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29 Jul 2005 - 19:12
It is possible for students to construct for themselves the mathematical practices that, historically, took several thousand years to evolve.*
That's just nuts. Still, I keep having these moments of recognition, stumbling over, however fleetingly, my own thinking & experience in constructivist texts. And from time to time this will have happened often enough that I'll have to stop and think: Wait a minute. Am I definitely against constructivism? The answer is yes. As it turns out, there is an obvious and simple resolution to the problem of When Bad Constructivists Say Good Things, which is that there is smart constructivism, and stupid constructivism, the latter, I've just this moment discovered, being known as radical constructivism in the trade, which is what I intend to call it from now on. Say it together, now:
radical constructivism! There isn't a parent on the planet who's going to enjoy hearing that his school is implementing a radical constructivist curriculum, which is probably why the words radical constructivism are nowhere to be found on the web site of the NCTM. So I'm going to be using radical constructivist from now on, and I'm going to say text instead of curriculum, for good measure. MATH TRAILBLAZERS: a radical constructivist text. That's gonna make me popular.
According to smart constructivism, all knowledge is constructed, period. There isn't active knowledge & passive knowledge, constructed knowledge and swallowed-whole knowledge, or any other kind of Correctly acquired knowledge versus Incorrectly acquired knowledge. Knowledge is knowledge; to get more of it, you have to build your new knowledge on the foundation of the old knowledge you already possess.
And this comment from a ktm guest beautifully expresses the essence of smart constructivism:
*Cobb, P., Yackel, E. & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23, 2-33.
** Bransford, John D., et al. (2000). How People Learn: Brain, Mind, Experience, and School Revised Edition. Washington, D.C.: National Academy Press. (pdf file)
keywords: radical constructivist radical constructivism lost in translation smart constructivism
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Even the title of that paper is nuts. A constructivist alternative to the representational view of mind in mathematics education. Will someone explain to me what a representational view of mind in math education might be? Catherine? Anyone? -- CarolynJohnston - 29 Jul 2005
(btw: I'm the person whose comments you posted here; right now, my browser and I are having trouble registering with wiki, etc. But my name is Allison, and in the future, I'll try to be a regular user.) You are spot on about smart constructivism. We are standing on the shoulders of giants, because if we didn't, we'd never get anywhere. Radical constructivism means that children, left to their own devices, will go down a bunch of wrong roads just like a billion other people did. Even Euclid got more things wrong in his life than he got right--are we supposed to let our kids get stuck at their inability to invent the concept of negative numbers? Because Euclid hadn't figured them out--and our kids are supposed to be able to construct them just by playing with some bricks? The best idea is to let kids do some of that creative play/learning, and then when they find something NEAT about the world that they don't understand, to be able to help teach them about WHY that thing is true. leaving them lost isn't teaching! Here's a lovely little example. I know a child who at the age of about 8 or 10, noticed something REALLY AMAZING about numbers! She noticed that 8*8 was 64. and that 7*9 was 63. That 6*6 was 36. and 5*7 was 35! And then she KEPT TRYING THIS thing she'd found-- 2*2 was 4, and 1*3 was 3! 11*11 was 121, and 12*10 was 120! Every time she tried it, she found this pattern! she didn't understand at all HOW THIS WAS Possibly true?!?!?! At that point, there was an amazing Kitchen Table Math moment of possibility in front of the girl. Someone could have used this to teach the idea of algebra,and of equations to the child. They could have taught factoring. They could have taught multiplication of polynomials. Because they could have taught the child that (n-1)*(n+1) = n^2 - 1 is a true statement for all numbers, and could have used that as a springboard for other mathematical truths. Left alone to do "radical constructivism" what are the chances that the child would have suddenly invented quadratic polynomial math? or even equations? About zero. The point of constructivism is how it leads us into the world of possibilities, how it allows our curiosity and experience to open the door to us. But sending a child into the world of possibilities without a guide is like telling them to walk around LA and figure out how to find clothing, food, and shelter all by themselves. So don't lose your belief in smart constructivism; you're right, it's invaluable. But it's not right to abandon a child to it. Even mother bears teach their children how to fish. -- KtmGuest - 29 Jul 2005
Wow, Alison. What that little girl was discovering was a fundamental principle of calculus!!! The observations about 6*6 vs. 5*7, for example, boil down to this: The expression x(12-x) is at its maximum when x=6 (half of 12). 11*11 vs 10*12 is explained by the fact that x(22-x) is at its maximum when x=11, half of 22. Another way to look at it (that's what I'm starting to think it's all about, knowing different ways to look at things): for a given fixed perimeter, the rectangle with the biggest area is a square. That's one smart little girl. I think your comment here is a great one. It's ludicrous to expect even the smartest kids -- even your very bright little girl -- to repeat the discoveries that Archimedes, Newton, Gauss, etc., made, even in a guided fashion. Learning this way seems actually to be fatiguing for many kids. Catherine found a great quote by Whitehead about intense thought being like cavalry charges. If it were shorter, I'd vote for that to be the KTM slogan... Alison -- I have a new registration by someone named alcibiades -- is that you? If so, you're in. If not, drop me a line at webmaster@kitchentablemath.net with the username and password you want, and I'll get you set up! -- CarolynJohnston - 30 Jul 2005
Ah, the rectangle with the biggest area is a square! funny, I knew that and had never connected it up either! Ah, what time, patience, and new ideas from others teach us! yes, and stuffing these ideas into someone's head doesn't work--the ideas have got to be allowed to sit there, and percolate, and rest. Resting is important. The neurons connect up when we sleep and when we eat cookies and milk. Different children, of course, are also different. Some children respond very poorly to failure--it scares them, essentially, and they don't want to think about something difficult afterwards. Other kids respond to failure as a challenge. Radical constructivism has got to be filled with 10000 failures for every success. The enormity of that, I think, is lost on us adults who have such immense conscious and unconscious knowledge so as to prevent those 10000 failures from occurring every second. If there's one thing that I want teaching kids to succeed at, it's making them LESS afraid and LESS timid. a billion failures in math isn't likely to do that. If people creating a math curricula started out with "First, do no harm", maybe we'd be better off :) -- KtmGuest - 31 Jul 2005
Will someone explain to me what a representational view of mind in math education might be? I just learned what it is last night! Now I've forgotten! I'll look it up.... -- CatherineJohnson - 31 Jul 2005
smart constructivism
For awhile now I've been noticing that not infrequently I'll read a constructivist text and think: OK, that idea does not sound actually insane. Then, with a sense of growing alarm, given the fact that I've just spent the last 3 months of my life banging on about constructivism here on the World Wide Web, I'll think: as a matter of fact, that idea sounds like an idea to which I myself subscribe. Fortunately, the cognitive dissonance never lasts long, because the next paragraphs invariably put forth wing-y observations and grand, looping conclusions that do not follow logically from any known principle governing the natural world. Such as, to quote my number one most demented peer-reviewed constructivist nonsense on stilts prose passage of all time:It is possible for students to construct for themselves the mathematical practices that, historically, took several thousand years to evolve.*
That's just nuts. Still, I keep having these moments of recognition, stumbling over, however fleetingly, my own thinking & experience in constructivist texts. And from time to time this will have happened often enough that I'll have to stop and think: Wait a minute. Am I definitely against constructivism? The answer is yes. As it turns out, there is an obvious and simple resolution to the problem of When Bad Constructivists Say Good Things, which is that there is smart constructivism, and stupid constructivism, the latter, I've just this moment discovered, being known as radical constructivism in the trade, which is what I intend to call it from now on. Say it together, now:
radical constructivism! There isn't a parent on the planet who's going to enjoy hearing that his school is implementing a radical constructivist curriculum, which is probably why the words radical constructivism are nowhere to be found on the web site of the NCTM. So I'm going to be using radical constructivist from now on, and I'm going to say text instead of curriculum, for good measure. MATH TRAILBLAZERS: a radical constructivist text. That's gonna make me popular.
What is smart constructivism, you ask?
A common misconception regarding 'constructivist' theories of knowing (that existing knowledge is used to build new knowledge) is that teachers should never tell students anything directly but, instead, should always allow them to construct knowledge for themselves. This perspective confuses a theory of pedagogy (teaching) with a theory of knowing. Constructivists assume that all knowledge is constructed from previous knowledge, irrespective of how one is taught (e.g., Cobb, 1940)--even listening to a lecture involves active attempts to construct new knowledge.**
According to smart constructivism, all knowledge is constructed, period. There isn't active knowledge & passive knowledge, constructed knowledge and swallowed-whole knowledge, or any other kind of Correctly acquired knowledge versus Incorrectly acquired knowledge. Knowledge is knowledge; to get more of it, you have to build your new knowledge on the foundation of the old knowledge you already possess.
smart constructivism at Kitchen Table Math
Probably all of the commenters on ktm assume or understand this, and have thought about it. Here is Anne Dwyer's wonderful story about connecting a lesson with her daughter's pre-existing knowledge:Erin is finally finishing up Primary Mathematics 1B. She is working on money. She has always had problems counting money. She can skip count by 5 and she can skip count by 10. But she couldn't skip count by 10 if she was on a number that ended in 5. I tried several methods to help her but nothing worked until we started working on the white board. I wrote out skip counting by 5 and underneath I wrote out skip counting by 10. And finally, in frustration, I suggested that when she had to increase the number by 10, she should skip count by 5 twice. Well, that just totally clicked with her. She got it and has never had another problem counting money. I never would have 'taught' it that way, but it totally worked for her.
And this comment from a ktm guest beautifully expresses the essence of smart constructivism:
Teaching is a puzzle. It's a puzzle where you must navigate backwards in a maze. A child is at point K, but they are supposed to be at point Z. If you just show them again how to go from A to Z, you are missing the point of how they got to K. And usually, kids made a rational mistake: they misunderstood something, or misheard something, and this thing is embedded in their minds. It leads them (Rationally) to this bad position K. Teaching is about figuring out how someone got into that position, so you can FIX that misunderstanding. It's not enough to tell them that K is the wrong place; you have to help them never follow that wrong path in the first place. The best way to help kids learn is to remember the typical misconceptions YOU had as a child, and ones similar to it, to try and understand why they would think what they think. Then, you can see how they are really very smart--just misguided.
*Cobb, P., Yackel, E. & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23, 2-33.
** Bransford, John D., et al. (2000). How People Learn: Brain, Mind, Experience, and School Revised Edition. Washington, D.C.: National Academy Press. (pdf file)
keywords: radical constructivist radical constructivism lost in translation smart constructivism
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Even the title of that paper is nuts. A constructivist alternative to the representational view of mind in mathematics education. Will someone explain to me what a representational view of mind in math education might be? Catherine? Anyone? -- CarolynJohnston - 29 Jul 2005
(btw: I'm the person whose comments you posted here; right now, my browser and I are having trouble registering with wiki, etc. But my name is Allison, and in the future, I'll try to be a regular user.) You are spot on about smart constructivism. We are standing on the shoulders of giants, because if we didn't, we'd never get anywhere. Radical constructivism means that children, left to their own devices, will go down a bunch of wrong roads just like a billion other people did. Even Euclid got more things wrong in his life than he got right--are we supposed to let our kids get stuck at their inability to invent the concept of negative numbers? Because Euclid hadn't figured them out--and our kids are supposed to be able to construct them just by playing with some bricks? The best idea is to let kids do some of that creative play/learning, and then when they find something NEAT about the world that they don't understand, to be able to help teach them about WHY that thing is true. leaving them lost isn't teaching! Here's a lovely little example. I know a child who at the age of about 8 or 10, noticed something REALLY AMAZING about numbers! She noticed that 8*8 was 64. and that 7*9 was 63. That 6*6 was 36. and 5*7 was 35! And then she KEPT TRYING THIS thing she'd found-- 2*2 was 4, and 1*3 was 3! 11*11 was 121, and 12*10 was 120! Every time she tried it, she found this pattern! she didn't understand at all HOW THIS WAS Possibly true?!?!?! At that point, there was an amazing Kitchen Table Math moment of possibility in front of the girl. Someone could have used this to teach the idea of algebra,and of equations to the child. They could have taught factoring. They could have taught multiplication of polynomials. Because they could have taught the child that (n-1)*(n+1) = n^2 - 1 is a true statement for all numbers, and could have used that as a springboard for other mathematical truths. Left alone to do "radical constructivism" what are the chances that the child would have suddenly invented quadratic polynomial math? or even equations? About zero. The point of constructivism is how it leads us into the world of possibilities, how it allows our curiosity and experience to open the door to us. But sending a child into the world of possibilities without a guide is like telling them to walk around LA and figure out how to find clothing, food, and shelter all by themselves. So don't lose your belief in smart constructivism; you're right, it's invaluable. But it's not right to abandon a child to it. Even mother bears teach their children how to fish. -- KtmGuest - 29 Jul 2005
Wow, Alison. What that little girl was discovering was a fundamental principle of calculus!!! The observations about 6*6 vs. 5*7, for example, boil down to this: The expression x(12-x) is at its maximum when x=6 (half of 12). 11*11 vs 10*12 is explained by the fact that x(22-x) is at its maximum when x=11, half of 22. Another way to look at it (that's what I'm starting to think it's all about, knowing different ways to look at things): for a given fixed perimeter, the rectangle with the biggest area is a square. That's one smart little girl. I think your comment here is a great one. It's ludicrous to expect even the smartest kids -- even your very bright little girl -- to repeat the discoveries that Archimedes, Newton, Gauss, etc., made, even in a guided fashion. Learning this way seems actually to be fatiguing for many kids. Catherine found a great quote by Whitehead about intense thought being like cavalry charges. If it were shorter, I'd vote for that to be the KTM slogan... Alison -- I have a new registration by someone named alcibiades -- is that you? If so, you're in. If not, drop me a line at webmaster@kitchentablemath.net with the username and password you want, and I'll get you set up! -- CarolynJohnston - 30 Jul 2005
Ah, the rectangle with the biggest area is a square! funny, I knew that and had never connected it up either! Ah, what time, patience, and new ideas from others teach us! yes, and stuffing these ideas into someone's head doesn't work--the ideas have got to be allowed to sit there, and percolate, and rest. Resting is important. The neurons connect up when we sleep and when we eat cookies and milk. Different children, of course, are also different. Some children respond very poorly to failure--it scares them, essentially, and they don't want to think about something difficult afterwards. Other kids respond to failure as a challenge. Radical constructivism has got to be filled with 10000 failures for every success. The enormity of that, I think, is lost on us adults who have such immense conscious and unconscious knowledge so as to prevent those 10000 failures from occurring every second. If there's one thing that I want teaching kids to succeed at, it's making them LESS afraid and LESS timid. a billion failures in math isn't likely to do that. If people creating a math curricula started out with "First, do no harm", maybe we'd be better off :) -- KtmGuest - 31 Jul 2005
Will someone explain to me what a representational view of mind in math education might be? I just learned what it is last night! Now I've forgotten! I'll look it up.... -- CatherineJohnson - 31 Jul 2005
| WebLogForm | |
|---|---|
| Title: | smart constructivism |
| TopicType: | WebLog |
| SubjectArea: | ConstructivistTeaching |
| LogDate: | 200507291511 |