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Like anything else, constructivism can be over-applied, though it certainly does have a lot of strengths. I read your blog post. I don't think it's accurate to make the sweeping statement, "Once a child learns an algorithm it is as if mathematical thinking stops". In addition I think the "fastest and most efficient way for [a] specific brain to work" will be more and more often a pre-defined algorithm as a child starts learning advanced math, since not all children will unerringly find the most efficient solution to a problem left to their own devices.
Lucounu - Well said; your description matches our personal experience. Unfortunately, the people we've spoken to in our school district seem a bit dogmatic when it comes to constructivism and math, and don't seem to think this way...
DH would go as far as suggesting some kids might get frustrated with the expectation that they rediscover the algorithms on their own, knowing that the grown ups were holding back known, efficient ones.
Me three. The key is the false dichotomy implied by contrasting constructivist teaching, in which students discover their own algorithms, with "blindly teaching algorithms". It is possible to sightedly teach algorithms :-) As a professional mathematician I have to develop ways to solve problems nobody yet knows how to solve, but if I insisted on solving for myself problems that do have known solutions, I'd never get to the new stuff. I've banged on before about how important it is for children to wrestle with mathematical problems that are hard for them, and doubtless I will again, so I appreciate the motivation behind constructivism, but I don't like the dogma.
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Good points amylou, ColinsMum. As usual I blathered on without much structure, but I think one key point is this: math students eventually have to learn how to take in prepackaged information, and do that efficiently and with as full an understanding as possible. It's like reading comprehension, but math-model comprehension. I don't think we'd get very far in any scientific field if people weren't well trained in communicating with each other.
We've also seen some posts here on how early math frustration led to an abandonment of math as an interest. But barring that, I'm guessing that one could be fairly inefficient with lots of bottom-level arithmetic concepts and procedures, or even fail to understand completely why they work as they do, and still be able to think abstractly at a high level very well.
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My comment about thinking shutting down once a child had been spoon fed an algorithm comes from what I experienced teaching third and fourth grade.
I would have (average) students, who were just beginning to really develop some number sense and mathematical understanding, and then they would go home and their parents would teach them to borrow and carry. The kid would come back to school and be totally confused, grasping onto the algorithm like a life raft and unable to think any deeper.
As a teacher I found that it was often the parents with the weakest mathematical thinking themselves, that are so passionately resistant to Constructivism. In turn their children were the ones who would be still trying to cross out the one in problems like "101-99 = ?" The children of engineers, mathers or scientists usually had a leg up, because they had been talking about numbers in a global sense, since they were very little.
Regarding Roman numerals... Yes, sometimes it is easier to just teach something directly. As with any educational philosophy, you can't be too hard-core all of the time. But on the whole, I think it is best to lean into children making learning discoveries for themselves, with you as the teacher or parent providing lots of structure and guidance to make this happen.
Jen, I'm concerned that you may be feeling a bit jumped on here, which is not how any of us want new posters to feel, I'm sure, but you need to understand that many people here have experienced the frustration that results when teachers with lots of experience teaching average children think that means they know better than we do how our children's minds work. Since you have a gifted child yourself I'm sure that isn't what you mean. How about you tell us some more about your child(ren?) and about what it is you're hoping to get out of the forum? And welcome!
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I want the life of Fred books, so many posts say so many kids really enjoyed reading them. Besides, it's supposedly great stories that features a precocious child doing math at a younger age than is often depicted. OTOH, hands on equations sells a set of manipulatives http://www.borenson.com/ that's a mini balance beam designed to illustrate algebra and teach what an equal sign means, because supposedly Americans don't know: http://kitchentablemath.blogspot.com/2011/05/equation.html?m=1 (I like the quote that says "quit trying to make it about manipulatives vs. Pencil work"), ie., use your words. Er, well I guess if you're into explaining then the visuals could just as easily be 2 or 3D. So, Jenbrdsly, how different is constructivism from unschooling? ?
I also like this lady on YouTube. She teaches structured drill. It's free. Use a whiteboard. Check out this video on YouTube:
Youth lives by personality, age lives by calculation. -- Aristotle on a calendar
Jen, I think we're actually closer than I thought to being on the same page, although obviously there's no need for us to be. I would call the steps of writing lines here and there, the mechanics of carrying, a "procedure" or something similar, but didn't think you were referring to that specifically by using the word "algorithm" (ex-computer science guy here). But I totally get what you mean and agree. Focusing on writing and rewriting symbols in particular ways could obscure understanding, especially if a kid's not taught the proper concepts in the first place.
Striving to increase my rate of flow, and fight forum gloopiness.
Sorry to get in the middle here...but I saw somewhere online a parent had taken exception to some of the material that is in the Fred books. Any one know what that might be about? I got the feeling it was about exposing kids to questionable material...
Fred, a 5yo, lives in his office at the university, where he's lived since he was 6 months old, carted around by various undergraduates when he was too young to walk. His parents are nowhere to be found, and he lives off of vending machine food, which is why he's so short. He also tries to open a box with a very sharp, very heavy knife, and nearly bleeds to death before Roman numerals save him.
If you have a working-on-5th-or-6th--grade-math child who might conclude from that that living like Fred is desirable, Life of Fred might not be for you.
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