Having earned an undergraduate degree in math I've often had similar thoughts about US elementary math education's emphasis on rules and mechanical calculation and memorization rather than more abstract concepts like pattern matching.
Turns out there was a thing called "New Math" (https://en.wikipedia.org/wiki/New_Math) in the 1960s where public elementary schools tried to teach concepts from set theory and abstract algebra. Basically everyone hated it.
Perhaps there was just a curriculum failure, or perhaps teachers weren't well-equipped enough to teach the material. But we can't ignore the attempt and need to come up with an answer for why New Math failed before we can try it again.
I was taught the "New Math" as a child and did not hate it. It was wonderful. When I look at math blogs today, they use notation and terminology I learned back then. This is a tremendous advantage (or so it seems to me).
And everyone else just curses when they open a Wikipedia article for smth that is supposedly (and actually) not that complex, but is explained in those terms, still :)
My high school math teacher used to give us extra classes that weren't directly connected to the curriculum. They were always about structure, elegance, and beauty. Quite a lot of time was also spent on history: who was studying what, when, and why. (Also there was a cult of Leonard Euler, which maybe is not so surprising if you did high school math.)
I found it to be the most important glue in my math education. In fact, all the natural science ought to be taught in this way. Kinda like Bill Bryson's Brief History of Everything, plus actual calculations.
We won a math contest in my last year. I was expecting it to stay interesting.
Unfortunately when I got to university math (and the rest of engineering) was taught in a very utilitarian way. There was very little context, just a lot of similar looking derivations.
I blame the exam culture. In a way it's good because it motivates you to learn something, but it's bad because you end up learning it in a way that isn't useful. At the end of your college days, you are unlikely to remember just how Stoke's theorem works or the coefficients in Runge-Kutta. That's just because the size of the curriculum is huge. But if you had a context, a set of stories about when and why something was studied, you'd have a much better chance of being able to recall that it even exists.
I had a college geometry professor who taught all of his classes this way. He taught me things about math, structure, elegance, and beauty that unfolded into a much better understanding of why math is so powerful in science. Studying history of math and Euclid did more to enhance my understanding of Physics and Math than any of the actual required courses, because they taught me the context in which the thinking is grown, which is far more powerful than the thinking itself.
That problem is actually very easy to solve. If you number the cells in binary, patterns fall out enough that it's easy to convert back and forth from the original numbering to the final numbering.
If he wants to just work it out by hand, well, maybe he doesn't understand math as well as he thinks. If you want to use math to "build hammers", it help to know what a nail is first.
I agree. I actually like this problem. I didn't solve it yet, but I guess that your idea to use binary base will be very useful. I'll give this problem to my daughter.
Suppose we are interested in the function f(N) such that a cell which originally has N squares to its left ends up with f(N) squares under it.
To calculate f(N), write N out in binary with K + 1 bits [where the tape has 2^K squares], including a leading zero. Reverse this bitstring [exchanging most- and least-significant bits, etc.]. Finally, if the result has a leading one rather than a leading zero [i.e., if N was odd], flip all the bits [turning ones into zeros and vice versa]; otherwise, leave them be. This yields the binary description of f(N) [with K + 1 bits, including a leading zero].
In the given example, N is 941; i.e., 01110101101, using ten + 1 binary bits. Reversing this, we get 10110101110. Finally, since this has a leading one, we flip all the bits to get 01001010001. This the binary description of 593, which is the answer.
(More generally, suppose at some point in the process, the tape has been folded to a width of 2^K * W squares and height of H squares, and we are interested in where the W-width, H-height block with N such blocks to its left ends up once we've gone through K more foldings to bring the tape width down to W and tape height up to 2^K * H. This block will end up with f(N) such blocks below it; if N is even, the block will be in the same orientation as currently, while if N is odd, it will be reversed 180 degrees from its current orientation (with its current top-left square becoming its bottom-right square, etc.))
Totally agree with the author. In high-school, I was interested in physics more than math and whenever I took interest in math it was because it would help me understand some specific aspect of the physics better. As I got into engineering, my interest in math again were only in areas where it was directly useful to me (like signal/image processing, operations research etc.) As a CS engineer, I've used very little math I was forced to learn by rote to write exams. Only use that came of it I think is training my brain muscles to do pattern recognition and fast memory recall.
But is there a reason why the maintainer of this blog hasn't shared his or her identity? I am very interested in your background. It looks like some of your earlier entries were made when you were in high school. In any case I am very impressed. Sent a few of your articles to my younger siblings to read.
Thanks! I'm actually still in high school. Shoot me an email or find me on IRC if you'd like to talk—I'd be happy to say hi.
I'm hesitant to tie my writing to a "real-life" identity because you get a lot of freedom when you dissociate yourself from what you publish. If I knew my friends/parents/potential-employers could find my blog, I would be much more hesitant to publish, which would have been very unhealthy in the long run.
Ok, I understand. I have multiple blogs and the ones that are not tied to my real life identity are a lot more active than the ones that aren't.
But all your posts are brilliant and I personally think you are doing yourself a disservice not putting your name out there. Like I said, I am very very impressed with the entries.
19 comments
[ 2.7 ms ] story [ 54.0 ms ] threadTurns out there was a thing called "New Math" (https://en.wikipedia.org/wiki/New_Math) in the 1960s where public elementary schools tried to teach concepts from set theory and abstract algebra. Basically everyone hated it.
Perhaps there was just a curriculum failure, or perhaps teachers weren't well-equipped enough to teach the material. But we can't ignore the attempt and need to come up with an answer for why New Math failed before we can try it again.
Wikipedia should be a resource for the average layman, not mathematically mature individuals who comprise the minority.
Also, like every other reform, it suffered because math illiterate parents (who know basic by rote by more) didn't understand their kids' curriculum.
Students with high aptitude (including many here on HN) thrived in New Math and became among the highest achievers in adult mathematical activity.
I found it to be the most important glue in my math education. In fact, all the natural science ought to be taught in this way. Kinda like Bill Bryson's Brief History of Everything, plus actual calculations.
We won a math contest in my last year. I was expecting it to stay interesting.
Unfortunately when I got to university math (and the rest of engineering) was taught in a very utilitarian way. There was very little context, just a lot of similar looking derivations.
I blame the exam culture. In a way it's good because it motivates you to learn something, but it's bad because you end up learning it in a way that isn't useful. At the end of your college days, you are unlikely to remember just how Stoke's theorem works or the coefficients in Runge-Kutta. That's just because the size of the curriculum is huge. But if you had a context, a set of stories about when and why something was studied, you'd have a much better chance of being able to recall that it even exists.
If he wants to just work it out by hand, well, maybe he doesn't understand math as well as he thinks. If you want to use math to "build hammers", it help to know what a nail is first.
Suppose we are interested in the function f(N) such that a cell which originally has N squares to its left ends up with f(N) squares under it.
To calculate f(N), write N out in binary with K + 1 bits [where the tape has 2^K squares], including a leading zero. Reverse this bitstring [exchanging most- and least-significant bits, etc.]. Finally, if the result has a leading one rather than a leading zero [i.e., if N was odd], flip all the bits [turning ones into zeros and vice versa]; otherwise, leave them be. This yields the binary description of f(N) [with K + 1 bits, including a leading zero].
In the given example, N is 941; i.e., 01110101101, using ten + 1 binary bits. Reversing this, we get 10110101110. Finally, since this has a leading one, we flip all the bits to get 01001010001. This the binary description of 593, which is the answer.
(More generally, suppose at some point in the process, the tape has been folded to a width of 2^K * W squares and height of H squares, and we are interested in where the W-width, H-height block with N such blocks to its left ends up once we've gone through K more foldings to bring the tape width down to W and tape height up to 2^K * H. This block will end up with f(N) such blocks below it; if N is even, the block will be in the same orientation as currently, while if N is odd, it will be reversed 180 degrees from its current orientation (with its current top-left square becoming its bottom-right square, etc.))
But is there a reason why the maintainer of this blog hasn't shared his or her identity? I am very interested in your background. It looks like some of your earlier entries were made when you were in high school. In any case I am very impressed. Sent a few of your articles to my younger siblings to read.
I'm hesitant to tie my writing to a "real-life" identity because you get a lot of freedom when you dissociate yourself from what you publish. If I knew my friends/parents/potential-employers could find my blog, I would be much more hesitant to publish, which would have been very unhealthy in the long run.
But all your posts are brilliant and I personally think you are doing yourself a disservice not putting your name out there. Like I said, I am very very impressed with the entries.