#math-pedagogy

1 messages · Page 57 of 1

stark pine
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oh I also really like Herstein's exercise to show that a ring is commutative if x^3 = x for all x. There are so many proofs, but one of them is pure computation

wispy jolt
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i think of computation in proof based math as an activity that takes place once you have setup the required justifications

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and can justifiably implement the computational method

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so like, this does sort of constrain the sort of reasoning the computation probably takes on, but it's also at a higher order of understanding your justifications, i think

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of course some computations you just always make, like basic algebraic manipulation of terms

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but that's long justified

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so i wrote a big long thesis on basic galois theory and just assuming some stuff from abstract algebra and basic algebra that may their selves be computational at times, i make a system of evaluating high order polynomials using modular arithmetic, which this final thing is definitely computational in implementation

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you probably cant even just find the computational method, you need to justify your way to it

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since you need to do stuff with the galois theory of specific groups relating to the polynomial

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once you have it all tho you can probably program algorithms to do some of it

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i guess it sorta puts it on the road towards a calculus

silent coral
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There's a lot related to Sylow Theorems. Permutations and classification of groups. I would say

meager bronze
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please read the rules

wispy slate
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sorry

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Where do I ask that?

meager bronze
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which it says in the rules

wispy slate
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Hello

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Any pedagogs present?

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Feel free to @ me or DM me

meager bronze
wispy slate
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A kid Im tutoring stutters when talking

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But not when solving problems and talking through the problem for himself

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I just noticed today. I would need to confirm. Should I tell the parents?

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Is this normal for people who stutter?

pastel horizon
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I'm not a psychologist but maybe he stutters because he's overthinking his speech but maths problems give a distraction?

wispy slate
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Either that or math is something he enjoys and talking to people isnt

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Or as you said, a distraction from focusing on speech

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I found it interesting anyways

astral laurel
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I think you should tell the parents if you think they are responsible for the kid. This kind of sounds weird in that you could assume so, but the thing is you should also respect the kid's wishes if you can discern them, and most importantly if you can sense that you do not think the parents are acting in the kid's best interests

wispy slate
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I think they want the best for him

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He's doing competition math for 4th grade

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His parents are my friends so I know them well enough

pastel horizon
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I think the key is not to tell him that he's not stammering when doing maths problems

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One thing I've seen is that you don't stammer if you can't hear yourself. So maybe when he's musing his thoughts he's not really listening to his own voice?

astral laurel
wispy slate
distant fractal
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what can I do to get students who dont really care about math to engage in the classroom (university)? im leading a discussion section of a freshman math class for non math majors. I did it last semester too, and had a much easier time in the calculus class. It got particularly droll when doing matrix stuff, like gauss-jordans method for solving linear eqs

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ill ask a question (and try to come up with the most interesting questions from the section) but frequently it feels like bleeding a stone

real mauve
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for this type of situation, maybe having very clear examples of where one could use it would be nice

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even stuff like "when you google this" or "when you use excel", etc, "this happens under the hood"

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i'd make it as concrete as possible in the early stages. when tutoring non stem students, one of the first questions i get is always "and what is this used for in real life"

long pelican
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Last semester of teaching was basically this exact situation with me for like 50% of the students all semester. My bold take is that these types of students are not merely uninterested; they have to be both uninterested and lack sufficient prerequisites to understand the material at a basic level

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Any fix quicker than solving their prerequisite problem will just be forgotten in the next semester

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Showing them a real life application will achieve getting their attention about 15% more than before, but doesn't solve the prerequisite problem

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(15% obviously not an exact figure)

real mauve
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that could very well be, but looking at linalg vs calc, i don't know how much the prerequisite for linalg is something you can quickly teach

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by that i mean you technically know the basics since hs, it's more intuition and maturity

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and no one likes grinding out GJ problems :p

long pelican
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I mean, if interest is all they lack, my intuition tells me they'll still spend the class time doing the assigned problems. Resistance to even starting the assignment usually comes from being so uncomfortable with the material so as to not even be willing to start

real mauve
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i do agree to some extent, but then what do you suggest is the missing background for linalg that was somehow not missing for clac?

long pelican
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A guess is that calc might have been computational and students could do fine understanding the stuff with their high school "test-prep" understanding of functions and variables, and it breaks down in linear algebra

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After all, all this test-prep algebra 2/precalculus is precisely to prepare them for what the educators or textbook writers think calculus class will be

real mauve
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very meta, but most certainly possible

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on second thought, systems of equations do seem to be a problem that kinda gets dropped along the way and suddenly picked up again in early linalg, and it goes downhill from there as the level of abstraction increases

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maybe reinforcing that first would be good?

long pelican
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Probably the issues will be more fundamental. I would bet at least 1 student in every linear algebra class cannot tell you what solving a system of linear equations accomplishes

real mauve
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that's what i mean by reviewing them though, maybe my wording was poor

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because many struggle understanding what they're for and how to set them up

long pelican
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I guess you're on the topic as a whole, I'm more on just the meaning of stuff

real mauve
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ah

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still, i would definitely toss in a good example early on. lot's of people are aware of things like noise cancellation, and at the very least, the memes regarding stocks. one can set up a simple linear prediction to show a cool application

long pelican
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That's a good motivational tool and works well when they have the ability (given the motivation) to plug in gaps in their knowledge

wispy slate
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@distant fractal address the fact that you're aware they dont care about math and that they would have been math majors if they did. Regardless, it's a requirement and the study habits and the logic used in math classes will help them finish university

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Empathy helps

distant fractal
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Thanks for the advice yall!

tawdry venture
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is there a way to teach the kind of intuition that is necessary for recognizing obviously-wrong answers?

tawny slate
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seems like a very abstract and broad question

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the most general answers that i can give basically revolves around like

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building good habits about being critical, not assuming that anything "obvious" is really that obvious

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asking yourself "why" and then repeatedly asking it

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challenging definitions (i graphed the equation, but what does it mean to "graph" something?)

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beyond that, i think it's difficult to answer without a clearer specification of what that intuition entails

long pelican
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Could be teaching to build the habit of making as many observations as possible instead of being hyper focused on the “correct” way to solve the problem

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Along with that, they also have to unlearn reliance on answer keys/teacher saying right or wrong

tawny slate
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it may help to learn to assign intuitive "probabilities" to confidence levels

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like when i say that some math statement I haven't proven feels correct, i might subconsciously assign it some condition or probability, like "as long as p lemma holds, I think there's an 80% chance this is right"

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so even for things you feel might be right, you're acknowledging that it could be wrong

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another way to answer the question might be in regards to just having a broad range of math skills and being able to apply them everywhere

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for instance, when doing multiple choice questions, you can easily eliminate many answers that are "clearly" wrong by using several techniques that utilize all kinds of different concepts

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maybe the last digit is wrong due to modular arithmetic

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maybe an estimation shows it's not the right magnitude

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maybe unit/dimensional analysis shows it can't possibly be the answer

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each skill you learn gives you another way to see how an answer might be right, but also why it might be wrong

charred silo
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I think this is like... recognising possible useful logical consequences of given statements, and knowing how to apply them?

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In terms of teaching it, perhaps inform the students what the useful checks are for some common statements, and have them use it to check answers, may be?

turbid zenith
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So I'm planning for next semester, and I'm trying to figure out what to do with kind of the opposite problem most people have

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My Mathematics and Human Nature course is a core course that all majors need to take at my school. For the vast majority of the students, this is their last math class and they're absolutely terrified of it. I kind of turn it into a "trauma healing" class a lot of the time, because I use it to show them the parts of math that most people DON'T get to see, encourage them to build number sense, etc.

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But recently I've been finding that I'm not quite sure what to do with the few students I occasionally get who do have a good math background or are occasionally a math major ... not to bore them

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So like the kind of student who already HAS heard of things like graph theory, non-Euclidean geometry, group theory, etc

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I introduce these things in my class in a way that makes them friendly to non-math majors

wise onyx
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Just get those students to share their knowledge and teach the less knowledgeable students

quasi musk
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So my pre-calc class is doing a lot of difference quotients

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Students are struggling a lot

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Taking derivatives by the limit definition

real mauve
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what is tripping them up? general handling of fractions?

next relic
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Introducing an entirely new concept might not be the best idea, but you can have a set of "standards" that you expect EVERYONE to achieve and a set of questions that goes beyond and above.

long pelican
earnest trail
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like if they have f(x)=x², they're lost as to what f(x+h) would be

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x²+h? x^(2+h)?

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so I like to show them a sort of substitution, since if I said, "okay what's f(t)" - they'll typically be able to say t²

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so then I say "well t=x+h, so replace t with x+h"

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sometimes that helps but sometimes they're still confused

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at that point idk what else to do so I straight up tell them, and then I think they understand, but yeah

astral laurel
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Well show them directly then?

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Suppose there is f(g(x)), and g(x) = ...?

earnest trail
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that might be helpful

long pelican
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I've grappled with students having similar issues last semester. Bottom line is that being lost as to what f(x+h) might be is good diagnostic information and is a symptom of something. If they knew how to read math, they could figure it out from parsing f(x+h), the same way a computer would parse an expression. If they can't do that, that means they don't have practice in parsing expressions.

Something that might also be true is that they think x is a "special type of number" (teachers say unknown, or variable) and don't read f(x) = x^2 as being a universally quantified "for all real numbers x" type of statement but rather a single statement that f maps the symbol x to the symbol x^2

Actually we do think of ring homomorphisms that way. If phi: R[x] -> R[x] is defined by phi(x) = x^2, then phi(x+1) is x^2+1, not (x+1)^2

earnest trail
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the last part is weird af lol

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anyway another thing I notice is that people think variables will always be the same

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like yesterday I was helping my friend with some integrals

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and I had to do two u-substitutions so I did a u and a v

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and he was confused with the v part because it wasn't called u

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so I had to explain to him that you can use any variable for substitution, and that u is just the most common

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and when we were first doing integrals I jokingly asked my teacher if I could put a "-C" instead of a "+C" and someone was like "you can do that?" lol

long pelican
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What'd the teacher say to the -C question?

earnest trail
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she said sure

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since it makes sense still

tawny slate
# turbid zenith But recently I've been finding that I'm not quite sure what to do with the few s...

I dunno how useful my response is going to be but I'm gonna try something

On an abstract level, I think telling good specific stories about the value of abstract and higher-level math, applications where the "invention" of new math concepts and objects is just as important as the deriving of formulas and proofs, will allow students to appreciate the value of these kinds of things while still being entertaining for more experienced students

More specifically though, if there is a concept I actually need to teach, I try to teach it from a different perspective than what most students have been taught, such as teaching algebra from a logic foundation (which is still pretty simple), geometry from a combinatorial perspective (which is still not necessarily that practical), or teaching combinatorics from a foundation of starting with bijections (which is still hard to apply practically). Maybe introduce subjects and topics based on how they should change how one views practical concepts, such as how being competent with probability allows one to more easily identify bias, or how being competent with game theory allows one to more easily understand how other people think and feel, or how being competent with asymptotic growth rates help understand facts relating to the nature of reality

quasi musk
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It's mainly all the different factoring tricks that they need to do

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Plus the fact that they're dealing w/ limits

earnest trail
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well maybe then first this is the progression you would need to teach this in:

  1. learning different factoring tricks for functions in general

  2. teach the difference quotient without ever mentioning limits, tell students to leave the h there but simplify as much as possible using the tricks you teach them from (1)
    side note: this might be interesting in itself since you can tell students to choose values of h that they desire to find the average rate of a change of a function where the x-values are a distance h apart

  3. then after this, introduce limits because now all they have to do is sub in 0 for the h's and boom

quasi musk
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It's not really an option due to the fact that I work for a private company

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And I have to go by their curriculum

earnest trail
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oh yuck

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suggest it to the company maybe?

quasi musk
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What happens is the parents push their students to a higher level than what they're ready for

earnest trail
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oh god

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that's annoying

quasi musk
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Then they get bodied in the pre-calc when we start doing limits

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I had a boy who just joined my class, 10th grader

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He's taking like algebra 2

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He's picked up on limits faster than the other students that have been in the class since the beginning

earnest trail
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bruh lmao

quasi musk
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The teacher for the first fifteen lessons had a medical emergency

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So I had to take over

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(We meet once a week for the academic year)

earnest trail
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oh I see

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damn the teacher was out for 15 weeks?

quasi musk
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So it's a combination of students are like wtf, I'm the new teacher, and I'm trying to get acclimated

quasi musk
earnest trail
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did he/she like almost die or smth holy shit

quasi musk
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Eye surgery

earnest trail
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oh damn

quasi musk
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I gave a diagnostic quiz yesterday and ppl got bodied

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I thought it'd take like 10 mins max

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And it ended up taking 25

earnest trail
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that's how it always goes

quasi musk
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Well it'd take me like 2-3 minutes to do the quiz

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So I just multiplied by 3

long pelican
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Did you investigate the students’ issues in the quiz yet?

quasi musk
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They're uncomfortable with the power rule, and analyzing graphs of piecewise functions to see if things are continuous or not

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I'm in a pickle because I just joined, so I don't know how the class is usually run

earnest trail
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oh god if they're struggling with intro to derivatives it's gonna be a looooooong year

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it shouldn't be too hard for them to get the concept of "hop the exponent to the front and subtract 1"

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but again with the things you said about the transition and stuff it's weird

quasi musk
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Yeah, I've been doing things by definition a lot

earnest trail
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wdym

long pelican
earnest trail
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^

quasi musk
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I guess I'll try to drill it in

earnest trail
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just do a bunch of problems

quasi musk
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Yeah, I think that's the best way to do it

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It's just one problem takes forever to address student questions

earnest trail
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my calc teacher just does problems all class, every class and it seems to work

quasi musk
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And I don't get to other enlightening parts

earnest trail
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sure, she doesn't really teach the understanding and intuition but it's not necessary for ap calc (which sucks but whatever)

earnest trail
turbid zenith
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Just got to read the responses to what I was posting earlier.

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@tawny slate I like the sound of that.

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And @long pelican that's a really good point.

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So ... I might explicitly preface things this semester by letting them know that for students who are "already on Team Math", they may have heard of some of the concepts before or even seen the "punchlines" of some of the results we're doing, so if that's the position they're in, I encourage them to (1) consider them in the larger context of the class, and (2) compare the active way the lessons are designed to the usual way these topics are introduced, and how that might help them become better "ambassadors of mathematics"

earnest trail
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ambassadors of mathematics

turbid zenith
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Seriously.

earnest trail
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please don't call them that, they'll probably cringe lol

turbid zenith
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Maybe 😛 It's all in how you pharse and deliver it probably

earnest trail
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lmao

tawny slate
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Also, it's not technically wrong

turbid zenith
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Math gets a bad rap because the math people get shoved down their throats is so different from what math really should be like. If someone managed to get through the system intact, I think in some sense they have a responsibility to make it less shitty for the next generation.

tawny slate
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Any dilettante that is passionate and vocal about a topic can end up being an "ambassador" of sorts, and therefore we should be more cognizant of what kind of message we are sending to society as a whole and to individuals

turbid zenith
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I already tell my students on the last day of class that it's their job to break the "I hate math" chain

tawny slate
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When I teach, my first priority is not teaching math skills or competency, but instilling a passion and interest in math

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If they like math, they will be motivated to work on it, and that is more efficient and more valuable than the specific skill they will learn in any particular class in the long run

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That doesn't mean the skills themselves aren't important, I'm just saying

turbid zenith
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Yeah that makes sense. You've got a metagame going on.

wispy slate
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just my personal experience though idk

earnest trail
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lmao

covert fern
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For some reason linear algebra topics are something that catches they attention.

astral laurel
long pelican
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I can definitely see linear algebra becoming a more "accessible" field as well as a field more people are paying attention to, due to it being one of the prerequisites of machine learning

left vault
strange river
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is there a name for a type of problem solving where you dissect a big problem to a bunch of smaller, easier-to-solve problems? in programming, I've heard someone call that "recursive problem solving" but is this term applicable across all scientific disciplines or does math have some kind of unique name to that principle?

charred silo
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I heard it being called "operationalisation" once as a general concept. Personally, I would just call it "breaking down a problem". Easier to understand.

vernal willow
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for algorithms ive heard divide-and-conquer

grand drum
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in ur guys’ experience, is the average eighth grader comfortable with the concept of a variable?

real mauve
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if you ask icy, the answer remains "no" well into undergrad

wispy slate
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Well at least de facto, although almost all are capable of being comfortable

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The piece on recognizing when something is happening linguistically/syntactically vs "mathematically" and how not making the distinction while teaching can suck, was enlightening

long pelican
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I’ve got a complementary hot take though: if you really wanted to take some average 8th grader and had some time available to spend with them and provided they have the motivation to learn, you could make them comfortable with the concept of variable by showing them well chosen examples of mathematical proofs

real mauve
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i don't see how that is all that different from the usual approach

inland wren
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the hard part is knowing when a variable is an unknown to be solved for, when it's an arbitrary constant, when it's implicitly dependent on some other variable, when its independent of another variable etc.

long pelican
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Usual approach in public schools is different, no? No examples of proofs shown, just drilling exercises of set types of problems (with shortcuts taken to their writing)

long pelican
real mauve
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but when one presents definitions of stuff like conic sections, for example, it is done this way

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though i suppose it is often not emphasized enough

long pelican
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I edited 2 messages ago to add that these set types of problems have shortcuts taken to their writing

real mauve
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ah

long pelican
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A good phrase to describe these shortcuts is implicit quantification. (I linked an article about that a while ago.) If you ask students to understand things written with implicit quantification without even presenting examples of explicit quantification, as millions of teachers fail to do, you can expect them to not get it unless they do math a lot in their free time. It’s kind of absurd actually

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Or maybe not absurd from a historical perspective. Emphasis on logical quantification and doing math from set theoretic axioms as a way to understand math really only started in the mid 1800s

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Public education just lags by more than 200 years I guess

real mauve
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i do agree showing well curated examples can be helpful. did you have any in mind? it is also a bit difficult to find something that very clearly conveys the different things a variable may represent without also requiring additional stuff that is often relegated to undergrad rather than HS

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though arguably there is no deep reason why it is done that way

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what depth were you thinking of?

long pelican
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In modern math everything is expressed like in the following examples

If x is an element of ... then ...
For all real numbers x....
For all real-valued functions f....
There exists a real-valued function f....
There exists a pair of real numbers (a,b) such that...
For all pairs of real numbers (a,b) ....
There exists a triple of integers (x,y,z) such that for all real numbers w...

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(not an exhaustive list)

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There is not a huge amount of variation to this type of syntax and it's very powerful, pretty much superseding the old "variables/parameters" distinction. Why not learn/teach it properly?

real mauve
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yeah, my first reaction was "maybe \forall and \exists is too advanced"

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but there is no real reason to think that way

astral laurel
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An actual issue is that the symbol soup sometimes is not explicitly taught. This is definitely true pre-uni from my experience.

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Then in uni I also never got taught, you basically pick it as you go along when you encounter more and more different cases requiring that you disambiguate or show specifics

long pelican
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Yeah that's probably the case for most math majors nowadays. Although the situation could be viewed as a weeder at the moment. Could be less of a weeder when this is made explicit

earnest trail
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you can talk about math to middle and high schoolers with the words for all and there exists but there isn't much of a need to introduce the symbols, perhaps until later after they understand the basics

long pelican
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You don't really use the symbols except in logic classes

earnest trail
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exactly

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it just makes it harder to follow and understand

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maybe the implication symbol and set membership and subset symbols can be used but that's about it

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in general the set symbols are okay but logic symbols are ehhhh

cosmic ibex
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The big trouble with introducing symbolic quantifiers too late is that students will eventually see them anyway and internalize them as just shorthand symbols for words with the usual natural-language grammar, which really nullifies much of the benefits of having symbols. Then they end up writing monstrosities like "∃n such that n>x ∀x".

earnest trail
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but they most likely wont see them again unless they go into math

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but yeah you're right

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not too late

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but they have to be comfortable with the logic in english first, only then can you introduce symbols

cosmic ibex
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Hmm, might be strategically useful to mention the symbols in passing while one is discussing that it matters whether you say "for all x there is an n" or "there is an n such that for all x". Something like, by the way, in higher math these chains can get so long that we need a symbolic notation to keep them straight, here's how that looks -- but we won't need that notation for this class.

earnest trail
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could be

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and then the year after introduce the symbols

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smth like that

cosmic ibex
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The challenge is to avoid feeding the "it's more mathematical with symbols" misconception.

tepid smelt
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I was doing some problems regarding maximizing profit given certain constraints or linear programming with my students. I didn't have a good intuitive answer to give them as to why does the maximum/minimum of linear programming occurs at a vertex? Can someone help give me a good intuitive explanation that a freshman can understand?

long pelican
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Try to come up with a mathematical proof then slowly turn it into an intuitive explanation

quasi musk
astral laurel
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When R^2 drawing fails then it will be very difficult to see intuition

pastel horizon
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The intuition is that the optimal solution is going to occur when one, or more of the variables is at the maximum value for its constraint. If you didn't have any constraint you could just push that variable up to infinity for the "optimal" solution

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The next part of the intuition is that say you have a group variables that aren't at the constraints. It means that you can increase/decrease them further improving your objective function

real mauve
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if you only want intuition, you can get away with the definition of half spaces and their intersections. you can show how the value of the objective function changes in a direction parallel to and perpendicular to the vector c in c^T x and overlay that on the feasible set, as shattered sunlight says.

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and then see what happens depending on the shape of the feasible set

quasi musk
real mauve
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that's predicated on the behavior c^T x though

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i think that's a better place to start

quasi musk
real mauve
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surely, but jumping straight to "if the variable isn't in the constraint..." leads to the false impression that this is always the case

quasi musk
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I guess it's just a difference of what level you're thinking of things at

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I've only ever taught these type of constraint problems to people in remedial math

real mauve
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they show up as the first steps in most convex opt courses, too

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which inevitably end up at how to find local optima for nonconvex functions

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and then local concavity/convexity changes whether it's true or not

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i assumed a different context :x i guess we'd need more input frm stephenruby8

pastel horizon
turbid zenith
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Good morning, I'm trying to stop procrastinating making my syllabi 😛

real mauve
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you're doing it wrong already 😛

turbid zenith
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DON'T JUDGE ME

earnest trail
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damn

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I'll make your syllabus!

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cheating on exams is highly encouraged

tawdry venture
turbid zenith
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I just don't give exams

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😛

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Joke's on you

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Would welcome ideas on this if anyone has any

turbid zenith
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I feel like I need one for engineering

real mauve
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arguably the accelerometer one fits there

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but maybe something like computer vision or somesuch

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"how machines learn", as 3b1b put it

turbid zenith
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I also added "…how computers approximate solutions when an exact answer is impossible to find?"

turbid zenith
real mauve
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it's a lot of different flavors of gradient descent

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engineering is a very broad umbrella, almost everything involves optimization in some way or another :x

turbid zenith
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oh how about like

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How your GPS figures out how fast you're going

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it doesn't have access to your speedometer

real mauve
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yeah that works

quasi musk
dark horizon
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So I have a question about studying math that I’ve been struggling with:

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I feel like I’m looking for a math course along the lines of ‘oh so you know formal logic, time to teach you how to do mathematics’. Because there’s a lot of things that happen when doing mathematics that you don’t learn about when studying logic: for example how to deal with limited functions, and how to formalize them and deduce their properties.

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And I’m looking for a course where I can practice doing that - making the switch between math and formal logic, and filling in any gaps (e.g. limited functions, dealing with adding definitions as you go along) - but I just cannot seem to find such a course. What comes close to this? ‘Intro to proofs’ courses don’t really scratch that itch because they don’t really touch formal logic.

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(let me know if this question is in the wrong channel, I'll gladly move it)

cosmic ibex
tepid smelt
# real mauve if you only want intuition, you can get away with the definition of half spaces ...

I should mention these are high school freshman in a public school. I tried mentioning the way I thought of it in that we are trying to maximize an objective function C=ax+by and for different values of C you just get different straight lines with the same slope but different y intercepts and the lines that fall within the feasible region satisfy our constraints. Since the feasible region is bounded by linear equations, as the constant-value line moves through and out of the feasible region, it last touches the feasible region at a vertex, so the optimal solution must occur at a vertex of the feasible region. Most were satisfied but I was not that satisfied. I was just happy that they were really engaged with this topic and the thought we could maximize profit using some simple techniques we have learned with systems. I know math should not be about showing when its useful but I definitely got a lot more engagement giving lots of simple business scenarios where we could maximize profit. I do appreciate the responses it has helped my understanding as well as this is my first time teaching math 1.

tepid smelt
long pelican
#

I feel weird being considered a preacher considering I'm probably going to have different ideas 5 years from now or even 2 years from now

earnest trail
#

preach!

quasi musk
#

And I think that's ok

#

Most people won't use math at all

long pelican
#

Going off of that, what do people think of removing some standards such as finding vertex form of a parabola and other things that most people will never use?

#

The number of specific skills teachers are expected to teach is pretty high and lends itself easily to skills-based teaching

#

Instead of skills-based standards how about problem solving and comprehension standards

earnest trail
#

I agree

#

because algebra class just feels like learning a bunch of specific problems

#

instead we should be taught how to approach a general set of problems so that we can learn general problem solving classes that can be applied to a wide set of problems

real mauve
#

there is probably a better way of covering it, icy, but i would say the goal is to become comfortable with objevts that are defined by a small number of features. like "this thing is highly structured, and so you only need very few parameters to completely describe it"

#

it's the sort of stuff that is often reviewed when covering statistics and estimation

#

that's one think. the other is that it's the sort of algebraic manipulation one does to more easily see compression, stretching, translation, etc. though focusing on that form as being a special thing with a fancy name really serves no purpose, it does facilitate becoming familiar with these transformations

#

maybe it could be done better as a quick section in early algebra?

earnest trail
#

I can do the simpler ones like flipping across the axes and moving a function up or down but I have no idea why (x-2)² moves it to the right by 2 instead of to the left, y'know

stark pine
# earnest trail I can do the simpler ones like flipping across the axes and moving a function up...

I like to imagine keeping the function fixed and moving the axis. The reason for this becomes clearer if you just try some examples. Plugging x = 2 into (x - 2)^2 is the same as plugging 0 into x^2; plugging x = 5 into (x - 2)^2 is the same as plugging x = 3 into x^2. What you’ve done is moved your input left by 2 - so you drag the x-axis 2 units to the left while fixing the graph. This has the same effect as moving the graph 2 units to the right

earnest trail
#

I see, so you fix the outputs but shift the inputs

stark pine
#

You may ask “why doesn’t x^2 + 3 then drag the y-axis up 3 units?”. The reason so think of is that x^2 + 3 is shifting your output - so you then move the graph. (x - 2)^2 is shifting the input so you’re moving the axis.

#

So I picture different things moving

#

But that way the directions are consistent

stark pine
earnest trail
#

oh okay that makes more sense

#

I can implement that reasoning in my teaching

#

thank you so much!

stark pine
#

No problem! That visualization was something I came up with back in grade 9 and I’ve had immense success with it when tutoring

#

The visual of dragging the axis really illuminated what’s going on - I’ve printed and cut out an x-axis and had my tutees just drag it to see what’s happening

earnest trail
#

is tutees a real word?

#

but nonetheless that's a good strategy for teaching it :)

long pelican
#

for function transformations you won't believe this method...

#

For each point (x, f(x)) on the graph of x->f(x) the corresponding point on the graph of x->f(x-a) is (x+a, f(x)). (x-a, f(x)) won't work: when you plug x-a into f(x-a) you get f(x-2a), not f(x)!

stark pine
long pelican
#

Act II: The corresponding point on the graph of x->f(x)+b is (x, f(x) + b)

stark pine
#

And the visual of the input axis being shifted is genuinely what’s going on, but it works for the students who are having trouble with the notation and helps them get more comfortable

long pelican
#

Let's teach function transformations but never mind what the definition of the graph of a function is anymore!

stark pine
#

But it is the definition. It’s a more visual representation, but the inputs being shifted left by 2 is what’s happening. I think that people should become comfortable with ordered pairs and function notation - and I think this kind of visualization is an incredibly helpful stepping stone

real mauve
#

i find that substitutions are the simpler way of explaining it

long pelican
#

Wait, what are you saying is the definition

#

That f(x-a) "means" shift right?

earnest trail
#

the visual representation is better for beginners, and then once they understand it they can be exposed to the abstraction of the function representation

stark pine
#

^

real mauve
#

i rather forget about "shifts" and the like and simply matroska operations like w = x-a, but the net effect is the same either way. keep the plot fixed and reimagine a new axis

stark pine
# long pelican Wait, what are you saying *is* the definition

I should have been more explicit - I’m saying that imagining a new/relabeled/shifted axis is a way to conceptualize the definition. There’s nothing incorrect about it. It’s not a lie for the sake of clarity, it’s an equivalent way to understand what’s going on for a function f : R -> R

long pelican
#

Pretty extreme viewpoint, unhelpful tbh. Yes it's a bit frustrating people haven't "made progress" on the topic of function transformations but have you helped?

real mauve
#

let's keep it civil here

stark pine
#

I don’t have to “progress”, I understand function transformations. I also understand tutoring math, as I’ve done it a lot and this has been super effective. You’re welcome not to use it, but barring strong evidence that it doesn’t work, I’ll continue doing it

#

@proven dawn sorry, I wasn’t aware that this conversation had happened before. I knew of a helpful visualization which I shared

long pelican
#

I wasn't referring to you actually

#

I know function transformations are a sticky point worldwide from the fact they are like the #2 question topic in the help channels

twin falcon
long pelican
#

What's "this"?

long pelican
real mauve
#

let's stop the useless banter and return to the topic at hand. it's certainly worth discussing. as icy points out, many people struggle with this and coming up with different ways to explain it is good

earnest trail
#

^

#

temperature was high in here for a fat minute

#

so anyway visualizations are always important especially at the beginning imo

#

then you can move to more abstract methods like with the substitutions via function notation

long pelican
#

Hmmm I might explain that I'm not against Nick's method at all, more that students are still confused even after seeing the visualizations a lot of the time

#

And that has to do with the missing connection between the visualizations and, well, what graphing f(x-a) even means, which takes a while to recall but it means to plot every point of the form (x, f(x-a)) in the plane

earnest trail
#

why do you say f(x-a) instead of f(x+a)

#

is it the same reason

#

that you're shifting the inputs and fixing the outputs

#

because to some students it may seem arbitrary

long pelican
#

I don't have a particular reason to not write f(x+a)

#

I'll just ditch the a from now on

#

f(x+2)

#

Riemann's thread is still ongoing, good data

#

🙃

earnest trail
#

riemann's thread

#

that sounds like a real thing

#

lol

long pelican
odd axle
long pelican
#

Welcome

tepid smelt
odd axle
long pelican
#

It has to be basic of course, because you can't ask for too much creativity on an exam

#

In summary I'd say that I'm focusing on the ability to do independent original sequential thinking about the objects they've studied (as opposed to copying a process)

pastel horizon
#

Omg yes, transformations are a place where many go wrong. I think it's because they get taught procedurally rather than working out from first principles

cosmic ibex
#

Regarding Icy's initial point, while it doesn't make independent sense to train "find the vertex form" as a quick-pull skill, being aware of the fact that a quadratic always has a vertex form (i.e. its graph always looks like y=x^2 with scaling and translation) is immensely useful when visualizing problems. And it is very easy for mathematics teaching to slide down the slippery slope from "this is always the case" through "we know this is always the case because here's how to do it" to "doing it yourself will be on the test" to "be sure to remember how to do it". (It wouldn't really be mathematics unless we deign to take the first of those steps).

#

The real tragedy is how social pressures and standardized testing tend to convert "good question for checking understanding" into "pointless nonsense skill to learn by rote".

astral laurel
stark pine
wispy slate
# tepid smelt I should mention these are high school freshman in a public school. I tried ment...

That's neat! After reading this I digested it a little further into something you might like since it's shorter, although you might think about it like this yourself in the first place: we've seen that for free choice of a and b we can recover, or create any point on the plane since a point on a line with such a slope and a y intercept determine a and b. So we can choose any point, meaning we can just pick out the highest point in the region for free

#

And if we are only allowing integer y-intercepts or some other constraint like that, we can just choose the highest point of any of the possible lines in the region because we can create any of those points too

long pelican
stark pine
# long pelican Ultraproduct sidetracked us here so I never got to ask this, but I am still conf...

function transformations; in particular, the visual of the coordinates being moved, not the graph, clarifies for many students why f(x + 2) moves the graph left, and not right as they often naively expect. I think the visual of the coordinates being moved, not the graph explains what's going on, since f(x + 2) is of course shifting your input not your output, and thinking of shifting the coordinate axis visually tells you what "shifting the input" looks like

#

so I guess precisely what I'm advocating for and what I do is teaching that f(x + 2) shifts your x-axis 2 units to the right, not teaching that f(x + 2) shifts your graph 2 units to the left, since I think the former is a more authentic representation of what's going on

#

and this visual helps students understand function notation/composition on a more visual level, which helps with abstraction

long pelican
#

Ok I understand. So it’s basically giving students the proper perspective to even begin understanding how it all works

stark pine
#

yeah exactly

#

because nearly every student I've worked with has had the intuition "adding moves it right because right is the positive direction", and when the reality is the opposite, instead of trying to understand they just give up and memorize

#

which does not set them up for understanding function composition in a more general setting in the future

long pelican
#

Yeah, perspective shifts are often the hardest parts of understanding something new and isn’t necessarily automatic from practice

wispy slate
stark pine
real mauve
#

that's similar to what i had suggested as well

#

by substituting stuff like x+2 by another variable altogether

#

then you can keep the curve fixed and draw a separate axis for the transformed variables

#

like overlay them in different colors and whatnot

pastel horizon
tepid smelt
# wispy slate That's neat! After reading this I digested it a little further into something yo...

I like this one I can show this visually also which helps the kids. I think after the linear programming I will show them how to use matrices next with systems. I just worry it might be too boring. How have any of you made this topic more interesting for students using it the first time? This is my last topic in algebra before switching to transformations in geometry which are much more fun visually for kids

pastel horizon
#

Maybe showing an example of how it's used for animation, that's surely a guaranteed win

#

I think any time you can link something with computing it's a win with the kids

real mauve
#

hmm i will admit this seems odd to me. if you were going to cover matrices anyway, linear programming would make more sense as a continuation of that, once one has studied matrices. e.g. by looking at them row wise as intersections of flats. then you can use the same geometric intuition fornlinear programming

long pelican
#

Random thought: I am waiting for the day this help channels on this discord provide someone with enough data to write a groundbreaking math education research paper about what works and what doesn't work when teaching/helping/tutoring

quasi musk
#

I think most people kind of ignore math education results

#

A lot of it is so regional/cultural based

long pelican
#

Are you saying that's a fault on the writers' side or the users' side?

quasi musk
#

Neither side, it's just a fact of math education

#

So many results appear to be geographical/cultural based

#

So findings in one year might not translate to other areas

#

e.g. findings on online software modules for math (Aleks, WebAssign, MyMathLab, My Open Math)

long pelican
#

So just the state of things. I admit to ignoring all the software findings too

#

A lot of published papers have sort of weak evidence too. But I imagine a comprehensive analysis of the entire math discord would be extremely strong evidence, objectively speaking

#

Maybe if it was done by some type of advanced AI (as a bonus)

tepid smelt
real mauve
#

absolutely, one always keeps refining the way topics are covered

#

the order, the explanations, which details are actually important, etc

molten urchin
pastel horizon
#

I think the reason why research is weak is because of the ethical implications. I doubt you'd be popular if you mentioned to parents that their children are guinea pigs to improve future education

pastel horizon
real mauve
#

in some sense, though, this is all after the fact. the systems have already been established and running for a long time

#

it's not the people doing the studies and collecting data that put the students in that position

pastel horizon
#

Well yeah. I was more about experimental studies on new techniques/systems. I think it's a hard sell for parents. Or what if you have to explain that someone's child is in a control group so they don't even get a chance to experience the new technique

real mauve
#

ah, indeed

pastel horizon
#

I'd argue even using data here is unethical since nobody has given consent for their questions to be reused as part of research

left vault
#

Most HRECs approve research with human data where said data is already publicly available

quasi musk
brittle crag
#

Every 8 years: PIVOT!

quasi musk
#

Covered the mean value theorem today. Asked the students if they wanted to see the proof or just memorize it

#

They asked to see the proof

#

Halfway through they regretted their decision

tepid smelt
# quasi musk A lot of it is so regional/cultural based

I am curious if you can elaborate on the culture. I started teaching at a title 1 school where most of the students parents were field workers and not educated. Many of the students struggled and I thought it was due to economic reasons. Yet I noticed many of my poor students from Hmong backgrounds did better and tried harder. Now working in the bay area with some very affluent families I notice still many rich white kids struggle but say those from Chinese backgrounds do significantly better and try really hard.

I still see overall that my students suffering economically do worse on average but its not as simple as I once thought

quasi musk
#

(These are HS freshman/sophomores)

long pelican
#

Are you in CA? You have school on Sundays? 😮

quasi musk
#

So a way out of poverty and into nobility/aristocracy was literally to learn how to read, write, and interpret poetry and calculate quickly & accurately

quasi musk
#

It's an after school & weekend program

long pelican
#

Oh so weekends too. By the way is RSM situated in CA in person or is it online and national/global?

quasi musk
#

It's largely in the US, head-quartered in Newton Mass.

#

They have like 30 or 40 locations in the US

#

You can't open one by franchising, how it works is that a good teacher/administrator moves to a new area and they see that the area would respond well to an after school math program

#

They only hire people with MS/MA degrees in STEM or Math education and up

#

You have to take a middle school math test, give a teaching demo, and interview in order to get hired so there's a lot of quality control on even teaching there

#

I had to take the middle school math test, and I think it took me ~ 90 minutes and I made a few arithmetic errors

quasi musk
tepid smelt
#

I wanted to enroll my daughter but is like 250 a month which was just too expensive for me. Plus I selfishly enjoy teaching her and she still is happy to do math with me. Do you know if its possible to purchase some of the curriculum? I have a Russian student who did it a little in junior high and he is my strongest student so anecdotally it clearly is good. I know they have regular/accelerated and honors track for each grade level along with contest prep courses by me.

quasi musk
#

The curriculum is good, but since I usually have 5+ students a class I can't give individualized attention

#

If you can get a job there stephenruby8 then you can get a discount on your child's tuition

long pelican
#

Back in my high school and college days, I hadn't heard of Russian school of math nor did I have any friends who used it, but I had many friends who went to "magnet" schools like TJHSST where a lot of smart math competition kids came from

quasi musk
#

When I was in HS I was poor, and everyone around me was poor

#

I had no idea people hired or could/would hire tutors for a stupid amount per hour

long pelican
#

Me neither, until one day I got emails from parents asking me to tutor their kid for things like AMC 10...

And reading these emails I'm always like "oh my gosh these parents are so transactional when it comes to math... they don't care about beauty of math at all... I'm never indulging in their worldviews if I can help it"

#

For these parents, the point of the AMC 10 tutoring is to increase their chances of getting into a good college which increases their chances of making $$$$ and that's pretty much it

quasi musk
#

I think the parents of my students at RSM have a similar mindset, but the kids genuinely enjoy being there and solving problems

long pelican
#

Always more hope for the kids than the parents 😛

quasi musk
#

I'm happy with my hourly wage at RSM

#

I'm really not a fan of private tutoring. Too much of a hassle to schedule things, talk to people, convince them to pay me, etc.

#

Plus I like the curriculum and the mindset of the students overall

tepid smelt
#

I mean the pay for private tutoring can be outrageous like 100+ an hour here in the bay area if you have a good background. So even a few hours a week can be nice.

quasi musk
#

Huge fan of the program ~ my only issues are the price & the fact that I don't have as much time to slow down and zoom in on topics as I'd like

quasi musk
#

I had the chance to go up to the bay area and work for mathnasium full-time, but decided against it

earnest trail
#

hey I work for them

#

part time tho

#

I definitely don't make 100+ per hour

quasi musk
#

I'm not getting 100+ per hour either. I could get $150/hr for private tutoring if I wanted, but I don't like private tutoring

quasi musk
#

Mathnasium won't hire me lmao

earnest trail
#

yeah I work for mathnasium

#

with my outstanding $11/hour

quasi musk
#

I've interviewed at mathnasiums like 5 separate times

#

and they've never hired me

quasi musk
#

I want to go work for Art of Problem solving, but I'm too dumb for them

earnest trail
#

I cant see why they wouldn't hire you

quasi musk
#

I don't get 99% on their test

#

And I ask too much to be a center director

#

($40/hr)

long pelican
#

@earnest trail Do you do tutoring there?

earnest trail
earnest trail
long pelican
#

How’s the experience?

earnest trail
#

well I do it online from home

#

I can definitely say that I would rather do it in person

#

I kinda can't do that consistently

#

since like transportation etc

#

but yeah other than that it's nice

#

I enjoy interacting with the students

long pelican
#

Are the students better or worse or about the same at thinking through things than, say, people asking for help here?

earnest trail
#

it depends, some students struggling with basic algebra while for others they don't even need tutoring lol

#

but they're not all a bunch of dumbasses or whatever

long pelican
#

Would you say getting all your students to like math is a feasible goal?

#

That’s something I think mathnasium says it wants to do

earnest trail
#

I believe their slogan is "we make math make sense"

#

but uh

earnest trail
#

most likely won't work

long pelican
#

Aww

earnest trail
#

I do it in kinda my own way, I understand that a lot of people won't get hooked but I do love to show the intuition and all that so that they understand clearly

#

like im able to use venn diagrams and set theory when dealing with probability, it's a good explanation and surprisingly effective

#

I mention basic set theory concepts (union, intersection, universe, etc) without mentioning their names

#

I use my deeper understanding of math to show what the average instructor might not, and when a student goes "ohhhhhh" I think my job was successful

#

I know you would be able to do something similar

quasi musk
#

From what I've seen at Mathnasium is that there's not a whole lot of quality control on who owns the centers, and they can vary a lot in experience

#

Especially in the type of students coming in

earnest trail
#

I believe that

quasi musk
#

Some of them seem to be good & committed, while others just seem like a shitty excuse for a cash grab

earnest trail
#

damn

#

but yeah it's a good first job for me

quasi musk
#

The average mathnasium I've visited seems good though, there's only like one I looked at and I was like

#

"you have no idea what you're doing"

earnest trail
#

oh shit lmao

#

what state do u live in

quasi musk
#

In the past 2 years I've lived in CA, WA, and TX

earnest trail
#

oh damn

#

I know they have them in TX but also the other two states?

quasi musk
#

They're big in California and Washington

earnest trail
#

oh alright

quasi musk
#

The guy I talked to in bay area seemed like he knew his shit

earnest trail
#

lmao

#

I didn't realize they were that big

quasi musk
#

He was a PhD in EE, and he retired cuz he wanted to teach math

earnest trail
#

oh sweet

quasi musk
#

He manages 3 centers, 2 of which are some of the largest in NA

earnest trail
#

that's crazy

quasi musk
#

What I love about mathnasium is that no matter where you are you can get help in the k-12 system

#

Something that RSM doesn't really support

earnest trail
#

except calc students

quasi musk
#

oh realy?

#

lmao

earnest trail
#

we have a couple students that are in calc

#

we don't have any pages for them

#

we help them with their hw

quasi musk
#

Are the pages good?

earnest trail
#

and obviously only select instructors who know calculus help them

earnest trail
quasi musk
#

To teach at RSM you have to have a MS/MA in Math, STEM or Education

earnest trail
#

yeah most of the instructors here are high school or early college

quasi musk
#

I think Mathnasium should do a bit more quality control

#

Less emphasis on "get a 98% or higher" and more emphasis on "can you do calculus"

quasi musk
earnest trail
#

hs

quasi musk
#

Oh you're chillin'

earnest trail
#

but yeah people practically have to get everything right on the mastery checks ("end of lesson" type pages) to move on

#

but that's fine because the mastery checks are fairly short anyway

#

what I hate is handling 2-3 students at once online, in-person that would be fine but not online

#

it should be 1-2

#

well I guess it depends bc sometimes 3 students each may not need a lot of help OR one student requires constant attention

#

idk

quasi musk
#

One thing that I learned was that I'd help a student for 5-10 minutes

#

And then I'd give them 10 minutes to practice on their own and I'd rotate through

#

(this was with college students, so you might shorten for k-12)

earnest trail
#

I try to do that but it often fails

#

maybe a student needs me to explain them a lot

quasi musk
#

Or maybe if they're in the same class/subject try pairing them

earnest trail
#

so im there for a lot more time

#

or sometimes people are smart and don't need help

earnest trail
quasi musk
#

Do you have a way of rewarding students who try to work on their own in silence for an x amount of time?

#

You can try to train them, like if you spend the next five minutes working on the problem then I'll give you a piece of candy

#

And you slowly extend that time frame

earnest trail
#

remember I work online

#

our reward system is basically: every page you complete, you get 2 points

quasi musk
#

oh Rip

earnest trail
#

what can you do with the points? absolutely fucking nothing

#

lmao

quasi musk
#

Online rip

earnest trail
#

yeah it sucks

#

but whatever

#

if I had a car I would 100% do it in person

quasi musk
#

When you transition to college you should tutor in the college center

#

The experience is invaluable

inland wren
earnest trail
#

alright

#

is it volunteer or paid?

quasi musk
earnest trail
#

you can discover things on your own and such

quasi musk
#

For me, my most productive group sessions are when I did prep work and brought parts I was confused or stuck on

#

and I can say why I am stuck

earnest trail
#

I see

quasi musk
#

Or what I need to do, what I've tried, and why it didn't work

earnest trail
#

is working at a college tutoring center paid or not?

#

or does it depend

quasi musk
#

then I say "Well I want to do this, but I can't see getting around this obstacle"

quasi musk
earnest trail
#

alright I'll look into that

#

ty for the suggestion

quasi musk
#

I started out volunteering at a college tutoring center

#

because I had 0 experience, and then at the end of the semester they hired me

earnest trail
#

oh okay

quasi musk
#

Also I met my wife at the college tutoring center

#

We've been together for 6 years, married for coming on 4 years now

earnest trail
#

damnnnnnn

#

that's awesome

#

alright so I saw that there are mathnasiums near both universities im interested in

#

so if I want to continue that then I can

#

but I will also consider the college tutoring center

quasi musk
earnest trail
#

that's true

quasi musk
#

And when you get that extra experience, it says a lot when you want to move on. Even if your ultimate goal is grad school, industrial work, or education

#

It's super helpful

earnest trail
#

alright yeah good points

#

tysm

tepid smelt
# quasi musk I don't get 99% on their test

I can't imagine there test is harder than the RSM one. Looking at the problem set from there international contest they run they are on par with the challenge problems from AOPS.

quasi musk
#

But what's more important than the test is the teaching demonstration that you do

turbid zenith
#

Anyone happen to be up?

real mauve
#

yep

turbid zenith
#

👋

#

I've been thinking of project ideas for my liberal arts class ... trying to think of how to modify them from what I've done to make the product better.

#

I usually make their last project to be a "pick a topic you're interested in and write an article about how math is related to it" assignment, but I often get really mixed results as to the quality of it.

#

Granted, the students aren't usually math majors, but it still feels like they only tackle it at a really superficial level. Like, they'll say "okay I'll do volleyball" and then it's like ... yay there are numbers that have to do with volleyball

quasi musk
#

what's going on

#

Oh I see

quasi musk
#

Maybe a few diverse set of examples, a rubric, and clear expectations of what you want in it and what you don't want in it

#

I'm sure you know that

real mauve
#

do you think proposing a pool of topics where you already know some accessible (both as in easy to find and not complicated) might make a difference? make it a little less open ended

turbid zenith
#

Yeah, I think I'm just not being clear enough with what I want might be the issue, but I'm not sure how to articulate it

real mauve
#

and if you can't easily come up with a rubric, provide an example or two

#

which shouldn't be all that complicated, something like a blog post

turbid zenith
#

I usually have a day where I have my students look at math articles from Quanta Magazine as examples of accessible mathematical writing

quasi musk
#

Yeah, when I did my excursions into math ed, we looked at an example of where a math educator planned this wonderful open ended assignment but students got frustrated

#

and it failed

#

Then the person enlisted help where they planned out where students are likely to get stuck and gave hints

#

Discussion leading questions, etc.

#

Massive success in that example

real mauve
#

students unfamiliar with a topic or uninterested in it will find open ended tasks to be terrible

turbid zenith
#

Lemme find the last iteration, I would love feedback on what could be better about it

quasi musk
#

I've always been told if you ask an ambiguous question, try to give them a frame for an answer

real mauve
#

that sounds about right

turbid zenith
quasi musk
#

Like "What kind of function is this"

#

"is it even, odd or neither" can help students pick out the important info for the context

#

because they can be thinking like "this is a quadratic" or "periodic"

real mauve
#

hmm i will say the task description is really very clear, you already had in there most/all of the stuff i had in mind

turbid zenith
#

I feel like what often happens is the math is barely there or it's a bunch of stuff the students don't understand and they're just typing what they found

quasi musk
#

What I would add is things you aren't looking for

#

or general things to avoid

#

Common pitfalls

turbid zenith
#

That's a good idea.

quasi musk
#

Like a "Common Mistakes on this assignment" page can help clarify a lot

real mauve
#

mhm, that sounds good

turbid zenith
#

I'm actually going to rewrite my Rubric as a set of specifications

#

I'm no longer going to do a rubric where it's like "well you got an S in this but a G in this", but rather just list "here's what I'm looking for, if you don't meet this you get a G and I give you specific feedback on what needs to be revised"

#

That's a movement I'm making across the board with my assignments

quasi musk
#

If you can, try to get a couple examples of diverse "stellar" things that you're looking for that previous students have done

#

That might inspire people about formatting choices, etc.

turbid zenith
#

Oh I definitely should be able to do that yeah

real mauve
#

if you can concoct explicit examples of bad stuff, too

quasi musk
#

That's about all the advice I can think of off the top of my head

real mauve
#

to show in the common pitfalls section

quasi musk
#

Assignments like this can be the most rewarding, but also the most challenging

turbid zenith
#

Sort of a "do this, don't do that"

real mauve
#

mhm

turbid zenith
#

One of the other things I'm finding myself torn in is to what extent I want to see "actual mathematics" in the article, and what that even means

#

Because one of the things I do in this class is get students to understand that math is not just about numbers and equations

#

But at the same time I feel like I have an implicit bias that if I'm not seeing at least a few numbers here and there, something isn't quite right

#

So I need a better way to qualify "this is showing enough thought about the mathematics involved"

quasi musk
#

I guess the last thing to do is to make sure that two students don't do the same topic

turbid zenith
#

I do have a way of doing that already. I use a Google Form/Sheets signup process.

quasi musk
#

AH ok

turbid zenith
#

Where they can see what topics have been picked already

quasi musk
#

Here's an outline from my knot theory final project

#

I thought it was good at conveying what is worth what, what we can do, etc.

turbid zenith
#

Oh nice

real mauve
#

if you want to see some equations and numbers, you can show an example of something like a verbal description of something that is then translated into some governing equation. that way you kind of force them to have equations with verbal explanation of real world phenomena. maybe?

turbid zenith
#

"MATHEMATICAL UNDERSTANDING and DEPTH: A clear understanding of underlying mathematical topics is demonstrated. Additionally, the topic has been investigated at a depth appropriate for an advanced graduate class."

So I'm not doing an advanced graduate class, but that does bring up a question. How do you actually assess whether something "has been investigated at a depth appropriate for an advanced graduate class", other than "I know it when I see it?"

quasi musk
#

Oh, yeah that was fun

#

So when you gave your presentation, he had a list of questions he would ask you

#

Like 5 or 6

#

You best be able to have a good answer for like 3 or 4 of them

#

Like I did my presentation on "Knot Distortion"

#

The definition of the distortion of a knot is incredibly subtle

quasi musk
#

So it wasn't enough to just sort of cite it and say yeah yeah whatever

#

You had to know

quasi musk
#

This is his rate my professor profile

turbid zenith
#

Oh wow

#

Mine is apparently pretty good right now ^^;; But I've only recentlyish started

#

But those answers make sense so far I think.

quasi musk
#

I'm so glad I don't have a rate my prof profile

real mauve
#

this is probably not the answer you were looking for, but i think open ended tasks like this one are inherently subjective past a certain point. especially here that you give an open ended task, sufficient depth could really be what we consider now HS level trig, just applied in a very clever way

quasi musk
#

My students would have ripped me lmao

turbid zenith
#

I just want to try to be as clear as I can for what I'm looking for

real mauve
#

what i could suggest, instead of saying depth, is "showing thorough understanding"

quasi musk
#

Like you're not going to bullshit your way talking to a topologist about topology you don't know

real mauve
#

because depth can be conflated with copy pasting random complicated-looking math

turbid zenith
#

I DEFINITELY had some students do that a few times and I had to be like "this is unacceptable"

quasi musk
#

lol

real mauve
#

yeah. a quick ctrl f search on your description shows no mention of "understanding" though 😛

turbid zenith
#

One was trying to do like ... fingerprinting software ... and was lifting a bunch from just like a single grad paper

#

I would rather see less math but explained thoroughly and intuitively

real mauve
#

you can probably make some 1 paragraph examples of that and put them in the do's and don't's

turbid zenith
#

Someone in another chat I'm in just now suggested that one way to think about it is not giving a laundry list of disconnected things but like picking one topic and really diving into it

#

I like this a lot I think

real mauve
#

ah, that's good too

#

to think of it as telling a story

#

i would also like to comment that students turning in shaky open ended assignments of this kind is not necessarily related to how well you describe the task and how many examples you give. this sort of cohesive writing is something many people struggle with even after doing it for a long time under supervision and with lots of feedback. it's good that you expose the students to it early, but it really takes a long time to master

#

so at the end, they can benefit from receiving (possibly anonymous) feedback from each other on how to improve

#

it tends to be difficult to catch one's own shortcomings in open ended writing tasks because one can hardly disconnect what was written from the thought process behind it, so the gaps are not evident

quasi musk
#

I'd definitely limit the options or topics to like 5 or 10 with a couple deviations

#

It'll be super overwhelming if there's like 45 things

turbid zenith
#

Each of my classes usually has 22 people in it

real mauve
#

i think you might see an overall improvement in the quality if you group them up in pairs or trios

#

just from their mutual feedback

turbid zenith
#

Hmm, that's a possibility

#

I do have students do a lot of stuff in groups but I've never had them do that for this

#

Though I usually have done a "peer review day" where they bring in their drafts and spend the day reviewing each other's work

real mauve
#

ah, that also sounds good

turbid zenith
#

So, say going from the volleyball example.
Bad example: Listing what shapes and angles can be found on a volleyball court
Good example: Explaining how volleyball statistics are kept and why those particular metrics might have been chosen
Good example: Investigating the spike strength and launching angle of a volleyball

#

Something like that

long pelican
#

My only input: in my high school (when I was in high school), the precalculus class had a similar project and 70% of the students do projectile motion every year, which is probably the easiest topic. So see if there’s a similar cop-out project and ban it or ban “easy” topics in general

tawny slate
#

one of the frustrating things about this kind of stuff for me is that it's often times difficult to find a topic that you sort of understand and care about

#

as soon as i found any topic i was into, it was mostly smooth sailing from there

#

the difficulty was just finding one

#

if say math was one of those subjects i just wasn't invested in or comfortable in, projectile motion might just be one of those topics id pick simply because i don't feel comfortable picking something else

#

i would've loved a laundry list of ideas, but not just bullet pointed, but with a quick explanation about what makes this idea interesting or promising

#

i would've gladly grabbed one of those

wispy jolt
#

accessibility is a serious issue in math education

#

i think a lot of students have issues finding ways to do interesting stuff with math that isnt answering textbook problems

#

which is really unfortunate

#

before i began studying math in a focused way i was attending this math and engineering club at my community college at the time

#

and we would often do these low level proof-y problems published by this university designed to be done with pre-pre-calc math basically

#

but they were kinda abstract because they were just engineered to be tricky and nonobvious, even to trained mathematicians potentially

#

i dont think i would have gotten into math without that experience, and also the positive social experience

long pelican
#

These low-level proof-y problems you describe sound very similar to USAMO/USAJMO and USAMTS problems

#

I think USAMTS was my first experience with proof-based math and boy was it hard for me (I was in like 9th grade)

wispy jolt
#

they were published by trinity college, i think in texas

#

tho there are a lot like them i bet

long pelican
#

O man, memory lane

unknown.png
#

I remember this problem haunting my car rides

#

I didn't manage to solve it

wispy jolt
#

yeah stuff like that

#

or weird geometry problems that have a simple but super unintuitive answer

#

especially at like a community college or a SLAC without a good program or uni without a good undergrad program math is just this wall of exams and homework

long pelican
#

What's a good way to get people to do them apart from just putting them in the required homework?

wispy jolt
#

yeah that's a good question

#

there seems to be a component of willing to do a club or whatever here

#

in that way it sort of ends up being cultural, since the culture accommodates the extra curriculars and stuff

#

in the US where i live i think the issues are more pragmatic rn than the artistic issue of catching the interest of students

earnest trail
#

g r o o v y

long pelican
#

Groovy! I challenge people to solve the problem

earnest trail
#

proof is trivial QED

long pelican
#

Oh yeah I live in the US too -- lack of interest is quite interwoven in all the aspects of the math education issue here

wispy jolt
#

yeah i have some experience helping teach and tutor calculus (especially calc 2) at a public uni that is very easy to get into

long pelican
#

Even in the state of Massachusetts where I live, which is according to some sources the smartest state

wispy jolt
#

and it's tough

#

in a pretty big city too so you get a big intersection of people

#

my uni has a very low math major rate but a pretty close private university has something kinda astonishing

long pelican
#

Private school here and also a low math major rate

wispy jolt
#

i cant remember the actual number but it was like up to 10% of the students were in a math major or minor

#

maybe higher, i just dont wanna give a ridiculous number

#

this is a pretty snazzy private uni

long pelican
#

10% isn't bad

#

If there's a lot of majors

#

after CS and mechanical engineering

wispy jolt
#

then it may have been higher, i remember being surprised by it

#

maybe that was just the major rate

long pelican
#

(Oh when I said low math major rate "here" I was not referring to MIT)

wispy jolt
#

however, the point is at my uni its like 2%

long pelican
#

2% is pretty low yeah

#

That's like the chemical engineering amount at MIT

wispy jolt
#

and also the number of majors is again lower, a lot of that is cs and engineering students

#

many math major classes do not run, people take the grad courses

#

no linear algebra 2, abstract 2, etc

#

like once in a blue moon

long pelican
#

dam

wispy jolt
#

the grad program is a stark contrast, not elite but decent and really good in a couple fields

#

big time topologists and some logic stuff

#

it's even R1, but its undergrad program is just some classes and the professors going "uh you might have to take the grad classes"

long pelican
#

Mmmm

#

Undergrads are poorly prepared from high school

wispy jolt
#

yeah that was what we got in those calc classes a lot

#

you get students who dont know trig, which could be explainable, but many just got out of the class

long pelican
#

When I taught calc 2 last semester, I wasn't able to care whether they knew trig or even logs

#

Would've been good if that was their most basic issue

wispy jolt
#

yeah i was doing a sort of undergraduate TA kind of position, though i was involved with a lot of tutoring in the pretty well developed tutoring center

#

some students really did have some dazzling problems

#

the uni has a well developed calculus program actually, with standardized exams across sections

#

it's like threading a needle because you cant give too conceptual of problems or you fail 50% of the class each hw

long pelican
#

Orly, my class was kinda conceptual all the way through

#

Unless your idea of conceptual is at a higher standard

wispy jolt
#

so problems students would have issues with would be taking a regular optimization problem and just translating everything to be say 3 units higher

long pelican
wispy jolt
#

turning it into a kinda quasi multivariable calc problem

#

but really just a normal problem where one value is fixed as usual, just not 0 as it would be in the background yknow

long pelican
#

So how much trouble did they have with it?

#

Did 0 people solve it?

#

50% of the class?

wispy jolt
#

problems of these kinds often had a less than 50% success rate

#

sometimes they would be like a quarter of the class if the HW writer got too clever, i was told

long pelican
#

oof

#

Sounds about expected based on my experience

wispy jolt
#

yeah there were a lot of students who retook calc 1 and 2

#

what actually saved students for a long time was too generous of HW proportion

#

they hadnt considered extreme values in some calculations and just sorta set them, then realized students were getting super low exam averages a lot and passing

#

flirting with F on the exams territory

long pelican
#

Did they curve exams generously in the end?

wispy jolt
#

well they simply changed the policy so that you also needed a baseline and still pretty generous percentage on the exams to pass, though i am not sure if it would crack you down to F or D. they also recalculated some final grade proportions etc

#

it's tough because they cant just keep lowering expectations yknow

#

the other departments will get political

long pelican
#

Expectations seem to have to be pretty low if 25% of students are getting conceptual questions right every year

wispy jolt
#

there were definitely kinds of problems that had consistent, almost stunning averages

#

by the time you get into calc 3 students are far more focused

#

they have done experiments in active learning here that were enormously successful tho

#

i was working decently closely with our faculty supervisors at the time and they were running a study that i think they wanted to publish or present somewhere

#

and changing to an active learning style improved exam scores like magic

#

it was a head to head thing, something like 200 traditional lecture students against 200 active learning

#

the active learning students had something like +15% or higher grades on both midterms

#

on average

#

how much was a component of fewer going into a vegetative state is hard to say

long pelican
#

Last semester there was also an experiment of sorts although we didn't talk about it-- my co-instructor used handouts with printed notes where the students filled in the blanks during class, and I didn't do anything like that, just wrote on the board like a traditional math class. I sometimes used polls and asked a lot of questions to the class although one of my sections was very uninterested in participating. My students got like 10% higher scores on the midterms than her students, but I'm not sure what to make of it

#

The conceptual short-answer section of the final had an even more pronounced difference: my sections outperformed the others by 39%

wispy jolt
#

that's pretty interesting

long pelican
#

25/36 average compared to 18/36 average

wispy jolt
#

there seems to be a lot of weight of psychology behind it

tepid smelt
# wispy jolt i think a lot of students have issues finding ways to do interesting stuff with ...

99% of textbook problems are absolutely garbage for higher students even at my public school. In order to get higher students engaged on the material I have to generally create them my own which are almost always inspired by math contests. I sprinkle them in hw and exams. I have found good projects that have low floor high ceiling potential are also good here. The problem is generally time to come up with them especially when 90% of my kids are barley holding on. Most public schools do have math clubs but I find our like robotics/engineering clubs pull are best kids as they have more funding and better recruiting strategy's.

wispy jolt
#

yeah there is this amazing capacity that students develop for knowing just enough to succeed

#

and textbook problems, while clearly vital, help keep students in that zone

#

you dont gotta be doing some big brain advanced topics but all kinds of math are developed to be used in this more hands-on way to solve things that arent obvious

#

just keeping track of geometric reflections or whatever is kind of cerebral for someone who isnt super far

#

some spooky stuff can happen and math education is low on the spook for a long time

#

and yeah when you are sorta in a spot that has many students with foundational issues the engineering stuff seems to be popular

#

more practical and concrete things, it's like that at my uni and was at my community college

wispy jolt
#

it's like a binomial thing, lots of unprepared people or super overprepared

#

(weighted toward the former)

long pelican
#

I wonder if next time I get a horde of unprepared people, I should just middle finger the curriculum and spend 1/3 of the class getting them up to speed in a way I calculate as optimal

wispy jolt
#

i will give my uni credit the structure to get students thru calc isnt bad

long pelican
#

Last semester the department sort of made a rigid calendar of topics and I followed it to the day

#

Probably shouldn't have done that

wispy jolt
#

here a single senior lecturer is coordinator for the whole class number

#

so calc 1, 2, 3, etc

#

they write the exams and HW with input from the other lecturers who are teaching the course number

#

tbh this sort of skews things somewhat toward the easier side

long pelican
#

How so?

wispy jolt
#

this could be because of our overall coordinator for calculus, who is a single clinical professor

#

she is actually pretty good at this stuff and did it previously at that big private uni i talked about

#

but it seems that because more people oversee everything stuff gets kinda cut down a bit

#

with more people able to give negative feedback on things they are getting the vibe for in the classes

long pelican
#

mm

#

The more people to rule out questions they think as being too hard

#

while not doing the same for easy questions

wispy jolt
#

it's not perfect, but it has the intended effect of standardizing the experience

long pelican
#

Does this make good questions rare on exams?

wispy jolt
#

yeah this seems to be the result is that people have a wide array of opinions of bad calc questions i guess

#

i think the exams are fine for the students often taking them, not making a value judgement

#

i had a pretty interesting problem on my calc 2 exam under this system

#

that semester also had low exam scores and mine squeaked out my desired grade

#

i think this curve following does probably hurt variety

#

in problems

#

they keep the final grades pretty well distributed

#

light on Bs tho

#

but people always fall in the middle

quasi musk
#

I think one thing is that you can even have the same material same prof, same time of day

#

Just different section

#

(e.g. M/W versus T/R)

#

And get huge differences

wispy jolt
#

yeah there is a difference between early and late classes here

#

early classes are usually students who live on or near campus, like 8am

quasi musk
#

You have to control for so many factors and run a "active versus traditional" experiment across like

#

3 or 4 years at minimum to get reliable data

#

And even then it might be bounded by your region

wispy jolt
#

my school is a large school and a commuter school

quasi musk
#

Although I'm a huge fan of active learning and use it regularly

wispy jolt
#

well right, but it was interesting that it had results so quickly

#

i believe they were running it for more semesters

#

they controlled for as many factors as they could, and i wouldnt call it nothing

long pelican
#

Are they testing one thing or a lot of things

#

In this "experiment"

wispy jolt
#

it was calculus 1 and calculus 2 sections i believe

long pelican
#

Is one group doing chalkboard lecture, facing the board

#

and another group doing like

#

hands on stuff and manipulatives

wispy jolt
#

yes so one group would have regular big lecture halls

#

with one lecturer consistently throughout the semester

#

big room, facing board

#

and then they had recently built these active learning facilities

#

that included nearby instruments for the students to work together

long pelican
#

Smaller classes?

wispy jolt
#

no the rooms were quite large

#

there were very large lecture halls but they didnt compare between those

quasi musk
#

but plz go on

wispy jolt
#

they had constructed special rooms to be used for active learning via a grant

#

they are these large rooms with whiteboards and stuff around, they had developed a plan for how to teach in it based on experience and outside information i believe

#

they also have a sound system and a camera system for coverage of the lecturer, but the intent is to have students working for at least half of class time

#

so they have a worksheet at the beginning of class, which the lecturer guides them through some stuff and then gives them a period to work on things and ideally gets them to write answers on the boards, etc

#

there was also a concurrent program called the learning assistant program which was an undergraduate TA position that ran study sessions, tutoring, and helped in both lecture hall format and active learning

#

in the lecture hall format they would answer questions for students so the lecturer didnt have to stop

#

and in the active learning room assisted the students who are arranged in groups at large tables

#

the lecture hall format you would be surprisingly busy, actually

#

once the students kinda warmed up to the idea

#

i think that was the semester before covid and then they planned on having it be an on going thing

#

if i recall the active learning rooms could hold over 60 people, maybe a lot more

#

they were huge, designed in brand new buildings

#

all with tables and sound systems it was pretty snazzy

#

to me this was all relatively elaborate

#

i think one could top out at 80 people and the other 100, the latter was very large and doubled as a lecture hall pretty easily

#

the idea is that combining their 2 discussion meetings per week, homework, "reading", and guidance they would largely work together

#

and this was true for like half the arranged tables of students, the tables would hold like 8 students max or something

#

so my impression of active learning is very positive but it has known limits

#

in my experience it seemed promising and anecdotally students at the right time of day liked it, though i would be interested in if they were continuing the study now

pastel sundial
#

So what actually is the point of K-12 math education. Especially in secondary/high school.

#

Like obviously it's easy for us to say "oh it's beautiful, it helps you think, etc" but I feel like most any expert in any field could talk about how cool it is and how the average person could benefit from learning some of it. Maybe not every field, but certainly a lot of them, including many we put far less emphasis on in school than math.

#

And yes, a lot of very important modern jobs do require math. But not all of them, and the people who are going to end up moving on to those jobs would probably end up choosing go take math in highschool even if they don't have to. Especially if they know they'll need it to study their subject of choice in college.

#

So are we just forcing everyone through math because we think they should all become engineers and scientists and programmers? It's not like those are the only good or important jobs out there. And there are plenty of people who are really passionate about a non-math subject that we still force through precalc. And I honestly see where those people are coming from when they ask "when are we ever going to use this." Like sure, as Icy has pointed out, a good math class could potentially have many of those people so distracted by how cool the subject is that they don't bother worrying about when they're going to use it. But why are we making them take math in the first place? There are a ton of cool subjects that I'm not being forced to take as a math student, nor do I want to go out of my way to learn, so why are future artists forced to take precalc when I was never forced to take a figure drawing class?

#

This was prompted, by the way, by an excellent article I read for my education class that argued that the main issue with the American school system is that no one can agree on what the purpose of school should be.

real mauve
#

it is certainly the case that many essential jobs don't really require that much training in math

#

in other countries, one can decide to do only until the end of the equivalent of middle school/year 9

#

and then do vocational training afterwards

#

the problem is related to how this is viewed in society, and that this type of job, though arguably more essential than many others, is seen as "low class" and often comes with a poor salary

#

i see this as a more systemic problem

pastel sundial
#

Right

#

Well it's like, "I want my kid to learn math so that they can get a higher paying job", "the nation's children should learn math so we can produce good engineers, scientists, and programmers", "we should teach math because it's beautiful and it teaches you how to think", and "it's useful to teach math because mathematical ability can be used as an easy proxy for academic ability when evaluating students" are all reasons people might support teaching math. But they each lead to different approaches to teaching math, so we should make up our mind on which one we actually care about.

real mauve
#

i don't know that it's necessary to pick only one, but schools should be clear on their methodology and readily accessible

#

esp at the k-12 level

#

beyond that, sure, make admissions as competitive as you want

#

but for children, the choice should be open

#

(imo)

long pelican
pastel sundial
#

Labaree?

long pelican
#

Here’s 2 additional reasons to learn math that I’ve found more appealing: 1: it provides an easy common arena in which to learn problem solving (as opposed to having to tailor problem solving situations to everyone’s daily lives) and 2 (related to 1): math is the art of abstraction

#

It’s been a while so I don’t remember the exact article

pastel sundial
real mauve
#

i certainly agree with that, but i also think one can practice these skills without doing it via math. this might just introduce an unnecessary hurdle

long pelican
#

I think the abstraction point is the most important one

#

How do we advance as a society without abstraction

pastel sundial
#

"Math is the art of abstraction" reads to me like just an expert trying to justify why their field of expertise is so important

real mauve
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my counterargument would be that anyway current math teaching is not helping with this

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you are facing this yourself with your students

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it COULD be a good way of practicing abstraction if done right

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but it currently isn't

long pelican
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It’s not just practicing abstraction, it IS abstraction

real mauve
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esp not at k-12

long pelican
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Like linear equations unite a lot of different phenomena together

real mauve
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phenomena that many people will never have to interact with at all

long pelican
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What about philosophy then

pastel sundial
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Like you're not wrong, but if you're resorting to "this subject is useful for everyone to learn" then a lot of subjects we care a lot less about in school than math would also have a pretty good claim to a piece of the pie.

long pelican
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No numbers, just logic and reasoning

real mauve
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what about it?

pastel sundial
long pelican
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If not math, philosophy should be required then?

real mauve
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some sort of abstraction, certainly. philosophy could be it. math could also be it if reformulated properly

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but just shoving the current k-12 maths down everyone's throat is not a good sol

long pelican
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What I said also carried with it that we should revamp what the curriculum is

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So yeah

pastel sundial
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I think philosophy, computer science, and general statistical awareness all have decent claims to take over what we're trying to with math in terms of teaching students how to think

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All for different reasons

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Imo a good philosophy class is going to be a lot more useful to the average student than precalc

real mauve
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then certainly. my argument was toward the present state. memorization of several seemingly unconnected concepts is neither needed nor leads to abstraction

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at present there would be little loss from cutting the system short a few years early

pastel sundial
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But this is getting into the weeds. Before we talk about the skills math (or these alternatives) build in students, we should first get clear on what skills we want to build in students in the first place.

real mauve
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from the get go, having people make life-changing decisions at age ~17 is absurd

pastel sundial
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About college?

real mauve
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mhm

tidal whale
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considering the fact that any useful understanding of CS concepts is itself on top of math

pastel sundial
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Yeah but like, I'd argue that the best way to learn functions is through programming

tidal whale
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not exactly

quasi musk
tidal whale
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recall that the following is true for a programming function but not a mathematical function
i) the same input may induce distinct outputs for a stateful function
ii) functions do not guarantee returns

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asking that we look at functions from programming is absolutely bizarre

quasi musk
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I'm also worried about exporting an understanding of something that is so innate to human nature to computers

tidal whale
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the most lax definition of a function in CS is "an organized code block that does something given some inputs"

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it's not general enough to translate anywhere else

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and it's not strong enough to analyze in a meaningful way

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then there's matters like signatures, voids, state, etc.

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at that point they aren't even the same thing

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it's not possible to replace math with CS, or to even teach math through the lens of CS

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you're trying to morph a field that borrows from math by definition into one that can serve as a substitute for math

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considering the fact that the first example of recursion in CS is almost always a factorial, it's not a reasonable idea in the slightest

tidal whale
quasi musk
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Indeed, it is a strangely disposed time that men may construe things after their fashion; clean of the very purpose of the things themselves

tidal whale
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and in response to the spiel about future artists having to take precalc, I'd like to ask this as someone that lives in an economy where jobs are incredibly scarce now - what will the artist do when the only alternative jobs involve that which they have not studied?

tidal whale
quasi musk
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Is that shakespeare?

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I thought it was Cicero

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oh

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Shakepeare's fictional version of Cicero

tidal whale
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I meant from a Shakespeare work

quasi musk
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Didn't know I'd been quoting shakespeare all this time