Re: Math is not a memoriter course.

In article <46CF787D.2050205@xxxxxxxxxxxx>,
Stephen J. Herschkorn <sjherschko@xxxxxxxxxxxx> wrote:
Herman Rubin wrote:

In article <1187966095.414568.184130@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<aatu.koskensilta@xxxxxxxxx> wrote:


Someone wrote:


Do they really believe that it could possibly do more harm to the
student than would be done over a lifetime by not knowing the single
digit multiplication facts?



Why do they have to know the single digit multiplication
facts? I often do base 16 computations, and I have made
no attempt to memorize its multiplication table.

In fact, one the touted methods of teaching, Chisenbop,
only assumes that multiplication by 1, 2, 5, and 10 are
memorized, and uses the distributive law for the rest.
Since multiplying by 5 is essentially dividing by 2,
not too much has to be learned. But the only thing
gained by learning tables is speed; why do we not have
children memorize the tables to 100 or 1000?



My experience is that students who cannot multiply without the aid of a
calculator have all the more trouble with later subjects such as algebra
and statistics. How can one learn to factor polynomials if one cannot
multiply? Very slowly, I guess.

They need to know what multiplication means. If they
know that, it is highly unlikely that they can do some
multiplications. You are right that if they are not
very familiar with the "expected" multiplication facts,
they will factor polynomials by hand rather slowly.
On the other hand, there is often the problem of
factoring polynomials in which hexadecimal arithmetic
turns out to be the key; how may will recognize 6144
as 3x2^11? I would factor x^2 - 6144x + 8388608 quite
quickly.


Only slightly off topic, it still annoys me if, when I ask a student
what half of 0.05 is (e.g., in the process of determining a rejection
region for hypothesis testing), s/he to pull out a calculator to give me
an answer

This does not bother me as much as multiplying .2 by .3
and getting .6. If a calculator is needed to get correct
answers to arithmetic, as long as the problem is understood
and correctly formulated, it does not bother me.

As for hypothesis testing, learning this seems to make the
learning of basic statistical concepts, of which using a
fixed significance level is definitely NOT one, very much
more difficult.



See the above. Napier's "bones" were devices for people
who could add but not multiply. But computers and calculators
are even faster.



There are real problems with mathematics education today -- some quite
widespread, some specific to certain countries --, and it would be
more sensible to concentrate on those, instead of ranting on imaginary
ills.



Students do not understand any of te concepts involved
with the integers, and cannot formulate problems. This
is FAR more important than knowing how to solve standard
problems; anything which is learned about solving problems
below the graduate level can NOW be done by machine.


This is not a matter of either/or. Students need to learn some basic
machinery by heart. And they should understand concepts as well.
Usually, a lack of the former really hinders the latter.


The evidence we have from children who could learn concepts
while their teachers, reasonably adept at numerical calculation,
do not, seems to indicate a problem there. It is seen elsewhere,
where calculus students who have been taught antidifferentiation
as integration find it VERY difficult to understand the real idea
of integration, millennia older than differentiation, and that
the "fundamental theorem of calculus" relates two DIFFERENT types
of objects.

When the old "Euclid" course was a standard prerequisite, and
students learned induction in "college algebra", things were not
quite as bad, but were not good. Now, neither of these is true.
--
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@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

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