Xref: utzoo comp.edu:1828 sci.math:5324
Path: utzoo!utgpu!watmath!clyde!att!rutgers!mailrus!iuvax!pur-ee!j.cc.purdue.edu!mentor.cc.purdue.edu!l.cc.purdue.edu!cik
From: cik@l.cc.purdue.edu (Herman Rubin)
Newsgroups: comp.edu,sci.math
Subject: Re: the why of math
Summary: Formalism is not understanding.  The WHY is not the proof.
Message-ID: <1088@l.cc.purdue.edu>
Date: 11 Jan 89 18:54:22 GMT
References: <605@ucrmath.EDU> <6578@killer.DALLAS.TX.US> <783@odyssey.ATT.COM>
Organization: Purdue University Statistics Department
Lines: 129

In article <783@odyssey.ATT.COM>, gls@odyssey.ATT.COM (g.l.sicherman) writes:
> In article <6578@killer.DALLAS.TX.US>, elg (Eric Green) writes:
> > in article <605@ucrmath.EDU>, marek@ucrmath.EDU (Marek Chrobak) says:
> > > Xref: killer comp.edu:1858 sci.math:5231 sci.physics:5475
> > > a proof. In Herman's terms, the question WHY does not cross
> > > their mind. They just want to know HOW. ...
> > > It would be silly to blame the students for this. This is the
> > > way they have been taught in school, it's no wonder this is
> > > what they expect from college. 
> > 
> > Sounds like you're blaming the teachers. You shouldn't. They're doing
> > the best they can, with what little knowledge they have. Blame college
> > curriculums which do not include any "Basics of Mathematics" courses,
> > only tons of courses in equation manipulation (Algebra, Trig, ...

Yes and no.  I am not convinced that the teachers are even capable of 
understanding after their "training".  But the "Basics of Mathematics"
course, as usually taught, are pure formalism without understanding, and
are usually taught mechanically, partly not to aggravate the students, and
partly because of the attitude, "These students cannot learn to think.  
Therefore, let us teach them the manipulations, so they learn something."
In addition, many mathematicians are only subconsciouly aware of the concept,
and take the attitude that the only intuition is geometric.

The Peano postulates (really due to Dedekind) contain the essence of the
structure of the integers.  They can be understood by first graders if 
appropriate language is used.  The method of "defining" integers by 
cardinal equivalence is bad; it is not at all easy to see the structure.
But after the structure is there, a child can be shown that cardinal classes
satisfy the properties.  Not with a formal proof; do not mistake completeness
(proving everything from first principles) with rigor (doing things right the
first time, so nothing has to be unlearned).

> I don't see any reason to cast blame.  Engineers are by far the
> biggest "consumers" of mathematics, and they need techniques rather
> than derivations for nearly all their work.  If they are interested
> in the derivations, they can take "honors" mathematics.  But I would
> rather cross a bridge built by an engineer who knows the formulas
> thoroughly than by one who knows some of the formulas and can derive
> the rest!

Engineers do not need the knowledge of the formulas, not the logical
derivation of them.  First and formost, and engineer needs the concepts. 
An engineer needs to know what a derivative and integral are, and how to
formulate a differential equation to express a physical problem.  It is
important that the problem of the bridge is set up correctly by the 
engineer.  Whether he knows how to solve the problem is relatively
unimportant; if the problem is correctly formulated and has a solution,
a mathematician can find the solution.  But if the engineer only formulates
problems that he knows how to solve, I am worried about the safety of the
bridge.

> There is also a philosophical issue here.  What we call "foundations"
> of mathematics are really built on top of practical mathematics, by
> cumulative abstraction.  Indeed, an application of mathematics may have
> alternative foundations; for example, calculus can be founded on Weier-
> strass logic or on nonstandard arithmetic.  It's something like the
> physicist's alternatives of Maxwell's equations or Lorenz's transform;
> either will serve as a model.  But mathematicians should beware of
> believing in their models!  There is no "why" of mathematics except
> in the art of the mathematician.  For example, why when you add two
> numbers does the order not matter?  A typical schoolteacher would
> answer "because addition is commutative." That's not an answer!

I question whether there is such a thing as "practical mathematics."
All of mathemmatics is a construct; there is mathematics which has been
applied, and mathematics which has not yet been applied.  In many cases,
the first applications of mathematics occurs decades after the mathematics
has been developed for its own sake.  But the formalism of mathematics is
not the "why".  The axiomatic development by Dedekind was at least partly
for the purpose of understanding.  Unfortunately, few books and papers give
much of the understanding.

As far as modeling, the mathematician does not produce a model as a mathema-
tician.  This shows up most in applications of statistics.  The user of
methods will apply methods which have many assumptions behind them.  Some
of the assumptions are important and some are not particularly important.
The user is likely to stress the unimportant ones.

The following are my five commandments for clients and consultants.

	For the client:

	   Thou shalt know that thou must make assumptions.

	   Thou shalt not believe thy assumptions.

	For the consultant:

	   Thou shalt not make thy client's assumptions for him.

	   Thou shalt inform thy client of the conseqences of his assumptions.

	For the one who is both (e.g., the biomathematician):

	   Thou shalt keep thy roles distinct, lest thou violate some of the
	   other commandments.

To apply mathematics correctly in science or engineering, one must understand
the concepts to formulate the problem.  There may be consultation with the 
mathematician as to what approximations can be made without messing things up
too much.  But the mathematician must not _as a mathematician_ make any
assumptions about the science involved, not should the scientist make any
assumptions merely because he can solve the resulting mathematical problem.
The scientist needs the why, so that he can make the translation; again, the
how comes into effect only after the translation has been made, and does not
have to be done by the scientist.

> > Is it any wonder why your typical bright and impatient student hates
> > "math"? It's BORING!
> 
> Patience is an effect of typographic culture.  As has been pointed out
> here and in comp.society, new methods are being developed for elec-
> tronic culture.  The real "new math" is yet to come!

I fear that elecronics will be used to emphasize what a computer can
"understand," namely, mechanics.  It can be used otherwise, but I doubt
that it will be.  But our educational system needs to be forced into
realizing that not all students should learn at the same speed, or even
in the same way.

But a teacher who does not know the how can teach it, I do not believe that
this holds for one who does not know the why.  And a student who can figure
out the why with the present teaching, even in first grade, is probably
PhD material.
-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907
Phone: (317)494-6054
hrubin@l.cc.purdue.edu (Internet, bitnet, UUCP)