3 ms·

...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 aspe

...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.)
pdhborges 15y ago
Will the student realize that standing against the wall continually would wield the same limit?

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.)

pdhborges 15y ago

Will the student realize that standing against the wall continually would wield the same limit?