From: jamess@athena.mit.edu (John A Mess)
Subject: Helping to Change Public Education (was Re: Is Math Hard?)
Organization: Massachusetts Institute of Technology
Date: Fri, 6 Nov 1992 22:12:30 GMT

In article <Bx9uy1.LB0@mentor.cc.purdue.edu> hrubin@mentor.cc.purdue.edu (Herman Rubin) writes:
>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.  
>
>>     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.
     You *don't* understand.  By phonetic symbols I mean as used by linguists,
*not* phonics.  But that is a separate issue.  The point which you missed is
that it was the public school administration, not the teachers comfortable
with phonics, who coordinated this change.

>>     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.
     You are misreading what I said.  Knowledge is accomodated differently
by students.  But some concepts cannot be mastered until there is an intellec-
tual maturing.  Which is why Baez, Chow, and others keep redefining a manifold
until one description will "click."  How would you explain general relativity
to a fourth grader?  Is that different from a high school senior?  A math
grad?  A graduate student in history?  We can speak truthfully but informa-
tively gauging it to our audience.  Those who are truly curious should be en-
couraged to learn more.  They will absorb, however, only what they are cap-
able of learning.  That changes with time.
>
>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.
     You're not clear to me.  Greek mathematics and geometry had a very
organized notation (cf. Heath).  The facts of mercury's precession did not
lead to general relativity (concept).  It was after the concept and its ap-
plication that the fact could be explained.

>>>>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.
     Oh!  It isn't important to understand the null hypothesis or the effect
of sample size to actually perform a statistical test.  You don't understand
the concept of mastery learning.  Mastery learning means you learn a funda-
mental concept thoroughly that you can then build upon.  It doesn't mean you
are only mastering one idea at a time.  Some concepts are disjoint.  Others
are not.  Discussing Hilbert spaces requires some background knowledge in
many areas.  Using it in quantum mechanics requires other experience.  In
the beginning though, you need to knuckle under to master the first step.
What is a topological space?  What is a group? etc.  You refine concepts,
not facts.
>>			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.  
     But eliminating it--even suggesting it--raises panic in parents' hearts;
my child is being "held back."  So we pass along someone who can't read to
let them play on the team, satisfy the parent complaining to the superinten-
dent, etc.  Your point is exactly mine.  Academics are told to teach and test,
let the students fend for themselves later.  If we don't challenges this atti-
tude, we are contributors to the problem.
>
>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.
     Apples and oranges!  A high school student is not an Asian graduate
student.  I certainly hope *any* graduate student is doing more than memor-
izing proofs.  But you also are wrong about proofs.  A truly beautiful
proof isn't simply operations.  The published proof is kind of a schematic
for the initiate to go from a to b concisely.  Proofs by contradiction are
a marvelous kind.  You must know why the contradiction holds.  Understanding
the spirit of a proof will allow someone to make more headway in rederiving
it than simply having memorized the steps once.
>
>>>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.
     That is because they learned facts and operations.  I have frequently
heard, "Give me the equation, and I can crank out the answer" remarks.  Of
course, the grades are "good" because they were fit to some curve.  If we
taught and tested on exactly what we wanted them to master and didn't penal-
ize them for taking the test over until the materials was mastered, the
grades would reflect knowledge not hide ignorance.
>
>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?
     So we failed a generation by letting them leave--turned off to mathe-
matics.  Here is where the system began to break down.  You can blame it on
the war and simpler times, but adolescents have different priorities and
needs.  It was this transition from pre-war to post-war education reform that
made its mark.

>>     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.
     I understand, but you are talking about *prospective* math teachers.  I
am talking about *practicing, successful, competent* teachers.  There are many.
It is hard to be successful as a teacher when on the job you must also be a
disciplinarian, foster parent, confidant, coach, club coordinator, and all the
other hats we force on students.  I am often shocked to hear of public school
teachers who must buy supplies for experiments out of their own pocket to the
tune of $250 a year. 
     Your colleagues want to *flunk* them?  Why don't they want to be *role
models* for them?  Having math teachers in your class is the easiest way in
the world to change the system from the inside.  So your colleagues turn off
more students to math.  Or worse, allow students to leave their class with
a substandard mastery of the material.  But as you rightly pointed out, if they
cared, they'd be in jeopardy.

>If we taught the children who want to learn, it would be a fantastic 
>improvement 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.  
     Wait, you have to stop jumping all over the place.  Age grouping is only
true for elementary education, which has its own woes.  There is no child who
wants to learn by sitting still for 8 hrs.  In every communication class, they
say the adult listens only to the first and last twenty minutes of a talk.  In
presenting we should have various visuals and action.  It's funny that they do
that for business and professional talks but not for kids in high school. 
Lord, I could have done without American History.  The important feature is to 
get competent teachers who can help each student learn at his/her own pace and
find satisfaction (i.e., enjoy) learning.  My World History teacher showed me 
that history can be interesting.  Teachers should not be penalized for involv-
ing students.  Sunny days are tempting to almost everyone.  I loved math, 
sciences and languages.  Good teachers helped me to love history, music, art, 
and literature as well.  Most people have one inspiring teacher or they don't 
graduate.  I don't agree with *most*.  I think most public teachers have big-
ger classes, longer hours, more obligations and can't give the attention we 
demand without breaking.  

>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.
                                 ~~~~~~~~~~~~~~~~~~~~~
     That *is* the failure of the American educational system.  *Bright*
does not mean motivated.  I think most American students are not challenged
to meet their full potential.  By ignoring them, we let thousands of students
turn off to math, science, and the humanities.  We tell them, "If you can't
cope, scoot along."  We tell them that only smart people take these classes,
and by implication the self-doubting don't.  We should say, "look, math and
science are easy if you try.  Here, help me with this."  Instead we lament 
that "They're not interested. How are we failing them?".  It must be the 
teachers; they're just not teaching the "bright ones."  Bit tautological isn't
it?  If we are truly concerned, we must motivate *all* students to learn.
     How?  One, reduce pressures on teachers: lower the pupil/teacher ratio,
reduce safety concerns, raise salaries to be competitive and reflect how we
value education.  Two, change societal perception:  studiousness isn't nerdi-
ness, heighten rewards for academic excellence not just sports, and recognize
that everyone can be intellectually motivated, as Lang's public lectures sug-
gest.  Finally, shift the educational model to mastery from memorization.  
     From the students I see, the public schools are not such monstrosities
as you contend.  They do reflect starkly societal expectations on our child-
ren.  Where parents value education and the community provides strong finan-
cial support, the education is highly regarded.  When you expect to be shot
or stabbed, parents joke about brawn no brain, and/or the schools have to
decide which teachers get layed off, education becomes less important than
sheer survival.