However ...Re: [Squeakland] Panel discussion: Can the
American Mind be Opened?
alan.kay at vpri.org
Mon Nov 26 06:33:04 PST 2007
Hi Bill --
I think the main thing in teaching "number" is to distinguish it from
"name" or "numeral" -- and I think the rush towards teaching "base 10
numerals" too early is one of the big problems in early elementary
mathematics. Numbers are ordered ideas that can be put in
correspondence and taken apart and recombined at will. Names and
numerals are symbols for these ideas that have varying degrees of
usefulness for different purposes. So one of the biggest questions
any math educator should ask is: what symbols should I initially
employ for numbers to help children understand "number" most throughly?
Most "child math" experts - like Mary Laycock, Julia Nishijima, etc.
- would argue that a wide variety of analog (both unary and
continuous) representations should be employed together (bundles of
sticks, bags of objects, lengths of stuff, etc.), and each of these
can have several labels attached ("one", 1, etc.). These can stay in
use for much longer than is usually done in school. E.g. Some really
great "adding slide rules" can be made from rulers, and then the
children can make some really detailed large ones (even using their
playground for baselines). These adding slide rules can add any two
numbers together very accurately, whether "fractional" or not, and
they can have scale changes to reveal what is invariant about two
numbers (their ratio), etc. This can be used to make multiplication
Another use of number that uses names in a non-destructive way is the
"equality game" of "how many ways can you make a number". First
graders are very good at this an even though they don't know what
"1000000" stands for (except it is large) they understand that they
can make this or any other number many ways by a combination of
additions and subtractions that add up to zero. This is a way to
start algebraic thinking without needing variables. And so forth.
Wu actually makes a point against himself when he argues that phonics
decoding is a good idea, even though no fluent reader decodes. This
is similar to how sight reading is taught, especially for keyboards.
Eventually the pattern results in a direct hand shape and mental
"image" of the sound (or for text reading, a mental image of the
idea). The question is how to get there, and teaching how to decode
seems to help a little in early stages (maybe even just for morale
purposes) rather than trying to teach either like Chinese characters.
It takes 2-5 years to get fluent at such learning, so there usually
need to be other supporting mechanisms (not the least is material
that can be dealt with successfully after a few months or a year).
So, what Wu should be asking is "what framework do children need to
get started in number and mathematical thinking about number?".
Another interesting example of what is not happening came out in a
Mary Laycock workshop in which I was a "floor guy" (literally since I
was on the floor with the children). One of Mary's games was to hand
out a series of sheets of 10 by 10 squares, each divided in regions,
with the question, "how many squares are in each region?" The 4th-5th
grade children start by counting the squares in the regions. As the
regions got more complicated, the children did not see that they
could switch over to geometrical reasoning -- to see what fraction of
the whole was occupied by each region and then divide -- instead they
kept on trying to count the little squares and fractions of squares.
Children who had learned to think mathematically would have had a
strategy to look for the best representations for the problems, and
these children had not acquired many (if any) math meta-skills.
To bring up a musical analogy again ... one of the best collections
of advice about how to teach children to play the keyboard is in
Francois Couperin's 1720 treatise "The Art of Playing the
Harpsichord". First, he says, keep the children away from the
harpsichord because it isn't musically expressive enough. And keep
away from sheet music because it "isn't music". Instead, take them to
the clavichord (loud, soft, and pitch modulation -- more expressive
than a piano) and teach them how to play some of their favorite songs
that they like to sing, and help them be as expressive in their
playing as their singing is. This is music. Play duets with the
children, etc. After they have done this for a sufficient time (from
6 months to several years), then you can introduce them to the
initially less expressive harpsichord (which, like the organ, can
only be expressive through phrasing). But they will have learned to
phrase very naturally from their clavichord experience and this will
start to come out in their harpsichord playing. Finally, now that
they have learned to "talk" (my metaphor), they can learn to read.
Now they can be shown the written down forms of what they have been
playing. And now they can start to learn to sight read music.
When I was teaching guitar long ago, I used this basic scheme as much
as possible, because "real guitar" has to be both music and
"attention out" (so that you can mesh musically in a conversation
with other musicians). Also, the guitar has some serious physical
problems which have to be addressed gradually over weeks and months.
Getting the students to play real stuff while all this is going on
makes a foundation for the next level of much harder work. Learning
to play patterns by ear allows the player to concentrate on their
musicality and accuracy. Then they can be shown the patterns as both
shapes and as decoded mappings in members of a key, etc.
The egregiously misunderstood Suzuki violin method also follows these
ideas. (It isn't mechanical -- read his books.)
Couperin's essay is a pretty good set of distinctions concerning the
general confusions between art and technique, and between ideas and
media. You eventually have to get to all of these, but leading with
art and ideas tends to preserve art and ideas, and leading with
technique and media tends to kill art and ideas. I think it is really
At 03:47 AM 11/25/2007, Bill Kerr wrote:
>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
>Some form of effective kitchen maths needs to come before algorithms.
>At this stage I'm left with more questions than answers.
>On Nov 25, 2007 3:28 AM, Alan Kay
><<mailto:alan.kay at vpri.org>alan.kay at vpri.org> wrote:
>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 -
>>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
>>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:
>>BASIC SKILLS VERSUS CONCEPTUAL UNDERSTANDING
>>A Bogus Dichotomy in Mathematics Education
>>BY H. WU
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