His other works are also worth checking: http://1ucasvb.tumblr.com/
(for example, he has a super-nice explanation of Fourier series).
After seeing that, trig really just "clicked".
Nowhere along the way I was shown that sine is actually how high above the ground you are when traveling around the circle. It was always some length of side in right triangle divided by length of another side. That was technically the same thing but nowhere close as intuitive and natural.
Even when at some point I got interested in math and start thinking about sine as y coordinate it was always in relation to angle of a triangle (rooted in the center of the circle, one point at radius, the other at x axis below it) and not in relation of actual distance traveled along the circle. At some point radians appeared and we were just taught that's other way to measure angles, so you know: 0 degrees is 0, 90 degrees is 1/2 of PI and 360 degrees is 2 PI. I thought it was kinda weird to this that way but w/e probably people have their reasons...
The concept is so beautiful and natural yet it was made as artificial as possible during math education I had.
No wonder so many kids don't get or like math :(
(a decent set of textbooks should have lots of these didactic gizmos in it, but that doesn't solve the problem of getting the student ready to look at it or making sure they look at it after they are ready)
That makes total sense, and I wish it would have been explained like that to me, too.
My math teacher in HS spent an hour and a half explaining how derivatives work by just working through stuff on the board after he insisted no one take notes, and it made everything so clear. And reading through the Tau Manifesto finally made me understand what was going on in trig at a basic level.
I totally agree with you about how artificial math education can feel, and I think if larger concepts were introduced more like that many more people would be able to get much better at it.
I read something a couple years ago about how some school was experimenting in a math program starting in 8th grade where they would teach algebra, geometry, and calculus concepts all at the same time. I guess the students were really picking things up fast, and understanding how everything fit together better.
I learned my trigonometry when trying to figure out how to rotate and move objects around in 2D space for a computer game. I learned then, that sin is responsible for y axis, and cos for x axis.
But I never ever phrased this as "how high above the ground you are when traveling around the circle". I like the elegance of that; things would probably have clicked for me immediately back then, had I heard a sentence like this (and ditto for cosine being "how far to the right are you").
Bottomline, I'm in agreement with both OP and GP here.
Did you (and everyone else) really never learn the (paradigmatic) parametric equation describing circular motion, (x, y) = (cos t, sin t)?
Here is the article if you want to take a look: http://nautil.us/blog/the-math-trick-behind-mp3s-jpegs-and-h...
I studied linear algebra in college and computed plenty of eigenvalues/eigenvectors, but until I saw that graphic I had no idea what they actually were. I can't believe in retrospect that a textbook would explain them without an illustration like this.
I now feel enlightened; too bad it has been more than 20 years since I last needed to calculate with them! However, it'll probably come in handy when I help my children with their math homework in the coming years...
The animation was helpful in that it explained everything very quickly and obviously, obviating the need for a numeric or verbal explanation.
Was a radian just defined as an angle of 57.3 degrees in your class? I imagine that would seem really arbitrary to students.
Well, the way I think of the basis for radians goes like so:
1. All circles are similar to all other circles (assumed without proof), so there's a relationship between the radius of a circle and its circumference.
2. Angles, being scale-free, ignore the size of a particular circle; if an angle captures 1/3 of a small circle, it will capture the same 1/3 of a larger one, or a smaller one.
3. We can use the combination of an angle ("How much of the circle are we talking about?") and a radius ("How big is this circle?") to specify an arclength along the edge. We do it by multiplying the radius by the angle, and then by a conversion factor.
4. The radian measure is chosen so that the conversion factor mentioned in step (3) is equal to 1, which is easy to multiply by.
Here's the problem: out of a normal-sized group of randomly selected math students, expect between 0 and 1 of them (generally 0) to be interested in hearing you say this.
There's another argument for using radians (related, but not obviously so, to the first), which goes like this:
1. When measuring in radians, the trig functions behave well when taking the derivative: d/dx (sin x) = cos x and so forth.
2. If we used degrees (e.g., call dsin x the sine function for degrees instead of radians), we'd get all these awful constants mucking up our computations. d/dx (dsin x) = (180/π) dcos x, and d/dx (dtan x) = (180/π) dsec^2 x. The fourth derivative of sin x is sin x, but the fourth derivative of dsin x is (1049760000/π^4) dsin x. So we use radians to measure angles for the same reason we use e to measure logarithms -- they are more correct.
That one has the double disadvantage of being something no one wants to hear and drawing on applications they haven't yet heard of.
There's a shorter, less accurate version of the first approach I presented, that goes like so:
1. A circle's radius and circumference are both linear measures; they both have units of length to the first power.
2. Therefore, the conversion between them is a dimensionless (= unitless) number, a constant.
3. Radians are the unique correct constant for that "conversion" (there can't be more than one, any more than there can be more than one constant that converts meters to kilometers).
This is short and easy to say, but is sloppier and unsatisfying to precisely the students who care.
I'm another story of someone who's good at maths, got excellent grades in University, etc. etc.
During High School I didn't study as hard, and while I knew how to do calculations with Radians, never "understood" them.
One day, while in University, someone explained Radians to me much as this gif does. It took one sentence, and something I'd been working with for years without understanding immediately became clear.
Why don't schools/textbooks offer this? I have no idea. But it's a shame.
Maybe somebody told me and I didn't take it in.
Shoddy math teaching was par for the course where I grew up.
I learned all the rules of logarithms in 10th grade, but didn't actually know what a logarithm was until chemistry in college. They just.. didn't really tell us. And how often are integrals explained as "the area under a curve", which is a definition that makes absolutely no sense?
That's just a cop out. We all had teachers and subjects that didn't connect, for whatever reason. Ultimately, the responsibility to learn is personal.
I don't mean to be harsh, but basically every comment on this subthread is someone saying "I was a great math student, but I never understood what radians MEANT until now!" That's crap. If you do 10,000 practice problems and get high marks and still don't understand the concept, you haven't done anything. Good teacher or bad, your job as a student is always, forever, to figure it out on your own.
If "teaching" was just a matter of reading and completing exercises then we could replace teachers with scantron machines. Yet college students pay teachers thousands of dollars per course to ensure that they receive more value than that. If they don't get it then they are being mis-served and cheated.
That's an excellent way to externalize personal failure.
A teacher cannot teach a student who doesn't make an effort to learn. Full stop. But it's a symptom of our ever-increasing sense of entitlement that we assume that our role as students is that of the audience for a play. We intend to lie back, and let the education wash over us. If it takes effort to learn, they must not be teaching correctly!
Let me flip around your supposition: if the people in this thread -- all of whom were apparently high-achieving math students -- cannot manage to understand a core principal of elementary trigonometry that is stated clearly in every textbook, the problem is more likely to be with the unwillingness of the students to think independently than the failures of a diverse group of teachers from all over the world to shove the knowledge into their head.
It's far easier to blame the teacher than to do the work of learning.
Quite so. But a poor teacher can do quite a lot to impair a student who does make an effort to learn, and, at least in the public schools of the United States, there is little meaningful incentive for a teacher to be other than poor. Not to say that all US public school teachers are poor, of course; there are at least a few really excellent ones, or were when I was a student. Excellence, though, is anything but a requirement.
I'd also argue that a really good teacher can elicit from her students the sort of effort you describe; I've seen it happen -- indeed, most of my basic facility with the English language, I gained under the tutelage of precisely such a teacher. That's a more complicated proposition, of course, but I start to wonder whether or not it's at the core of the current argument.
It's not hard, or so the US public school system would have us believe, to teach someone enough of a given subject that she can qualify to teach it herself. But is that the essential feature of pedagogy? Or is it instead to lead a class full of students, in such fashion as to overcome the basic distaste, for complex and difficult learning without immediate reward, which is part of the human heritage?
The problem here is that it's so common for students to be able to glide through course after course using rote memorization and a willingness to do busywork without attaining understanding and in so doing not only passing courses but also getting high grades. And that's a failure of student and teacher alike, but it's a failure that the students have been trained for and guided to by the system.
Again, the problem is that schools do not reward or generally even bother assessing learning. And certainly part of the fault for that lies on the student, but given that they are being passed through the system and actively rewarded for not learning I think the bulk of the blame lies on teachers and institutions. If the value of an institutional education is to provide an opportunity for auto-didactism and whether one takes that opportunity or not is not reflected in grades or degrees granted then I have to question the value of such education.
No. College students pay colleges and universities thousands of dollars per course. There's a difference.
Should an education concentrate on short term knowledge of trivia that fades away rapidly or on long-lasting understanding of fundamental concepts? Is it even remotely reasonable to charge tens of thousands of dollars to students for something that is only temporary?
This is a 30 second animation that could be shown to a junior high school class room and most would get some of the idea (if not all).
The real beauty of this is that it could just as well be broadcast to the stars as a quick primer in linking earth based, fundamental mathematical symbols with universal constants.
Things like this were often the downfall of math classes: Yeah, everything was explained, somehow, but if you didn’t really pay attention that one time it was easy to miss the subtleties. If something is then additionally not frequently used it’s easy to get by not really understanding it (and maybe only knowing that radians are an alternative way to represent degrees, where 2π are 360°).
It’s not that hard to understand, but if you don’t need to understand it to succeed, why bother if you aren’t inherently interested in the topic anyway? (Ok, the percentage of people who are inherently interested in math is probably higher on this website than elsewhere. Or at least of people who needed to work with radians to the degree that they had to really understand them.)
This being 'Hacker' News, I think this kind of material is an awesome way to 'hack' the 'understanding' of concepts and way better than spouting convoluted terminology used by textbooks and teachers who care shit about making sure students 'get' the concept rather than just show how smart they are and how dumb all those are who don't just memorize and regurgitate whatever they say by taking it on face value as some gospel truth. Not everyone sees (literally!) the world the way you do and just because they cannot see what you see does not make them any less smart or incapable than you (and that road goes both ways). If there is a way to enable others see the world your way, it empowers them to bring their abilities to solve such problems and enriches the world by tapping into a previously untapped resource.
I say, bring more such hacks to HN.
Just to give an anecdote of the inverse.
but you are... there's nothing wrong on trying to make concepts more understandable.
How? I simply asked whether people had trouble with this material because I don't remember it being an issue with any of my classmates in high school or in my 24 hours of undergraduate math.
Saying "anyone who doesn't understand this is an idiot" would be condescending and dismissive, but I fail to see how asking an innocuous question is equivalent in your mind.
Practical: take a 15cm ruler and drill small holes at each end. Thread a 1.2m ish loop of string through the holes so there is a radius of 57cm. Tie the string together about 20cm from the ruler.
The result lets you measure angles on the horizon, e.g. width of Moon (not Sun with children please!) and the altitude of navigation stars near rise and set.
I don't think it's worth actually changing things, but if we can accept that 2pi is a more natural constant than pi, then it should help us write more sensible formulas, even if the symbol tau isn't used.
Who was it that said something along the lines of " If you can describe a complex thing to someone briefly in simple terms that shows knowledge" Feynman? Einstein?
Whoever said that I find it profound because most people can learn something eventually but to know something, knowledge, I find close to impossible, very hard.
Perhaps this one? I pulled it from memory, even though I couldn't remember the author. Google attributes it to Einstien. But the interwebs misattributes quotes all the time, so take it with a grain of salt.
edit: I still think the most confusing thing about radians when learning them is the 2 pi. This seems so arbitrary.
We now use Tau at work and it makes everything easier to understand.
The early use of pi was calculating area and it was derived based on that. At least as far as history can tell us. Records are a little sketchy that far back.
edit: forgot that * would make everything italicized.
Sphere surface area: 4pi * r^2
Sphere volume: (4/3) * pi * r^3
Both would still have a coefficient if tau replaced pi...
Incidentally, I recently read an article about how the creators of The Simpsons actually held math degrees. They hid mathematical easter-eggs all over their episodes. One gag was a "solution" to Fermat's last theorem where 1782^12 + 1841^12 = 1922^12. It's valid up to like, the twelfth significant digit.
The only part that doesn't make sense.
Wouldn't surprise me if the minutes and seconds relates to navigation time in the past.
This being HN, how about someone whipping up a Web application where teachers can post 'want ads' for small things like this and creative visual people can respond by posting links to bits of work? For the lutz and to help?
radians = arc length in a circle with R=1
I love when something is explained in such an excellent simple way and was a good refresher for me. This is how things should be taught.
so it's the .141... part