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#341
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cover article in Time magazine on gifted education
toypup wrote:
Yes, I find writing is one thing that does improve with practice. With math, as long as the concept is understood, more practice in that area does not improve anything. Eh, I'm not sure I buy that at all. Even with something as simple as multiplication tables, practice gets you speed (which is useful in some cases, not so much in others). When you get to anything beyond the basics, each problem is a little puzzle to solve, and there are lots of wrinkles to be thrown in. Each different perspective broadens one's problem solving repertoire and one's ability to think about the issues (up to a point, obviously). I'm not suggesting there's much value to doing the *same* sort of problem over and over again once you've got the concept, but it'd have to be a pretty unimaginative teacher who couldn't come up with nearly endless varieties of challenges once past the basics. Best wishes, Ericka |
#342
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cover article in Time magazine on gifted education
On Sat, 01 Sep 2007 20:23:49 -0400, Ericka Kammerer wrote:
toypup wrote: Yes, I find writing is one thing that does improve with practice. With math, as long as the concept is understood, more practice in that area does not improve anything. Eh, I'm not sure I buy that at all. Even with something as simple as multiplication tables, practice gets you speed (which is useful in some cases, not so much in others). When you get to anything beyond the basics, each problem is a little puzzle to solve, and there are lots of wrinkles to be thrown in. Each different perspective broadens one's problem solving repertoire and one's ability to think about the issues (up to a point, obviously). I'm not suggesting there's much value to doing the *same* sort of problem over and over again once you've got the concept, but it'd have to be a pretty unimaginative teacher who couldn't come up with nearly endless varieties of challenges once past the basics. Just thinking from my own experience. Once I got the concept, no more practice did I do. Problems with a twist didn't seem to be very difficult. If I didn't understand it, the practice helped. |
#344
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cover article in Time magazine on gifted education
In article ,
Ericka Kammerer wrote: toypup wrote: Yes, I find writing is one thing that does improve with practice. With math, as long as the concept is understood, more practice in that area does not improve anything. Eh, I'm not sure I buy that at all. Even with something as simple as multiplication tables, practice gets you speed (which is useful in some cases, not so much in others). When you get to anything beyond the basics, each problem is a little puzzle to solve, and there are lots of wrinkles to be thrown in. Each different perspective broadens one's problem solving repertoire and one's ability to think about the issues (up to a point, obviously). I'm not suggesting there's much value to doing the *same* sort of problem over and over again once you've got the concept, but it'd have to be a pretty unimaginative teacher who couldn't come up with nearly endless varieties of challenges once past the basics. If one knows the concept of multiplication and its relation to addition, leaning the multiplication tables only produces speed. If one teaches multiplication by way of the tables, only MISconception is produced. I think it would be a very good idea after the concept of multiplication is learned to have the students produce their own copies of the multiplication table, and point out how it can speed up the process. But there is no concept there. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University Phone: (765)494-6054 FAX: (765)494-0558 |
#345
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cover article in Time magazine on gifted education
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#346
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cover article in Time magazine on gifted education
In article , Ericka Kammerer
says... toypup wrote: Yes, I find writing is one thing that does improve with practice. With math, as long as the concept is understood, more practice in that area does not improve anything. Eh, I'm not sure I buy that at all. Even with something as simple as multiplication tables, practice gets you speed (which is useful in some cases, not so much in others). When you get to anything beyond the basics, each problem is a little puzzle to solve, and there are lots of wrinkles to be thrown in. Each different perspective broadens one's problem solving repertoire and one's ability to think about the issues (up to a point, obviously). I'm not suggesting there's much value to doing the *same* sort of problem over and over again once you've got the concept, but it'd have to be a pretty unimaginative teacher who couldn't come up with nearly endless varieties of challenges once past the basics. Well, the pure *concept* might be understood quickly; it's the application that is practiced. Banty |
#347
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cover article in Time magazine on gifted education
In article , Herman Rubin says...
In article , Ericka Kammerer wrote: toypup wrote: Yes, I find writing is one thing that does improve with practice. With math, as long as the concept is understood, more practice in that area does not improve anything. Eh, I'm not sure I buy that at all. Even with something as simple as multiplication tables, practice gets you speed (which is useful in some cases, not so much in others). When you get to anything beyond the basics, each problem is a little puzzle to solve, and there are lots of wrinkles to be thrown in. Each different perspective broadens one's problem solving repertoire and one's ability to think about the issues (up to a point, obviously). I'm not suggesting there's much value to doing the *same* sort of problem over and over again once you've got the concept, but it'd have to be a pretty unimaginative teacher who couldn't come up with nearly endless varieties of challenges once past the basics. If one knows the concept of multiplication and its relation to addition, leaning the multiplication tables only produces speed. If one teaches multiplication by way of the tables, only MISconception is produced. I think it would be a very good idea after the concept of multiplication is learned to have the students produce their own copies of the multiplication table, and point out how it can speed up the process. But there is no concept there. Right. (Good example, BTW.) Concept can be grasped quickly; it's the application that's practiced. And application before concept is confusing. Like in high school when I was "taught" trigonometry by opposite-over-hypotnuse, etc. It wasn't until I picked up a college pre-calculus textbook that had a simple graphical illustration of the unit circle that I understood all that. And pretty much in an instant. Actually creating the multiplication table is a great way to practice the application and to internalize the concept. But if those facts aren't at some kind of easy recall, further application is hindered. Banty |
#348
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cover article in Time magazine on gifted education
toypup wrote:
On Sat, 01 Sep 2007 20:23:49 -0400, Ericka Kammerer wrote: toypup wrote: Yes, I find writing is one thing that does improve with practice. With math, as long as the concept is understood, more practice in that area does not improve anything. Eh, I'm not sure I buy that at all. Even with something as simple as multiplication tables, practice gets you speed (which is useful in some cases, not so much in others). When you get to anything beyond the basics, each problem is a little puzzle to solve, and there are lots of wrinkles to be thrown in. Each different perspective broadens one's problem solving repertoire and one's ability to think about the issues (up to a point, obviously). I'm not suggesting there's much value to doing the *same* sort of problem over and over again once you've got the concept, but it'd have to be a pretty unimaginative teacher who couldn't come up with nearly endless varieties of challenges once past the basics. Just thinking from my own experience. Once I got the concept, no more practice did I do. Problems with a twist didn't seem to be very difficult. If I didn't understand it, the practice helped. Yet it is the case time and time again that people who *think* they have the concept *don't*. And while I have nothing to base this on other than my experience in my kids' gifted classrooms, it seems that quite a number of gifted kids are prone to thinking they've got stuff before they've really thought it through all the way. Many of them have minds that seem to leap to conclusions, creating shortcuts that work with a subclass of problems but not other subclasses of a problem. Or, sometimes they're just so used to getting things amazingly quickly and so they assume they have something solidly when they don't quite. So, in my opinion, it is helpful for kids to go through a reasonable set of exercises that hit upon different variations of the problem to verify that they've really got it. After that, those who need more practice should do more problems and those who've demonstrated mastery should move on to more challenging variations. After all, math is mostly used as a tool. The goal isn't just to understand the mechanics of a particular operation. The goal is to be able to solve problems. There's a nearly endless supply of increasingly challenging puzzles to solve, and continuing to work on those *does* improve a person's thinking and problem solving abilities. Best wishes, Ericka |
#349
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cover article in Time magazine on gifted education
Chookie wrote:
In article , (Herman Rubin) wrote: If one knows the concept of multiplication and its relation to addition, leaning the multiplication tables only produces speed. If one teaches multiplication by way of the tables, only MISconception is produced. I think it would be a very good idea after the concept of multiplication is learned to have the students produce their own copies of the multiplication table, and point out how it can speed up the process. But there is no concept there. ITA. I had to learn my times-tables by rote (as in, "Learn your three-times tables for homework". It was extraordinarily difficult and I never really got them down properly (I find it difficult to learn something that makes no sense). Frequent times-tables races and so on did not improve matters; continually rewriting wrong answers will not improve performance OR speed! I am actually fascinated by this idea that there are people who were simply taught multiplication tables as memorization exercises with no discussion of what multiplication *is*. While I was expected to memorize my multiplication tables so that I could solve problems quickly, I certainly was taught what multiplication was and could have generated a multiplication table anytime I needed to. My children were all taught the same way. They knew what multiplication was, and could figure out a multiplication problem in a number of different way prior to memorizing their multiplication facts. They were just slow until they had them memorized. And, of course, it is *very* painful to do long division or all sorts of mental arithmetic that one encounters every day if one doesn't have those multiplication facts memorized. I fail to see memorizing multiplication facts as a Bad Thing, though obviously it's important for children to understand multiplication as well as memorizing their facts. It's hard for me to fathom the notion of someone memorizing a multiplication table without having been taught multiplication. I certainly wouldn't gainsay anyone's personal experience in that matter, but is it really the case that there are loads of classrooms in which this is the way teachers set out to teach multiplication (i.e., by rote without actually teaching what multiplication is)? Best wishes, Ericka |
#350
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cover article in Time magazine on gifted education
Banty wrote:
In article , Ericka Kammerer says... toypup wrote: Yes, I find writing is one thing that does improve with practice. With math, as long as the concept is understood, more practice in that area does not improve anything. Eh, I'm not sure I buy that at all. Even with something as simple as multiplication tables, practice gets you speed (which is useful in some cases, not so much in others). When you get to anything beyond the basics, each problem is a little puzzle to solve, and there are lots of wrinkles to be thrown in. Each different perspective broadens one's problem solving repertoire and one's ability to think about the issues (up to a point, obviously). I'm not suggesting there's much value to doing the *same* sort of problem over and over again once you've got the concept, but it'd have to be a pretty unimaginative teacher who couldn't come up with nearly endless varieties of challenges once past the basics. Well, the pure *concept* might be understood quickly; it's the application that is practiced. Yes, but in the practicing, one often finds that there are nuances to the concept that weren't initially grasped, especially once one gets beyond the very basics. I'm very suspicious of this "all you need is the concept" argument, because time and again I see students who "got" the concept quickly, but it turns out, they didn't "get" it well enough to apply it in a wide variety of situations. Best wishes, Ericka |
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