From: hrubin@mentor.cc.purdue.edu (Herman Rubin)
Subject: Re: Is Math Hard?
Organization: Purdue University Statistics Department
Date: Fri, 6 Nov 1992 02:01:12 GMT

In article <1992Nov5.180244.27364@athena.mit.edu> jamess@athena.mit.edu (John A Mess) writes:
>     I apologize this is very long, but it is a response to the numerous
>postings recently criticizing, unjustly, most high school teachers.  

>In article <Bx79Lo.LG1@mentor.cc.purdue.edu> hrubin@pop.stat.purdue.edu (Herman Rubin) writes:
>>In article <1992Nov4.044300.15766@cbfsb.cb.att.com> wa2ise@cbnewsb.cb.att.com (robert.f.casey) writes:
>>>In 5th grade, we did set theory (at least the simple stuff).  At the time,
>>>I couldn't believe that we were doing such easy stuff instead of hard
>>>long division and such.  But it seemed that the teacher and the rest of
>>>the class had a hard time figuring it out....

>>From the experiences when people who had an understanding of mathematics
>>taught it, most of the children could get some grasp of the subject.  But
>>failure of the attempts to teach the teachers was what killed the new math.
>>Teachers who can only teach rote can only do harm to those who want more.

>     This was a very common problem in the 60's and 70's when public educa-
>tors wanted to "modernize" the process.  The idea of teaching reading by
>using phonetic symbols, by leaving out words, and other garbage was often
>*forced* on public teachers from above.

This happened partly, but you have the whole thing wrong.  Before the 40's
reading was ALWAYS taught that way.  It was the educationists and school
of education reading theorists who used a few correct pieces of information
to jump to the incorrect conclusion that correct reading could only be taught
by the whole word method.  This led to the decline in reading ability of 
children, and produced a large set of criticisms by scholars, and before
the schools reacted, the scholars produced books by which the parents, even
with no training, could teach their children to read.  It was this which led
to the pressure, still being resisted by those who do not realize the power
of the alphabet, to teach the other children to learn to sound out words at
a stage before the 7th grade.  It was that bad.

>>>I also had a science teacher who didn't know sh*t about science, and I
>>>could tell it then (didn't dare _say_ anything about it).  He was trying
>>>to explain about atoms and molecules, "molecules of pencil sharpener"
>>>not "atoms of iron and carbon" making up a pencil sharpener.  Wasn't too
>>>hard to ask questions he couldn't answer, but he had ways to retaliate
>>>for making him look dumb.

>     This is a recurrent complaint.  We've all had one or more "incompe-
>tent" teachers.  Public teachers must teach the class, a very heterogeneous
>group.  If the idea is to sell basic science ideas, it should be presented
>simply.  Now before you hit F, let me state clearly:  the teacher must help
>the students learn basic ideas which are basically correct (there are few
>facts which don't fail as you learn more, be it physical--consider optics
>where you begin assuming straight lines and end up considering electromag-
>netic waves-- or biological--the failing of the Central Dogma of genetics.)
>I do not advocate teaching patently false science as in the above case but
>beginning with basic assumptions which can be refined as the student masters
>other fields of knowledge.  Relativity is a classic example.

THIS is another piece of baloney which the educationists brought in during
the 30's.  The idea that the level of the class is to be adjusted to those
who are in the class, rather than the child being put into a class at his/
her level, and that children should be kept according to their age, rather
than according to their ability.  This social adjustment change has prevented
the bright, and this is especially true for the "disadvantaged," from having
a chance to learn.

As far as learning concepts from facts, someone who can do that is research
material.  Euclid and Archimedes could not symbolize mathematics; it was not
until Diophantus came up with the idea that this simple concept, easily 
taught to first graders, but apparently incomprehensible to most college
students, and most high school mathematics teachers.

>>>But grammer school was mostly based on one's ability to memorize random
>>>facts, not on one's ability to figure stuff out.  

>     This is the failing of public education.  To teach the gcd of students,
>most schools teach facts.  Mastery learning requires that a student must 
>master a concept. 

Sorry to contradict you again, but it is not necessary to master a concept
to use it.  The idea that one should not go on to something else until a
concept is mastered is also antithetical to education.  There are also 
many places where the concepts are not yet known, and all that can be
done is operational, or at most the concepts are very incompletely known.
Yet we can proceed.

			This self-paced study doesn't conveniently fall into 
>"yearly advance" scheme.  Most of us use this approach, doing outside reading
>to come up to speed.  However, teaching facts and using standardized tests
>to measure retention is more convenient.  College faculty are no less culp-
>able.  Few faculty devote time to helping students learn the concept.  In
>mathematics, we teach operations.  It's no wonder when students come to an
>advanced class and are told prove this they panic.  Proofs aren't facts or
>operations.

So get rid of the yearly advance scheme!  And college faculty are responding
to pressure in most cases.  Even a distinguished scholar who tried to teach
understanding rather than facts might very well be removed from teaching
such courses.  Someone without tenure, or at teaching schools even someone
with tenure, is in a very risky position if the students are asked to think
rather than vomit up the textbook.  

Proofs are not concepts, and memorizing proofs has nothing to do with
learning concepts.  And if you think that students cannot memorize the
proofs in the texts, your experiences with Asian graduate students are
not extensive.

>>I believe that most of us do not think that such teachers should be in
>>the classroom.... The system is broke.  There is no way that we can get 
>>enough competent teachers NOW.... That is why there is no hope of getting 
>>an adequate public school system.
>>Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399

>     We often think of public school teachers as having the same curricular
>flexibility as college academics which is untrue and unfair.  In a high
>school class, a student wishing to take the ACT or an advanced placement
>test must have covered certain materials.  School boards often tell the 
>teacher you must cover all of this.  Second, when the student enters college,
>s/he has had some exposure to the discipline presumably.  I ran up against
>this problem trying to explain to a bright middle school student what a nega-
>tive number is.  They "knew" that it came before 0 on a number line.  He
>wanted a "real" meaning.  Why was it positive when multiplied?  Wouldn't it
>be a "bigger" negative number?  Given the number of students of my genera-
>tion whose parents couldn't help them with their homework--simple algebra and 
>geometry--I question the adequacy of education in the 40's and 50's!

The students I see in our undergraduate courses are mainly totally incapable
of formulating a word problem, or understanding what a function is.  They
probably have received good grades in high school algebra and geometry.

Now in the 40's, it was only the better students who even took high school
algebra.  And if they could not do word problems, they usually did not get
a grade higher than C- and dropped out.  Now essential illiterates get B's
in algebra and geometry.  But does that mean that they know it?

>     The increasing education of parents and support of kids will increase
>expectations which will have to be met.  The image of the nerd is a cultural
>failure of America where studiousness is reviled.  Students need to be reas-
>sured that you can be bright and popular.
>     I deeply resent all the epithets thrown at public school teachers who
>try hard to cope with the drugs, violence, crowded classrooms, and asinine
>requirements.  There are some teachers who are incompetent, most are not.  
>Most the students here at MIT and around the country do not come from private
>schools.  If you are deeply concerned, contact a local high school and offer
>to teach for a week.  I promise you, it will challenge you.

>     Is the system broke?  I would ask, has it ever worked efficiently?  
>Schools are forced financially to choose between hiring a new science teacher
>or outfitting a new laboratory taught by a former English teacher who had a
>biology course in college.  School boards tend to see facilities as image en-
>hancers and one-time capital expenditures, hence favorable.  We don't have a
>dearth of competent teachers but a dearth of jobs.  When industry pays more
>competitively, you are left with the old, tenured teachers and those whose
>abilities don't meet industrial standards.
>     When my wife was studying at IU to teach Biology, her advisor told her
>to learn a sport to coach so she could supplement her salary.  She was told
>to expect starting at $19K.  With coaching, get $22K.  Today she works at
>Harvard Medical School as a technician where she earns $26K.  She would still
>like to teach, but now it would be a drop in salary, benefits, and extended
>hours away from home.
>     I've ranted longer than I wanted, but I feel that those who point nebu-
>lously at public school teachers do everyone a great disservice.  If a stu-
>dent doesn't want to learn, there is nothing that can be taught.  Changing
>that attitude can sometimes be done by the teacher, but it requires the sup-
>port of the environment--peers, parents, society--as well.  If it is broke,
>now is the time to fix it.  If it's not broke, now is the time to make it
>better.  

I have occasionally taught courses taken by prospective mathematics teachers.
Many of my colleagues teach those candidates more often.  They all would like
to be able to flunk most of them.  So they have had calculus; but they can 
only do routine, and not even that well.  Does an undergraduate who has taken
two full years of calculus have time to forget it?  I have had 16 out of 21
undergraduates taking a probability course with such a prerequisite unable to
use their calculus to do reasonable problems on a take-home part of the final;
from what had happened earlier, I knew that there would have been no hope on
the normal exam.  Most of them were prospective high school teachers of
mathematics.  This was not the weakest program, either.

If we taught the children who want to learn, it would be a fantastic 
mprovement over the current situation.  Most of the tenured teacher
are from the generation of teachers who are steeped in keeping children
with their age group, and stifling all attempts to really learn.  The
system is worse than broke; it is worse than if we had no schools at
all; in that case, there would be, as there has been throughout most
of history, at least an attempt to educate the bright.
-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@snap.stat.purdue.edu (Internet, bitnet)  
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