From: jac@ds8.scri.fsu.edu (Jim Carr)
Newsgroups: k12.ed.math
Subject: Re: math enrichment at elementaray level
Date: 28 Sep 92 16:07:39 GMT
Reply-To: jac@ds8.scri.fsu.edu (Jim Carr)
Organization: SCRI, Florida State University

In article <3577.2AC627B3@puddle.fidonet.org> Dave.Hunter@p0.f23.n105.z1.fidonet.org (Dave Hunter) writes:
>I'm teaching math at the middle level and have had occasion to meet 
>a 5th grader from one of our feeder schools who, in the short 
>conversation I had with him appears to have a very good conceptual 
>and practical grasp of most of elementary algebra and many geometric 
>concepts.  His math skills and interests are beyond most of his teachers 
>and will no doubt surpass mine before too many years pass.

I was such a kid, so I can imagine what he feels. 

My lucky breaks were twofold: (1) we had an "advanced instruction" 
program in 4th and 5th grade that offered some very nice abstract 
problems in addition to what I was doing in my Dad's college 
algebra book (2) my HS geometry teacher recognized my talent and 
set it up so that I could double in algebra/trig and take calculus 
at the JC my senior year.  Without this, I suspect I would be a 
dropout wino intellectual in some college town somewhere.  

My biggest problem was that junior high was a waste of my time.  We 
had a good honors program, but it was all stuff I already knew and 
none of it was real mathematics.  If my parents had not respected the 
education establishment so much, I think there would have been more 
opportunities.  It sounds like this kid is lucky to have found a 
mentor at a very early stage.  

Do not forget that mathematical power comes early and starts to  fade 
early, often by the age of 20.  He should be allowed to move at his own 
pace, since it is a pity to waste his powers on proving the same old 
things if he is capable of proving new things.  Too much pressure to 
move fast can be bad psychologically (unless he is the type to thrive 
under such pressure), but not letting him move on is worse.  

>I'd welcome any ideas on how to help this guy explore mathematics. 
>I've got to say that its a new experience for me to listen to a person 
>who looks and acts like a normal fifth grader in other respects talk 
>about math with understanding and excitment that surpasses many high 
>school kids I've worked with.

The best way would be to help him explore real mathematics, abstract 
reasoning and logic in particular.  I remember how much fun I had 
in 4th grade with a self-teaching text (new stuff back in 1962, this 
was probably a prototype from some Ph.D. project) that covered the 
Egyptian number system.  Some of these things are taught today, but this 
had not been watered down to my grade level.  There are some great 
books out there (see below). 

>I'm after two kinds of ideas:  1)  What can his present teachers do 
>(other than hand him a high school text and say "go for it") to expand 
>his math exposure.  2)  Any ideas on what I can be lining up for him 
>next year.

I hope his present teacher "does no harm".  I remember being given various 
types of busy work -- writing numbers from 1 to 10 million or so (what 
an awful waste that was) -- and reading library books (not so bad, since 
I have encyclopaedic knowledge now, but did not develop my talent) to 
keep me out of trouble.  (I had been in a split 4th/5th class and so I 
had learned all of 5th grade math in 4th grade.)  

I recommend he be allowed to learn algebra and trig (HS or college text) 
and calculus at his own pace.  He should also be exposed to abstract 
algebra and number theory at some point.  One good way to see if he 
is interested in the latter would be to find a copy of _1-2-3 Infinity_ 
by George Gamow, or perhaps _G\"odel Escher Bach_ if he has reading and 
reasoning skills at that level.  Maybe in a year or two.  He may end up 
wanting to read about Georg Cantor, for example, or Evariste Galois.  

 [Galois should be the 'hero' of any teenage mathematical genius.  ;-) ]

For next year, when he enters your school, you should look into what can 
be done to test him through the middle and high school courses at his 
own pace and get him into college level material.  This might mean 
finding a way for him to cover two or three years of material per year, 
and would definitely require coordination with your principal! Best if 
a university is nearby, but JC level calculus would not be bad as a 
next move.  Some very good math students have started work on upper 
division or grad level courses by 16.  He can do a lot on his own, but 
direct work with a mathematics professor/mentor would be the best if 
he proves to have real talent.  

A very good set of logic problems exist at the back of any collection of 
all of the works of Lewis Carrol.  These are kid oriented and, like 
1-2-3-infinty, can be a lot of fun to read and work on.  

>Also, if you are working with a kid of similar abilities, it would 
>be great for us to get them in touch via this network.  One of the 
>be needs he expressed to me was the feeling that he had no one to 
>talk to about the things that interested him!!

I can relate to this!  It can be hard to find *adults* to talk to about 
sufficiently abstract mathematics, even at age 10.  We have a 10th grader 
here working (for the second year) as an equal partner in a human genome 
research project involving computer simulations; also a 16 year-old 
sophomore working on a project involving chaos in brain waves while 
pursuing a triple major.  These kids already have publications!  Their 
parents are generally clueless about what they are doing, but supportive. 

--
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)