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It's always puzzled me how many of these interesting things are not in a standard math course. I recall doing a 4th year "advanced mathematical techniques" cou

It's always puzzled me how many of these interesting things are not in a standard math course. I recall doing a 4th year "advanced mathematical techniques" course where they went over the basics of graph theory and RSA algorithm. Descrete maths. This kind of thing is not the intensive calculus that people get told is math in high school, but a high schooler could do it if you showed them.

IanCal 1y ago
There's a load of things I think are things many kids could do much earlier in maths, though my personal feeling may be off. My son seems to be taking very quickly to the feeling of maths so I'm trying to share some parts that I only realised drastically later on. Rules you learn in maths are often based on much simpler, truer (I'll get back to this), things. I learned loads of "move this denominator up to this side, like an escalator!" and such and only when I was something like 13 I nearly shouted "EQUALS MEANS THE TWO SIDES ARE THE SAME THING" as I realised why so many of these god damn rules existed. It also explained suddenly why some things are OK to do and some aren't (notably taking roots). The other key thing was finding out that these things are decisions. We can choose whatever rules we want and see what happens, and keep whatever is useful. Lots of the more interesting things I've seen later in life have been "well there's no number you can square and get a negative number. What if there was?" or "What if there was a way to raise to a real number, what would that mean?". There's no magical reason we can't divide by 0 but there's not a neat useful answer really, but you can totally define a setup where you can do that and have fun if you want. Imaginary numbers and Argand diagrams are things my 6 year old can mess about with. He's still developing reasoning so certain things trip him up about seeing the process for solving a thing (fascinating to see where his comprehension is up to and the leaps that happen, I remember the first time seeing him pick up a chess piece, go to move it and before putting it anywhere say "no, if I do that you'll do X") but lots of things are very accessible. Powers are fun because you can make big things quickly, factorials too. Basic solving of equations I was only taught in secondary school yet I was taught "two bananas cost 30p and one banana and one apple costs 40p, how much is one apple" years before and that's just swapping fruit for letters. I was always good at maths, but I think it's far more fun now I've understood more about what you can mess about with and how simple parts are at times.
jerf 1y ago
This is one of the reasons whenever the topic of math curricula comes up on HN I've tended to argue to remove some things. The system is full up at the moment and you can't really add anything without removing some things. But I can't even get HN readers to agree that some things should be removed, let alone normal people. Everyone wants to add but no one is willing to remove anything, no matter how useless it is for all purposes practical, mathematical, scientific, and every other purpose. With the curriculum highly centralized and standardized, change to what is taught has proved to be on the scale of multiple decades if not centuries, and not in a good way. Changes to how The Holy Curriculum is taught can occur on decades, but what has changed hardly at all, despite the fact that frankly the curriculum was audibly creaking when I was doing it in the 1980s... at least, if you tried to hear the creaking. Even then it was obvious that what we were being taught had been frozen circa 1930 or 1940 and needed updating for a number of reasons. It's even more out of step with the world now, in every way, but here we are.
- acoustics 1y ago
  What do you propose to remove? Personally, I'd say that geometry and trigonometry seemed the least useful out of the high school math sequence. Trigonometry seemed like "Algebra 3: this time with identities to memorize" and geometry seemed mostly like an homage to Euclid. In my day-to-day life, algebraic thinking comes up much more often than geometric thinking, although of course I know this is not universally true. Geometry is the classic introduction to proof and rigor, but if I were the benevolent dictator of the math curriculum, I would try to accomplish that with basic combinatorics, graph theory, or even a tiny bit of abstract algebra (proving basic arithmetic facts from the real number axioms, maybe modular arithmetic).
  - jerf 1y ago
    A lot of trig. Trig identities is my bête noire, though I would save off one day for them, after Taylor polynomials, to derive a few through practice that way. But I remember grinding on them for quite a while; I would just have one day and use them as Taylor fodder. (Taylor polynomials are not practically useful for most people but they are extremely important as "the simplest thing that converts transcendental functions into the world of the elementary arithmetic functions"; it is good to demystify how one might compute sine or cosine "from scratch".) But I wouldn't nuke trig out entirely as classical geometry is an almost unparalleled place to learn basic proof techniques and mathematical thinking in a playground where you are not distracted by a lot of arithmetic. I think I still like that better than graph theory, and simple trig/geometry is in fact practically useful for a chunk of the students we're pushing through these courses. It would just get toned down. I would cut out a lot of integration grinding. The concept of integration is extremely important, and I want the students introduced to the ideas of doing it symbolically, but the details of all of the manipulations are much less important than the concepts. I would retain only things that are of mathematical interest, like integration by parts (useful as an exercise in how much fluidity you have in doing math and a good check you understand the concept). Symbolic differential equations I'd cut down a lot. I think there's a sense in which they are useful but the utility is not revealed until a couple of semesters in. Even my college semester was frankly not that useful, you really need to dedicate yourself to them to get the value out of them. I'd put in some more work on numeric integration, and working with computers to get them done. The concepts of differential equations are super, super, super important; the exact knowledge of how to solve this super precise form of this exact differential equation is not. More practical experience to teach intuition, less grinding on symbolic details. The net time spent on them if I were writing the curriculum might even go up, because I'd tip in some (very) basic chaos mathematics here instead of all that grinding. Matrices I'd have a hard think about. HN is at the epicenter of their utility so it might be hard to see that for most people it's not all that useful in any sense. I might like to move them on to a very explicitly STEM-focused track. If that sounds horrible, remember that my whole point here is that there's a lot of stuff that should be in the curriculum that currently isn't, or is just glossed over very briefly. Like, I'd like more work spent on basic financial math, both for personal finances and doing things like a bit of running stochastic financial scenarios (integrate this into the stats curriculum, for instance). Which also brings up the idea of doing more simulations of somewhat larger statistical scenarios than you can solve as a closed form by doing Monte Carlo simulations; statistics and probability is very important but for most people it spins off into the symbolic weeds when in fact most problems people will have in real life are not cleanly amenable to such things. (This might also help erase the implicit belief that stats tends to teach that everything is uniformly distributed.) The curriculum as it is is also structured as an unmotived series of solutions to problems the students don't have. I would try to give them the problems before the solutions; e.g., if I'm building a stats curriculum around simulations that go into symbolic math rather than just handing them symbolic math from the beginning, imagine covering the Central Limit Theorem from the point of view of me giving out a couple of different simulation assignments that all "happen" to converge to it despite very different scenarios, and we can discuss & teach why the seemingly totally different scenarios produced such a similar outcome, rather than just handing it down from On High as a Solution to a problem none of them have. You can't have things like that if you aren't willing to cut something else because we are full up right now. There's a lot of room now for people to get more intuitive grasps of these things by virtue of working with them through practical numeric integration, Monte Carlo simulations, maybe we'll do a brief section on enough 3D geometry to do a useful introduction to 3D modelling/CAD/CNC/3D printing/the whole complex of modern tools available that use some form of 3D modelling, there's so much useful stuff that's just aching to be let in that the culling of the old needs to be a bit aggressive after a century. To be honest I'd not cover this dimensionality stuff, except perhaps as a special-interest side day or something. It's not bad to do a few of those, just to expose students to the diversity in math. I got graph theory like that in my own high school, and it can stay that way; I wouldn't amp it up any. Dimensionality does pivot nicely into fractals and fractals have notoriously pretty pictures associated with them, so maybe I'd sneak it in with that.

IanCal 1y ago

There's a load of things I think are things many kids could do much earlier in maths, though my personal feeling may be off. My son seems to be taking very quickly to the feeling of maths so I'm trying to share some parts that I only realised drastically later on. Rules you learn in maths are often based on much simpler, truer (I'll get back to this), things. I learned loads of "move this denominator up to this side, like an escalator!" and such and only when I was something like 13 I nearly shouted "EQUALS MEANS THE TWO SIDES ARE THE SAME THING" as I realised why so many of these god damn rules existed. It also explained suddenly why some things are OK to do and some aren't (notably taking roots). The other key thing was finding out that these things are decisions. We can choose whatever rules we want and see what happens, and keep whatever is useful. Lots of the more interesting things I've seen later in life have been "well there's no number you can square and get a negative number. What if there was?" or "What if there was a way to raise to a real number, what would that mean?". There's no magical reason we can't divide by 0 but there's not a neat useful answer really, but you can totally define a setup where you can do that and have fun if you want. Imaginary numbers and Argand diagrams are things my 6 year old can mess about with. He's still developing reasoning so certain things trip him up about seeing the process for solving a thing (fascinating to see where his comprehension is up to and the leaps that happen, I remember the first time seeing him pick up a chess piece, go to move it and before putting it anywhere say "no, if I do that you'll do X") but lots of things are very accessible. Powers are fun because you can make big things quickly, factorials too. Basic solving of equations I was only taught in secondary school yet I was taught "two bananas cost 30p and one banana and one apple costs 40p, how much is one apple" years before and that's just swapping fruit for letters. I was always good at maths, but I think it's far more fun now I've understood more about what you can mess about with and how simple parts are at times.

jerf 1y ago

This is one of the reasons whenever the topic of math curricula comes up on HN I've tended to argue to remove some things. The system is full up at the moment and you can't really add anything without removing some things. But I can't even get HN readers to agree that some things should be removed, let alone normal people. Everyone wants to add but no one is willing to remove anything, no matter how useless it is for all purposes practical, mathematical, scientific, and every other purpose. With the curriculum highly centralized and standardized, change to what is taught has proved to be on the scale of multiple decades if not centuries, and not in a good way. Changes to how The Holy Curriculum is taught can occur on decades, but what has changed hardly at all, despite the fact that frankly the curriculum was audibly creaking when I was doing it in the 1980s... at least, if you tried to hear the creaking. Even then it was obvious that what we were being taught had been frozen circa 1930 or 1940 and needed updating for a number of reasons. It's even more out of step with the world now, in every way, but here we are.

acoustics 1y ago
What do you propose to remove? Personally, I'd say that geometry and trigonometry seemed the least useful out of the high school math sequence. Trigonometry seemed like "Algebra 3: this time with identities to memorize" and geometry seemed mostly like an homage to Euclid. In my day-to-day life, algebraic thinking comes up much more often than geometric thinking, although of course I know this is not universally true. Geometry is the classic introduction to proof and rigor, but if I were the benevolent dictator of the math curriculum, I would try to accomplish that with basic combinatorics, graph theory, or even a tiny bit of abstract algebra (proving basic arithmetic facts from the real number axioms, maybe modular arithmetic).
- jerf 1y ago
  A lot of trig. Trig identities is my bête noire, though I would save off one day for them, after Taylor polynomials, to derive a few through practice that way. But I remember grinding on them for quite a while; I would just have one day and use them as Taylor fodder. (Taylor polynomials are not practically useful for most people but they are extremely important as "the simplest thing that converts transcendental functions into the world of the elementary arithmetic functions"; it is good to demystify how one might compute sine or cosine "from scratch".) But I wouldn't nuke trig out entirely as classical geometry is an almost unparalleled place to learn basic proof techniques and mathematical thinking in a playground where you are not distracted by a lot of arithmetic. I think I still like that better than graph theory, and simple trig/geometry is in fact practically useful for a chunk of the students we're pushing through these courses. It would just get toned down. I would cut out a lot of integration grinding. The concept of integration is extremely important, and I want the students introduced to the ideas of doing it symbolically, but the details of all of the manipulations are much less important than the concepts. I would retain only things that are of mathematical interest, like integration by parts (useful as an exercise in how much fluidity you have in doing math and a good check you understand the concept). Symbolic differential equations I'd cut down a lot. I think there's a sense in which they are useful but the utility is not revealed until a couple of semesters in. Even my college semester was frankly not that useful, you really need to dedicate yourself to them to get the value out of them. I'd put in some more work on numeric integration, and working with computers to get them done. The concepts of differential equations are super, super, super important; the exact knowledge of how to solve this super precise form of this exact differential equation is not. More practical experience to teach intuition, less grinding on symbolic details. The net time spent on them if I were writing the curriculum might even go up, because I'd tip in some (very) basic chaos mathematics here instead of all that grinding. Matrices I'd have a hard think about. HN is at the epicenter of their utility so it might be hard to see that for most people it's not all that useful in any sense. I might like to move them on to a very explicitly STEM-focused track. If that sounds horrible, remember that my whole point here is that there's a lot of stuff that should be in the curriculum that currently isn't, or is just glossed over very briefly. Like, I'd like more work spent on basic financial math, both for personal finances and doing things like a bit of running stochastic financial scenarios (integrate this into the stats curriculum, for instance). Which also brings up the idea of doing more simulations of somewhat larger statistical scenarios than you can solve as a closed form by doing Monte Carlo simulations; statistics and probability is very important but for most people it spins off into the symbolic weeds when in fact most problems people will have in real life are not cleanly amenable to such things. (This might also help erase the implicit belief that stats tends to teach that everything is uniformly distributed.) The curriculum as it is is also structured as an unmotived series of solutions to problems the students don't have. I would try to give them the problems before the solutions; e.g., if I'm building a stats curriculum around simulations that go into symbolic math rather than just handing them symbolic math from the beginning, imagine covering the Central Limit Theorem from the point of view of me giving out a couple of different simulation assignments that all "happen" to converge to it despite very different scenarios, and we can discuss & teach why the seemingly totally different scenarios produced such a similar outcome, rather than just handing it down from On High as a Solution to a problem none of them have. You can't have things like that if you aren't willing to cut something else because we are full up right now. There's a lot of room now for people to get more intuitive grasps of these things by virtue of working with them through practical numeric integration, Monte Carlo simulations, maybe we'll do a brief section on enough 3D geometry to do a useful introduction to 3D modelling/CAD/CNC/3D printing/the whole complex of modern tools available that use some form of 3D modelling, there's so much useful stuff that's just aching to be let in that the culling of the old needs to be a bit aggressive after a century. To be honest I'd not cover this dimensionality stuff, except perhaps as a special-interest side day or something. It's not bad to do a few of those, just to expose students to the diversity in math. I got graph theory like that in my own high school, and it can stay that way; I wouldn't amp it up any. Dimensionality does pivot nicely into fractals and fractals have notoriously pretty pictures associated with them, so maybe I'd sneak it in with that.