Hi Bill --
I just read Professor Wu's paper. I agree in the large with his assertion that the dichotomy is bogus, but I worry a lot about his arguments, assumptions and examples. There are some close analogies here to some of the mistakes that professional musicians make when they try to teach beginners -- for example, what can a beginner handle, and especially, how does a young beginner think?
Young children are very good at learning individual operations, but they are not well set up for chains of reasoning/operations. Take a look at the chains of reasoning that Wu thinks 4th and 5th graders should be able to do.
Another thing that stands out (that Wu as a mathematician is very well aware of at some level) is that while people of all ages traditionally have problems with "invert and multiply", the actual tricky relationship for fractions is the multiplicative one a/b * c/d = (a * c)/(b * d) which in normal 2D notation, looks quite natural. However, it was one of the triumphs of Greek mathematics to puzzle this out (they thought about this a little differently: as comeasuration, which is perhaps a more interesting way to approach the problem).
A few years ago I did a bunch of iconic derivations for fractions and made Etoys that tried to lead (adults mostly) through the reasoning. One of the best things about the divide one is that it doesn't need the multiplication relationship but is able to go directly to the formula. So these could be used in the 5th grade.
But why?, when there are much deeper and more important relationships and thinking strategies that can be learned? What is the actual point of "official fractions" in 5th grade? There are many other ways to approach fractional thinking and computation. I like teaching math with understanding, and this particular topic at this time - and provided as a "law" that children have to memorize - seems really misplaced and wrong. Etc.
Cheers,
Alan
At 05:53 AM 11/24/2007, Bill Kerr wrote:
David:
Further, but perhaps drifting off topic for squeakland, is it provable that 'back to basics' and 'progressivism' are equally as inadequate?
Alan: I said above that the simplistic versions of both are quite wrongheaded in my opinion. If you don't understand mathematics, then it doesn't matter what your educational persuasion might be -- the odds are greatly in favor that it will be quite misinterpreted.
David,
I read the original maths history http://www.csun.edu/~vcmth00m/AHistory.htmlhttp://www.csun.edu/~vcmth00m/AHistory.html that prompted your initial questions about constructivism and agree that it critiques the cluster of overlapping outlooks that go under the names of progressivism / discovery learning / constructivism - fuzzy descriptors
But more importantly IMO it also takes the position that the dichotomy b/w "back to basics" and "conceptual understandings" is a bogus one. ie. that you need a solid foundation to build conceptual understandings. The problem here is that some people in the name of constructivism have argued that some basics are not accessible to children. (refer to the H Wu paper cited at the bottom of this post)
I think the issue is that real mathematicians who also understand children development ought to be the ones working out the curriculum guidelines. This would exclude those who understand children development in some other field but who are not real mathematicians and would also exclude those who understand maths deeply but not children development.
This has not been our experience in Australia. I cited a book in an earlier discussion by 2 outstanding maths educators documenting how their input into curriculum development was sidelined. National Curriculum Debacle by Clements and Ellerton http://squeakland.org/pipermail/squeakland/2007-August/003741.htmlhttp://squeakland.org/pipermail/squeakland/2007-August/003741.html For some reason the way curriculum is written excludes the people who would be able to write a good curriculum -> those with both subject and child development expertise
For me the key section of the history was this: "Sifting through the claims and counterclaims, journalists of the 1990s tended to portray the math wars as an extended disagreement between those who wanted basic skills versus those who favored conceptual understanding of mathematics. The parents and mathematicians who criticized the NCTM aligned curricula were portrayed as proponents of basic skills, while educational administrators, professors of education, and other defenders of these programs, were portrayed as proponents of conceptual understanding, and sometimes even "higher order thinking." This dichotomy is implausible. The parents leading the opposition to the NCTM Standards, as discussed below, had considerable expertise in mathematics, generally exceeding that of the education professionals. This was even more the case of the large number of mathematicians who criticized these programs. Among them were some of the world's most distinguished mathematicians, in some cases with mathematical capabilities near the very limits of human ability. By contrast, many of the education professionals who spoke of "conceptual understanding" lacked even a rudimentary knowledge of mathematics.
More fundamentally, the separation of conceptual understanding from basic skills in mathematics is misguided. It is not possible to teach conceptual understanding in mathematics without the supporting basic skills, and basic skills are weakened by a lack of understanding. The essential connection between basic skills and understanding of concepts in mathematics was perhaps most eloquently explained by U.C. Berkeley mathematician Hung-Hsi Wu in his paper, Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education.75"
Papert is also critical of NCTM but is clearly both a good mathematician and someone who understands child development - and has put himself into the constructivist / constructionist group
I followed that link in the history to this paper which is a more direct and concrete critique of discovery learning taken too far, with well explained examples of different approaches:
http://www.aft.org/pubs-reports/american_educator/fall99/wu.pdfhttp://www.aft.org/pubs-reports/american_educator/fall99/wu.pdf
BASIC SKILLS VERSUS CONCEPTUAL UNDERSTANDING A Bogus Dichotomy in Mathematics Education BY H. WU
cheers,
Bill Kerr http://billkerr2.blogspot.com/http://billkerr2.blogspot.com/
Good discussion :-)
To be honest I've never been certain about the best way to teach "number" and have tended to try a smorgasboard in practice
Perhaps Alan is correct David?
Professor Wu (good mathematician) is making a brave attempt to make the teaching of algorithms to young children more concrete but his approach still puts too many demands on most children. From my experience of teaching of maths I feel that for disadvantaged students too many eyes would glaze over for some of the steps. It might work for his children but not for 90% of children.
I still feel that he makes some valid points and criticisms. I like the transparency and open-ness of his paper, as well as the conceptual position put by its title.
Another paper by Ellerton and Clements identifies the main issue as this: "... many children who correctly answered pencil-and-paper fraction questions such as 5/11 x 792 = q could not pour out one-third of a glass of water, and of those who could, only a small proportion had any idea of what fraction of the original full glass of the original full glass of water remained" - Fractions: A Weeping Sore in Mathematics Education http://www.aare.edu.au/92pap/ellen92208.txt
Some form of effective kitchen maths needs to come before algorithms.
At this stage I'm left with more questions than answers.
squeakland@lists.squeakfoundation.org
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Alan Kay -
Bill Kerr