🚀 go-pugleaf

In article <l51jv0hvrqbedbchhbjomt34p411aasa1n@4ax.com>,
 <racqueteer@killspamverizon.net> wrote:
>On 27 Jan 2005 13:49:21 -0500, hrubin@odds.stat.purdue.edu (Herman
>Rubin) wrote:

>>In article <jay-0EC958.09290427012005@news.stanford.edu>,
>>Jay Kadis  <jay@ccrma.stanford.edu> wrote:
>>>In article <ctb6i3$1stu@odds.stat.purdue.edu>,
>>> hrubin@odds.stat.purdue.edu (Herman Rubin) wrote:

[snip]


>>>> Even this is not enough, as few teachers understand the
>>>> basics of their subjects, but proceed by rote.  
>> -- 

>>>OK, got some statistics on this?

>>How would one get statistics on this?

>Why would someone who teaches statistics think that his opinions are
>fact in the absence of any objective data?

Statistics is the theory of decision making under
uncertainty.  If you think it is a collection of
recipes, you are sure to misuse it.  One of the
results is that objective procedures are at best
reasonable in situations where there is plenty of
the type of data needed for them.

>>I have taught a few prospective teachers, and my late wife 
>>taught many of them.  Often coming home after a day of
>>teaching these, she complained of illness from that.  Not
>>all of them were bad in the advanced classes, but the one
>>time she taught prospective elementary teachers in a required
>>class, it was miserable.

>What did she teach?

Mathematics.  Her research specialty was set theory and
logic, in why I also worked, and am known for my work.
These are the "basics of the basics".  I can assure you
that I have found good use of understanding those abstract
concepts, which someone getting the typical doctoral 
program will no longer see, in the "applied" subjects
of probability and statistics, and even the applied aspects.

>>The best piece of clear evidence we have is the "new math"
>>debacle of more than 40 years ago.

>Here we go again..  Something no one wanted except the mathematicians
>who were pushing it.  Something OTHER mathematicians argued
>strenuously AGAINST.

One can get along in some fields without knowing the basic
concepts, except intuitively.  In particular, classical
analysis and related areas are those in which the properties
of the integers are rarely used, except for induction, which
they can easily use, but do not recognize as basic.  The
basic properties are properties of COUNTING, not computing.

Euclid reduced geometry to five geometric axioms, but used 
"arithmetic" axioms.  Peano reduced the integers to that
number of axioms or fewer, using nothing but logic.  

Unfortunately, the new math people were more familiar with
the cardinal concepts of integers, which are somewhat more
apparently simple, but really harder to use.  However, not
that much harder, and they developed and tested it on
tens of thousands of children before getting it adopted.
Nobody thought that such an intuitively obvious approach
would cause problems for the teachers.  But while all
mathematicians, including those against it because they
believed that knowing how to calculate was more important,
would have had little problem with it as "intuitively obvious",
the teachers could not replace the computational intuition
with cardinal number intuition, or grasp the formalism.

	 Something PARENTS strongly objected to.

There are those parents who think they know the elementary
material when they do not.  I suspect that we still have
parents who learned by "Dick and Jane", and never really
got the idea of phonics, who object to that.  Or learning
foreign languages, saying (this is no exaggeration) that
knowing English, which is the language Jesus spoke, should
be sufficient.

		Forced
>down teachers' throats from above.

Lots of bad things were so forced, such as teaching reading
by the whole word method, ignoring spelling and grammar
errors, teaching social adjustment, holding bright children
back, lowering the level of their classes to the "least
common denominator".

 		Elementary teachers, with
>essentially NO love of math, and (usually) limited math skills

The "skills" are irrelevant.  This is one of the problems with
the teachers, and this is why the problem will only get worse
until it is reversed.  Concepts may help in understanding skills,
but rarely is the converse true.

The old high school mathematics curriculum, with its formal
geometry course, helped.  We really need the understanding 
much earlier, BEFORE the computations.  Those mathematicians
who do not see this were among the few who were able to
incorporate, at least intuitively, the concepts to a sufficient
extent, and there is still quite a bit of research into methods
of computing answers to standard problems which has not been 
done to enable them to flourish.

A big mistake was made by the leaders of the new math movement;
when the teachers could not get the ideas, they weakened the
program assuming that the teachers could handle that.  Instead,
they should have gone public, and said,

	Your children can learn this; their teachers cannot.
	Regardless of how mathematics should  be taught, 
	should THESE be teaching them?

			being
>poorly trained over (maybe) a summer on math concepts and then
>expected to impliment AGAINST the wishes and objections of parents.

You cannot TRAIN someone to understand concepts.  Twenty
years of training will not accomplish this.  Twenty minutes,
or even less, would be adequate to get the concept across;
the details then have to be developed.  A child who can 
understand the alphabet (not even know it all) with normal
intelligence can understand the concept they used involved.

	If one can match the elements of sets 1-1, the
	sets have the same number of elements.

	If two sets have nothing in common, the union of
	the sets (all those elements which are in at least
	one of them) has the number of elements equal to
	the sum of the numbers of the sets.

I will not write down the definition for product, but it
is also not difficult.

There is much which remains to be done, and this requires
knowing what is a proof; it is a sequence of statements,
each of which is an assumption, or follows from the 
preceding ones by using a set of rules; the old geometry
course, with statements on one side, and reasons on the
other, is a good example.

This is what they could not do; they could not give up 
the idea that a number is a string of decimal digits,
with the usual algorithms for addition and multiplication.

You have given an excellent argument against a state-run
school system.  Things were bad enough then, but at that
time, the educationists had not done that much damage to
the high school curriculum.  We are at the point where we
have to say that it is no longer possible.

>Yeah, fat chance THAT had of going anywhere.  It's chief cheerleader
>argued against rushing it into wide usage!

He was not a mathematician, but a mathematic educator.

		 For Pete's sake, Herman,
>it's been 4 decades - Get over it!  It was of questionable value,

We could have done better, by using ordinals to start out.

This is one of the problems; there can be multiple concepts
connected with the same entity.  If you ask me to define 
"one", I cannot do it, as it has many meanings.  However,
the positive integers behave the same way for all of these.

>rushed into play FAR too quickly, and essentially unsupported (too
>little training over FAR too short a time, and no effort made to "sell
>it" to anyone.)

Even the educational establishment did not foresee that the
teachers would not have found it "obvious".  It is they who
were "sold", as they had been sold on so many other ideas 
which unfortunately were implemented.

		 It was just one more thing in an endless string of
>"good ideas" forced into play by people who didn't have to make it
>work.

Sound mathematics cannot be introduced gradually; neither can
the use of sound grammar or sound science.  With a much more
abstract approach than the new math used, we can teach those
capable of learning it, more mathematics than is now taught
through high school, and most of the concepts in a good 
college program, in elementary school.  One can start science
with a sound foundation, instead of with leaf collections, etc.
Language learning can be speeded up bay a factor of nore than
two by using grammar, which scientists have shown is learned 
by young (much pre-school) children.

The typical high school graduate 50 years ago who took the 
college preparatory program was ready to learn good mathematics.
Today, even with adding calculus in high school, the answer is
no way.  I question whether the typical student with a BA in
mathematics is ready to get into thinking conceptually; their
preparation is worse, and they only want to be taught how to
get answers.  I include today's PhD students in mathematics
and statistics in that group.

-- 
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