>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.
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.
Agreed. And they didn’t do this. If a physicist says that A is obvious to physicists then this isn’t an insult to others. It’s expected that having expert level knowledge in something indicates that some things are obvious after all the training that are not obvious to those without training.
Your comment convinced me to revisit the quotation. It's even worse than I thought. The author contrasts people who 'understand' because they 'did maths' (i.e. studied maths formally in school) with those who don't understand because they haven't. The implication: formal study in school is the only, and a guaranteed, path to understanding.
Clearly they were not saying this. It is a fact that the overwhelming majority of people who grok mathematics were trained to do so. The set of people who are self taught in math is small enough that one ought to be forgiven for assuming this. Similar to how people who know how to perform brain surgery almost certainly got that knowledge from training.
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
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.
I have training in math and wasn’t consciously aware of this a few years ago when I first saw this. I didn’t then and don’t now think the language used was insulting.
I feel that categorizing people into math-doers and people that just merely apply formulas is insulting in general. Mathematicians can too use other people's theorems without working out all the details, not just people the working in other fields. (Also mathematicians use results from other fields in their daily lives all the time without understanding.)
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.
is not lovely.
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.
Agreed. The writing is quite prolix, affected, and pretentious, too. This is apparent even in the first sentence, where the author inexplicably puts quotation marks around "social network feeds" (while incorrectly failing to hyphenate the compound adjective).
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.
I expected there to be something interesting in this loveliness. Instead, I found boredom. I wanted to share in the joy, but found despair in both the original comment and the shocked responses.
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.
Maybe they meant that, if you were to sit down and figure out whether that statement is true, someone with a math education would find it is and not be particularly surprised. Not necessarily that you should know it beforehand.
I don't think the author explained it well. What % means, is, "out of 100" so like X% is X out of 100. % is a unit, just like meter or candela. You can do dimensional analysis with %. To remove a % you do like, X % * 1 / (100%) and then the two % (remember % is a unit) cancel out so you get X % = X / 100.
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
> Given the unalterable underlying
ground truth that people don't
want to be educated, they want
to be entertained - what should
we be doing?
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.
My wife teaches Algebra 1 and this is something she’s encountered with a few of her students. They’ve been able to get by more or less by using what we refer to as “advanced arithmetic” e.g. figuring out the variable in an equation by inspection rather than solving the equation. But of course that only gets you so far, and she’s worried they’ll bomb Algebra 2 next year. But no matter what she does they just can’t/won’t learn algebra.
Basically. The questions were easy enough that they were doing it in their heads. She figured it out when she made them show their work, and their "work" turned out to be just plugging the right number into the equation.
This is not uncommon. My education was in math and, although there are also polyvalent people, I'd say quite a few of my comrades were either mainly Analysis or mainly Algebra people. In a sense, Algebra is similar to high-level languages and OOP, thinking of structure and how things articulate because of their nature, Analysis is like closer to the metal programming, how does this thing behave locally, controlling approximations and such. I believe programmers are equally split along this axis, are they not?
Not sure which one I am in. It really depends on the project and who I am working with, I’m kind of a chameleon and sometimes I change to fit in with my organization and other times I change to be what the other people around me aren’t.
When I was 12, I'd been BASIC programming for two years, and of course I used variables constantly (variably?) yet I didn't understand this simple crucial fact about algebra, which is that the letters are variables. I had a bright yellow pocket book titled "Calculus" and I used to carry it around in the hope that people would notice, but I couldn't understand page 1. I was into substitution cyphers and I devised a few, and I read about numerology, and at one point came up with the theory that the letters in algebra stand for numbers based on their position in the alphabet. Despite being steeped in all these instances of symbols standing for other things - and maybe even using x and y as variables in my programs - I just didn't imagine that algebra might be doing the same thing. It had a sort of stony, inscrutable face, permanently mysterious.
It really goes both ways. Attempts to completely supplant all rote math education with exercises to build intuition haven't had good results.
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.
Hell, I understand the example formula in the sense the author means, and some things a fair bit more advanced than it (thought not a ton more, in the scheme of things) and have never once seen the beauty some others find in math.
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)
I am one of those who does naturally find the beauty; nevertheless, I think your claim is indisputably true. It matches the empirical evidence, as well as my personal experience teaching math.
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.
I think the problem with the kind of very-selective appreciation that I, admittedly, probably could cultivate for some small niches of math if I tried, is that it's kinda useless for anything but a hobby-level interest having about as much value, however you like to measure it (productivity, education) as solving sudoku puzzles or any other somewhat-mentally-active pastime. Thinking Euclid's a little fun to read doesn't readily translate into feeling an urge to work through an analysis curriculum, nor create FOMO that I might reach a place of greater joy if I gritted my teeth and did it anyway, you know?
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 l...
I agree with most of what you've written, particularly the sense of hopelessness around getting most students to "see" the beauty -- most simply won't.
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.
Interesting story, I bet that was a gut-punch, haha. Hadn't seen the needle problem, cute and immediately clear (when it's pointed out—it wouldn't have been immediately clear if you'd asked me to solve it myself, of course) why the probability would have a relation to pi, thanks for introducing me to it.
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.
Out of curiosity, what sorts of things do you find beautiful? Clearly maths and model trains don’t make the list. Surely you find beauty in other aspects of life?
Forests, flowers, mountains, rivers, oceans, deserts, dry grasslands and a flat horizon and an anvil cloud 100 miles off.
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.
Perhaps the "o"s are the eyes of a melting face with a fake mustache glued on between them. The melting face being an appropriate image for how much many people fear math, as if they've seen the face of Cthulhu or another Great Old One.
Or those Henry Moore statues that the author says are "moving". They all look like scenes from The Thing, with a twisted amalgamation of ambiguous body parts, except they're enormous and weigh two tons. I very much hope they won't start moving. Liking those statues goes along with thinking mathematics is lovely.
For me it is lovely because it exploits the fact that the order does not matter mathematically, but it matters in our heads.
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.
I think the reason why it's lovely, because it gives you a perspective shift. When we get used to thinking of something as having "this one nature", and we find out there's another "equivalent and consistent nature" think about it. It gives color to the world.
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 prio...
Several people in the comments have provided explanations of this percentage trick. That is, they're trying to impart perspective shifts to the audience, just like you are. It's interesting that they're all different perspective shifts. There's no one true way to internalize it.
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.
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[ 2.6 ms ] story [ 109 ms ] threadThis 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
- - -.. this is just a reasoning of why
is not lovely.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
* 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.
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.
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 l...
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.
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?
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.
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 prio...
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.