You don't understand something until you think it's obvious.
mebassett.blogspot.com
mebassett.blogspot.com
My friend was TAing a freshman calculus class in college while working on his PhD. He clearly remembers that his (and everyone's) biggest problem with calculus was limits. Hands-down, limits were the most frustrating and un-intuitive part of the class. So he sat down to try to figure out a good, intuitive explanation of limits that would help his students avoid the frustration that he had.
He immediately ran into the problem that they weren't hard to understand. They were clear and intuitive. He could not for the life of him remember why he thought they were difficult when he was first learning them. He asked me for help, and I had the same response. I remember clearly banging my head against them and screaming in rage and frustration at those goddamn limits. But I look at them now, and I can't figure out how I could fail to understand something so obvious.
And thus, the cycle of pain continues, with limits remaining the eternal bane of freshman calculus students.
Sal's "Introduction to Limits": http://www.khanacademy.org/video/introduction-to-limits
Explaining the end result -- the epiphany -- to someone else is often difficult because most will have gaps in their understanding that prevents them from seeing it they way you do and therefore they will have limited ability to verify and understand the significance of what you what you are saying.
Looking back at issues I've had, learning it right the first time has been very important. If I've skimmed over some concept, and learned enough to apply it simply, when it comes to applying it more generally, I'm struggling. Going back and relearning it correctly takes time, and generally has to be done more than once.
Teachers should be careful to explain things simply, but ensure that the understanding really is correct. This can't really be assessed by exercise sheets, but discussion.
(Well it can by exercise sheets, by asking students to apply the concept to a different problem, but personal experience is only a few people make the jump without help. And when they get the help, that short circuits the learning that should have happened to allow them to do it themselves.)
A (good) teacher looks at a problem from as many aspects as possible and works with students to understand them. A (good) teacher collects multiple ways to understand a concept, so if one doesn't sink in, the next may. It's not an inborn skill; it takes practice.
The best way I've ever seen limits explained, incidentally, is taking a student 10 feet away from a wall. Ask the student to go half way to the wall. Then again. Then again. Eventually, the student is taking centimeter steps. Ask the student if he's at the wall. The student says yes or no, but usually, there is an "aha!" moment, the idea being that you can figure out where "there" is without ever technically being "there."
Understanding the basics of the derivative was the first place I banged my head, but that was because I didn't adequately understand the secant. Once I understood that, the derivative fell into place. (In fact, calculus was where I realized how much I didn't understand trigonometry.)
One flip and you can have 1or 0. Three times it's 1,2/3,1/3,0. Five times it's 1, 4/5,3/5,2/5,1/5,0 As long as you do odds you will never see 1/2 as a option. However, if you divide ever larger odd number of flips you are going to see numbers really really close to 1/2 without ever really getting there.
(Teaching a group of N pupils is hit and miss, since a teacher can't go in N different directions at once. Instead he explains something in different ways until the common misconceptions have been covered. He doesn't know what these are until he has taught many classes on the topic.)
I also think it relates to why teachers cannot reach a great depth in any subject they teach. Their students "hold" them back.
1. We introduce limits. Which are blindingly artificial, and so it is hard for us to form a concept about why people want this.
2. We introduce the definition of the derivative. Which instantly results in a practical reason for limits. But typically we make the mistake of introducing the derivative as a function, which means considering the limit of a lot of things at a lot of points.
There really needs to be an intermediate step, which is to introduce the definition of the tangent line. And how to calculate it. Then work up how to calculate tangent lines of various combinations of functions at a single point. And after the person is well and truly comfortable with that, then introduce the derivative.
I don't see limits as artificial as all. For example, Zeno's Paradox[1] is resolved thanks to the limit of a succession. And the concept of the infinitesimal, which was something I had thought about before I learnt about limits, is another rather intuitive thing that is explained by limits.
Derivatives are only one of the most common uses of limits, they are definetely not the only reason they are useful.
[1]http://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_and...
The more you learn, the more you come to realize that the nature of the real numbers is deeply mysterious and not to be treated lightly. That's not to say you cannot convey many useful intuitions to students without the somewhat abstruse formal machinery.
That's more or less what I was trying to say. There are some deeply intuitive concepts for which limits are, at the very least, a very good aproximation. It may occur in some cases that the actual case and the intuition that one has about it doesn't match, but that doesn't mean that the model is arbitrary or artificial.
People understood the solution to Zeno's paradox centuries before we had a rigorous definition of limit. And limiting processes were understood both before, and after, without necessarily referring to limits. (In fact in computer science people prefer the more flexible Big-O/little-o notation for that purpose. Knuth has in fact suggested replacing limits in introductory Calculus with that notation and I'd be quite interested to see how well that would work.)
The idea of infinitesmals were thought up well before limits. In fact the idea of a limit came out of Cauchy's work. Cauchy tried to define an infinitesmal as a function that went to 0. (His idea of an infinitesmal became what we now call a Cauchy sequence.) When limits came they did not explain infinitesmals - they replaced them. It was not until Abraham Robinson came up with non-standard analysis in the 1950s that anyone came up with a sound explanation of infinitesmals and properly explained the connection with limits.
And finally I was trying to explain what things are like from the perspective of the student.)
In particular the idea of approximation seems inherently useful. All approximations take the form something = approximation + error, so the idea of having a useful language for the error seems useful. Once you have that language, the tangent line is just a particularly useful example. Only after working with tangent lines for a while would you get to the definition of the derivative.
The nicest thing about the approach is that all of the derivative rules are easy to remember for tangent lines. Product rule? Multiply the tangent line approximations, throw away the high order term. The chain rule? Plug one function into the other. And so on.
The only thing that I found did not flow naturally was L'Hopital's Rule. However that rule is a source of confusion for students who don't understand the importance of the precondition that the limits be of the form 0/0 or infinity/infinity. And furthermore all standard cases where it is useful are obvious upon inspection for anyone who knows Big-O/little-o notation. (Don't believe me? Crack open a textbook and look at the problems that you're supposed to use the rule for. They are all obvious.)
I lack the resources to make or find such a graph myself, so please let me know if something like this exists, or if you intend to create one. My parents are both teachers, and I'm sure they'd benefit from it.
I think the brain must do some sort of cross training - learn python and improve your C++, so that when I came back to limits they seemed intuitive and not as difficult as I remembered.
Back in vanilla WoW, when Naxxramas was new and wonderful, my guild hit Thaddius. He was the big Frankenstein looking guy with lightning shooting everywhere. His big gimmick was polarity. If two people were near each other and had opposite polarity, we'd shock one another to death pretty quickly. However, if you had the same polarity, you'd increase the amount of damage you could do. This meant you wanted to stand next to people with the same polarity while keeping away from people with different polarity. Of course, this changed every 30 seconds, and ~10 people had their polarities changed. If you didn't move fast enough, you'd shock people. And with 40 people in the raid, all it took was one person to be slow on their feet and they'd shock the entire raid. Of course, this is going on while everyone is trying to maximize DPS and keep the raid healed up.
Anyways, the strategy was simple: positive on the left, and negative on the right. The debuff icons we had were blue for postive and red for negative, and the original idea was people would equate red with right. Association. This work well when people had to get into their initial positions. However, once we were engaged and everyone got to their initial spot, polarity changes were troublesome. People would move as soon as they knew they had to move. However, they had to think about it. The thought process worked like this.
1. Polarity changed happened.
2. Did my polarity change?
3. Yes. What color?
4. Red. What side am I on?
5. Left. I need to move to the right.
6. Move to the right.
That's a lot of work. A second delay, and considering you only had 3-4 seconds to react, coupled with a second of lag, and you'd burn people.
The problem wasn't people reacting. The problem was the thought process. How to get people to react quickly.
I finally caught on to the solution. The problem was thinking too much. What I told people to do was to initially setup on the left or right like normal. However, after that, ignore sides. Instead, follow one simple rule:
Change sides when your polarity changes.
Change when the icon changes.
Simple enough. You don't need to worry if the polarity doesn't change. You don't need to worry about going to the left or the right. You simple move to the opposite side.
The difference was dramatic. The very next attempt, we got him so low we thought we were going to kill him, but someone lagged out and caused a wipe. Regardless, the effect was clear.
Understanding now only the problem, but how to approach that problem and make it easy to learn is important. It's hard to understand where the problem lies with people. I was proud of this. Oh, it was just a game, but it was a clear example of being able to understanding a problem with how other people thought, and being able to come up with a solution.
But that doesn't make the article wrong. While you don't understand something until it seems obvious to you, it does not imply that you understand everything that seems obvious to you.
I don't think that's quite true, but developing familiarity with a difficult concept is tantamount to coming to understand it, in my experience.
For me, the process seems to go like this:
1) assemble the disparate pieces of a concept, and try to wrap my head around the novel ones;
2) work out how Piece A relates to B, and B to C, and ... ;
3) forget what the hell Piece X is;
4) repeat steps 1–4 several times;
5) convince myself that all the pieces work together as they should.
And only later, after I've worked with this collection of pieces several times, do I stop seeing the pieces and start seeing something new, the emergent phenomenon of the new concept. And it's not until I've forgotten the details of how Piece X fits into Piece Y, and maybe why we need Piece Z in the first place, that I feel like I really grok this new concept.
Obvious to you. Amazing to others.
"Every mathematician worthy of the name has experienced the state of lucid exaltation in which one thought succeeds another as if miraculously. This feeling may last for hours at a time, even for days. Once you have experienced it, you are eager to repeat it but unable to do it at will, unless perhaps by dogged work." -André Weil
My guess is that when you finally understood the subject, your brain has completed the route for that particular problem. It's now clear how to solve the problem for the brain, since a specific route exists. It's obvious and very easy for the brain to do that with the route shortcut.
For you, the problem becomes obvious and simple. It's no longer complicated and requires less brain-processing-power to solve. You think you were stupid you didn't figure out that from the beginning, but you are not!
On the other hand, I think the things that become obvious once you've understood them tend to be (though are not always) the things that have already been fairly well digested by others, and so are presented to you in a smooth, flowing way to which you just have to accustom yourself.
My experience with (mathematical) research is that understanding has a roughly equal possibility of meaning that you find it trivial, or that you finally understand all the (apparently) irreducibly complex difficulties. Indeed, my feeling if anyone tells me that, say, Deligne–Lusztig representations are obvious is that he or she hasn't probably fully understood them (disclaimer: neither have I, not even close, which may mean that I'm just illustrating the author's point).
I don't mean at all by this that you shouldn't go on searching for the 'obvious' simplification—that way lies great insight. (As someone said much more elegantly, if things aren't already obvious in mathematics, we tend to make them obvious by changing the definitions.) What I mean is that you shouldn't drag yourself down by saying “I thought I understood that, but it's hard, not obvious!”
Another corollary: understanding brings peace.
But then I get into conversations with my peers and realize that some of the stuff I do apparently isn't as obvious to everyone as I'd thought.
As a researcher, who has also TAed classes, it takes a lot of effort not only to teach a particular topic, but also simultaneously present the crux of the idea as well as the right way to think about it and its place in the context of other information. And learning to think about everything with these in mind will make you understand things much much better. In particular, one of the most challenging things is not to explain breakthrough ideas to cutting-edge researchers but rather explaining breakthrough ideas to complete outsiders and laypeople.
Here's the thing, though: if you make an honest effort to explain something like that to a lay audience, you may fail. Or you may not. Or you'll give them a workable, but incomplete and strictly wrong idea. Either way you yourself will end up with a deeper understanding.
A huge part of "real" mathematics is finding isomorphisms. Teaching is more or less finding isomorphisms between new concepts and concepts that the student already has.
The interesting consideration, to me, is how much we can get done with hardly any understanding at all. That's human nature. We do things, then we wonder how, and then we eventually come to understand what the hell we just did. It seems bass-ackwards, but that's the way it goes most of the time.
So yes it cuts both ways.
You could (presumably) construct axiom systems under which this is not true, and I believe that some logicians do this sort of thing, but this is rather far removed from mainstream research mathematics.
The Banach-Tarski paradox comes to mind first. I don't think anyone argues that the proof is wrong, but you'll easily find people to argue that that very fact means that the full axiom of choice should be held in deep suspicion.
The 4-color map coloring theorem is more interesting in this thread, though; I think it's the best known example of a proof that offers little to no insight into why the theorem is true. I don't think any solution that requires breaking a problem into 1,936 special cases, and then mechanically checking each one of those cases, will ever lead to an understanding that makes the theorem "obvious".
Looking for insight is what many people do in mathematics, but nonetheless believing that every true fact is true for a reason is naive, just as naive as believing that every problem has to have a solution. So far, most of the time we succeed in explaining why true things are true, but it may be the case that some facts are true merely by combinatorics, and not for human-imaginable reason.
Sure, I would like to believe that there is a crucial observation we haven't made yet that makes 4 color theorem easy, just as it is for 6 color theorem. Nevertheless, I accept the fact that there may be no such thing and 4 color theorem holds just because the constraints force it to be.
I remember reading something of Chaitin's a couple of years ago that drove home the point that most mathematical facts are, in some fairly-well defined sense, true for no reason at all.
(Alas, I don't remember the argument well enough to re-cap it here.)
No, the problem that those who dislike the axiom of choice have is not inherent with the axiom but more the law of excluded middle. Constructivism has no rooms for such imaginings hence those who follow it are not happy with existential proofs. This more practical viewpoint will prove more profound IMHO just as bayesianism is winning the day. Just too many things connected.. types and terms in programs, intuitionistic logic, parts of physics, topoi, CCC.
It's, um, obviously quite hard to explain why you find something obvious, but I think in the first case it's because you can just think about rescaling the number line (using a bigger unit) or shifting the origin, which to me makes it clear that bigness is completely arbitrary and implies that there can't possibly be any biggest number in this sense; in the second case, there clearly is no "next" number in the reals as there is in the naturals, and that insight gets right at the heart of the concept of countability.
I think you missed the point. There is no "next number" concept in the rationals either, and yet they're countable. There is "next number" concept in the first uncountable ordinal, and, as the name suggests, it is uncountable.
So I think I've given credence to the notion that those who claim to find something obvious might not fully understand it. Or at least not at four in the morning.
(Oh, and a point that Stephen Abbott made in his wonderful intro analysis book now springs to mind: proofs are useful also as a check on our intuition.)
- When someone says that something is obvious, we cannot claim that he/she actually understands it well.
- But, if someone understands something well, it will seem obvious to him/her.