was late 40s early 50s)

[jolly mulch]

I am writing a lesson for year 11 students (age 15-16) who are thinking of doing A-Level (16-18) Comp Sci or maths about logic & boolean algebra. How advanced should I go? (Note, some students will be there out of pure interest and will be as young as 13/14.)

I know some of them are very advanced, and can handle a lot, and most of that won't be covered in the lesson but I'll add it all to the notes I distribute. I'll tell them beforehand the notes are very in depth, and not to worry if some stuff seems really hard -- it's ok. it's just there to provide a wholly comprehensive look at logic.

Although I'll say some of it is extra and over the top, there must still be a limit. Here is my current list, what should I just remove altogether?

• Introduction
• Elementary concepts (and/or, implies, iff)
• propositional logic
• first order logic
• boolean algebra (formulae, laws, etc)

[tardy raptor]

First order logic

[jolly mulch]

probably wise, no need to mention it really. i can put it in a 'see further' section and note that propositional logic is sometimes called 'zeroth order logic'

[distant fractal]

So I'm finally teaching a class where I have complete control over the grading scheme. I was wondering what are some alternatives to written exams being a huge portion of the grade. This is a calc 1 course.

Last semester this was taught it was 40% midterms and 20% final. That's a huuuge portion in exams. I'd like to dial that back, but also offer two options. One in which the students take the tests and everything is graded as normal. But I'd also like an alternative for those who do not perform well on tests or feel comfortable taking them. I considered myself a horrible test taker so I sympathize. I just don't know the alternative. For me an oral exam would solve my problem with tests, but I can see how that would be even worse for others. It would also open me to bias accusations if there isn't a rigid marking scheme for it.

I also don't want to make the grade entirely things which can easily be cheated, like homework and take home exams.

I like the idea of projects instead of exams but I'm not sure what would be appropriate for calc 1 if not walking them through proofs of the theorems we learn. I'm not sure that would be appropriate

[lethal leaf]

Courses here sometimes do weekly / biweekly "examlets" where they cover less material (essentially unit tests). However, I've seen some of this just lead to "I will cram this week and forget it for the rest of the semester" which ruins them when a final comes at the end

if the class is small, oral exams would be a cool option

[distant fractal]

It's a class of 30

[lethal leaf]

I've also seen some classes do "corrections" where you can get X% of your missed points back by correcting your work and resubmitting

[distant fractal]

Oh yeah I've done that before and liked that

[lethal leaf]

I've always liked the "here are 8 questions, solve 6 of them" style tests

[lethal leaf]

this makes it so that a mistake on a mini exam weighs less on the grade but then of course they have more exams to do, but then again these exams would probably be shorter anyways

[long pelican]

I do exams but designed in a way so that 50% is a B and 75% is an A

[lethal leaf]

interesting

[onyx grail]

I’ve seen letting the final replace a midterm score if it’s beneficial (while still being part of the grade) being a good strategy to help people who may crack under the greater time pressure that midterms have

This has some cons like intrinsically adding greater weight to the final though

But I don’t think it’s extreme

[pale wasp]

I saw a syllabus where the final was optional, in that you could just take the grade with hw+midterm, or you could take the final and if you did better it would replace a midterm grade

(there were 2 midterm exams)

[lethal leaf]

the mini exam system I said has the benefit that if you do it like biweekly, that's 6ish mini exams? You could easily just say you'll drop the grade of the lowest exam

[onyx grail]

Yeah, dropping something or having pressure reduced in some fundamental way is the most important thing here I think

[tribal tiger]

My calc 3 professor did an interesting thing where each exam (besides the final) is worth 100 points but your highest exam is multiplied by 1.5 and your lowest .5 we weren't told until the end of the course but it was nice. We also had bonus points for attendance and effort we would do in class works which would be collected you'd be given corrections and we would get like 1 bonus point, you'd get less if you didn't try though

[lethal leaf]

Interesting

I kinda like that system

[cold trench]

from my ECO teacher's syllabus

you can use a similar system and replace one of the exams with an oral exam

this way, everyone does the written & oral exams but if they aren't good at one of the exam types, it's not detrimental to their grade

[merry meteor]

i like the idea but i also think you should always be up front about the grading scheme

[cold trench]

I agree

[wise onyx]

i did this for my class

20% HW + 30% Midterm + 30% Final + 10% Class Participation + 10% Quiz
Lowest HW grade will be dropped

[light glade]

Our math classes are typically 20% assignments (From 4-8 over a 12 week trimester), 25% each for a in-class test and 30% final exam

[raw cipher]

Lots of cool physics problems you can do with Calc 1 knowledge.

For lessening exam impact, one thing my Calc prof did was if you had an A or above, you couldn't take the final. Thought was it can only hurt you.

If you had a B- to A- you could elect not to take it and keep your score.

Removed a lot of end of semester pressure.

[distant fractal]

i like that idea

[long pelican]

I find the final is very important to have if your class is the kind where most students have never seen proper math before before coming in, because gaining the first steps to mathematical maturity takes time

[raw cipher]

I'm not convinced there's any real benefit for students in exams. I'd love to see some data from the ed side on their efficacy as a learning tool.

There's certainly benefit as an instructor. I can't adjust to meet my students where they are without some form of evaluation. Exams can help close that gap. I don't think they're as good of a metric as quizzes, homework, or exit tickets.

I really don't see value in finals. Data for me and nothing for the students. Especially those students who have already demonstrated B- to A understanding by week 14.

[long pelican]

Grad level math classes where you can basically trust every student to be honest on homework and to already know what doing math means, regularly have no exams and I like that

Can't say the same for intro level undergrad classes though

[winged urchin]

I definitely don't think homework is a good metric of their understanding. Students routinely collaborate for anything that isn't in-class. I think it's perhaps good for teaching the students as I think collaboration is good for that but yeah

[raw cipher]

I can see that. Homework is really "forced practice".

[long pelican]

Practice should be just one part of homework. The other part is that making insights while working on challenging problems is how you really learn the material in a lasting way

Lecture just isn't gonna do that no matter how good the explanations are

And if the extent of your engagement with the material outside class is just practicing skills, that's also not going to do it

[raw cipher]

reasonable characterization of homework.

[lethal leaf]

I have a loaded question

So I started a math and algorithms club last semester

and the goal has been pick a topic each semester and run like presentations on it

for people to come and listen to and learn about

it was pretty successful last sem but my main issue was interaction

no one wanted to ask questions

this also happens in the classroom so like what can I do about this?

how can I encourage more questions

[devout fable]

In the classroom I don't ask questions because I have no idea what the heck the lecturer is talking about past like 20 minutes into the class

[winged urchin]

At my school it was common for the organizing professor to be expected to ask a question of their guest

Perhaps if you're the main organizer of these then trying to have one question at least to open the floodgates a little to others perhaps?

[lethal leaf]

Hmmm ok

I'll try to get others who help me run the club to ask questions when I'm the one presenting

[lethal leaf]

Yea this seems to be an issue also lol

People get a little lost / confused

Which I guess is to be expected but also I would like to remedy that

[winged urchin]

Also also! Sometimes people ask after the talk itself, a little less stressful

So I would leave some time after the talk to just stand around and talk, drink some coffee or whatever aha

[lethal leaf]

This would involve me getting money to buy snacks / coffee lol

I wish

[devout fable]

Can't say much about how to remedy student confusion

[lethal leaf]

Also yes I encourage people to stay and talk (I'm usually the last to leave)

But we meet from 5-6 so people want to eat dinner

[devout fable]

How long is each talk?

[lethal leaf]

30ish minutes, sometimes more

I try to have the presenters include some time for example problems they have

[devout fable]

Hmm if it's a new field for the audience, then I can't imagine they would be able to understand much

it would be hard to introduce something new and also show interesting results in only 30 minutes, unless the topic is pretty simple

[lethal leaf]

Oh there's no interesting results

These are all like introductory talks

[devout fable]

Oh like what's an example talk

[mint lark]

Try and ask the department. They love to fund stuff like that generally

[lethal leaf]

Here are some slides I made

[lethal leaf]

Hm true. I'll see what I can do about getting some funding

[devout fable]

I guess one (is this extreme?) thing you could do is to ask the speaker to prepare some notes before the talk and the audience could read them prior

Oh I don't know anything about computability theory 😅

[lethal leaf]

We post the slides before hand but I don't want to give people HW 💀

[devout fable]

Yeah that was probably a very unrealistic idea lol

You might get some good responses by asking on math education stack exchange @lethal leaf

[lethal leaf]

Oh true

I'll ask there

I mean I'm less worried about people not understanding

And more worried that when people don't understand

They don't ask questions

[cosmic ibex]

Hmm, how about rotating the "session chair" duty of asking a pity question among regular attendees?

[lethal leaf]

Yea I plan on doing that

I think someone above mentioned a similar suggestion. It's a good one to get the ice broken

[pale wasp]

I think it can help if one of the organizers asks simple questions during the talk, like even just clarifying questions if they think something may be confusing to the audience

then people in the audience may be more willing to ask questions when they don't understand

[grim spindle]

have you approached your department for funding?

oh this was already brought up

it's always good to ask

[lethal leaf]

So uh

it's not really through the department

This is a school club aimed at teaching math / TCS to people, and it's exists as essentially as a sub-organization under this massive club at our school

so if I want more funding it would have to come from them and they're a little tight on funds

not sure how getting full department funding would work

[grim spindle]

idk about full funding

but you could also ask the head of the club to approach the dept for funding

departments like funding student-led events even if they're not officially organising them in my experience

[lethal leaf]

I am the head of the club 🤡

[lethal leaf]

Hm I see

[grim spindle]

of the massive club?

[lethal leaf]

oh no not the massive one

I think they already get some funding from the department

[grim spindle]

you could ask them to approach the department about your initiative and see if they can get more funding specifically for your thing I mean

[lethal leaf]

Hmmm maybe

I'll look into that

[grim spindle]

It could very well be your dept isn't even aware of this thing

or the people that have control of the money aren't at least

worst case scenario they say no. Anyways this discussion has escaped the confines of pedagogical discussion so if you wanna keep discussing this (I am the head of the student council at my dept so I may have some insight) we can go to discussion or dms

[lethal leaf]

I think I'm fine

I will try next sem to implement the stuff about trying to get the people who run the club with me to ask questions to break the ice

and hopefully that helps

[carmine flame]

solve iota for iota?

[steep fox]

Is there a follow up to a Mathematicals lament by lockart?

[long pelican]

Yeah there’s something by Keith Devlin

https://www.maa.org/external_archive/devlin/devlin_05_08.html

[turbid zenith]

Anyone have plenty of experience with WeBWorK?

https://twitter.com/solidangles/status/1610377155359264769

Bill Shillito - now at @[e-mail] (@solidangles)

Anyone who's familiar with #WeBWorK: Is there a way to get fractions to not turn into decimals in a Formula object?

I'd like it if Formula("1/2*x")->reduce() stayed as ½x instead of 0.5x.

Twitter · Jan 3, 2023, 8:46 PM

[cold trench]

I can't wait for the semester to start so this channel becomes more active lol

[lethal leaf]

I got some content for this channel

anyone have tips for gauging how long a presentation where I'm teaching material will actually take?

I feel I vastly overestimate or underestimate

[cold trench]

just do a full rehearsal, pretend you're in the classroom with the students

and also account for students asking questions

[quasi musk]

I usually do a 3-5x multiplier

if it takes me 5 minutes to get through all the major details by writing down at full speed

It'll probably take 15-25 minutes to explain

Even technical definitions can take a long time

I remember doing a presentation in knot theory where I just tried to speed run a 15 page paper

And my friend in the audience was still stuck at the definition before I started anything

I could have done an entire presentation just building up intuition for the result, and spent a minute or two hand waving my way through the technical details

[winged urchin]

Also I always have a couple extra questions that I can go through in case I get through material quicker than expected

[meager bronze]

my advice for your first class is to just prepare for a whole week and make notes of good stopping points

[tribal tiger]

I think part of it depends on you as a person personally I know I tend to talk fast during presentations so I try to overshoot a bit but leave enough time for questions and parts where people look confused and I'll have to be more thorough

[wispy slate]

does anyone know any online non-profit organisations which accept math tutors to teach underprivileged children?

[strange lynx]

Anyone know if there is some online library of Europe math books for all pre university education? Want to do some comparisons with how they teach it directly from the exercises and in what order. I mostly find books to buy but I have no idea how often they are used or if they switch in the middle of the series in some schools.

[noble hare]

A nice summary of ChatGPT and its education implications I was emailed that might interest some of you

[rotund sandal]

I’ve never given a class but please please don’t go without that week of preparation. I did that for a talk and I ended up being way too ambitious for the timeframe I had so it was a disaster

Make sure you’ve got something coherent that works in the time you’re given and leaves space for questions

[meager bronze]

That’s why I said make notes of good places to stop

I always have little annotations in my notes of like “if youre running out of time, skip x and y and go straight to z”

Think about that beforehand so you dont have to think of it on the spot

Also, people can tell when you are just rushing to get to the end, so if you prepare ahead of time what youre going to do if youre short on time, people wont notice and you’ll seem less rushed

[tawdry venture]

do any of y'all know a better name for the rule $\int f(kx) \dd{x} = \frac{1}{k} F(kx)+C$ from integral calculus other than ``reverse chain rule''? it pains me to see people misinterpret and overgeneralize that rule so much

[burnt vessel]

Ann

[lethal leaf]

it's a substitution

I don't know why it would need a more specific name for the fact that it's a constant

rather than something like f(g(x)) in general

[shadow flower]

instead of calling it a rule i learned it as how to “undo” the chain rule but that could be interpreted as the same thing i suppose

[gray smelt]

there's this student of mine (i tutor sometimes) who struggles with a lot of the algebra 1 and arithmetic stuff (my guess is that she got lost early on and a combination of too-nervous-to-as-questions and teachers not being patient enough led to her not learning a lot of the stuff beforehand). however, it seems that if i give dressed down versions of the grown up versions of things (e.g. she didn't understand that 0*x = 0, but did understand the proof of that statement in any ring (the dressed down part here is that i didn't give the definition of a ring).

the astounding part that happened today was with polynomials. she just didn't understand what a polynomial was...but after explaining linear combinations and spans/spanning sets/basis vectors (hiding the actual definition of a vector space), she now knows what a polynomial is.

really makes me wonder if there are a bunch of other students like that

[brave trench]

@gray smelt Are you teacher?

[gray smelt]

no

[cloud zealot]

If it were a requirement to construct each number system from the naturals to the complex numbers in a real analysis class, when would be the best time to present this material, and why?

also, i've heard of people using either dedekind cuts or cauchy sequences to construct the real numbers, but i'm curious if anyone has adapted moschovakis' construction of the real numbers that mixes both dedekind and cauchy's method. it can be found in his book Notes on Set Theory

[cold trench]

this is actually extremely interesting. for your first example regarding rings, what specific things did you mention?

[gray smelt]

i mentioned associativity of addition and multiplication, the distributive property, and identities. i then did the following:

x = 1*x = (1+0)*x = 1*x + 0*x = x + 0x. So 0x is an additive identity. but if a, b are both additive identities, then a+x=x=b+x means that a=b. so 0*x = 0

[gray smelt]

perhaps after the first 10 or so proofs that used the completeness of the reals, as then the students will see why that's the property you want to construct the reals to have. to me, this is also why the cauchy construction of the reals as completion of the rationals under euclidean distance feels more natural

[frosty pewter]

idk if anyone has a more informed theory on this but I feel like maths is taught horribly at lower levels. Stuff is sometimes given with little explanation - instead of explaining something and then talking about mnemonics and computational tricks, early on it felt like they just went straight to the computational tricks which left me unable to tell why things worked. Like I remember being taught FOIL and grid methods but not that you're just writing (a + b)(c + d) = a(c + d) + b(c + d)

though I find the example you give quite extreme and impressive on their part? lol

[austere inlet]

yes, I think a lot of people benefit from knowing why things work rather than just memorizing a technique or algorithm or getting a surface-level understanding of the subject

[austere inlet]

I think I had a similar experience when first introduced to proofs; when learning math in HS and freshman year things never seemed to click for me and I made very dumb mistakes, I even once wrote a/b + c/d = (a+b)/(c+d) in 7th grade hahah. I never was particularly good at HS math, but as soon as I got a gist of why things work or are defined that way (which in this case would be e.g. the proofs that 0*x=0 and the slightly different definition of a polynomial) it started making sense in a paradigm-changing way pretty much -- this was what made me decide to major in math

[frosty pewter]

ya i didn't know how to add fractions properly when i was first taught @ 10 or 11

[gray smelt]

i've found that clearing up algebra confusions by deriving most things does work for multiple people. the polynomial one probably wouldn't work as well for the others that i tutor

[tepid smelt]

I actually think most students don't care and it distracts most from just being able to solve a particular problem. I have encountered few students who care about the why where the vast majority just want a technique to apply to solve problems. I still of course will continue to show the why but teaching will depress you on how many students could care less.

I will say students do enjoy solving non routine problems but it has to be just beyond what they are comfortable doing. I do feel at the secondary level their should be more of a push to give harder problems. The textbooks do try to show through exercises and questions why things work but the problem sets are not challenging for many students

[lethal leaf]

I've had multiple students tell me they don't care why something works they just want the points and to do well in the class via pattern matching

[strange plover]

Am I curious --- has anybody seen or designed a successful first-year calculus curriculum, geared for science and engineering students (not pure math), that does NOT spend large amounts of time on closed-form or algebraic manipulations, e.g. analytic evaluation of antiderivatives or definite integrals, clever changes of variables, calculations by rearranging expressions, etc?

[long pelican]

I think I can say I have, just this past semester

I think it was hugely successful but still about 1/3 to 1/4 of the students struggled to an extreme extent with it

[strange plover]

do you have any general materials concerning that course that could be shared, e.g. a syllabus, or even just a textbook that was used?

[long pelican]

I used Strang's calculus (integrals, applications, probability) for the first half and the textbook I used for the 2nd half of the course was written by another professor in this department and I think I would need to ask his permission to share it here

Here's a syllabus

Final review sheet

Final exam

[strange plover]

that is more help than I expected already, thanks

[long pelican]

You're welcome!

[tepid smelt]

Thank you for sharing. I always appreciate the great problems you come up with.

[gray smelt]

especially that last problem

[prime kindle]

yo, i recently got a job teaching math to kids, does anybody have any tips for explaining mathematical concepts or just explaining my thought process in general?

[vagrant meadow]

i've been tutoring for about 3.5 years (which isn't that long compared to most of the regulars in this channel), and that's still something i struggle with. it does get better with practice, though. the more students/people you work with, the better your explanations get. you can also accumulate more and better analogies for things. pay attention to the way your teachers teach, too. i've stolen borrowed a lot of explanations/analogies for certain concepts from a number of my professors. youtube videos can be good resources too.

one thing i'd be careful of is to try and keep it simple, especially for kids. sometimes i use more advanced techniques for basic stuff (like matrices/linear algebra for a basic system of equations), and it generally only confuses them if you even mention it.

[cold trench]

why would you use matrices for a basic system lol that sounds inefficient

[hollow kernel]

It's not efficient, true, but I see why you would try. Solving three equations at once purely arithmetically is less less visually intuitive then thinking about a linear transoformation and searching for those vectors transformed onto some other.

[cold trench]

well this is getting off topic but I see your point

[shadow flower]

a student trying to solve a system of 3 equations in an algebra class isn’t going to be able to understand things in terms of a linear transformations

there’s not enough mathematical maturity to explain it in that way

[covert plaza]

I've kind of realised by helping people around, and also by talking about what makes me enjoy helping people in maths with my gf, that I do it by passion. She had some arguments against the concept itself, which is about the person helped potentially not respecting the helper for the time that the helper is giving them freely. Money was brought into the conversation as well. I personally want to be a teacher after my Master's degree in teaching. And I used of my personal time to voluntarily help people because I truly wanted to, because I felt like I could help the person and it felt satisfying to do so, it made me have a smile on my face. But I sometimes did it when I had exams afterwards. On my side I know the reason, it's because I was scared of these said exams, helping voluntarily was kind of like an escape from this fear of getting my head back into the maths I had real trouble with.

So my question arises from this discussion :
Where is the great balance for a voluntary helper between the passion he has for it and the job he will have in the future ?

[vagrant meadow]

cramer's rule is fast for a 2x2 system. or i multiply by the adjugate. less brainpower and scratch work required, at least for me. helpful to quickly check their work to see if they got the right answer.

[vagrant meadow]

i have definitely felt the same way. but i used to go too far helping others which was really draining (i still do sometimes). i've been guilted into tutoring (which ended up being more like teaching) classmates for free, and i never feel good afterwards (note that there is a distinction between tutoring a classmate and studying with/helping them). it's important to establish boundaries and take care of yourself, especially if you're not getting paid to do something people often pay a lot of money for. the feeling of satisfaction from helping someone doesn't always make up for the mental toll it can take. helping people with math isn't easy! it requires a lot more energy and thought than just solving the problem yourself.

i'd say if you feel like you "gave more than you had" after helping someone for free, that's a sign you should have said "no" earlier. remember that you have no obligation to help anyone for free, even if your passion for it makes it not feel like work. it sounds harsh, but self-care is way more important. it takes experience to establish a sustainable balance, but it's something you can figure out in time.

i will say that i have also found myself helping people on this server instead of studying for my exams/doing my homework haha. there are worse ways to procrastinate than honing your ability to explain and help people with math. i think it's okay as long as you take care of yourself and don't feel incapable of doing the stuff you need to do afterwards.

[covert plaza]

Thank you a lot, this is exactly what I needed

[winged urchin]

As a funny store related, when I first started tutoring I was still a 3rd year undergrad and I kid you not, I asked for only $5/hour and STILL felt bad about it ahaha

Thankfully I've always enjoyed helping students or even my classmates. As long as it's a course I'm very familiar with

There has been times when I've... Stepped out of my depth a little and tutored something I didn't feel as solid with

That's not fun and I do not recommend ahah, but sometimes you have to try to see I suppose. Eventually you build up experience to be honest with yourself

[gray smelt]

i'm pretty new to the whole teaching others thing, how do i encourage students to ask questions when confused? some of those that i tutor have the habit of asking when confused, but some don't.

if i explicitly ask the not-as-asky people things, they will then be much more likely to ask a question (e.g. "well, why is this true?" or (after defining "x") "so, what is an 'x'?" will prompt things like "why is [subpart of thing] true though?" and "what does [subpart of definition] even mean?".

however, i am pretty bad about remembering to do that consistently, and about knowing when is optimal to do that. it would be good if the people would ask unprompted or with less prompted (e.g. with the "does this make sense?" kind) these questions. how do i do stuff to make that happen more?

i am not sure if the students in question don't notice when they are confused, or just don't want to ask their questions. (i would like to think the latter is false, but it is a possibility)

[winged urchin]

Get used to asking questions. In some sessions the majority of what I say to the student are questions

Also if you do the sessions in person or online and they have a tablet then you can also get them to work through problems and just watch how they work through the problems

Sometimes I'm watching for blatant mistakes and sometimes I'm watching for poor notation or steps where they might be just pushing symbols without understanding what they are doing

Though even if I see a blatant mistakes I won't always remark immediately and instead wait to see whether they realize their mistake. Sometimes their answer is ludicris for the context which is what I'm really trying to get them to notice

To be critical of their own work as they are working. Thinking about their answers

And other times it's not blatant mistakes but I'll ask them how they think they got from line X to line X+1

If they use 'slang' like 'i moved the 3' or 'i cancelled the 2s' i may try to press them further and eventually show them what they mean when by the slang

[vagrant meadow]

seems difficult for teaching an upper level class, though. most of my professors lecture the whole time, and we still end up having to skip a lot of material.

[vagrant meadow]

had a professor that did that last semester, and it was actually really uncomfortable. everyone working individually and silently and him coming and looking over your shoulder. it probably would have been better if he had asked us to get into groups.

[hollow kernel]

I'm tutoring a Calc 1 exercise group of roughly 12 students this semester - it's been pretty cool so far, have had the feeling to be able to help their understanding and learn to improve my teaching aswell. - But the last few exercise sheets they had to solve were both super long and rather technical, so they're mostly confused now and stopped submitting most exercises. Really don't know how to deal with it, because honestly even I don't see how to get the idea of most computations. I feel like the professor wanted to foreshadow a lot of cool analytic number theory and analysis trivia, and I'm struggling with that even as a tutor. They just started with differentiation, but the tasks are on... Minkowski inequality, prime number density, and stirling formula?
How do you deal with this kind of situation? I am overwhelmed by the technicalities myself, but still somehow want to help them understand the topics and exercises.

[long pelican]

All that in Calc 1 damn…

[winged urchin]

I am predominantly a 1 on 1 tutor so I'd hope me sitting beside them and looking at their work is a little less intimidating than professor walking around leaning over you every now and then ahah. But it is something I keep in mind

But yeah, second Icy. Wow that is a BEEFY Calc 1 course

It might feel like you're outing yourself somewhat but the professor does have an obligation to give his TAs sufficient direction to help them be effective

So if you feel out of your depth or even just a little confused with what the professor intends with the assignment then absolutely you should talk with them

I'd certainly like to know if my students were starting to dis-engage due to technical obscurity or just general confusion ahah

Oh! And of course do not be afraid to say you don't understand something in my opinion. Ideally you follow that up with reassurance that you'll look at the problem/concept later and come back or email/message/whatever with an answer

Though of course that might flow over your allocated teaching hours and all that... but it's certainly worse to try and explain something you aren't confident with and potentially mislead your students

[vagrant meadow]

once i was assigned as a course embedded tutor for linear algebra when i was originally trying to sit in on the class, since i didn't really understand it all that well. and it was a very difficult professor who had very difficult problems kind of like that. it's tough because, as a tutor, you want to give your students confidence and trust in your ability. most of the time i was able to feign confidence, because the subject is relatively intuitive to me, personally, and usually i was on the right track. but when you don't know... then you don't know. best to just be honest with them.
"these are extraordinarily difficult questions for a calc 1 class, so i'm not 100% sure how to solve them. but let's try to figure them out together"

also ask the professor if you can for help, and for any topics you should look into that may come up on future exercise sheets. also, just giving the prof feedback that the questions are very difficult and the students are struggling with them is good. if the assigned tutor is having a hard time, then that's a sign that the difficulty settings are probably a bit too high... definitely don't give up though!

[tacit adder]

Ah, this hits home for me because there have been plenty of times where I've been assigned difficult courses to tutor/teach and even to this day, it's still been a struggle for me. As other people have said, please do not hesitate to admit that you may not be able to do the problem(s) -- students would rather you be honest about it than to try and force yourself and the students towards a solution that's unsatisfactory. If you can, sit with the student(s) and bounce ideas off of your students in the event that you're stuck on a particular problem. That's usually my goto strategy, and more often than not, they tend to hit a eureka moment along the way.

If your professor prepares tutorial sheets in advance, it's absolutely okay to ask your professor if you can have access to them in advance so that you can prepare -- either by doing the problems or just skimming through the problems yourself. In fact, your professor should be very willing to give you the tutorial sheets in advance (and possibly with solutions). If your professor prepares tutorial sheets on a whim, then have a look through it before you begin your tutoring if you can. It'll give you some sense of what to expect when taking a tutorial. I've had professors who gave me the tutorial sheet 1 hour before my class and, especially with the more difficult concepts, it's definitely a rough situation -- but even just looking at the tutorial sheets will give you slightly more confidence than not having looked at it at all!

Above all, don't stress too much if you find yourself thrown in the deep end. Be truthful to your students and your students will understand -- use it as an experience to collaboratively work through the problems instead of instructively. And if you can't finish a problem in the allocated time, take some time to think about it and let your students know that you'll (try to) solve the problem for the next tutorial session. Good luck, and hope everything goes well! It's a tough situation but you'll make it through :)

[vagrant meadow]

so a student came into tutoring today. she said it was her third time with her professor taking algebra (both in person and asynchronously! this time is asynchronous). she asked for help with her online homework, and oh my god. i just can't believe how awful asynchronous math classes are, and how lazy the professors get. the online math homework had 44 problems, and each one had about 4-15 parts to them (one actually had 15). she said she's a math major (wants to be an actuary) and she's considering dropping college altogether.

it really seems like she's trying. she's nice and funny, and it's just really hard to see someone like that set up to fail. i can seriously imagine my love for math not being sparked if i had taken shitty online math courses. enrollment in the math program is significantly lower at my university. is it just me that thinks the increasing popularity of asynchronous math classes is going to be incredibly detrimental?

[lethal leaf]

44 problems with multiple parts???

jeeeeez

[vagrant meadow]

one prof i talked to said that colleges are offering/outsourcing online math classes because pass rates are higher than in person classes. but the people taking the class end up not learning the necessary material from it.

[lethal leaf]

yea I know alot of people who prefer online async for that reason

(tbf that's why I did an online async spanish class over the summer to get out of a foreign language req)

async classes seem like a godsend to lazy professors but they seem to add so much more work to profs who want to actually teach

[shadow flower]

my mom teaches at a community college and her in person classes aren’t getting filled though

the students are choosing to take the asynchronous classes instead of in person

i don’t think the professors are necessarily choosing for the classes to be online, but rather the demand for them to be online is higher than in person

[tacit adder]

44 problems, each of which containing 4-15 subparts?

or 44 problems altogether including subparts?

[shadow flower]

the latter

[tacit adder]

ok that makes more sense. If it was the former, I'd be dead by the time I get to the end lmao

[shadow flower]

oops i reread what you said, the former sorry.

too much is the point

[tacit adder]

that's an unreasonable amount to give to the students

[vagrant meadow]

i am 95% sure there were over 100 subparts total, and i wouldn't be surprised if it got to 130-150.

[vagrant meadow]

yeah the math classes at my uni aren't getting filled either

[vagrant meadow]

and i've totally done the same. but i worry for all the math majors who will never be because of this online stuff. apparently high schoolers did horribly on the last standardized test. it's hard enough to get a motivated college student interested in math, how the heck are you going to get high schoolers to care when they're online? ugh idk

[shadow flower]

my college is lacking math majors all together

we have 4

3 seniors and a junior

[vagrant meadow]

holy shit that's it extremely small. how big is the college?

[shadow flower]

1300

2 pure math professors, one who is pure math but taking more leadership roles, and then one that doubles as math and cs

[tacit adder]

all of our math majors pair their degree with another degree (me included) and then we go off to do something either adjacent to math or just completely drop off math after year 3

well, not all but most

[shadow flower]

that is how i ended up being the only one taking abstract algebra

and the junior will be the only one taking real analysis when it is offered next year

[tacit adder]

mmm, our university doesn't really offer too many undergraduate pure courses sadly so I end up taking the graduate level courses and get completely obliterated lol

our undergrad pure is basically: a first course in algebra, a first course in real analysis, a first course in topology and diff geo

and then everything is basically left to postgrad

[shadow flower]

we have like intro to proofs, lin alg, abstract alg, diffeq, real analysis, geometry, calculus based statistics, numerical analysis, and then a research class

[lethal leaf]

there's a huge selection bias to math majors being the ones that skipped through calc in high school

rather than doing it in college

and high school has more individual instruction

[prisma hearth]

I'm assuming non-proof based algebra?

I guess D&F does have a hundred or so exercises per chapter but I can't imagine that many problems all being assigned as HW

[hollow kernel]

@winged urchin @vagrant meadow @tacit adder I want to thank you three kindly for your detailed answers, they helped me a lot in feeling better. 😊 I had a talk about it with my students this morning, and they decided they'd much rather have more time to talk about their questions regarding the lecture and examples for definitions and applications of the theorems, than go through every of the exhausting and technical exercises. So I'll focus on discussing those exercises that help understanding the lecture and leave out the rest for a more interactive tutorial, which I highly welcome anyway. 🙃

[hollow kernel]

I applied to tutor Calc 2 next semester with the same prof, so I'll want to address this to the professor soon, in hopes next semester the tasks will be more inviting to make the students actually want to try them at the very least. (Yes, also to make life less frustrating for me, secondarily, hehe.)

[severe walrus]

This is something I’m pretty unhappy about. I’m attending a cc atm, and just took intro LA last semester and will start Calc3 in a few days. Last semester I signed up for in person classes for LA, but because not enough ppl wanted to go in person, they canceled the class and made it asynch. Even though I got great grades, I learned next to nothing (I will admit though, I didn’t utilize the internet as much as I should have, but still, it was just not a good experience). The class consisted of textbook, homework assignments, and tests only, no lectures or anything. Definitely not the way I like to learn. A few days ago, I received notice that my calc3 in-person class had also been canceled. I so very much wanted to study it in person, but rip I guess, no point getting upset over it now. Hopefully its better than my experience with asynch LA.

[shadow flower]

i completely agree. i took diffeq online bc of covid and it felt like just memorization of the methods instead of actual understanding of the theory, which is why i’m auditing it rn in person.

[severe walrus]

Oh yep ikwym, also I’ll see if I can do something like that as well

[gray smelt]

this exact kind of thing made me legitimately dread math for a year. my grade school wouldn't let me move up to the algebra class, so for some really fucking stupid reason thought having me do it online during the usual math class would be fine. teacher wasn't to blame here - she just didn't have class time to teach me!

(i'd later not hate online classes, because i'd later become interested enough to just find another resource to self teach from)

[silver blade]

isn't "memorization of methods" just 99% of first courses in diffeq anyways? Unless you go to an elite enough school that treats pure math properly 🤷‍♂️

idk at my current school (generic state school) I took the honors section of diffeq in-person and it was still just that

meanwhile at Caltech they had an "analytical" section of diffeq that was a mix of undergraduate and graduate level diffeq which gasp....actually required you to do proofs!

Too bad my crippling bipolar depression forced me to leave that environment after two trimesters cause I really enjoyed how they treated math courses over there ;-;. But yeah, I've seen similar things from looking at the syllabi of schools like Princeton and MIT. At most schools though there aren't even enough undergrads who give a shit about pure math for there to be a reason to offer these types of courses. I guess trying to skip straight into their graduate version of diffeq is a possibility though? Tbh I tried that but my advisor at current school wouldn't let me lol

[shadow flower]

well maybe. but personally i’d rather be able to derive the methods and “formulas” from scratch if need be, because then at that point i’ll remember it forever, if needed.

[tribal tiger]

I can't blame professors for teaching it as memorization since most people taking it are engineers who will forget the course within a year

[silver blade]

the worst part was that my "honors" section was for math and applied math majors only

and it was still all memorization

worst math class i've ever taken

[lethal leaf]

Also teaching memorization is easier

Also if you ask a student what they want, understanding vs memorization

Alot of them will say they want to understand

But then they'll say they understand through rote memorization

Or if you actually ask questions that test understanding they'll complain it's too hard and not applicable

[long pelican]

Haha yeah 100% the above, unless you convert them

[shadow flower]

i guess we’re in the minority then

[silver blade]

luckily for me i had enough credits from high school and Caltech that diffeq was the only non-proof based class I had to take

unluckily for me the math major at this school has very few degree requirements and financial aid only applies to "degree applicable units"

so i technically ran out of math classes I could claim financial aid for last semester

I had to do some financial aid backflip tricks to get aid for 3 of the classes I wanted to take this term and I had to give up on taking complex analysis ;-;

i don't even know what i'm gonna do next semester 🤷 this trick I did will only work once

My only options are to either get enough scholarships to cover tuition, return to Caltech (which apart from its fun math classes is an otherwise hellish school that destroys my mental health even more than it's already been destroyed by my genes), or i guess not go to school at all, apply to grad schools and treat next year as a gap year

[zenith slate]

A bit late but I've not found any of this to be true in my experience

It depends on the age of the students to a degree

[lethal leaf]

I've found this to be true a nontrivial number of times

But I TA a discrete math for CS course and many of them just wanna program

Which is understandable

[zenith slate]

Yeah

I find this to be more true with undergraduate students

Though I think this is to be expected

The pressure is much higher for UGs versus HSers

[lethal leaf]

For sure

I totally see where they come from

[zenith slate]

And memorization by rote is a much safer option when for whatever reason you find yourself unable to really understand something

Lord knows I simply rote memorized my way through several undergraduate modules

But I think the idea that people want to learn things by rote versus really understand things doesn't really ring true

I think it simply arises out of necessity

[long pelican]

If you have not found any of it to be true in your experience but you have found it to be more true with undergrads, does that mean you have mainly worked with high school or younger students?

[lethal leaf]

The want comes from self awareness

They realize it's easier to an extent

[zenith slate]

I've taught undergraduates for 6/7 years

[lethal leaf]

And just gets them more of them than not a good grade

So they optimize for that

[zenith slate]

Highschoolers on-and-off for the same period of time

[lethal leaf]

Because a lot of early math classes (calc, numerical Lin Alg, intro diff eq) you can do alot of rote memorization and it's easier and gets you a pretty good grade

And only bites you in the ass if you go further into math and often they (engineering and CS types) don't go further into pure math

[long pelican]

They lose whatever opportunities they could gain from learning to think mathematically

(But often times the instructor for the low level classes teach it like memorization and don’t try to teach mathematical thinking anyway, in which case their desire for memorization is what the professor wanted)

[left vault]

IMO if you are up-front with your students about your expectations & exam conditions and respond in an understanding way to their concerns you will have students who love you

The only time when I've had bad experiences as a student and a teacher is when student expectations are not met

For example, if you walk into an exam thinking it's going to be very computational and it's very proof heavy, that will leave a bad taste since your grade doesn't reflect the effort or ability.

You're never going to be able to make students "not care about grades" and "focus on understanding" since students have other classes where grades are more central

[left vault]

It's a bit of a contrarian take, but I don't like assignments very much for this reason. Since I don't think they set expectations well for exams/tests

Lots of small weekly graded quizzes (at the same difficulty level as the exam) easily beats assignments imo

[topaz steppe]

Hi 👋 ,
I develop a math worksheets generator https://mathrelay.com

Unfortunately, I am not a teacher and I need a feedback from you

• does this worksheets generator seem useful for you? if not what can I change?
• what type of exercises do you need?
• what functionality do you want?

If you can help me here is the site channel https://discord.gg/eFzgzeXQAC
Thank you!

[left vault]

looks quite comprehensive, but your competition is the worksheets I made last year

tailored to my teaching

which is the problem with all these worksheet generators lol

also probably competing with massive online question banks (which have interesting word problems too)

I wonder if you could use chatgpt to generate word problems

[topaz steppe]

Hi, thanks for your feedback - could you post links to online question banks, please?

[long pelican]

The first question I saw already exceeded my expectations — but they are still confined to algebra with a fixed set of instructions or question types. ChatGPT technology might be able to help generate more varied types of problems pretty soon though (see math contests for what I mean by variety, but even math contests are “practiceable”)

[topaz steppe]

Ok thanks, I will check ChatGPT

[long pelican]

Oh no, oops I meant in the future

[topaz steppe]

ah, ok - so, in general teachers need world problems first? @left vault @long pelican

currently, I look at the solutions proposed by teachers on Youtube

and try to group them on the site

for example, here are world problems https://www.mathrelay.com/topic/algebra/word-problems

[long pelican]

Oh wait, it looks like these aren't generated algorithmically because each one links to a youtube video about that exact problem

[topaz steppe]

this process tooks time - so I would like to get a direction from the teachers

to collect useful exercises

[long pelican]

Hmmm ok here's a concrete idea: logic problems
Is it true that for all pairs (x,y) P is true?
Does A imply B?
etc

[topaz steppe]

ok, I see - do you mean Propositional Logic ? something like https://www.youtube.com/watch?v=6FPpv_A8GpE

[long pelican]

Nope I don't mean that, I mean related to algebra/geometry etc

I also want to clarify that I really mean problems that require logic to solve, not "logic problems"

Ok I came up with an example:
Supposing you know that 1 + 2 + ... + 100 = 5050
Use this to find out what 1 + 3 + 5 + ... + 199 is

The logic here is writing the sum of 2n-1 as twice the sum of n, minus n

[topaz steppe]

does this make sense for you? https://www.mathrelay.com/topic/algebra/find-expression-value

[long pelican]

Hey those are the problems that caught my attention in the beginning, which is good, but is only one example

[topaz steppe]

I think, I have a problem with my main page

Currently, it shows last added problems

Many thanks! I will rethink the presentation of the main page.

[thorn lantern]

e

[topaz steppe]

Ok, I redesigned the homepage to show the topics https://www.mathrelay.com what do you think?

[cloud depot]

Hello everyone, how does this channel work? I’m currently teaching trigonometry to high schoolers

[cold trench]

do you have any specific questions?

[tardy quartz]

teach me too

[cloud depot]

Like in general, I feel like the students have a lot of gaps in their foundations, so even for example, rationalizing a denominator, not all of them know how to do it

So idk, I guess how you can deal with these differences in backgrounds during class, while not getting the test of the class bored, because I have noticed there are some others who actually get everything quite fast

[long pelican]

Very true, and before forming solutions must first ask the question of what you want all students to get out of the class

For me I want all students to experience math as a creative activity, while someone else might just care that they understand the material on the syllabus

[stoic python]

This class I am teaching has gotten the dubious honor of having test scores 20% lower than the other section again. I have never taught a course where the average student was so unpleasant to work with.

This is multivariable calculus for engineers, but due to the way this class is filled, it is essentially a runoff section for people that barely made it into the program after better students turned down an offer. Their non-mathematical antics are giving me grey hairs.

[cold trench]

well u gotta explain shit in a way they better understand

[cloud depot]

So if you have one high school student who stands out in class, and actually wants to pursue a degree in math, is it okay/ethical I guess to push a little harder that student?

[lethal leaf]

Push then harder how?

[cold trench]

hold them at gunpoint when they answer questions

[lethal leaf]

[winged urchin]

That seems totally fine. Yeah. Do it

Pressure makes diamonds... Right?

[turbid zenith]

Any suggestions for what to do with a Calculus II class on the first day to ease into the semester?

Especially some kind of group work to set norms

[austere inlet]

grading and coursework should be the same for them all, but it's fine to have more personal interactions (e.g. office hours if you have such a thing?) and suggest further reading or problems or just math talk in general

[lethal leaf]

review of calc 1 stuff?

in a group

[turbid zenith]

Okay this greatly narrows it down ;P

[lethal leaf]

hopefully the first sentence narrows it down more 😭

but yea review activities seem to help alot and are good first day things

[turbid zenith]

Any kind of review activities you think work well?

I was considering making an "I have, who has" set

[lethal leaf]

oh that could be nice

I was just thinking a worksheet but I may be boring

[turbid zenith]

It depends what you mean by worksheet XD

My students do "worksheets" a lot but they're like ... group work and often exploratory

Just so I have a place to put the questions down, etc

[lethal leaf]

exploratory or just "here are more basic review questions, work to answer them"

exploratory would be better

but "I have, who has" sounds cool also

[turbid zenith]

I definitely want to try to do group work better this semester though

[lethal leaf]

group work is hard

[turbid zenith]

Have you ever done that?

[lethal leaf]

people don't want to talk

[lethal leaf]

nope

[long pelican]

Not review, but my first days are used to talk about a fun piece of math history related to how the subject was invented

Or, not quite history, paradoxes

[lethal leaf]

I am a sucker for history so that sounds cool

[turbid zenith]

You make like a bunch of cards and the interaction goes like...

Student 1: Who has the derivative of ln x?
*pause while everyone looks at their cards*
Student 2: Oh! I have 1/x! *flips card over* Okay, who has how to tell if a function is concave up?

etc.

[lethal leaf]

yea ik what it is but I've never done it

[turbid zenith]

Oh okay

[long pelican]

I've done: Q(sqrt2) for linear algebra, estimating the area of a circle by inscribing and circumscribing regular n-gons for n = 2 x 3^k and 0.999... = 1 for calc 2, and Cardano's cubic "paradox" for complex analysis

[turbid zenith]

Ooooh that's a fun idea

By Cardano's paradox do you mean cubic equations having real solutions but the cubic "formula" involving imaginaries?

[long pelican]

Yes!

[turbid zenith]

Nice!

I'm hella excited about this semester btw ... I'm trying something new in my liberal arts math class and teaching them more combinatorial game theory than I usually would

Like I'm gonna use it as a consistent theme for the first half of the semester, culminating in an axiomatic approach to Hackenbush

[long pelican]

Damn, combinatorial game theory. I always say that the "lower level" recreational math classes teach more actual math than the standard honors calculus track

[turbid zenith]

Can't quite go full-on Winning Ways with them but I'm gonna give them a taste ❤️

[turbid zenith]

I know right XD It's kind of a shame really

Though at our school, all students take the liberal arts math class, even math majors

[long pelican]

Well that's a win

[turbid zenith]

COR 314: Mathematics and Human Nature

Students in this course will explore the mathematical method through logical and quantitative reasoning. Through an in-depth study of the tools of abstraction, generalization, and axiomatization, students will learn to solve problems and communicate mathematics. A central theme is the difference between evidence-based and axiom-based argumentation, engendering a discussion of the commonalities and distinctions between mathematics and science.

Readings:
How to Bake Pi / Eugenia Cheng
The Unfinished Game / Keith Devlin

[long pelican]

Through an in-depth study of the tools of abstraction, generalization, and axiomatization, students will learn to solve problems and communicate mathematics

I integrated all these ideas into calc 2 (especially in my problem sets) and I could tell half my students in their head were thinking "wtf [abstraction, generalization, ... problem solving, communication] is part of Calc 2?"

They eventually got in the groove though

[cosmic ibex]

(Quick, name any mathematical topic that doesn't have abstraction, generalization, problem solving, and communication in it).

[left vault]

Googology

pre-emptive rebuttal to communication: when was the last time you watched a seminar on googology

[cosmic ibex]

Googology is entirely about communication: if you cannot communicate to your peers exactly which giant number you're thinking about, it doesn't win you any bragging rights!

[cold trench]

if you can, I would suggest writing it with the proper audience in mind: the students. avoid saying "students will" and instead write "you will" unless your department doesn't allow that

also a pro tip: when doing group work, mention the word "group" after you explain the directions; otherwise students will be distracted looking for partners and such, so for example, say "okay so today we will be doing this activity where you do this thing, so get into groups and you can begin"

and also idk if you do this but try to minimize how often students choose their partners bc friends always go with friends and the weakest students always suffer

I read all of these in a book called teaching college" by norman eng, would recommend

[turbid zenith]

That's from the official course bulletin, not my syllabus XD

[turbid zenith]

This on the other hand has my own language

[turbid zenith]

Also, ooh I might need to try to get my hands on a copy of that

I kind of want to have students in "base groups" of 3-4 students all semester and sorta gamify it ... leaderboards, etc 😂

(Rewarding good work instead of just right answers ofc)

[cold trench]

wow I love this syllabus and the class structure, I'll transfer over just so I can take it lmao

[cold trench]

oh ok that's good

[cold trench]

ofc ofc lmao

by the way this book was given to all students in the course I'm in which trains us to teach the intro to business class as a "peer leader"

could I share your syllabus in class today if I get a chance? it's a class of 24 students

@turbid zenith

because the course is really well structured and reflects a lot of the tips and advice given in the book I mentioned so it's like a real world example

[turbid zenith]

Oooooh

Can I PM you about that actually?

[cold trench]

definitely

[turbid zenith]

school math

😛

[cosmic ibex]

Puhlease, I said "mathematical topic".

[turbid zenith]

Point! XD

[turbid zenith]

Something I’m really struggling with … anyone ever done like … group roles in high school/college?

https://nrich.maths.org/7908

This sort of thing … I always wrote it off as hokey in the past but I’m wondering if it might be a better way to get people all interacting

Because the issue I’ve been having is that people don’t want to discuss the problems — they just start on them independently and happen to all be at the same table

[gray smelt]

i mean, starting or middling independently makes sense. to me, try doing the group think after 5-10 minutes of individual work

[lethal leaf]

did this in a class

usually leads to the reporter / facilitator doing all the work

and the others doing nothing

(source: was usually the reporter / facilitator, did all the work)

[cold trench]

damn

[turbid zenith]

Oof

An idea was to encourage different group members to take on different responsibilities each time

[cloud depot]

What is an optimal way to assign exercise during class?

I am seriously struggling on getting the entire class to solve exercises. Like I have my outliers who do everything super easy

Like today I tried some sort of competition. So the first team to get the right answer would get a point and so on. But quickly noticed that not all the team members would actually do their own scratch work so ugh idk I’m frustrated

I’m talking about high school kids btw

[vagrant meadow]

i think it's mostly unavoidable for a group assignment to have people who don't contribute at all (especially in high school). i had one prof who gave "quizzes" where everyone turned in their own work, but were allowed to work/discuss with the people sitting around them. it was over a short enough time to prevent one person doing the whole thing and letting their neighbor copy.

the way the seats in the class are set up would have an impact though. in that class, all desks facing the front (the "normal" way?). i can see clumps of tables making that less effective though

[long pelican]

Non-contributors might overlap significantly with people with low mathematical confidence in starting problems?

And/or people who are lost

[quasi musk]

A lot of times the outliers doing everything quickly often times mis-steps, use guess and check to get through lengthy word/algebra problems

So I tell them that they need to develop a process to get to the answer other than just "seeing it"

I also come prepared with bonus problems, challenges, etc.

If all else fails I tell them to draw and not disrupt others

I don't let students read books in class as that's become an issue

[cloud depot]

This strategy sounds good. I might give it a shot on Monday

[cloud depot]

I hate the distribution of the classroom. It’s not desks. We have like movable chairs, so “desks” are individual

I do have like some challenging problems and assign them as extra credit, and they actually solve them. So that’s cool

[winged urchin]

Something about this bugs me a little. Not trying to be personal with you MoonBears of course but using this as a discussion point.

Because aren't step-by-step procedures to problems sometimes seen as a bad teaching practice? And, in fact, exploring the problem with guesses and trying to build some intuition as to what the answer might be considered a generally good way to approach problem solving?

Now there's a pretty big gap between 'seeing the answer' and 'step-by-step procedure' of course but ya.

I suppose if you're referring more to like... if someone solved an algebra equation by just 'seeing the answer' then they aren't practicing the manipulation of the equation...

Hmmm... I think this is just a bit of a rambly thought train from your comment ahah.

[vagrant meadow]

yeah that's really great. i love challenging extra credit problems. for the people who want to learn, it's an opportunity and incentive to feed that curiosity. and for people who don't care, it's no big deal. i'm glad you have students who are going for them 🙂

[cloud depot]

I actually have one girl who wants to pursue an undergrad in math, so that’s awesome

[vagrant meadow]

nice! that's always so good to hear. we need more math majors in the world haha

[cloud depot]

Tbh yes haha

Also, one difficulty I have found is that, I guess it’s all because of having online classes, there a lot of gaps. It’s challenging

[vagrant meadow]

imo it's a balance. you don't generally want to tyrannically force people to write out things they don't need to. some people can pretty easily rearrange algebra equations like
4x-7=6x-21 -> 2x=14
in just one step. but it's also beneficial to practice showing your work, and getting into good habits.
i've seen students who flew through algebra without having to try, but then came to a class they couldn't do that in. they hadn't developed their ability to persist and get discouraged or frustrated because math doesn't feel "easy" anymore (they can't just see the answer), and they give up.

[vagrant meadow]

oh yeah i've heard about that being especially bad in high schools. if you had to give a rough percentage, how many of your students do you think lack a severe amount of foundational knowledge?

[quasi musk]

The issue is when I have more complicated digit problems, motion problems, or something

Suddenly their approach of guess & check and plugging in numbers stops working

So it works for very simple problems

So they say "I don't need to write down steps because I just see the answer"

Then they get to hard problems, they don't have the executive function skill to take notes, write down their ideas in a systematic way, and solve complicated problems

So they say "I don't know how to do this, so I'm not going to write anything down"

It comes to a point where you need to learn how to write things down & solve problems; for students that have trouble writing down their ideas & steps I help them build themselves up to it

For those that write down too many steps & don't do anything in their head, I make them skip steps so they learn how to decide which steps are important to write down, and which ones aren't

[quasi musk]

And you touched on it precisely "if someone solves an algebra problem by seeing the answer, then they didn't do any algebra"

That's exactly it

My goal is to develop their skills

In class when I'm teaching I make them do that. On tests I say anything goes and get the answer

So I have different ways embedded in my class to help students suceed

[vagrant meadow]

so on tests, you would give full points to someone who wrote the correct answer with no scratch work? theoretically, i mean.

[quasi musk]

More or less, as long as the directions don't say otherwise

e.g. if it says use completing the square to solve this

Then you best you completing the square method

[vagrant meadow]

interesting

[turbid zenith]

@cloud depot Maybe the “group roles” thing I’m working on this semester could help you as well

I’ve decided on four:

Manager — reads problem out loud, keeps group on task
Articulator — asks questions if groups get stuck, shares group results at the board
Technician — checks accuracy and reasonableness of answers
Historian — records group’s work, refers to previous material

[quasi musk]

I think if you're going to have difficult problems that require group work to solve, then for each role you can have a list of tips, or pre-loaded hints into the lesson

[turbid zenith]

I’ve seen some sites give example sentences for each role

The “problems that require group work” thing is very difficult tbh

Something I’ve struggled with is that all the education sites give the impression that the best teachers are giving rich, complex problems every single class, addressing real-world issues and working in conjunction with community stakeholders, etc

And I’m like “uhhh … I need to teach u-substitution okay”

[cosmic ibex]

Shake it up a bit and teach t-substitution or y-substitution instead.

[turbid zenith]

XD

[long pelican]

"working in conjunction with community stakeholders" hah

[lethal leaf]

idk if I said this earlier to you or someone else

but I've done this as one a student (i.e. my prof implemented this)

and 90% of the time, no one cares about the group roles and whoever are the strongest in the group do all the work

source: did all the work in my computer architecture course until I dropped

it's an unfortunate reality

[long pelican]

Wow this actually reminded me of something that one of my TAs (who led labs) wrote up for a 2021 fall instructor report, exactly same thing observed here. In Fall 2021 the labs had group roles exactly as was being discussed here

[lethal leaf]

"roles should rotate" is a nice thought

but in reality it's just a name

didn't matter what the role was called, someone was carrying in the group

[long pelican]

Ya, the report pretty much confirmed to me that group roles aren't it

[lethal leaf]

In theory if it actually happened as prescribed it sounds nice

[turbid zenith]

It seems like they were like “hey guys do this and figure out how to do it”

I’m going to be prescribing them myself

I’ve got 16 students in each class. Doing groups of 4, changing it up every two weeks for a total of 5 rotations, and I managed to find a way to make it so that every student gets to work with every other student exactly once. (I’m actually curious about the mathematical problem behind doing this but that’s a different story). So each group of 4 meets for four days, so I’ll basically assign the roles myself for those days.

[vagrant meadow]

not always, but i've found that in some particularly lazy classes, group assignments lead to a sort of game of chicken. had a friend who was forced to do a group presentation in a core class last semester. she is a senior, and absolutely needed to pass the class that semester to graduate on time. her groupmates were a bunch of sophomores/juniors who didn't care what happened. they'd ghost her or choose not to respond to requests to meet, and she was forced to do everything because they knew she had far more to lose than they did.

[winged urchin]

Thank you @quasi musk for your response to my ramble thought train ahah. And this goes to anyone who comments on my comments that I don't get to respond to in full.

I do read through your responses and I love hearing you elaborate on it.

❤️

[left vault]

Doing an engaging activity on like a friday afternoon is good

Something to look forward to

alternatively could do stuff on mondays and it acts as motivation for the rest of the grindwork students have to do

[gray smelt]

honestly i feel that these roles only end up teaching actual class content/skills to the technician

[long pelican]

I just thought of this: Our main problem is students don't care, for several possible reasons, ranging from bad experiences with math to being too confused. To find solutions that help them to care, shouldn't we focus on trying to understand what is missing in the students' minds mathematically and what insights they require to start caring? If it's even at all possible. Another way to say it is, if e.g. advances in groupwork helps them care, what is the mechanism behind that, given that these advances are layered over the same curriculum?

[turbid zenith]

I disagree, because I consider things like mathematical communication and relating what you’ve learned to previous material as being just as important as, say, learning integration by parts.

[white forum]

probably even more important, as you can always go look up a formula or memorize one but building the intuition and problem-solving ability isn't something you can just read about or learn overnight.

[burnt vessel]

stéphane

[meager bronze]

@wispy slate when i teach calc 3 in the US, we make a distinction between vectors $\langle a, b \rangle$ and points $(a, b)$. vectors and points are both pairs of real numbers, they just have different geometric interpretations

[burnt vessel]

Buncho Bananas

[gray smelt]

what distinction are you referencing here?

(also, from what i've seen, this isn't a common US vector teaching thing)

[tawdry venture]

affine vs vector space, or something

[marsh thistle]

It is common in the US, Stewart uses it for example
I think it can be a bit confusing since after calc3 students will very possibly use () for vectors and <> for inner products, but I don't mind it too much since I think it's less confusing if you don't conflate the space with the tangent space at a point, even though in calculus class you could if you really wanted to

Like, you want to be drawing vectors anchored at whichever point is physically meaningful, rather than from the origin. So the <> should represent the coordinates with respect to that point, unlike the ()

[cosmic ibex]

Hmm, do we need a third and fourth kind of brackets to distinguish displacement vectors from tangent vectors, and gradients from either of those?

[winged urchin]

Ooo

What do they want to do? Thinking of going into some kind of mathy field?

I feel like logic is a pretty nice topic that's somewhat generally applicable even outside math

[white forum]

basic linear algebra would be good I think

[cloud zealot]

https://www.reddit.com/r/math/comments/110ocuq/if_you_were_incharge_of_designing_a_mathematical/

[long pelican]

I like this topic

[median iris]

Hey, does anyone have recommendations of a good book or video series on math pedagogy? Apologies if this is a common question here lol

I’m interested in tutoring but sometimes have difficulty expressing concepts in the most understandable way

[winged urchin]

How to solve it by George Polya

[pale wasp]

Knot theory is cool and you can do stuff like Adams' knot book with high school knowledge

[winged urchin]

Ohh. You could try to just like... Do group work on math puzzles or competition problems. The more nebulous problems that involve some playing around to figure out even what the heck is going on in the problem. Of course not just the calculate blah blah blah problems.

[winged urchin]

Oh! And of course if you go down the logic route you can do knights and knaves puzzles

Or other sorta logic brain teasers. Like here's a bunch of statements can you derive from that more information or figure out who did what or what colour house someone had

Etc etc

[cloud zealot]

https://www.amazon.com/Symmetry-Mathematical-Exploration-Quantitative-Critical-dp-3030516687/dp/3030516687/ref=dp_ob_title_bk

i recently found this book

maybe this could be cool

basically a book discussing symmetry groups and such

lots of pictures and pretty much no calculations till a discussion of matrices in the very last chapter

of course it doesn't start with any technical language

[modern mirage]

Curious what book would you recommend first year math students to learn calculus from. Say they’ve never seen calculus before. Would a book like Stewart be recommended, or is the book a bit too handwavy and should be avoided?

[rain rain]

What do you guys think of Lockhart's Lament? What are your issues with the math education system?

I'm not a professional educator, I just love math and love teaching people. I think a lot of what Lockhart says is valid, but I think he's leaning a bit too hard into treating math as an artform. Math should be taught like it's exploring puzzles and coming to conclusions on your own, but that doesn't mean that it's not important to learn some fundamental formulas, algebra, and basic trigonometry. It's the same reason we teach science: just to give students a bit of a background in the world right? Prepare them for the future if they decide to pursue college or other forms of higher education. Math is just too important practically (same with science) to not teach it (at least partially) as a "practical subject"

https://mysite.science.uottawa.ca/mnewman/LockhartsLament.pdf

Though he seems pretty self-aware that that's how the reader might interpret it, going off of his conversations with Salviati and Simplicio

[long pelican]

I think Lockhart's essay is as dramatic as it is because it's a response to the current prevailing beliefs about math education

When I read it, I didn't focus too much on the extreme-sounding-ness for that reason

[rain rain]

Oof I do love this statement though. I think a lot of people forget about that 💀 people just write proofs to be clean and show off how clever they are. I don't have much experience with them but that's the general perception I got reading others' proofs and having them critique mine

[rain rain]

He's being dramatic because he believes others are being dramatic about how "practical" math is?

Having difficulty understanding what you mean, sorry

[long pelican]

Yes, most people believe the purpose of math classes is to teach them the basic tools that are prerequisites of study in STEM subjects

As opposed to as a means to learn problem solving, critical thinking, creative thinking, principles of argumentation, and so on

(OK, most people would acknowledge the latter but it takes a backseat, and they usually don't have the time or training to emphasize it)

[rain rain]

Where do we draw the line? How much can be taught from that perspective?

By that I mean like

[long pelican]

If you ask me, it should be completely from that perspective

[rain rain]

As far as I'm aware the standard US HS math curriculum goes up to Algebra 2 as a minimum

I feel like a lot of the ideas before that you can develop intuitively

Especially up to linear and quadratic functions

But if you want to explore other types of relationships, exponential, logarithmic, trigonometric, whatever

How can you approach that intuitively and from that puzzle-solving approach?

Genuine question, not a rhetorical one

I have no clue how to do that

So where do we draw the line between how much is practical and can be taught purely from posing problems and guiding students to developing solutions, how much is enough to build those skills without shoving down math that only accountants and STEM majors use

[long pelican]

Don't understand that last sentence

[rain rain]

Like obviously I think everyone can see that these two lines are symmetrical, with or without the notation. I agree that the notation is stupid and unnecessary to understand that opposite angles are congruent. But how can you prove that, or, like, show that the lines are "symmetric" without at least some notation or proofy stuff? How can you teach students to definitively show that the lines are symmetric instead of just saying "oh I mean they look symmetric" or "oh it's obvious"

[rain rain]

So like

Linear relationships

I think those can come up very naturally when you're problem-solving

Same for quadratic, same for exponential and higher powers and everything really

[long pelican]

Hmmm I think Lockhart did miss something actually

He missed that what happens in proof-based geometry class, no matter how flawed, is closer to the (i.e. his) ideal than what happens in a typical algebra, algebra 2, or calculus class

I remember a good amount of people (non-math majors) say that geometry was their favorite class because it was so logical and focused on proofs

[rain rain]

But do we have enough time to derive those relationships through problems and then teach students how to go about solving the puzzles created by those types of relationships (by that I mean for a variable) for all the various elementary functions? Or is that even necessary in the first place? Is it just necessary to teach up to quadratics, then let the STEM majors & careers that actually use math all the time learn those other things in college or in optional higher classes?

[rain rain]

I hated geometry 💀 but I didn't like math in general back then

I love geometry now, but I don't like how HS geometry was, not even looking back on it

I think geometry is where you get the most intuitive puzzle-solving artsy deductive reasoning essence that Lockhart wants

I think proving those things can be fun, it's just the way the proofs are handled is lame

[long pelican]

Ya

In my first 3 complex variables problem sets so far I asked them to mention anything they wanted to about the problems such as what they liked or what took a long time, and there were a lot of responses of the form "I liked problem X, it was very elegant and surprising"

Now think about how in the typical college freshman college student fresh out of high school has seen 0 elegant and surprising math problems in their 12 years of math education

It's all been, learn this method, practice this method, test this method and repeat

[rain rain]

Yeah :/

What I'm trying to say is

Like I agree that that's how it should be taught

But the issue I see with it is in time

Students only have so much time to learn K-12

And teaching from that perspective takes more time

[long pelican]

You'll find 100% of teachers say they don't have time too!

[rain rain]

And it takes more time to do problems and explore solutions yourself too

So if we want to be able to shift in that teaching direction but still maintain 12 years of pre-uni educatio

[long pelican]

But then again, Lockhart's audience is mainly not the teachers in fact

[rain rain]

We need to cut the curriculum short somewhere right? Short relative to where it is now

[long pelican]

He knows many teachers are passionate

He's actually addressing the higher-ups, for example curriculum designers and standardized testing people

[rain rain]

Yes

[long pelican]

Like, the curriculum is still a huge laundry list of skills

Even in the common core

The presentation as a laundry list of skills already frames math incorrectly, poor teachers

[rain rain]

Yes

I feel like I’m coming across wrong

My point is like

How can curriculum designers redesign or shorten the curriculum to teach the basic skills and build logical thinking & deduction

Where do they cut that list short

They can then redesign it so it’s no longer a list

But if you redesign it rn and still try to cover all the content it’d take longer classes or more time in some way no?

[long pelican]

I think they can delete everything specific of the form "Students should be able to solve [this] type of problem" and put the higher order skills in, it'll be a strict improvement

A proper math education will allow students to solve specific types of problems with general tools anyway

e.g. solving equations with exponentials doesn't need to be on the list

[rain rain]

So what higher order skill(s) would students need to learn in order to solve those types of problems with general tools?

How can students solve an exponential equation without knowing about the logarithm? How can students solve an exponential equation if they’ve never even heard of or seen one before? And it just naturally comes up in their attempt to solve a larger problem

[rain rain]

What are these general tools?

[long pelican]

Why would logarithms be cut out altogether? That doesn't follow from what I suggested

Well some general tools: treating a sub-expression as a variable and seeing a simpler equation beneath the structure, using inverse functions

[rain rain]

They don’t, I just misinterpreted

[rain rain]

So you'd teach about the various kinds of functions and their inverses but leave the actual problem-solving up to students?

Or like

You don't have to force them to just solve a bunch of exponential equations to get the hang of it

Or you don't need a whole unit on just exponentials

You just kinda frame it within a larger real-world problem or puzzle (or a contrived one that's interesting) and let it come naturally and ask them how they'd solve it using what they know?

Would that be a good solution?

[long pelican]

More or less, yes

[rain rain]

Gotchya

Thank you :3

[cosmic ibex]

My impression, purely as an outsider, is that the real problem is testing. Everybody seems to agree more or less what students ought to learn, despite minor spats about how to word those goals. Everybody agrees that students should end up understanding what they're doing. But it is hard, time-consuming, and expensive to test understanding -- and twice as hard to do it in a way that you can argue is robustly objective, which is pragmatically important in a society that (for other excellent reasons) is permeated by a focus on preventing systemic, possibly unintentional, ethnic or racial biases in education. So the temptation must be extremely strong to specify measurable goals in terms of skills: students who understand the things we want them to understand will be able to use their understanding to solve such-and-such problems, so let's test on those problems as a proxy for understanding.
The trouble is that weakish students then demand to be taught the problems instead of the understanding because that is more efficient in the short term. And harried teachers are all too incentivized to give them that, not least because it's the teacher's best hope of not being declared to be bad at their job when the test results are in. And the long-term damage such teaching does will only show up later when it is some other educator's problem to deal with.

[noble phoenix]

I'm TAing a linear regression class and many of the students (~25%) have not taken a linear algebra class, despite it being a listed prerequisite. To what extent should I be modifying the material so that these students can keep up?

[austere inlet]

you can do a few classes of review, I wouldn't change the syllabus if it's clear that linalg is a prereq though

[winged urchin]

I do fear that if you taught highschool or lower classes as Icy mentions would produce mostly negative reactions

Heck even university level. You really need students who are willing to explore and try and try and try again and observe their own process

[long pelican]

Been there 😉

I can always improve my communication and psychology skills as well as understanding what's missing better

[winged urchin]

As a tutor sometimes when I try to guide them as they explore through problems there are some students who just say they're stuck and don't even try to reorganize the expression or equation or whatever

But again that particular problem can be from learning algebra as just formulas and not as a creative process

Yeah we strive for the ideal no matter how daunting it seems haha

[long pelican]

It's the same with doctors treating patients who don't want to be treated

You either get good at connecting with them, psychology, or get a psychiatrist to talk to them, or whatnot

in the doctor scenario

Something different obviously but analogous in the math teaching scenario

in the end whether they get treated is their say

A lot of times we fail to convince students that they should be learning math in the proper way

Not their fault as it's our word against all the teachers they had before them

You have even less leverage if you're a tutor or TA

[winged urchin]

Actually I thankfully have a few younger students whose parents have given me pretty much free range to teach them what I think is appropriate

So I'm showing students in grade 4 to 6 algebra and what I call 'mathematicalizing' word problems.

They can sometimes write the equations representing the word problem statements and then solve those basic systems of equations

I really try to stress the use of variables to be a placeholder when they don't know what the value really is.

You know those kind of inverse problems students struggle with

They can do:

A) If a can costs $4 and I buy 30 cans how much does it cost?

But they struggle with:

B) If I paid $500 for my cart of cans and each can costs $4 then how many did I buy?

Maybe not quite that simple but I'm sure you know what I'm referring to

I try to get them comfortable with just using a variable and make an equation then solve that

[cosmic ibex]

Hmm, I wonder if there'd be any difference if you instead ask

If a can costs $4 and I have $500, how many cans can I afford?

[winged urchin]

In this case that might work but I feel that's going down the kind of rote memorization of steps that we try to avoid

They shouldn't remember "Oh, it's a problem type A where I take the bigger number and divide the smaller" because that doesn't work in more complicated word problems

I'm hoping I teach my students how to read and write mathematics so they can write the equation that's suitable for a given statement and then just solve that

[cosmic ibex]

Hmm, as long as you keep a firm connection to the underlying intuition. It wouldn't do if they just end up seeing "solve the equation" as a magical black box.

[winged urchin]

I think part of mathematical literacy is being able to try different things in an equation and solve them through a kind of creative process

And some of the more difficulty systems of equations require some creativity I think

You can't just expect to use substitution to solve all the time and sometimes there are weird little tricks

When I get them to solve I get them to tell me what they're doing in each step. What are you doing to both sides? What are you doing with that fraction or in completing the square?

Then they should be able to suggest things to do and be able see whether it does what they want it to do

If we have 2x+3=1

And the student says.. oh we can remove the 2 on the x

You say 'tell me what operation you're doing on both sides'

Perhaps

Then they retort sometimes with.. 'ill subtract 2 I guess?'

Then I tell them to do that or write out what happens and ask if it does what they want it to do

Does 2x+3-2 = 1-2 actually remove the 2 on the x?

[wispy slate]

@meager bronze : thanks. So the same mathematical object is noted differently according to the geometric interpretation that we wish to give it. Why not, if it helps students understand better. In France in teaching for pupils up to 18 years old, we are unfortunately not there, where we still have to work in affine spaces of dimension 2 or 3 without explaining the structure to the pupils. Up to 14 years, it is not very serious but then it is problematic. On the other hand, after 18 years, things are done seriously: introduction of the vector space structure, then of the affine space structure.

[cloud zealot]

https://math.berkeley.edu/~wu/

i read a few of the linked articles and slides

https://math.berkeley.edu/~wu/CommonCoreIV.pdf
https://math.berkeley.edu/~wu/CommonCoreVI.pdf
https://math.berkeley.edu/~wu/AMS_COE_2011.pdf

some of the slides that stood out to me

i'm also curious if anyone here has used hung-hsi wu's k-12 books in a classroom

[cloud zealot]

according to the preface, although hung-hsi wu's six textbooks are meant for adult instructors, they can serve as a basis/model/guide for future student-oriented textbooks

[long pelican]

I think Wu has the closest to an accurate diagnosis of the underlying fundamental issue behind why K-12 math education is not working as well as it should, out of everyone who has written anything about it. People criticize his solution as something that won’t help and these criticisms may have some merits, but they don’t give his diagnosis enough credit

Although to take an example, his conception of rational numbers (one of his main focuses solution-wise) is much better than the status quo in terms of coherence as well as understandability

[long pelican]

This isn’t quite an answer to your question but a current college student who also happens to be a friend of mine, and has had the typical crappy math education in school, has told me that he really enjoyed reading the Wu books

[cosmic ibex]

Can you give a super-brief summary of what that diagnosis is?

[long pelican]

Sure, in one sentence, it's: Students have trouble making sense of the math we teach them because the prevailing presentation of the math we (most teachers) actually give them is unnecessarily hard to make sense of

That might have been too brief, let me know if you want something slightly longer

[cosmic ibex]

Hmm, yes, that borders on being a truism, doesn't it?

[long pelican]

Hm I might have left too much out

Very well, 2 sentences: The "prevailing wisdom" says students find math hard to understand because it's too disconnected from real life. Wu thinks that maybe real life examples aren't the main issue and that we're not giving students' logical abilities enough credit and we should instead look at how the material is presented logically, or lack thereof

How familiar are you with his criticism of how we teach rational numbers?

[cosmic ibex]

Not at all.

And for that matter I'm not terribly familiar with how rational numbers are taught in the US.

[long pelican]

OK, it's a good example to illustrate. As of 2020 at least, the strategy of teaching rational numbers in the US is to not define rational numbers at all but instead use analogies: pie slices, parts of a whole, and something else I forget right now, then teach rules for how to add, subtract, multiply, and divide rational numbers, sometimes using the analogies to intuitively justify these rules.

From Wu's perspective, this is unnecessarily taxing on the students' memory and cognition and does not make enough use of their already existing logical skills. They should instead be taught rational numbers with a single precise definition: as (I'm paraphrasing here) points on the number line whose some multiple is an integer; have all the analogies tied back to that definition; and prove all the rules using that definition rather than the analogies.

[cosmic ibex]

That sounds fair.
Just the other day I had a conversation about 0.999...=1 that went rather off track when I said something like "the meaning of an infinite decimal expansion is the limit of the rationals you get by truncating it" -- and I was prepared to need to explain limits, but it turned out they had no idea what a rational number is. (They later stated they were 14).

[long pelican]

was the conversation in this server?

[cosmic ibex]

Yes.

(Some of it spilled into DMs).

[marble oriole]

Are there any channels for teachers?

[austere inlet]

this one; it's meant for discussions about pedagogy as the title and description say

however there are no channels for students to look for tutors/teachers if that's what you mean @marble oriole

[cloud depot]

Today one my high school kids told me that the tutoring sessions we had were actually helpful, and it just feels so great

[long pelican]

Sorry never got a chance to reply, but I think it can be quite scary how people are in algebra, precalculus, calculus, and are still uncomfortable with rational numbers conceptually. It takes you into the mode of "I know how to do computations I was shown and practiced but I am too scared to do computations outside of what I have been shown and practiced"

[cosmic ibex]

But that's more of a fall-out from never having learned that things can and should make sense, yet being force to practice procedures they don't understand, for the sake of grades.

[long pelican]

Yeah that's the underlying theme

[cosmic ibex]

Sometimes it feels like a series of increasingly desperate attempts to teach kids the right things takes the teaching farther and farther away from what the right things are.

The kids understand somewhat when we get down to pie slices, but they struggle with learning the rational numbers as an abstraction ===> stop teaching the abstraction at all and make everything about pie slices, hoping that will get them through the test at the end of the year.

Or: kids struggle to express word problems as symbolic algebra ====> double down on that and try to teach them already in grade 3 to write C for the number of cookies and P for the price of one cookie when asked how much a dozen of them cost ... even though the real problem might be they are insufficiently comfortable with the concrete calculations that the symbolic expressions stand for.

[winged urchin]

^^ Tropo your last point is pretty much what I was talking about last time I spoke with you. About tutoring younger students on algebra and 'mathematicalizing' word problems =p

I definitely think students can be introduced to variables very early as well as the algebra involved

In fact I think sometimes doing word problems without knowing the sufficient algebra to solve equations is perhaps part of the problem

[cosmic ibex]

We may have to agree to disagree about that ...

[winged urchin]

Most word problems students do at the younger grades have to be so simple as to be explicit calculations like the $4 per can of blah.. buy 52 cans... etc etc

Great! I love disagreement =p

What do you mean though?

[cosmic ibex]

I mean, symbolic equations make sense to me because I know they encode the pattern of a computation I could do with actual concrete numbers without involving variables.
I can't imagine how I would have any intuition about them if I were not first allowed to build an intuition about how numbers work and how to calculate with them to solve problems.

Suppose I have a word problem about fencing in the maximal possible rectangular area with such-and-such a total length of fence.

Of course I'm going to write down an expression for the area if the side length of my rectangle is x.

But the reason I'm able to write down that expression is that I know that if I already knew what the right side length is, in numbers, I would be able to compute the area without any algebra. Just subtract this from that to find the other side length, divide by two (or not, according to whether there's a wall that doesn't need fence) and then multiply them together, that gives me the area.

If instead I had been taught from the beginning that "the way to find the area in this case is to write down such-and-such expression and afterwards plug in such-and-such value for x", then the expression would have come from nowhere and I couldn't have any intuition about why that expression is the right thing to write down at that stage.

[winged urchin]

Hmm... I know that if I were to work with a student on that kind of a problem at some point I'm going to read the sentence talking about "the area of the fenced in space" and ask the student how we might write what that is mathematically

I do agree that when we introduce formulas we should use some solid numerical examples first instead of jumping straight into variables

[cosmic ibex]

But the student has no chance of being able to write it mathematically before he knows how he would calculate it with numbers.

The point of writing down symbolic expressions is that they encapsulate a lot of possible calculations with numbers. But if you consider the actual numbers to be secondary, merely motivating examples, then there's nothing left for the symbolic expressions to mean.

[winged urchin]

There's a couple points there I don't know if I agree with or if I understand them completely. To be honest.

Like I do think it's possible for a student to derive a formula with just variables without having to take specific examples. Although I would agree that considering specific examples with actual numbers is a great stepping stone to making sure the formula is right

I also don't know exactly what is meant by your sentence at the end about if numbers are just secondary then there is nothing left for the symbolic expressions to mean.

It does sometimes feel like simple symbol pushing but I do think students need some ability to manipulate equations, solve systems, write equations to describe a situation.

I'm also sometimes bad at describing my thoughts or what I actually do during sessions. To be fair ahah

[tardy ember]

being able to do the symbolic computations is definitely useful
but so is understanding what the symbolic computation means and why it works

optimally they would both reinforce each other: being able to actually do symbol manipulation means you have something to look at and understand, and being able to understand it means you can remember it and won't make the kind of mistake that's obvious if you know what it means

but what it means is, numbers

if you never look at numbers then the symbols are just symbols that follow some rules just because they do

[winged urchin]

As an example, if a student adds 1/3 and 4/3 together and gets 5/6 because they misremembered the symbolic rule. Hopefully they would use their numerical intuition and say 4/3 is more than one whole yet it added up to something less than a whole!

Although personally I find a lot of students I tutor do not have that mindfulness to be aware and thinking of their answer. If I put them on it specifically they can notice at times but getting them to really consider every step can be difficult and tedious for them

And sometimes I try to get them to use numerical results they are confident in to figure out how the symbolic rule goes again

They will be quite confident that 1/2 plus 1/2 is 1 and then I can usually get them figure out how to add those together correctly in order to get the right result

[cosmic ibex]

My experience is only helping people in places like this server, not any "real" teaching, so what I say should perhaps be taken with a grain of salt.
But that rudimentary experience is that when I try to help someone at a beginning level, and they can't write down the formula they need, it is generally not because they don't understand formulas -- but much more basic than that: They don't understand which calculation it is they need a formula for. Even if I try to be very concrete and say something like

Suppose I claim that a side length of 19 m gives you a fenced-in area of 45 m², how would you then find out if I'm lying about that?
(with the plan of then explaining how the calculation they describe can be expressed as a formula) they still have no idea what to do. They can't calculate things! For those learners it is not the algebra that gives them trouble, it is calculating anything at all. It seems impossible to me that the cure for that can in any way be that they should have been confused with letters as a symbolic representations of calculations even earlier, instead of simply figuring out a plan for making a straightforward calculation -- which they can then express symbolically.

[winged urchin]

If I had a student with that problem my brain is telling me I'd ask them how we can connect the facts. We know the area and the supposed length of one side. I'd get them to draw a rectangle if they haven't already drawn a figure. Then I'd ask them how they think the side length of a rectangle may be related to the area

Eventually I think I would try to lead them to saying A = l*w and then further asking what they can fill in from this equation

This requires very little algebra but I have even seen mistakes in solving this simple equation of 45 = 19w or 45 = 19l whichever one they called 19

Then they get an answer but of course there is the caveat implicit in the question that we want integer side lengths

I think this kind of thinking stems from my own problem solving where I might not know how to get from question to answer but if I just start 'mathematicalizing' statements or drawing figures then eventually it clicks and I see how to get the answer

How would you approach this problem with a student?

I can see possibly getting them to try out various values for the other side length maybe

Then they could maybe be led down a path where.. oh if I pick the other side length to be 1 or 2 then it's area is less than 45 and if I pick 3 or more it's more!

And of course getting them to do specific examples with known side lengths they might be more comfortable with generalizing that to 45=19w

I appreciate our dialogue Tropo. I too say what I say with some amount of salt even though I've been tutoring for ten years plus now ahah

I am always a little suspicious of my own teaching methods and love discussing specifics so that I might improve

[long pelican]

@winged urchin I think you misunderstood Troposhere's thought experiment (he meant square, not rectangle), but nevertheless I have a question for you. Would you be surprised if an algebra (or even let's say high school algebra 2) student shows that he's unable to determine whether a square of side length 19 has area 45 or not?

[winged urchin]

I should be surprised ahah

But I feel like no I wouldn't be entirely surprised I suppose

Though I would ask more questions then. Like can they tell me what the area would be with a side length of 2 or 3

If they are confused there I guess I'd need to have a discussion with the student about what area means to them

I do think that sometimes students would say they don't know but really they just mean they can't do the 19*19 without a calculator

Although wait nevermind. In this problem that wouldn't be an issue really. They don't even need to do the calculation. Just have a vague idea of how big the area should be

Hmmm.. maybe I would be surprised?

Like if they genuinely can't tell whether a square with side length 19 has area 45 then what..

Hmm..

I feel like that would less surprising with a different shape. I suppose

[long pelican]

Well different, more complicated, shape would be further from my point

Here's another question

Suppose the student does have this insane gap in their mathematical knowledge, how long would it take for you to find it if he presented to you with algebra 2 homework he needed help with?

[winged urchin]

It wouldn't come up unless we specifically talked about an area problem I'm guessing

[long pelican]

Do you believe that it doesn't affect how he'd do in algebra 2 and beyond?

[winged urchin]

Not knowing the area of a square? Definitely would affect them

[long pelican]

Yeah, so you'd say it's very hard to find out that issue (which is probably a very important issue to resolve) without knowing it beforehand, right?

[winged urchin]

Sure. That's the kind of thing that's so basic I would generally assume the student knows it

Because to not assume the very basics means you would need to explain everything from basics which parents or they wouldn't pay for.

[long pelican]

Yes

Well, no

It would just mean you have to find out his knowledge gap some way without having to ask every single basic question in the book

[winged urchin]

Oh yes that is true

And my strategy. It's too demeaning to explain fraction addition without seeing them screw it up first

But once I see that then I take a minute to review fraction addition

For instance

[long pelican]

I'd imagine if they screw up a fraction addition, you'll just think 99% it's a careless error right?

[winged urchin]

Well.. one of my favourite things to do when I see a mistake is to let them keep going and (hopefully) the answer they get is ridiculous in the context

And I'm trying to see if they notice

If they don't, or if I'm in a different mood that day, I might tell them there was a mistake somewhere and ask them to find it

Eventually if they can't spot the error then my thought will be it wasn't careless

And I might ask them what 1/2 + 1/2 is

As a teacher, I suppose, if I was just marking a paper I might just think careless error I suppose

If I don't have the opportunity to interact with the student like a tutor

Though of course some mistakes are more easily identified as careless errors

[long pelican]

This is an aside but I think it'd be cool to encapsulate the common tutoring tactics with names or phrases, because I'm seeing some universal tactics

Well.. one of my favourite things to do when I see a mistake is to let them keep going and (hopefully) the answer they get is ridiculous in the context
This one for example lots of people use and it's very effective in many contexts but obviously there are preconditions

Less cognitive load makes it easier to make connections and see what we didn't see before, you know?

when talking about this stuff

kinda exactly like how mathematical progress is made across history too

[winged urchin]

Yes I do agree this would be interesting ahah. I often talk about a 'mathematical' toolbox that students should have to 'open up' whenever they have problems. What tool works here hmmm!!

Could have a 'tutor' toolbox of techniques ahah

With regards to cognitive load are you referring to considering simpler examples?

Like adding together 2985/34 + 48293/45
vs
1/2 + 2/3

They might be making the same fundamental mistake in each but in the first example they have essentially no hope of catching the mistake and the calculation itself takes some amount of their brain

[long pelican]

No no, cognitive load for us when talking about teaching

[winged urchin]

Sorry I am enjoying the discussion but I will be busy for the next hour or so. I'll check in after. ❤️

[long pelican]

See you in a hour, I may or may not be asleep then

[cosmic ibex]

Not that it matters much here, but what I had in mind was a problem like "you have 100 meters of fence and you want to use it to enclose a rectangle with the maximal area". So the calcluation I would have hoped for would be something like "19 m to the horizontal sides means those sides use 38 m of fence, which leaves 62 m of fence to the two other sides, so they are 31 m long each, which makes an area of 19·31, which is 589, and that is definitely not 45".

After talking about this plan I would hoped to show how it corresponds to $x\frac{100-2x}{2}$ and start simplifying that.

[burnt vessel]

Troposphere

[long pelican]

oh I see

[cosmic ibex]

The overall point I was trying to make is that there's a skill of seeing that, oh, all these quantities are connected such if we know this one, we'll be able to calculate the rest of them. That's something different from knowing how to write down the finished plan abstractly as an equation, and if students lack the former skill, attempting to introduce the latter to them at an earlier age will not help with learning either.

[fallow siren]

Thanks for this. I hadn't thought about it in this way.

[fallow siren]

Not exactly the term you're hoping to have, but this situation revolves around the sanity check.

[long pelican]

A sanity check could also describe the situation where you immediately stop them and ask them to do a sanity check on their answer, which is very distinct from the strategy of not saying anything

[fallow siren]

Yes. At some point the student has to realize a mistake was made.

The student is unlikely to notice right away. The next best thing is for the student to notice that the final answer makes so sense. But the student has to actually check whether the final answer makes sense in order to notice that something is wrong.

Too many students see an answer, any answer, as the end of thinking.

The only difference in seeing is whether you tell the student to do a sanity check or you wait for the student to do a sanity check on their own.

But maybe I'm missing something.

[long pelican]

What are you missing? We can just coin the terms "immediate sanity check" and "delayed sanity check"

[gray smelt]

this does make me realize that i should probably explicitly teach some of the automatic sanity checks that i do.

i guess it's most salient for highschool physics problems. if e.g. the problem is a setup with an angle theta and a distance d, i could advice students to check if the answer they got matches with what intuition says would happen at 0° and 90°, check for the aymptotic behaviour (e.g. "hey wait if d is big this thing should have no force but my answer goes to infinity")

[quaint shale]

i want to make a youtube course on commutative algebra but im not totally sure which book to use

atiyah-macdonald is too short & i dont love the way they do primary decomposition, altman-kleiman is huge and comprehensive but i'll have to adapt to their notation and the breadth of topics i'll actually select from, i know milne has a set of notes up as well

[long pelican]

That's very advanced and I support this goal a lot! I once asked 3b1b in person if he ever considered doing animations for representation theory but by his response I think that's a bit out of reach for him. I suggest incorporating Eisenbud's Geometry of Schemes in that somewhere because I think algebraic geometry elucidates commutative algebra a lot

[quaint shale]

don't think i've ever gone through geometry of schemes - i am familiar with eisenbud's book on commutative algebra but i also am aware that richard borcherds has a huge youtube series following that book

definitely would like to make explicit the connections to alg geo as much as possible

[cold trench]

when did you have an opportunity to talk to 3b1b in person?

[long pelican]

He gave a talk a couple of years ago :0

[cold trench]

that's awesome

[long pelican]

Ye

[prime kindle]

I was helping someone with math but i kinda encountered a moral dilemma. he had a exam in 1 hour and he had a review. he was begging for answers but i knew he had no idea how to do the problems in his review. i deemed it right to not give him any answers and instead help him through some of the questions. after i told him that i will not give him answers and nobody in this server will either he left the server. do y'all think it was right to do so or was it wrong? i feel like it was wrong cause that would be academic dishonesty if i just handed out answers.

[lost raptor]

Well handing out answers would be non-productive in general since they’re still lost on how to do it

[prime kindle]

thats also what i was thinking

i still feel bad about doing so

[lost raptor]

Though whether it’s academic dishonesty depends on if it’s just a study guide or something actually meant to be submitted

[prime kindle]

true actually

[winged urchin]

Imo that sounds a little harsh to me.

If you suspected it was for marks or something then that's one thing

But if it truly was just study questions (or you know, you could modify the questions a bit so it isn't the exact same and answer those) then I would show the student how I would solve them if they were really pressed for time

[long pelican]

It's fair to just reject them outright if they are seeking help in time pressure... not the correct motivational environment in the first place

[austere inlet]

if we're talking about this server, then you did the right thing -- handing out answers isn't the point of this space and there's the whole issue with possible dishonesty

[prime kindle]

true i see what you mean. i did suspect it was for a grade.

[austere inlet]

also , it's your own time they're asking you to invest

(not entirely sure if this channel is the place to talk about server meta, but of course there's pedagogical justification for why you shouldn't just give away answers when helping someone learn)

[winged urchin]

I'm imagining this more from a paid perspective I suppose. If they've paid for my time but there simply isn't enough time to properly go over the concepts then I will just show them how I would navigate through the problems to give them something at least

Like near finals a lot of university students get tutoring sessions like last second and there's simply not much choice but to work through the practice final with them

[prime kindle]

no, it was in one of the help channels i had plenty of time to help

[winged urchin]

Ideally I try to get them to do the problems but for some students it seems they just drag their feet or aren't motivated whatever the case may be

Basically just writing out solutions to question is a last resort

The other thing too is I guess my 'answers' aren't just like.. the raw final answer or something

My answers are more expository than that. Showing steps and justification and going into details if the student is confused at a particular step

[prime kindle]

ok i see, thx so much for the input on the situation

[proven wadi]

what r yall thoughts on bonus questions on high school math tests?

[prime kindle]

I think they are good to implement in some tests. However the questions should be more challenging than the rest of the questions on said test.

[lethal leaf]

But also if they're more challenging right

Then the only people getting the bonus are those who probably don't need it cause they're already doing well in the course

[cold trench]

make the rich richer 😎

[cosmic ibex]

Intuitively I think including challenge question in a test would risk the mid-range students feeling their inadequacy was being rubbed in. If you want to keep the bright students engaged and thinking, by all means propose some challenges to them, but I'm not sure a test is the right place or time.

[long pelican]

Since it’s a high school test, I’m guessing the non-bonus problems are just basic skill checks with textbook problems and the bonus problem is the only outside the box problem

I think this type of design will send the message that outside the box thinking is for the geniuses

[shadow flower]

you could also build in extra points into the test, such as 105 or 110 and take it out of 100. I’m not sure what your goal is with the bonus questions

[fallow siren]

I include bonus questions in part as sympathy for the students who are finished quickly. I haven't had any of the issues outlined above, and I already have questions that are slightly beyond basic skills checks, but I might rethink.

[lethal leaf]

I think a better version of this is like

give 6 questions, make them answer at least 5, then 6th can be bonus if you want

but that gives them some choice and bonus but isn't a "rich gets richer"

I include bonus questions in part as sympathy for the students who are finished quickly.
sympathy for what?

[shadow flower]

how is this any different? the same students that will answer the bonus questions on a standard test will also answer the 6th question

[lethal leaf]

Because it's not an explicitly harder question necessarily

and students who may not be as strong may be strong enough to answer 5/6 questions

the difference is more about the question not being an explicitly harder challenge question

[shadow flower]

why does it necessarily have to be harder on a standard test though? would you not be able to achieve the same thing? give 10 problems and an 11th as a bonus

i guess that’s kinda the same thing idk

[lethal leaf]

they also have free choice over which problem they want to omit

it is not "you must answer questions 1-5, do 6 as a bonus if you want"

it's more "here are 6 questions, pick and solve any 5 for full credit"

I mean TBH I've had classes do this and you got no bonus for doing the extra which I can also see as nice but that's a slightly separate topic than this current conversation about questions for >100% credit

[shadow flower]

this was how my analysis exams were structured. 12 choose 10

[sand valve]

I need help regarding a student of mine. He's in the 4th grade and already solving Algebra 2 problems. I don't believe he should be doing algebra 1/2 worksheets without understanding the basics, like what a slope is and what the y-intercept is. He's spectacularly fast at mathematical reasoning/computation, however his parents don't help him with defining parts of a function and the new concepts that he's never seen before. His answers are almost all wrong without guidance from someone with strong algebra 1/2 knowledge. How should I teach him? Or should I just ask that the parents give their child a break from higher level maths?

I'm just a tutor btw, undergraduate aiming for a major in mathematics. I'm very familiar with pre-uni maths however I've never had to teach a child such advanced maths before and I'm unsure how to approach it

[copper gyro]

One approach you could take is to incorporate review of algebra 1/2 topics into your lessons with the student. This could involve going over the basic concepts they may have missed, like slope and y-intercept, and working on building their understanding of these concepts before moving on to more advanced topics. You can use visual aids and real-life examples to help make the concepts more relatable and understandable for the student.

[gray smelt]

for someone spectacularly fast at this, and if the student is also motivated, i'd considering doing a rapid fire tour of all the content.

if it is too fast, slow down of course.

[sand valve]

my biggest issue is motivation. he told me that he hates math, which was disheartening to hear. I assume that he hasn't had anyone to teach him in an engaging and interesting way, from what I heard, he literally just does worksheets :c

[sand valve]

I'll try that. I hope after working with him more he'll feel more comfortable talking to me about the material

[gray smelt]

oh.

the best thing to do, if you think you getting him to like math is a possibility...is to entirely ditch the curriculum and show him some cool advanced stuff.

i realize that's not as possible in the tutoring context

[winged urchin]

Depends on the parents. There are two kids I'm helping right now whose parents are absolutely fine with me showing their kid more advanced stuff

It helps that the curriculum here is less advanced than they wished ahah. But yeah cool math stuff like logic puzzles or cool animations. Chaos and the double pendulum is nice. Cryptography offers a nice activity too

[violet haven]

My high school mathematics is struggling to teach our class. We are all lazy teenagers and of course we do not always do his homework or participate in class.
He is asking for help and I want to help him.

What is actually the best way to teach mathematics in a engaging and fun way?
How to keep young students (16-18) engaged and make them actually learn?

[long pelican]

There's a lot of standard ways to make math "engaging" by using real life examples or being entertaining, but I don't completely agree with that being the solution. I don't know the full solution but here's something I came up with just now that I haven't mentioned before: Think of a time you saw something cool or potentially cool but the math didn't make sense, and you wished you could make sense of it. It could be something like how something works in real life, or why a certain problem's magical-looking solution actually works, or why a certain pattern in numbers holds. Maybe the teacher can try to somehow tap into that desire

[gray smelt]

honestly...i have no clue how to make the vast majority of people like math.

there is a certain type of person that finds my explanations of math concepts interesting. i think the commonality here is a neurodivergence with hyperfixations - even if they don't hyperfixate on math like i do, someone prone to hyperfixating is more likely to get sucked in and engage with an enthusiastic explanation. for this kind of person, i'd try to lead them in with enthusiastic explanations of cool shit, and then slowly introduce the rigorous details.

there is also a certain type of person that likes the kind of thing that is actual mathematics. this person may also hate the high school mathematics class (my calc 1 teacher reportedly was like this). the best way to spark this students mind is to show them some abstract algebra or real analysis, taking time to explain things in detail but also not dumbing it down.

there is also the kind of person that just doesn't like that thing. i'm not sure you can get those people to like math - it's not like you choose your interests.

[turbid zenith]

This is interesting to see this asked here

I have to commend your teacher for having the guts to ask y’all for help, and you for coming here to be part of that.

What math class are we talking about? Precalc, etc?

[violet haven]

Algebra 2 and Pre calculus

[hexed sand]

Hi everyone. Im teaching introduction to representation theory and my midterms are usually difficult but this time the mean grade was very low. One problem was only solved by one person [they are the only ones who scored points] (proving frobenius reciprocity for infinite groups). I feel that I taught what was needed in order to take the exam, and felt that when I taught it online my midterms were actually harder than this.

How should I approach preparing the class for the final? Quizzes and assignments are usually done well.

[cloud zealot]

https://www.poshenloh.com/quadraticdetail/

has anyone specifically taught this method to students?

were you more or less successful than with teaching the traditional approach to quadratics?

[hexed sand]

Which grade are you teaching to?

[long pelican]

How low? For representation theory I wouldn't sweat it if the average was 60-70. Also averages in online exams are always going to be significantly higher than averages in in-person exams because online exams have a nontrivial percentage increase of cheating compared to in-person exams

[hexed sand]

The average was just over 50

[long pelican]

Mmm, that's not too alarming. The two largest components of variance in an exam average is the problems themselves and the cohort of students (which you have no control over). Problems tend to be more of the variance in my experience

Predicting how hard a problem is going to be is nearly impossible

Anecdote time! This was the hardest problem in my first complex analysis midterm. No way I would have predicted that beforehand

[hexed sand]

Ah thats reassuring

[cloud zealot]

i'm not teaching to anyone. rather, i'm wondering if anyone has tried teaching quadratics this way instead of the usual method of factoring via guess-and-check, completing the square, or applying the quadratic formula.

presumably the audience that could be taught such a method would be at minimum middle school students.

[cosmic ibex]

Isn't that just the same as the quadratic formula without having the formula itself written down explicitly?

(Oh, and with the overall factor of ½ moved inside the square root, so you take the root of ¼b²-ac instead of b²-4ac).

[cloud zealot]

yeah ofc, but you would never memorize x=-b/a as the solution to ax+b=0. the main advantage is that the method is purely procedural, no tricks or memorization required.

[hexed sand]

hmm this is interesting since I was thinking that you wanted to teach vieta's formulae which are useful

solving quadratics is never the most enlightening thing since you still have to memorize the procedure so if theres not something else you wish to go to afterwards id steer clear

(of course it leads to nice things when you extend it but if youre teaching someone who isnt interested in maths it seems to be all for naught)

[cloud zealot]

I'm not interested in teaching Viete's formulae. Rather, I'm interested in an improved way of teaching students how to factor quadratics. Obviously, some baseline level of memorization is required. But the motivation that we need to memorize is actually a very intuitive geometric observation. For example, consider $x^2-4x+3=0$. Its roots are $x=1$ and $x=3$. We know that parabolas are symmetric about an axis of symmetry. The axis of symmetry is simply the midpoint, or \emph{average}, of the roots. In this case, it's $x=2$. Notice that $1=2-1$ and $3=2+1$. The axis of symmetry is our invariant, and to find our roots, all we need to know is the distance from the axis of symmetry to our roots, which in this case is 1. This is precisely what motivates Po-Shen Loh's algorithm.\
\hfill \break
Let's use Loh's algorithm to find the roots of $x^2-4x+3=0$. Consider $(x-r)(x-s)=x^2-(r+s)x+rs=0$. We want $-(r+s)=-4\Rightarrow r+s=4$ and $rs=3$. The critical observation to make is that $a+b=c\Leftrightarrow\dfrac{a+b}{2}=\dfrac{c}{2}$. In words, the sum of two numbers equals $c$ if and only if its \emph{average} is $\dfrac{c}{2}$. With this interpretation in mind, we can choose $a=b=\dfrac{c}{2}$. Also, observe $\dfrac{c}{2}+\dfrac{c}{2}=\dfrac{c}{2}+z+\dfrac{c}{2}-z=c$. This motivates our choice of $r$ and $s$. Choose $r=\dfrac{4}{2}+u=2+u$ and $s=\dfrac{4}{2}-u=2-u$, where $u$ is our unknown. Notice the 2, our axis of symmetry, in $r$ and $s$! Additionally, from the discussion above, $u$ can be interpreted as the distance from our axis of symmetry. Then $rs=4-u^2=3\Rightarrow u^2=1$. Without loss of generality, we can choose $u=1$. So $r=3$ and $s=1$, which means our roots are $x=1$ and $x=3$. We have also constructed the factorization $(x-1)(x-3)=x^2-4x+3=0$.

[burnt vessel]

Sour Drop

[fallow siren]

Is that really different from the "standard" way? We were (x-r)(x-s), then note the sum and product requirements. Taking r=s will give the latest possible product and then you adjust it down using difference is squares.

[narrow nest]

this is literally just the quadratic formula

axis of symmetry = -b/(2a)
distance = +-sqrt(b^2-4ac)/(2a)

but i do think it's a good derivation

using vieta's instead of completing the square

[long pelican]

Reading this discussion, a fair conclusion sounds like we should be aiming to teach students neither one formula nor the other but the sufficient understanding to independently be able to see and explain (i.e. do the necessary calculations with variables) why the two methods are doing the same thing

[narrow nest]

here's my derivation: