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And now that it's obvious to you, it's suddenly really hard to explain to someone else. My friend was TAing a freshman calculus class in college while working

by lmkg 15y ago

And now that it's obvious to you, it's suddenly really hard to explain to someone else. 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.

haberman 15y ago
That's what makes guys like Sal Khan special. And why his videos are so valuable. Sal's "Introduction to Limits": http://www.khanacademy.org/video/introduction-to-limits http://www.khanacademy.org/video/introduction-to-limits
espeed 15y ago
"If you can't explain it simply, you don't understand it well enough." -- Albert Einstein
- waqf 15y ago
  And the tragedy is that nobody else understands the simple explanation once you understand enough to give it.
  - espeed 15y ago
    Yes, true insight requires a level of understanding usually obtained through a focused effort to understand all the components that make up the idea, several levels deep. 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. See http://jamesthornton.com/blog/how-to-get-to-genius http://jamesthornton.com/blog/how-to-get-to-genius
  - Swannie 15y ago
    Well that sounds like the problem of "I learnt it wrong the first time". 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.)
ebiester 15y ago
...And that is the difference between a (good) teacher and a researcher who is conscripted into teaching. 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.)
- aristus 15y ago
  Indeed, you can explain it to 9-year-olds: http://carlos.bueno.org/2011/01/tortoise.html http://carlos.bueno.org/2011/01/tortoise.html
- Retric 15y ago
  I think a much better way to understand limits for a HS student is coin flips. Say heads = 1 tails = 0. 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.
  - klipt 15y ago
    That's not a simple limit, that's the Law of Large Numbers from probability theory - a more complex beast altogether.
- debatchment 15y ago
  The way I see it is that a teacher may help a pupil to understand something by identifying the pupil's misconceptions. These are unique to the pupil and the teaching process includes to-and-fro questioning aimed at identifying and blasting them away. (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.)
jared314 15y ago
The curse of knowledge strikes again. This is one of the reasons teaching is a dedicated profession. Teachers practice more on how to teach a subject, than the subject itself. I also think it relates to why teachers cannot reach a great depth in any subject they teach. Their students "hold" them back.
btilly 15y ago
The problem with limits is simple. We force people to learn multiple concepts in a poor order. 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.
- Fargren 15y ago
  1. We introduce limits. Which are blindingly artificial, and so it is hard for us to form a concept about why people want this. 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_tortoise http://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_and...
  - psykotic 15y ago
    The limit concept was introduced by Cauchy but lacked a rigorous definition until Weierstrass came along. In the absence of such a definition, Cauchy committed some infamous errors involving nested limits, e.g. he ascribed properties to continuous functions that actually require the stronger assumption of uniform continuity. Take into account that Cauchy's treatment of calculus was by far the most rigorous for its time, so he was hardly careless in these matters. If everything was as straightforward and obvious as you imply, neither Cauchy's nor Weierstrass's work on the foundations of calculus would have served any meaningful purpose. 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.
ahoyhere 15y ago
To teach effectively, you have to get inside other people's heads. That is a skill most people don't ever spend any real time trying to develop. So it's no wonder that they are bad at it. It's not about the state of an individual's understanding, but whether they've bothered to cultivate an entirely different, unrelated skill.
- jasonlotito 15y ago
  It's a bit of a story, but it's got a point that relates to this. 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.
Typhon 15y ago
I disagree. In my opinion, if you can't explain something to someone else, you don't really understand it. 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.
SatvikBeri 15y ago
It took me about a month to really get limits. Once I did, I started complaining about how horribly unintuitive the definition the class had used was. So a friend asked me how I would define it. After a few tries I had the perfect definition...and it turned out it was word-for-word exactly the same as the one in the book.

by lmkg 15y ago

haberman 15y ago

That's what makes guys like Sal Khan special. And why his videos are so valuable. Sal's "Introduction to Limits": http://www.khanacademy.org/video/introduction-to-limits http://www.khanacademy.org/video/introduction-to-limits

espeed 15y ago

"If you can't explain it simply, you don't understand it well enough." -- Albert Einstein

waqf 15y ago
And the tragedy is that nobody else understands the simple explanation once you understand enough to give it.
- espeed 15y ago
  Yes, true insight requires a level of understanding usually obtained through a focused effort to understand all the components that make up the idea, several levels deep. 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. See http://jamesthornton.com/blog/how-to-get-to-genius http://jamesthornton.com/blog/how-to-get-to-genius
- Swannie 15y ago
  Well that sounds like the problem of "I learnt it wrong the first time". 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.)

ebiester 15y ago

...And that is the difference between a (good) teacher and a researcher who is conscripted into teaching. 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.)

aristus 15y ago
Indeed, you can explain it to 9-year-olds: http://carlos.bueno.org/2011/01/tortoise.html http://carlos.bueno.org/2011/01/tortoise.html
Retric 15y ago
I think a much better way to understand limits for a HS student is coin flips. Say heads = 1 tails = 0. 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.
- klipt 15y ago
  That's not a simple limit, that's the Law of Large Numbers from probability theory - a more complex beast altogether.
debatchment 15y ago
The way I see it is that a teacher may help a pupil to understand something by identifying the pupil's misconceptions. These are unique to the pupil and the teaching process includes to-and-fro questioning aimed at identifying and blasting them away. (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.)

jared314 15y ago

The curse of knowledge strikes again. This is one of the reasons teaching is a dedicated profession. Teachers practice more on how to teach a subject, than the subject itself. I also think it relates to why teachers cannot reach a great depth in any subject they teach. Their students "hold" them back.

btilly 15y ago

The problem with limits is simple. We force people to learn multiple concepts in a poor order. 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.

Fargren 15y ago
1. We introduce limits. Which are blindingly artificial, and so it is hard for us to form a concept about why people want this. 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_tortoise http://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_and...
- psykotic 15y ago
  The limit concept was introduced by Cauchy but lacked a rigorous definition until Weierstrass came along. In the absence of such a definition, Cauchy committed some infamous errors involving nested limits, e.g. he ascribed properties to continuous functions that actually require the stronger assumption of uniform continuity. Take into account that Cauchy's treatment of calculus was by far the most rigorous for its time, so he was hardly careless in these matters. If everything was as straightforward and obvious as you imply, neither Cauchy's nor Weierstrass's work on the foundations of calculus would have served any meaningful purpose. 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.

ahoyhere 15y ago

To teach effectively, you have to get inside other people's heads. That is a skill most people don't ever spend any real time trying to develop. So it's no wonder that they are bad at it. It's not about the state of an individual's understanding, but whether they've bothered to cultivate an entirely different, unrelated skill.

jasonlotito 15y ago
It's a bit of a story, but it's got a point that relates to this. 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.

Typhon 15y ago

I disagree. In my opinion, if you can't explain something to someone else, you don't really understand it. 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.

SatvikBeri 15y ago

It took me about a month to really get limits. Once I did, I started complaining about how horribly unintuitive the definition the class had used was. So a friend asked me how I would define it. After a few tries I had the perfect definition...and it turned out it was word-for-word exactly the same as the one in the book.