I am a professor at a university. Unbelievably, all the classes offered by our math department (with roughly 100 different teachers) are scheduled by hand.
This takes a tremendous effort, and then when one class gets cancelled and one professor has to be reassigned to a different class, but she can only teach on Tuesday/Thursdays, it sets off a giant stack of dominoes...
Seriously, the amount and sheer drudgery of thankless admin work in my department makes me want to cultivate a reputation for disorganization and irresponsibility.
But will your university pay for a better solution? Most small universities, public or private, won't shell out the cash to solve that particular problem. Same thing goes with small hospitals, high schools and probably dozens of other public and private institutions.
I wouldn't trust my university. Their IT department has the collective IQ of a toad.
But individual departments have their own budget. If one faculty member says "I'll do the scheduling if you pay $500 for this software", and everyone else says "I'm too busy", then that $500's gonna get spent.
That's fascinating. The company behind it has a web site. They had this working in 2002. Their web site is so outdated that it uses RealVideo and QuickTime. Their standard system costs $90,000. Their web site was last updated in 2004.
The name "AutoAuditorium" is licensed from Telcordia, which used to be Bellcore, which used to own Bell Labs. So this may be a Bell Labs spinoff that was commercialized, but not very well.
Somebody in the online education business should buy this technology and modernize it, so it costs about $9000, or $900, instead of $90,000.
Have a look at https://en.wikipedia.org/wiki/Basel_problem . I'm not a mathematician, and I'm probably wrong, but I think it has to do with infinite series that "circle" around a value, and asymptotically approach a value, only accurately expressed in terms of pi.
"Euler's original derivation of the value π2/6 essentially extended observations about finite polynomials and assumed that these same properties hold true for infinite series."
Sorry, not very convincing or geometric, and I'm sure someone else can provide a better answer, but that's how I visualize it.
It would take an extraordinary effort. I doubt there is anyone who could learn this material given anything less than a year of solid study -- and I am thinking of the leading researchers in the world, who are already experts in everything related. I am a professional mathematician myself, and I doubt I could manage it even within a year.
This is dictated by the difficulty of the subject, and it is even more so since Mochizuki's papers are notoriously difficult to read. What I have heard from other research mathematicians is that Mochizuki doesn't make much of an effort to make himself comprehensible, to answer questions, or in general to explain his results to the community.
Also please be warned that the consensus is generally that Mochizuki's theory is probably incorrect.
That said, if you want to learn the stuff, there is no reference other than Mochizuki's papers themselves. If you would like to learn some general background theory I would recommend learning some algebraic geometry, for which see Ravi Vakil's book:
If you find that incomprehensible, start with Dummit and Foote's Abstract Algebra (read it cover to cover), Neukirch's Algebraic Number Theory, Atiyah and Macdonald's Commutative Algebra, and Fulton and Harris's Representation Theory.
If you indeed want to read all that, you may as well enroll in a Ph.D. program in math....
After reading many articles, posts, communication channels, etc. on the Internet about IUT Theory of Shinichi Mochizuki, I think I am starting to understand why some people believe that the theory doesn't prove the ABC conjecture. It can be summed up by a phrase I found on some blog: "I believe in a proof if I understand it.". I have done some research, and came upon the source of that idea:
Pierre Deligne was first to express: "I don't believe in a proof done by a computer ... I believe in a proof if I understand it."
[George G. Szpiro, Kepler's Conjecture, John Wiley, 2003. p.21]
"I don't believe in a proof done by a computer, says Pierre Deligne of the Institute for Advanced Study, an algebraic geometer and 1978 Fields Medalist. “In a way, I am very egocentric. I believe in a proof if I understand it, if it’s clear.” While recognizing that humans can make mistakes, he adds: “A computer will also make mistakes, but they are much more diffcult to find."
"the consensus is generally that Mochizuki's theory is probably incorrect", "I am a professional mathematician"
- these two claims contradict each other: no professional mathematician would make such false claims
>no professional mathematician would make such false claims
Well, that's obviously incorrect, because the previous commenter did in fact make those claims and is in fact a professional mathematician. It's not like you can make some sort of no true scotsman argument about his profession: http://people.math.sc.edu/thornef/ .
The consensus is that Mochizuki's theory, at least insofar as it claims to prove ABC, is probably incorrect. (Otherwise this workshop would be a lot more popular.)
But not definitely incorrect. Mochizuki is certainly not an obvious crank. His theory may be right, perhaps only partially -- in which case it will be enormously consequential. So it's still worth paying attention to. (In Minhyong Kim's judgment, which carries a hell of a lot more weight than mine.)
I do stand corrected a little bit: I see that a couple other reputable mathematicians have already invested some time digesting his work, and further that Mochizuki has decided to be more open to answering questions.
> Chalk is on it's way out. It's a rapidly shrinking market.
I am a math professor. Most of us prefer chalk, and have no desire to go to whiteboards, smart boards, Powerpoints, what have you. I think that there is a fairly durable market for chalk.
Moreover, over the last five years I think that awareness of Hagoromo (and more generally, of high-end chalk) has been growing. You see it for the first time when visiting some other university and think "Wow, what is this? I must have some!" This led me to buy several boxes of it at $35 a pop. My out-of-town guests, who give lectures in the research seminar I help to run, get treated to the good stuff.
As a student, I preferred chalkboards to whiteboards because they were easier to read. It seems that most of the time, unless the professor had an exceptionally well working dry-erase marker, the lines were too thin and light for me to read them, no matter the color.
I remember thinking that the whiteboards were an improvement at first, but that quickly changed when I realized I couldn't read what was written on them anymore.
Unfortunately there simply aren't very many math professors. An average American school district has 200 teachers, all of whom would have used chalk until recently, that's a lot more than there are professors in a university math department and there are many more school districts than universities.
I agree that there is a durable market for chalk, but not at this kind of volume.
> How do you know if the proof is valid, or if they're just scratching each others' backs and pretending they're all brilliant?
(Professional mathematician here)
Such a conspiracy could never happen. Mathematicians are, in general, scrupulously honest to a fault. There are exceptions of course, but such a scam would need everybody's cooperation.
Moreover, if you have tenure then you would have no incentive to participate in such a fraud (even to the extent of keeping quiet about it). Once you get tenure you are basically working for pride and for the sheer joy of solving problems, and maintaining a big lie would do nothing for either.
I would like to both broaden and qualify that statement:
scientists are, in general, scrupulously honest where it concerns their work, probably because their work is about 'truth'. Nevertheless, scientists are still people and many people lie and cheat when it suits them.
> Trig is a bunch of arbitrary formulas if you don't have the calculus behind them.
Why do you claim this? You don't have to have seen calculus to appreciate how trig functions are defined, how to manipulate them, or how to use them in applications.
As a math professor, I personally like the fact that we teach trig and exponential/logarithmic functions before calculus. They are (as you well know) exceedingly rich examples which illustrate why calculus is interesting and useful, and knowing them already enables the student to study calculus without excessive digressions.
I don't know, I agree with the parent. I hated trig in my first encounters with it; the thing that made it practical was writing videogames and graphics demos, and the thing that made it interesting was calculus. For many people, it's the last math they're taught, they don't get the applications, and it's no wonder they don't hunger for more.
How did you get that information? How did you, in a non-magical way, go from information about an angle to information about a ratio?
If students don't know this information, then perhaps they are studying applications. So, what applications are students taught in typical trigonometric texts? Periodic behavior perhaps? Like sound? Only perhaps a brief blurb in the text that application is even possible. Perhaps they look at something about an incline plane. It is unlikely that they will touch projectiles.
It appears that trigonometry is there to give students some sense of mild comfort for future work in physics or engineering. This makes me think, "Why not statistics instead?"
> How did you, in a non-magical way, go from information about an angle to information about a ratio?
By having a right triangle?
The rest of your post seems to show that you want trig to be about periodic behavior, when it really is about triangles. That's what trigonometry means - measuring triangles.
Yes, trig has applications to periodic behavior, projectiles, differential equations, inclined planes, and all kinds of other stuff. But the point of a trig class is not to teach the applications. The point is to teach the tools, and maybe touch on the applications.
The problem is this. Using compass and ruler constructions there is a set of angles you can construct, and you can calculate sin and cos for those angles. You can even write the values for those out explicitly. However no part of this construction sheds light on how to find sin and cos for angles that you don't know how to construct. Or even gives good intuition that no matter how you do it, you can define it in a way that makes sense for all angles.
In fact we draw a picture, people look at it, and their intuition tells them that things will work out. Very few students will notice the logical gaps.
But to close the logical gaps, you need to start with Calculus first, and then derive trig formulas from that.
(Yes, I'm aware of the history here. Euclid presented trig reasonably rigorously a very long time before Calculus. Newton invented Calculus in the 1600s, and then used it as a heuristic to figure out answers that he then rederived using trig in The Principia. Leibniz reinvented Calculus in part based on inspiration from Newton's work. None of this was made formally correct until the late 1800s.)
(I have no opinion on pedagogical arguments about which is best to present first. I believe that we present trig first as a holdover from a curriculum where The Elements was the standard textbook until very recently.)
Well... you can use the half-angle formulas and the angle addition formulas to calculate sin and cos for angles that are arbitrarily close to the ones that you want. Add to that the idea that sin and cos must be continuous (I consider that intuitively obvious from a unit circle, but I don't know how to make that argument rigorous), and you can start to interpolate. You can in fact use these methods to calculate sin and cos for an arbitrary angle to any desired degree of precision... if you have the patience. It will be shorter to use the series derived from calculus, I'll admit.
I do not believe that there is an argument for continuity without starting with Calculus. Certainly starting from ruler and compass constructions it is not obvious.
That said, if you have enough Calculus to define how to measure the arclength of a segment of the circle, you can quickly prove that sin and cos in radians exist, have a nice power series, and so on.
It is like x^y with x positive. We can manually define it every rational y. But the easiest way to get a rigorous and straightforward definition is to prove the algebraic properties of the integral of 1/x, use that to define the logarithm, define its inverse function to be the exponential, prove its algebraic properties, then define x^y as e^(y*log(x)). And it all just works.
It's easy to show more or less directly (by comparing arclength to straight-line length), and certainly without calculus, that the absolute difference between sin(x + delta) and sin(x) is at most |delta|.
Replying really late, just in case anybody reads this. (I went on vacation, and this occurred to me then.)
If you don't have calculus, you don't have anything like a delta-epsilon proof of continuity. But without calculus, you also don't know that you need it. So you just assume (correctly) that you can interpolate, and it works just like you expect, and life goes on.
As young students, we are taught that these functions are ratios of sides on right angles. It is fairly intuitive for them that provided they can draw a right triangle, they just need to measure the sides. And it is easy to accept that someone computed sines and cosines for a lot of angles and put it in a table.
My point being, that we didn't need (in the past) calculus to compute trigonometric functions, and I don't see how it is a burden for students to be introduced to those functions without the Calculus definition.
And the reality is, that the definition of sine as a ratio of the catheti and hypotenuse is a rigorous definition of the function. Strictly, this sine is different from the sine of calculus. The first, the sine from Euclidean geometry, assigns a real to pair of rays, while the calculus sine, is function from the real numbers to the reals. And it does take some work to link them formally.
What kind of pedagogical or pragmatic relevance do you see trig as a building block for? I would answer that question by saying that it most likely comes up again either in physics or engineering contexts, or in a standardized exam like MCAT. And only in the sense of familiarity with the unit circle and trig functions.
What other foundation or learning pathway do you see trig serving as? Somebody else mentioned that trig serves use by teaching students that calculus has rich applications. So then I question, what kind of applications are students learning in trig? And if students are to learn rich examples of calculus applications, then why not statistics, which is also relevant to the bio / social sciences? Also, couldn't we mash trig inside calculus?
It seems to me that if I know trig, I can use it to solve the set of geometric problems to which it is applicable, without having been taught any special applications of trig. That's valuable in itself.
Then I take physics, and I find a whole bunch of other applications. I take calculus, and I find a bunch more uses. I take mechanics, and I find a bunch more. But it is not the job of trig to teach me those applications (though hints would be useful). It's not trig's job to teach me physics - that's a job for physics. But I need trig as a foundation.
I'm not sure that I answered your question, though...
It is about triangles. The idea that sine is the solution of this ODE for f(0)=0 and f'(0)=1 is quite modern.
I would say a course on trigonometry usually covers (my experience):
exact value of them for the angles 30, 45, 60, 90 ... degrees
Trigonometric formulas for the sum and difference of angles. A formuka for the double and the half angle.
Law of sine and law of cosine
Lots of relations derived from the Pythagoras theorem (sin^2+cos^=1)
how to solve trigonometric equations
With all this, you are equipped to completely determine a triangle, knowing some of its and the length of some its sides. As as application, I was taught, how to measure heights and distances provided you can measure angles.
Thus, without trigonometry, it would be fairly hard to take a course on analytic geometry.
Now, how would the course be enhanced by introducing sine as the solution of an ODE?
I think I am missing something because, I am unable to see why it is huge burden to introduce sine and cosine without their rigorous definition. At which age, are students taught trigonometry? And what does a course on trigonometry covers? What would you think they would be able to do without it?
When we were introduced the sine and the cosine function, we were already familiar with Thales theorem, so therefore we could show that this ratio was a constant.
I am quite sure historically as well sine and cosine predate the more formal construction of those functions, be it as a series, solution of an ODE or inverse of arc sin (and this defined as an integral)...
I think I wasn't clear in saying that I believe the current pedagogical value of trigonometry is in giving students a brief familiarity with the trig functions when they see it again in the context of physics or engineering. Or standardized testing. I think those are the likely scenarios where students are going to be seeing relevance in trigonometry.
What other foundation or learning pathway do you see trigonometry serving as? Somebody else mentioned that it gives students a sense of applications, so they know that calculus is not for nothing. So then I question: what applications? And I pose, how about statistics?
It may be because of your math education but I was introduced to trigonometry in junior high in China. I had, and my classmates had, no trouble understanding them as functions of angles coming from ratios. It is the glue that binds circles and triangles and squares and ... . By high school analytic geometry greatly expanded their scope and use. This is the problem I saw in American high school when I moved to US: shallow introduction to mathematical topics made them vapid and jejune. Ancient Greeks were enthralled by trigonometry, ancient Egyptians built Pyramids and ancient Chinese built great dams with trigonometry. Calling it practically useless and pedagogically only useful as a prep for calculus is going too far.
My introduction was through model rocketry. Somewhere along the line, Al-Biruni's method for finding the radius of the earth was brought up. That was in the early '70s, in what would be "junior high" in the US (elementary school in my part of Canada). It was always about right triangles, not periodic functions, at the beginning, which makes a whole lot of sense - trigonometry was both useful and used for a whole lot of years before calculus was invented. And like logarithms in the pre-scientific-calculator days, there was a point where one turned to tables for practical reasons without thinking of the table values as "magicical" - we were taught how to calculate intermediate values to the limits of practicality. Is there any practical sense (a sense that would be useful for people who would be entering the trades track) in being able to calculate much more accurately than you can measure angles?