K (189?–?) Soviet pioneer. From Kazan, Tartarstan, USSR, K was diagnosed as a ‘transvestite’ in 1937.
She was given permission by the People’s Court to wear female clothing, her identity papers were changed to her female name, and her name was removed from the military recruitment rolls.
She was featured in a 1957 gynaecology textbook.
M.G. Serdiukov. Sudebnaia ginekologiia I sudebnoi akusherstvo. Moscow: Meditsina 1957: 47-8.
Dan Healey. Homosexual Desire in Revolutionary Russia: The Regulation of Sexual and Gender Dissent. Chicago and London: The University of Chicago Press, 2001: fig 24.
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My hot take is that math class doesn't do the one thing that justifies its place in core curriculum. We're told that the reason we all have to take math is to strengthen our logical thinking skills so that we can apply it to life. Sort of like how you can go to the gym and lift weights, and the ability to lift them isn't inherently useful to you, but you're keeping yourself healthy and building muscles that you can use elsewhere in life. Everyone needs to know how to think, regardless of what you do in life, and in theory mathematics is a great "gymnasium" to practice that skill.
But is it doing that? In my experience, it doesn't. Not even in the slightest. Being told a formula or algorithm, lectured on exactly how it works and how to use it, and then being given homework that drills your ability to apply this new concept 20 times or so does nothing for a person's ability to think. All this requires is blind rule following with essentially no real purpose in sight. This ends up alienating many students who don't care, have no reason to care, and have a hard time learning things they don't care about. Math class is boring as shit, if we're being entirely honest, but it doesn't have to be that way.
Fundamentally, above all else, mathematics is a creative art. Painters are artists who express themselves through the medium of pigment. Musicians are artists who express themselves through the medium of rhythmic sound. Mathematicians are artists who express themselves through the medium of thought. As such, math class should function like an art class. Students should be given the reigns to be able to do their own thinking. They should be able to create their own ideas and make their own art. Genuinely fuck the quadratic formula, fuck the Pythagorean Theorem, fuck difference of squares. So few people have any reason to care about these things. Are they wonderful pieces of math that have their own wonderful nuggets worth studying? Hell yeah. But if a student has no reason to care, in my opinion they should be allowed to think about what they want to think about, to express themselves through mathematics in their own way, and in doing so, strengthen their thinking muscles the way math class should be helping them to do.
Finally getting to your actual question, the only thing I care about is if the thing the student is using is something they've thought about themselves (almost certainly with instructor assistance). They need to have done some of their own thinking. FOIL is just a series of letters in an order, and I don't imagine a student is going to divine that from thin air. The box is closer to what I'd want to see, since it demonstrates actual mathematical conceptualization, but it ruins everything if the teacher just shows the box and tells the student to learn it.
One progression that a student may go though to discover this type of stuff for themselves is by thinking about multiplication. Say I want to do 6*32 in my head. That's kind of a pain in the ass, so what can we do? Maybe with a little help (and a few drawn rectangles) they can see that it's a lot easier to do 6*30 and add it to 6*2. That's pretty neat. We've turned something I probably needed a pencil and paper or calculator for into something that's quite easy. But there's nothing special about this number problem is there? Thinking about this, a student may discover distributive property for themselves. In standard mathematics, we symbolize that as a(b+c) = ab + ac for all a,b,c, but the student may symbolize it another way. They may even just say it as a sentence or paragraph, and that's okay because they have an actual concept in their mind that is tangible to them. Okay then what about (a + b)(c + d)? Well a+b is just a number, so we can use our neat little property:
(a + b)(c + d) = (a + b)c + (a + b)d = ac + bc + ab + bd
Three applications of our fancy new trick and what do we have? FOIL!
Okay but what about trinomials?
Using the fancy trick 4 times spits it right out. A student may notice they're just multiplying everything in the first bit by everything in the second and adding everything together and can explore whether or not this works in general.
Would this style of math class slow content learning down big time? Ohhh yes, and there's a conversation to be had about how to fit this into a world where more advanced math is useful for a lot of people, but for everyone else, there's no reason anyone should be graduating high school with calculus under their belt. If we took our time with math ed instead of trying to over standardize an art form, we'd have better thinkers.
Math class kills the very thing it aims to build up, and I'm sick of it.
Does this even count as a hot take? Whether someone hates math and just thinks that because they don't want to do it or they love math and wince at seeing how much pain math classes cause, it seems there's a lot of agreement that the current system isn't great.
Love that approach to math. Feeling like "I'm bored and want to be able to solve this question type without writing any work" and creating your own way to solve a problem that's very disconnected from the way teachers teach things. So frequently, the method taught are things that people can memorize for a single test, but without any underlying understanding, they won't remember a day longer. When they come back a decade from when they learned it, there's almost no chance they're gonna remember much.
Having real world problems you want to solve (not just wanting the answer, but wanting to go through the process) is also a nice way to start to learn the intuition of mathematical concepts. Playing Sonic is how I started to learn to think about infinite sequences back in elementary school. I didn't even know algebra, but I was thinking about things like questioning if a sequence would converge or diverge before I had that language to describe it or the tools explore it as much as I'd have liked to and trying to get an idea of whether it looked like the sequence was plateauing or growing infinitely by just doing calculations by hand. Didn't have a calculator to do such high-precision calculations nor did I know anyway to visualize the data.
I'd say that should be something some people should be exposed to to varying degrees. But its practically expected that you've taken such at competitive colleges, at least if you want to go into STEM.
PS: Is your username a Euler joke?
I had never heard of foil and am amazed it’s an actual acronym used by math teachers. I agree with you completely, it’s redundant and removes the connection to general multiplication rules.
Your area must have a much better relationship with math than mine. I and so many of my peers learned FOIL as the way you multiply binomials, with no discussion as to why. Math ed where I'm from is littered with this kind of thing. It's incredibly frustrating.
I won’t say mathematics education in my country is great (Netherlands), but it seems to be a lot better than some places in the US. Sometimes there will be core concepts that are not explained well and just given (like the quadratic formula and limits in functions). Overall I think that there’s way too much focus on algebra while other subjects like logic are neglected.
Personally I love math because my autistic brain makes fun logical leaps making math easy. Many of the neurotypical accelerated math class takers seem kind of miserable and perfectionistic doing the memorizations and stuff. And the rest just don’t like math. I agree that math should be figured out more logically and people who aren’t interested shouldn’t have to learn as much bs.