alt.education

In article ,
toto wrote:
>On Thu, 20 Mar 2008 20:19:33 -0700 (PDT), Dom
>wrote:

>>Although Lockhart decries the sterile formalism in
>>which mathematics courses have been and continue to be taught, he
>>makes absolutely no reference to the fact that the traditional
>>mathematics curriculum was demolished by the excessive formalism and
>>abstractions of the SMSG new math, as incorporated in the Houghton
>>Mifflin series of books co-authored by Mary P. Dolciani. This apparent
>>ignorance on Lockhart's part is likely due to the fact that he was
>>educated with Dolciani-type books, and he may not be aware of the
>>preceding textbooks.

>Gee, Herman thinks the new math was wonderful, but could not be taught
>to teachers only to students who had never learned to compute in a
>rote way.

I did not say the new math was wonderful, and in fact I
criticized it at the time. It was better than teaching
arithmetic with no idea of what integers were, but the
cardinal approach, which is apparently very simple, is
of some difficulty, and leaves out important parts.

Also, the SMSG texts were not that well formalized, and
the abstractions were only half way abstract. The SMSG
"Euclid" was not recommended, and Lockhart seems to
fail to understand why formal proofs are needed. They
are not often used as such, but an informal proof is
one which can be expanded to a formal one. The idea
of formal proof, freed from the restrictions of geometry,
can easily be taught in elementary school; it has been.
Back in "ancient history", the only real mathematics
course most got was the formal Euclid.

However, teaching concepts to teachers seems to be very
difficult, as a high school head of a mathematics department
found when he asked candidates to prove that 2+2=4, not
caring which proof was used. Some even questioned whether
this was mathematics.

Indeed, the only real weakness I can find in Lockhart's
Lament was that he was unaware of the vast part of
mathematics not reachable by cute methods, and requiring
the need for really abstract thinking. This part of
mathematics is also beautiful, and getting new results
requires a type of intuitive approach about abstract
ideas. Peano's axiomatization of the counting numbers,
and showing that it was categorical, is such, and it
is not something to "lead up to"; we do not need to
reinvent the wheel, and teaching the integers from the
counting approach can be done for kids.

We do not need them to learn arithmetic to get ready
for algebra. Most of high school algebra is adequately
summarized by the general use of variables, not just
for numbers, and the general rule of equality, and
these can be learned as soon as the child can recognize
and reproduce letters. If it is then used to teach
arithmetic, the use of algebra to solve the usual
problems needs just a little practice.

The Greeks were able to carry out recognizable proofs
in geometry because they could use variables for points,
lines, planes, etc. As soon as multiple variables in
general was produced by Viete in the late 16th century,
everything "took off"; it could be written understandably.

We can teach this understanding if we do not swamp it first.

--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558