From: jac@ds8.scri.fsu.edu (Jim Carr)
Newsgroups: k12.ed.math
Subject: Re: Three points about times tables (was Re: MOTIVATION, LACK OF . . .)
Date: 25 Sep 92 23:52:58 GMT
Reply-To: jac@ds8.scri.fsu.edu (Jim Carr)
Organization: SCRI, Florida State University

In article <19lr21INNd19@agate.berkeley.edu> bh@anarres.CS.Berkeley.EDU (Brian Harvey) writes:
>
>It's a subtle distortion to say "mathematics is (in part) about
>multiplication," which is true, as if it meant "mathematics is about the
>results of specific multiplication problems," which is not true.  It's
>not just a matter of elementary versus advanced topics.  The same thing
>could be said about calculus.  Mathematics is (in part) about the idea of
>taking integrals.  It's NOT about knowing the integrals of specific functions.

Ah, but it *is* about knowing how to *obtain* the integral of a function. 
Given what I consider a "knowledge" of calculus, someone should be able 
to obtain (either by derivation or memorization, but if memorized, should 
know enough of the rules to double-check that the answer one remembers 
is actually correct) the integral of a fair range of functions. 

This is, actually, a rather curious example because the number of integrals 
one needs to know to do engineering, chemistry, or physics is much smaller 
than the number of facts in the times tables.  The basic functions are 
like the times tables, and the rest is like multi-digit multiplication 
with carries etc etc.  It is rare to encounter anything beyond x^n, 
sin{x}, cos{x}, or e^x in introductory courses, and after that you 
use the computer.  But if you use the computer and do not know some 
elementary facts, you will never know when your answer is garbage. 

Actually, this is a bit off the point, but my biggest problem (as someone 
who has taught both calculus and physics) is that calculus is taught as 
a theoretical "mathematics" subject and not as a practical "arithmetic" 
subject in the first go-round.  The result is a group of students who 
fail because they are not abstract enough in the Piaget sense to handle 
the theory, but only needed to know a few elementary rules (that are 
almost the equivalent of addition) to do engineering problems, and a 
group that passes the class but has no retention of the basic facts 
and has to learn them all over again.

>The danger of emphasizing "number facts" is not merely that the kid might be
>bored right at the moment.  If that were all, I'd agree that kids could be
>taught to be self-disciplined a little.  No, the danger is that the kid will
>learn to think (as MOST adults, including MOST school teachers, think) that
>mathematics is entirely about calculations.  Most people think that
>mathematicians are sort of like actuaries; they sit around all day adding
>up columns of numbers.  If a kid learns to think that, how is s/he ever 
>going to get interested in joining the profession?

At the elementary level, and we are talking about skills that should 
be known before leaving elementary school, it is more important to 
know the skill.  Mathematicians have that skill.  A student who has 
interest and lacks the skill will never be a mathematician or an 
actuary.  (Also a curious example, since many BS math grads who do not 
go on to grad school do get jobs as actuaries!)  Of greater concern 
is the student who wants very badly to be a doctor or an engineer 
and will not succeed because of an inability to work with numbers. 

>If you are going to support the torturing of kids with arithmetic facts,
>you have to own up to the fact that the overwhelming result of your
>position, which is the position taken by most schools, is that MOST
>kids and MOST adults HATE math.  If that's what you want, teach
>arithmetic.  If it's not what you want, you'd better be more flexible.

If you think it is torture, the students will also.  They pick those 
things up quickly, and I think many students are turned off to math 
and science by teachers who are not comfortable with either one.  
Even a 9-year-old kid can ask questions that challenge the explanatory 
skills of persons with a BS or PhD in the subject area.  

--
J. A. Carr                                    |  "The New Frontier of which I  
jac@gw.scri.fsu.edu                           |  speak is not a set of promises
Florida State University  B-186               |  -- it is a set of challenges."
Supercomputer Computations Research Institute |   John F. Kennedy (15 July 60)
Newsgroups: k12.ed.math