From: hrubin@pop.stat.purdue.edu (Herman Rubin)
Newsgroups: misc.education,talk.politics.misc,alt.individualism,talk.politics.theory
Subject: Re: High costs of  education
Date: 13 Oct 92 19:30:19 GMT
Article-I.D.: mentor.Bw2rIK.8GJ
Organization: Purdue University Statistics Department

In article <dlhanson.10.718911196@nap.amoco.com> dlhanson@nap.amoco.com (David L. Hanson User) writes:
>In article <Bw0K9z.G6K@mentor.cc.purdue.edu> hrubin@pop.stat.purdue.edu (Herman Rubin) writes:

>>This might be the case if what you have stated was taught in K-12.  Basic
>>mathematics is not taught at all to most of the children, and in many places
>>is not even available to the best students.  Basic mathematics does not
>>consist of learning how to manipulate, and at this time, most high school
>>mathematics teachers do not understand basics.

>>Also, the other subjects you have mentioned are hard to get in the schools.
>>Few ever learn any world history, and even American history is badly done.
>>Civics is likely to be the Demopublican line, and political science is 
>>usually garbage.  Basic science is not taught, but a few trivia and some
>>manipulation and facts.  

>>Now the basic material can, and should, be taught earlier.  I maintain that
>>anyone capable of getting an honest bachelor's degree in a science should
>>receive all the basic material taught in all of the sciences at an under-
>>graduate level now BEFORE the present age of graduating high school.  But
>>this is not going to be done in the near future.


>Dr. Rubin,

>Do you (or any other reader) have a reference for what "should" be taught
>at various ages?  What are above average children capable of in math and
>science with the right kind of instruction and encouragement?  

There is far from agreement on this.  I can give some information on what
I considered reasonable for my children, not that it was totally successful.
I am also using some information which I obtained from Suppes and his group
at Stanford.  What will work in a school environment might not work taught
separately, especially if the child is going to school, and for home
schooling, I suspect the results will be still different.

1.  Development of the understanding of a formal system and the use of
variables.  I cannot overemphasize this last; I would make the mathematics
requirement for entering college the ability to FORMULATE word problems at
the high school algebra level, in many variables.  Knowing how to solve the
problems will not help in understanding how to formulate, and almost all
of the current algebra books make the mistake of trying to minimize the
number of variables used.

There is what I consider good material here.  The book, _Introduction to Logic_,by Suppes, is excellent but suffers from sesquipedalianism.  It might or might
not work by itself; Suppes was able to teach it directly to the top 20% of
6th graders.  There is also a book, _Mathematical Logic for the Schools_, by
Suppes and Hill.  This was intended for the top half of 5th graders.  In some
places, Suppes is easier to understand, and Suppes goes into the predicate
calculus, which Suppes and Hill does not.  In the second half of Suppes'
book, there are also applications to mathematics.

A caution is in order.  As is typical in books, the student is asked to prove
things.  It is not necessary to be able to carry out the process of creating
proofs to understand the concepts and their uses.  This is universally true;
one cannot do certain mathematics without being able to derive, but those who
have difficulty in deriving can still understand.

2.  The structure of the integers, rational numbers, and real numbers.  Here,
I do not know of anything written for this age group, although others may be
able to give some references.  There is a small book by Landau which was
written for graduate students in mathematics at a time when these foundations
had not yet gotten into the undergraduate courses about a half century ago,
and it does have the concepts there.  I do not know how easy it would be to
teach.  Set algebra can be included here, and there are approaches, such as
the "new math", which start out with it.  If the new math materials are used,
use the ones before the watering down.

3.  Arithmetic and algebra.  These can properly be combined, arithmetic being
taught with algebraic notation, already present in part 1.  This should not
take too long with the above background, and should reach the highest levels
of the current high school courses.

4.  A good strong geometry course, at least at the level of the old Euclid
books.  There are better books of this type.  Trigonometry can be added in
a short time.

This is not intended to exclude other topics, but this will give a good
basis for taking honors mathematics courses in college.  Calculus can be
added when and if desired.

As for the sciences, at least get the mathematics through item 3. above
and then use some of the junior and senior honors material, at least in
physics, chemistry, and astronomy.  Astronomy will call for geometry and
spherical trigonometry.  As for biology, the approach still seems to
go by looking at leaves and bugs, instead of looking at biological 
structure using physics and chemistry.

This can be used also in the "social sciences."  There used to be a 
fourth grade physical and political geography course, and a fifth grade
world history course based on it.  Now the old materials are certainly
dated, and I do not know what is available now.  But our views of world
history up to the 1930's have not changed that much, and a knowledge of
world history makes understanding US history much better.

>Have you looked at the Saxon math series for elementary and high school?
>If so what do you think of it?  If you don't like it, what would you 
>recommend?

I am unfamiliar with these.  Considering what the schools teach, I must be
highly suspicious.

>We have home schooled our two boys (ages 9 and 11) for almost a year now.
>We find that particularly in the area of reading and vocabulary we are
>having to fix up a lot of damage done by the public school (we didn't pay
>attention that they were reading very little so we are just as much at
>fault). 

-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@pop.stat.purdue.edu (Internet, bitnet)  
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Newsgroups: misc.education