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Not to be dismissive or condescending, but do a lot of people have trouble understanding radians? I was under the impression that they were standard material in

Not to be dismissive or condescending, but do a lot of people have trouble understanding radians? I was under the impression that they were standard material in high school trig/physics.

javert 13y ago
I got through calc in high school and college and I really wish somebody had told me that radians were based on the radius like this. I had no idea. Maybe somebody told me and I didn't take it in. Shoddy math teaching was par for the course where I grew up.
- robryan 13y ago
  Yeah, you could easily get though by knowing the method of calculation without ever having understood a model of what they actually are.
- judk 13y ago
  It wasn't a guarded secret... "Radian" is almost the same word as "radius", and 2pi * radius = circumference = 2pi radians.
  - Jtsummers 13y ago
    But unless you take a moment to think about it, that connection isn't so obvious. And in HS, unless math is your interest (and even if it is) there are hundreds of other things running through your brain that are probably more interesting than pondering the relation between the words "radian" and "radius". There are also many other words that people use frequently that they don't realize are connected until much later. And as others have said, HS math (in the US) often presents them as: radians = degrees * pi/180. The calculation has zero relation to the radius of the circle, as presented.
deevus 13y ago
When I was in high school in the late 90's/00's in Australia we were taught in degrees. It wasn't until university that I relearned in radians.
- arrrg 13y ago
  Yeah, I think for many students radians are that thing you have to turn off if you want your calculator to show the correct result. 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.)
- Khaine 13y ago
  I learnt radians in Australia at high school in the same time period.
InclinedPlane 13y ago
Do you want to know a disturbing secret? Most people don't understand the fundamentals of most of the subjects they have studied. People get by because the perhaps even more disturbing fact is that cargo cultism often actually works in practice. A lot of the time you can get away with just going through the motions.
- carlosaguayo 13y ago
  And I believe it's a problem of the education system. Back in high school they had me solve dozens, if not hundreds, of derivates and integrals, it was not until college that a fantasic teacher had the guts to come up with his own syllabus, threw away all the "traditional" calculus and taught us what was a diffeeential, what were we doing when getting a derivative and an integral, completely changed my mind.
  - timr 13y ago
    "And I believe it's a problem of the education system." 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.
    - InclinedPlane 13y ago
      The responsibility of a teacher is to instill and assess knowledge. Repetition of busywork does not require a teacher, it only requires a textbook. The purpose of having a teacher in the loop is to ensure that students are actually understanding the material, which is why they are graded. If a student can get through a class without acquiring any degree of competency or basic understanding of fundamentals that's a failure on the teacher's part. 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.
nsxwolf 13y ago
I just remember radians being taught as a "thing", an alternative to degrees. We learned how to convert between the two and that was that. Seeing this animation, and seeing that radius bend to wrap itself around part of the circle was pretty breathtaking for me and now I must study it more.
nlew 13y ago
High school math is often taught as simple facts, without proper explanation of the significance or how it's derived. It's entirely possible to learn how radians correspond to degrees and what you can do with them without understanding where they come from or what they mean. 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?
- anonymousDan 13y ago
  Can you elaborate on your understanding of logarithms and integrals? Curious to know if it tallies with mine!
reuven 13y ago
I got excellent grades in high-school math. I took a year of math in college. I knew how to calculate with radians, but never understood their basis until viewing this animation, despite asking people on a number of occasions what the benefits were vs. degrees. (They never gave me a real answer to that question, and certainly never derived the basis for radians.) 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...
- derekp7 13y ago
  So here's a question -- is it just the animation that helped, or is it the combination of the animation and being 20 years older? Or, more specifically, having the luxury of being able to throw brain power at the subject without having to worry about learning material for several other subjects, and many sub-topics within all those subjects, all at the same time.
  - reuven 13y ago
    I'm sure that it all helped. But I definitely remember asking, on several occasions, why we would ever want to use radians rather than degrees, and why they seemed to be in such weird increments. And no one ever really told me. The animation was helpful in that it explained everything very quickly and obviously, obviating the need for a numeric or verbal explanation.
- dbbolton 13y ago
  I believe when we were first introduced to the radian, it was explicitly defined in terms of the radius, i.e. 1 radian = an angle whose arc length is equal to the radius of the circle. Was a radian just defined as an angle of 57.3 degrees in your class? I imagine that would seem really arbitrary to students.
- thaumasiotes 13y ago
  > I knew how to calculate with radians, but never understood their basis until viewing this animation, despite asking people on a number of occasions what the benefits were vs. degrees. (They never gave me a real answer to that question, and certainly never derived the basis for radians.) 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.
edanm 13y ago
Yes. 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.
- Someone 13y ago
  Have you checked whether they do? In my experience, textbooks often contain more information than you remember they do. Also, for the radians thing, how else would you introduce them? As a magic scaling factor, in the same vein as explaining the difference between degrees and grads?
k-mcgrady 13y ago
I studied Maths to the second highest level available in high school (so higher than the majority of students) and although I used e.g. 2PiR all the time this is the first I've actually understood how it works. It isn't something we were taught. We were taught how to get the answer, not why/how that is the correct way to get the answer.
Intermernet 13y ago
New ways of explaining things should always be welcome 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.
ipedrazas 13y ago
"Not to be dismissive or condescending" but you are... there's nothing wrong on trying to make concepts more understandable.
- dbbolton 13y ago
  >but you are 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.
bberrry 13y ago
It actually is quite condescending to think that people upvoting this actually have trouble understanding the concept. It's an elegant animation and young students would benefit from seeing it.
runarberg 13y ago
I didn't do trig until after I had finished BS in psychology. Rather I learned it from Khan academy, Wikipedia, and the like. During my second BS (this time in math) a colleague was amazed at my inside and intuition to trig, even though he had taken as much math prior to Uni as he could, and I hadn't. Just to give an anecdote of the inverse.
shiven 13y ago
These animations make more sense than any math teacher or textbook ever did in my entire life. There, a data point of one. 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.

Not to be dismissive or condescending, but do a lot of people have trouble understanding radians? I was under the impression that they were standard material in high school trig/physics.

javert 13y ago

I got through calc in high school and college and I really wish somebody had told me that radians were based on the radius like this. I had no idea. Maybe somebody told me and I didn't take it in. Shoddy math teaching was par for the course where I grew up.

robryan 13y ago
Yeah, you could easily get though by knowing the method of calculation without ever having understood a model of what they actually are.
judk 13y ago
It wasn't a guarded secret... "Radian" is almost the same word as "radius", and 2pi * radius = circumference = 2pi radians.
- Jtsummers 13y ago
  But unless you take a moment to think about it, that connection isn't so obvious. And in HS, unless math is your interest (and even if it is) there are hundreds of other things running through your brain that are probably more interesting than pondering the relation between the words "radian" and "radius". There are also many other words that people use frequently that they don't realize are connected until much later. And as others have said, HS math (in the US) often presents them as: radians = degrees * pi/180. The calculation has zero relation to the radius of the circle, as presented.

deevus 13y ago

When I was in high school in the late 90's/00's in Australia we were taught in degrees. It wasn't until university that I relearned in radians.

arrrg 13y ago
Yeah, I think for many students radians are that thing you have to turn off if you want your calculator to show the correct result. 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.)
Khaine 13y ago
I learnt radians in Australia at high school in the same time period.

InclinedPlane 13y ago

Do you want to know a disturbing secret? Most people don't understand the fundamentals of most of the subjects they have studied. People get by because the perhaps even more disturbing fact is that cargo cultism often actually works in practice. A lot of the time you can get away with just going through the motions.

carlosaguayo 13y ago
And I believe it's a problem of the education system. Back in high school they had me solve dozens, if not hundreds, of derivates and integrals, it was not until college that a fantasic teacher had the guts to come up with his own syllabus, threw away all the "traditional" calculus and taught us what was a diffeeential, what were we doing when getting a derivative and an integral, completely changed my mind.
- timr 13y ago
  "And I believe it's a problem of the education system." 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.
  - InclinedPlane 13y ago
    The responsibility of a teacher is to instill and assess knowledge. Repetition of busywork does not require a teacher, it only requires a textbook. The purpose of having a teacher in the loop is to ensure that students are actually understanding the material, which is why they are graded. If a student can get through a class without acquiring any degree of competency or basic understanding of fundamentals that's a failure on the teacher's part. 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.

nsxwolf 13y ago

I just remember radians being taught as a "thing", an alternative to degrees. We learned how to convert between the two and that was that. Seeing this animation, and seeing that radius bend to wrap itself around part of the circle was pretty breathtaking for me and now I must study it more.

nlew 13y ago

High school math is often taught as simple facts, without proper explanation of the significance or how it's derived. It's entirely possible to learn how radians correspond to degrees and what you can do with them without understanding where they come from or what they mean. 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?

anonymousDan 13y ago
Can you elaborate on your understanding of logarithms and integrals? Curious to know if it tallies with mine!

reuven 13y ago

I got excellent grades in high-school math. I took a year of math in college. I knew how to calculate with radians, but never understood their basis until viewing this animation, despite asking people on a number of occasions what the benefits were vs. degrees. (They never gave me a real answer to that question, and certainly never derived the basis for radians.) 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...

derekp7 13y ago
So here's a question -- is it just the animation that helped, or is it the combination of the animation and being 20 years older? Or, more specifically, having the luxury of being able to throw brain power at the subject without having to worry about learning material for several other subjects, and many sub-topics within all those subjects, all at the same time.
- reuven 13y ago
  I'm sure that it all helped. But I definitely remember asking, on several occasions, why we would ever want to use radians rather than degrees, and why they seemed to be in such weird increments. And no one ever really told me. The animation was helpful in that it explained everything very quickly and obviously, obviating the need for a numeric or verbal explanation.
dbbolton 13y ago
I believe when we were first introduced to the radian, it was explicitly defined in terms of the radius, i.e. 1 radian = an angle whose arc length is equal to the radius of the circle. Was a radian just defined as an angle of 57.3 degrees in your class? I imagine that would seem really arbitrary to students.
thaumasiotes 13y ago
> I knew how to calculate with radians, but never understood their basis until viewing this animation, despite asking people on a number of occasions what the benefits were vs. degrees. (They never gave me a real answer to that question, and certainly never derived the basis for radians.) 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.

edanm 13y ago

Yes. 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.

Someone 13y ago
Have you checked whether they do? In my experience, textbooks often contain more information than you remember they do. Also, for the radians thing, how else would you introduce them? As a magic scaling factor, in the same vein as explaining the difference between degrees and grads?

k-mcgrady 13y ago

I studied Maths to the second highest level available in high school (so higher than the majority of students) and although I used e.g. 2PiR all the time this is the first I've actually understood how it works. It isn't something we were taught. We were taught how to get the answer, not why/how that is the correct way to get the answer.

Intermernet 13y ago

New ways of explaining things should always be welcome 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.

ipedrazas 13y ago

"Not to be dismissive or condescending" but you are... there's nothing wrong on trying to make concepts more understandable.

dbbolton 13y ago
>but you are 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.

bberrry 13y ago

It actually is quite condescending to think that people upvoting this actually have trouble understanding the concept. It's an elegant animation and young students would benefit from seeing it.

runarberg 13y ago

I didn't do trig until after I had finished BS in psychology. Rather I learned it from Khan academy, Wikipedia, and the like. During my second BS (this time in math) a colleague was amazed at my inside and intuition to trig, even though he had taken as much math prior to Uni as he could, and I hadn't. Just to give an anecdote of the inverse.

shiven 13y ago

These animations make more sense than any math teacher or textbook ever did in my entire life. There, a data point of one. 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.