However ...Re: [Squeakland] Panel discussion: Can the American Mind be Opened?

Alan Kay alan.kay at
Sat Nov 24 08:28:11 PST 2007

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.



At 05:53 AM 11/24/2007, Bill Kerr wrote:
>>Further, but perhaps drifting off topic for squeakland, is it provable
>>that 'back to basics' and 'progressivism' are equally as inadequate?
>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.
>I read the original maths history
>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
>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:
>A Bogus Dichotomy in Mathematics Education
>Bill Kerr
-------------- next part --------------
An HTML attachment was scrubbed...

More information about the Squeakland mailing list