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 machines, etc.

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 that simple.

Cheers,

Alan

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
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.

--
Bill Kerr
http://billkerr2.blogspot.com/




On Nov 25, 2007 3:28 AM, Alan Kay
<alan.kay@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.




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.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.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.pdf





BASIC SKILLS VERSUS CONCEPTUAL UNDERSTANDING



A Bogus Dichotomy in Mathematics Education 



BY H. WU




cheers,



-- 



Bill Kerr




http://billkerr2.blogspot.com/