Why Is It Lovely
solipsys.co.uk
solipsys.co.uk
This comes off as obnoxiously superior. I was excellent at math in school. I was the guy who understood the theory and could extrapolate. And yet this surprised me. Not because the theory is surprising, but because I just hadn't thought about it before - or maybe I did, and it didn't stick. I don't work with percentages every day.
It's sort of like implying that only people who are bad at programming don't know about a particular API or pattern. That's not what makes somebody good or bad at something.
It's the meaning of what the author wrote. Even if the author doesn't believe it, I think I'm accurately interpreting the overall posture and attitude. Given the pretention of the rest of the post, I'm not willing to cut the author slack or give the benefit of the doubt
Yet I wasn't aware of it, and I've never used it (as a mental math trick). It was definitely surprising that it is true--and I wasn't aware of it.
I find that very surprising.
Maybe it was just the way I was taught percentages ( "% means /100, 'of' means multiply), but it seems too trivial a result to even be something worth commenting on.
> but it seems too trivial a result to even be something worth commenting on.
Yes, but I also think it is memetic and lovely enough to remembered it if you've encountered it. Calculating 52% of 25kg of something as 52/4=13 sparks joy.
Also using this random factlet to sort them out seems especially weird, since
- as I am an example, people with education in math might find it surprising
- it is standard middle-school stuff, so everyone *understands* it, regardless of a degree in math
.. this is just a reasoning of why
> For those of us who did maths this is obvious, but for those for whom maths was just a list of formulas and processes, applied without any real understanding, I can see that this can come as a surprise.
Okay, I'm done. I've probably spent too much time on a random 2 sentences on a random internet stranger's blog, I don't find this topic that interesting.
This would, of course, be obvious to anyone who has studied languages or writing. For those for whom language is a mere instrument or vehicle of thought, while receiving no thought or appreciation itself, I can see why this may come as a surprise.
Understanding the definition of percentages and expecting them to commute is more basic than some random API. For someone trained in math it's like a "programmer" not understanding what a function is, and vibecoding their way through.
I had to stare at Euler's identity for a while to recover from this.
It's pronounced "YEW-ler," right?
This dimensional analysis makes more sense with multiple terms, for example X % * Y = X % * 1 / 100% * Y = X * 1 / 100 * Y = X * Y / 100. (And note "of" means "times")
It's the same thing as saying, 5 * 2cm = 10cm = 5cm * 2 (hopefully the statement 5 * 2cm = 5cm * 2 qualifies as "obvious")
If this entire explanation is as automatic to you as breathing (which yes it is if you did a significant amount of technical courses as a student, although admittedly dimensional analysis tends to come up in chemistry not math) then the statement "X% of Y = Y% of X" is vacuously true and I would feel comfortable saying this is more or less "By definition of %" with no further explanation, if I were talking to someone who I know has an equivalent background to me.
Doing the conversion can often be helpful. For example say I'm thinking to myself, "what is 4% of 50" and I'm like ugh idk but half of 4 is 2 so the answer is 2
This here is the key question, and I’d argue all we can do is keep talking about how great the view is from the mountainside, and how easy it is to reach the trailhead. Maybe people only take a few steps, maybe they give up when the trail gets rocky, maybe they don’t even get out of the car, but they’ve made an attempt, and the mountain is still there for them.
When I was homeschooling my son I got him between 2-3 grades above grade level at math. He was so good at mental arithmetic that I had to improve my skills to keep up with him. On the other hand I was not able to get him to do algebra at all and he did not master it in high school although he did OK with geometry.
From my viewpoint it was really puzzling, almost like he had some kind of psychological resistance or mental block to the whole idea of algebra.
There's a 3rd type of programmer, besides the first two you mentioned.
Multiplication commutes (a fact pointed out by everyone) but our human brains are better at multiplying some numbers or fractions than others. Computing 50% of 34 is easier than 34% of 50 because we are more accustomed to cutting things in half than multiplying two-digit numbers.
And it's even more lovely when you realize that this applies equally to computers where you may have to rearrange terms or rewrite formulas to get better numerical stability.
It's syntax sugar for multiplication!
Yeah I got that, but I don't see how it looks like a mustache! Is the "/" a crooked filtrum and the each "o" one half of the stache?
Intuitive understanding of math and similar processes gives you access to these bits of symmetry and beauty. "Rote" and process based understanding lets you take in and use processes and algorithms that are outside of your intuitive reach by treating them with a little bit more rigor.
I suspect the beauty in question is the sort that only those predisposed to appreciate it, ever will. Like I'm never going to look at a model train set and go "oh, how lovely!", at best I'll express "oh, that's really... intricate?" while mostly thinking about how I'm glad I'm not on a life-path that involves ever having to work with or store such a bulky, fiddly thing that's largely just an exceptionally hard-to-clean dust-collector—which is basically how I feel about math. IOW I think those trying to get 2nd graders and such to see the beauty are on a fool's errand—most of us just aren't gonna start oohing and aahing over the quadratic formula or PDEs or even Euclid (the only source I've seen that even gave me a glimpse of a glimpse of what people must mean when they say "math is beautiful") no matter how inspired and brilliant the teacher, and perfect the lesson plan (and, practically, those are never going to be so good for the median case)
(I don't mean to pick on people who like model trains, that feeling of "thank god I'm not responsible for that thing" I experience when I see one, and a total lack of desire to admire it for more than a few seconds no matter how many thousands of hours of work went into it, was just the first analogy that came to mind for something that some folks can love extremely while most others just don't get the appeal)
With that said, there is a spectrum. And even people like you (you alluded to this with Euclid) will have tidbits of appreciation. And it matters if you have a good teacher doing good curation -- you might see 2 or 5 times as many of those tidbits. And there might be 1 or 2 more students in a class of 20 who get pushed over the edge into really liking the subject. But yes, these are improvements at the margins, and your point stands.
Math-fans who (and I don't fault them for this! I get it! I just think they're barking up the wrong tree) urge attempts to adjust curricula in an attempt to get students to find the kind of appreciation they have for math are hoping, I think, that it'll drive a rather broader-reaching and more productive engagement with mathematics through the rest of school resulting in much better overall understanding of math across the public, rather than to get more people to engage in casual and occasional Sunday morning stepping-through of simple geometry proofs while sipping tea, or solving a few easy diophantine problems or puzzling over a couple pages of Martin Gardner while winding down before bed. I'm sure they would be happy (or, at least, not unhappy) to see more of those latter effects, too, but I don't think that's why they advocate changes in pedagogy.
I think they want—for good reasons—to try to help students find intrinsic motivation to work up to the point that they can appreciate and enjoy higher math, and I (weakly—it's not like I've done a study on this, it's mostly impressions from talking to people over a lot of years) believe they're attempting something doomed to fail. I think they'd have better luck focusing far more on practical applications (Yuck, not even really math! Yeah, I know, I know) than traditional k-12 math instruction has, to motivate most students, and putting up less resistance to the deadly-effective (if also deadly boring) rote memorization in early grades that some in the math-lover contingent seem to especially dislike. Their fellow math-lover youngsters will mostly (I speculate) find their way to higher math and to their joy at a finely-constructed proof without much more than a little exposure to it to get them started (which parts of the curriculum the rest of the students will probably find very unpleasant, LOL)
I think (again, I very much could be wrong on this, I'm no expert) their fundamental mistake is a belief that school is somehow making students dislike math because it's not showing them the beauty of it, or because they take too long to get to proofs (you're not even really doing math until then, after all! Is a sentiment I've often read and heard) or what have you. I do believe there's a motivation problem with mathematics in school, I just think we're in an unfortunate position where most of the folks best-positioned to drive change in how we teach math have a perspective on it that's also the reason they did so well and went so far with it, and want to transfer that perspective to as many students as they can, but I just doubt there are a lot out there who didn't happen on that perspective by some combination of birth and fortune but who would acquire it if you could only present them watered-down Velleman and some toy demonstrations of integrals as second graders. I worry that's not a useful route to making the median student care at all more about math than they already do, and may (MAY! This part's an especially weak claim) displace instruction that would benefit them more.
To put it another way: students who like reading, or at the very least are motivated enough to address it with an attitude more-present than extreme reluctance and avoidance, mostly like it due to what literacy does for them. They like what it lets them accomplish ("I can understand this video game now! Hell yeah!"), they like that it unlocks the ability to enjoy stories, including maybe some that would be off-limits in visual media (the reading is just the medium, it's not like they're sitting down with joy to do reading exercises in their spare time, they're reading because they're getting something else out of it) or to communicate ordinary everyday things among their friends. You do get the kids who nerd out over language itself—these are like the natural math-lovers—but that's not the usually motivation for most students who manage to do at least OK at becoming literate.
Literacy's superpower in a school setting is that its benefits and the value of putting some effort into it are immediately apparent to any student who isn't badly deficient in it, and natural opportunities for practice accessible to young students are everywhere in normal life, with potentially dozens of well-motivated opportunities for practice per hour if you're not even trying to seek them out. Math's handicap in that same setting is that its immediate benefits are much more scattered and limited—it's a very powerful tool, but you don't need it to gossip with your friends over instant message, or to read that slightly-too-naughty-for-your-age book you managed to get ahold of, or to figure out the early puzzle that's been keeping you from getting very far in some video game you want to play, or what have you. Occasionally it's relevant to those kinds of things! Sort of! (mostly the video game example) But not so consistently as reading and writing, and when it is, it's not proofs, generally, it's basic arithmetic and maybe some light fractions or percentage work.
We don't motivate young readers to stretch themselves by going "look how fucking sublime this passage in King Lear is! Oh, oh, and look at that aptly-employed gerund over there! Oh and that pun there may have roots in this fascinating, obscure, middle-English work, which suggests that..."
No, we go, "here's this story that's for older kids, but I think maybe if you try, you can read it, and who knows what might be in there, for older kids..." (meanwhile, we know the worst thing in there's a mild dick joke, few relatively tame swear words, and an oblique reference to a sex act that they definitely won't notice unless they're already familiar with it, all of which are probably less-bad than things they've heard from their peers this week—but man, will they feel like they got to do something forbidden!)
What's the equivalent for math? For god's sake, literacy basically gets to employ all the seven deadly sins to get kids to read and write! Literacy has Satan, lord of darkness riding shotgun! Reading is metal! What is poor mathematics to do? I do not know if there is anything like that for math, but I don't think trying to get kids to appreciate the beauty of proofs or delightful symmetry in some seemingly-unrelated pair of equations is likely a productive path to explore. We wouldn't expect something similar to work in literacy—and fortunately we don't have any strong reason to try it—so I don't know why we'd expect it to work for math.
What I'd like to see is an attempt to all but completely remove math per se from the curriculum of most grades, rolling it into other subjects, including literature but especially (of course) science, to include social science, plus practical and fine arts, with most of the remaining dedicated mathematical instruction in early grades devoted to bad ol' "math facts" (memorization via drilling) and the use of those and teaching of methods occurring mainly in other subjects, to try to demonstrate utility alongside the instruction in methods. Would it work? I dunno, but I think it's worth trying. The closest thing I'm aware of (and it's only kinda close) is aimed mainly at postsecondary students, under so-called "great books" programs. I don't know whether something akin to that's been attempted in a school setting (home schooling's so very different, for a bunch of reasons) with kids as young as primary school. I'm not sure there are a lot of alternatives, aside from accepting that most of k-12 math education is going to be a widely-hated mostly-waste-of-time for the majority of students (and hell, maybe that really is just the best we can practically achieve, and we have to live with it)
Quick story: When I first TA'd a stats class at age 23 or so, I was sure I could transmit my own sense of excitement. I was a good teacher and got good feedback. But I got a dose of reality one day when I told the students about Buffon's needle problem, one of my favorite puzzles (that a needle as wide as one of the slats, when tossed onto a hardwood floor, has a chance of 2/pi of landing across a line). To me it was mind-blowing, but the class's reception was lukewarm. So I said I'm just curious, please answer honestly: How many think this is really cool (few hands); kind of interesting (maybe 7 hands); don't care (rest of class, like 15 hands). I thought, "Ohhhh, now I get it."
> maybe that really is just the best we can practically achieve, and we have to live with it
I think this is the answer honestly, but I think the incremental gains of a really good teacher are much greater than you're suggesting (eg, watch an Eddie Woo video on youtube and compare to your HS math teacher). It can be the difference between "I hate this" and "This is not so bad." Or, for that naturally apt, between "This is ok" and "I love this."
I don't think "focus relentlessly on the practical applications" is the answer. Once you've accepted the frame of "What will I use this for?" you've already lost, because the truth is most kids won't need to use, say, calculus in their daily lives very often. That's not the reason to learn it, or most math subjects. The frame given here is the right answer imo: https://www.youtube.com/watch?v=RqQlXG__vGs. The goal is to see the world differently and internalize an approach to reasoning. While developing this requires actual study of specific subjects, the particular sub-fields you study are less important. Moving the needle even a little here matters.
And even if you focused on, say, basic financial literacy (which would be practically useful), they'd hate it just as much, and I doubt it would stick any better. Really we're just saying that it's hard to get people to do things they don't naturally care about.
Agree that teachers can make a big difference, and my ideas for system-level reform are absolutely half-baked and may actually be awful. The closest thing I've got to a claim I'm fairly sure is true on this topic, is that trying to get kids to see the beauty of math before school has prejudiced them against it is unlikely to achieve much, because I don't think schools' deficiencies are the key reason for students failing to see that beauty in the first place :-/
> Really we're just saying that it's hard to get people to do things they don't naturally care about.
QED, LOL.
Women in a bunch of ways; men in a somewhat-different set of ways; well-executed personal style.
A tight and effective lecture; a perfectly-selected shot in a film that communicates something that hits you right in the gut in addition to being visually striking; expert realization of a fictional character; that time at the end of the play after the curtain closes when all the murderers and tyrants and lovers and victims come out and smile and bow, holding hands, like they're settling into an eternity in some kind of Heaven and that tragedy was just one bad week in a blissful infinity of sublime humanity and loving and they want us to see how happy they all are now before we leave them behind, so we don't worry about them, and I tear up every damn time.
That moment when your sip of a new whisky develops, incredibly, a fifth distinct flavor out of nowhere, the fourth somehow fading to let it shine.
And books are often nice.
X**Y=X**Y
X\*Y=X\*Y
* 1 / 100 % is calculation of percentage, * 100 % / 1 is calculation of absolute amount.
And, as stated in a different comment, remember that the commutative property applies to these numbers.
In this context, we often think of percentages as parts of stuff. I think it's because it was often taught alongside fractions. So in our heads, we intuit and think of percentages as breaking stuff apart. Typically, when we break stuff apart, there's often a low chance that for any number of divisions we pick, there will be combination of parts that sum up to the same amount. With that framing, it's surprising that by virtue of switching the percentage to the other number, we get the same part of the whole! It betrays our intuition.
However, if we re-frame it with our abstractions in math, it makes more sense. Multiplication is often taught as repeated addition. This isn't wholly correct. It's a way to scale a value. But scale goes both ways! You can both scale up to make the value bigger, but you can also scale down! By making the scale a fraction, we can scale down, or break stuff apart. And because we can now rely on the properties of the abstraction, namely commutativity of multiplication over natural numbers, it becomes much easier to see that it must be true that by virtue of switching the position of the percent sign.
It's kinda like when you get that ah-ha moment. A moment of insight. Other people call it making connections. But essentially, it's a new way of seeing something that was there all along.
It's like that game, The Witness. On the surface, it seems like it's about solving line tracing puzzles. But in fact, the entire theme of the game is to see with a new perspective what had been in front of you all along. I won't spoil the game for those of you that haven't played it yet, but if you want a guide tour of repeated ah-ha moments and insights, I suggest you try the game.
Which on the other hand is strange, because its creator Jonathan Blow really rails against functional programming and category theory. To me, as I learn more about it, I see the connections that were there all along by giving me apparatus to think about things in a new light.
The Curry-Howard Correspondence was completely surprising to me--that there's a relationship between logical proofs and programs. Propositions correspond to types, proofs correspond to programs, and proof verification corresponds to type checking.
It was also surprising to me that we can think of functions as exponential types, and the algebra works out!
Alan Kay laments that we build software like ancient Egyptians build pyramids--by piling on rocks. We don't build software with arches. I think that's starting to change as mathematical ideas like monads filter into mainstream programming. A lot of the concepts in functional programming and category theory filter down into language features of mainstream languages. Result and Optional types are prevalent now, and they're monads. But it took leveraging the mathematical apparatus to find them, despite decades of stabbing at it by programmers.
In a practical sense, having the insight and ah-ha moment before you need to use the connection is helpful. In the midst of being busy trying to find product-market fit, you often don't have the state of mind or the time to see the connections. But if you already know them, you can take advantage of the properties and shortcuts that it gives you. This is the answer math teachers give you when you ask "when am I ever going to use this?"
But I don't think that's really not why mathematicians do math. I think having that insight and ah-ha moment feels like an expansion of the mind, a perspective shift feels out-of-body. And it can get addicting, especially if you had to work hard to get it. I think if you want to live a life of insight and ah-ha moments as a priority compared to everything else, then there are probably little other vocations better than mathematics. But you don't need to be a pro-mathematician to have those little ah-ha moments to enrich your life. You just have to be willing to work at it a little, and get that occasional perspective shift. And that's why I think it's lovely.
I mean they might all be mathematically equivalent, sure, but in any mathematical problem, the initial information is equivalent to the result. The result is just the same thing, expressed differently. With sufficient insight, it wouldn't be necessary to do any mathematics at all, because all results would be implied already.
The author of the article worries about people learning "formulas and processes" and "magic words" instead of gaining understanding. That is, rote learning. It's a valid concern, but every gain in understanding, every insight, remains like a magic trick. The more you internalize it the less surprising the trick gets, but it's forever a trick, and you are building a catalog of tricks, even as you seek greater understanding. We never get close to the above-mentioned perfect and total level of insight, and learning is a process of continually adding clutter and low-key mysteries.