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Commentary_ Volume 47_ Number 4

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Commentary_ Volume 47_ Number 4 Powered By Docstoc
					Commentary
In My Opinion

Face to Face: Teachers
and Mathematicians
Fame rarely visits the mathematician. While the names                   Too often the good mathematician turns out not to be
Beethoven, Rembrandt, and Einstein are household words,             the good teacher. Neither is the good teacher always the
only those most interested in our field recognize the names         good mathematician. Oddly, the two often work in close
of Cauchy or Weierstrass. We have no Hall of Fame, no Nobel         proximity, but back to back. Can we turn them around? Only
Prize, and when a team does well in the International Math-         if each community recognizes that it has something to learn
ematical Olympiad, the press hardly blinks.                         from the other.
   Why is the work of the mathematician so hidden from                  For example, mathematicians can sometimes learn about
public view? Part of the reason is that it is difficult to ex-      teaching from people in their own education departments.
plain most advances in mathematics to people with only              Now it is certainly not true that anyone on the faculty of
general knowledge. But we can go a bit deeper. It’s not just        a school of education will know how to teach: those fac-
the results of mathematics that the public has trouble re-          ulties suffer from publish-or-perish as much as others, and
lating to. It’s also the act of creating mathematics. This is       the good teacher of teachers sometimes will not get tenure.
partly because, of all the arts, mathematics is perhaps the         But there are many people in schools of education who un-
best at concealing itself. Mathematical thought is ripest           derstand well and have thought deeply about how to com-
when all traces of the struggle to achieve insight have been        municate with students.
obliterated, when the result appears as if it has sprung full           Another place to look for knowledge about teaching is
grown from its creator’s head. It is difficult to recognize         the local high school or elementary school. Teachers on the
the product as the result of an intellectual struggle.              precollege level are not used to talking about what they
   This is one of the most significant obstacles faced by           do; they’re rarely asked. But good teachers possess deep
the students of mathematics, who are striving to assimi-            wisdom about what it means to know mathematics and how
late results created by others. They must unfold the care-          to recognize whether the student has heard what you in-
fully wrapped package that is a mathematical idea.                  tended to say.
   The classical work of Poincaré, Hadamard, Polya, Hardy,              Better still than simply taking a lesson would be seek-
and others provides us with examples of mathematicians              ing—or constructing—situations in which mathematicians,
who were also gifted at revealing their art to the public. But
                                                                    teacher educators, and teachers can sit together and learn
there is no particular reason why someone whose talent
                                                                    from each other. There has been much talk recently about
lies in obtaining mathematical results should also have a
                                                                    the participation of mathematicians in education, and
talent for illuminating their results or taking others through
                                                                    there have been some rather heavy-handed attempts by
the process by which they were obtained. For these are not
                                                                    mathematicians to dictate curriculum and even method-
mathematical processes. They do not partake, for exam-
                                                                    ology. More success would come from a more open ap-
ple, of the ineluctable nature of mathematical truth. Dif-
                                                                    proach, one through which mathematicians and educa-
ferent people acquire an understanding of mathematics in
                                                                    tors might learn from each other, working together to
different ways.
                                                                    explore aspects of mathematics that emerge in the class-
   Happily, mathematicians are not always called upon to
                                                                    room.
make their work accessible outside the profession. How-
                                                                        For we are not talking just about fame here. And we are
ever, there is one activity which most mathematicians en-
                                                                    not talking just about classrooms. We are talking about the
gage in almost daily and which involves helping others ac-
quire an understanding of mathematics. This is the activity         role of the mathematician in society. We can let this role be
of teaching. If it is to be more than the delivery of cut-and-      assigned by our institutional structures and unconscious
dried results, teaching must involve the stimulation of a           prejudices, or we can choose it carefully and consciously.
creative process in the learner. To be a good teacher, one              I hope we follow the latter course.
must examine one’s own thought processes and how they
relate to those of the students. This sometimes involves a                                                         —Mark Saul
recapitulation of the creative process. We must recognize                                                       Associate Editor
that a knowledge of mathematics does not automatically
imply the ability to convey that knowledge to others. This
is the task of the teacher, and it requires an effort sepa-
rate and distinct from that of thinking mathematically.


APRIL 2000                                            NOTICES    OF THE   AMS                                               437
Letters to the Editor


                                            lates to others’. An edited version of      der to illustrate the obvious progress,
                                            this material (of course emphasizing        one can mention a variety of examples.
                                               its “interdisciplinary” aspects), to-    Among these are the justification of
                                                  gether with an adequate bibliog-      the Boltzman ergodic conjecture for
                                                     raphy, might be collected and      certain classes of systems, the discov-
                                                     made available on the Web un-      ery of Anosov systems, time series
                                                     der the auspices of the AMS or     analysis, etc. Moreover, on the basis
                                                    NSF or both, thus serving as a      of deterministic chaos theory it is
                                                    first approximation to the doc-     possible now to describe such physical
                                                   ument that I have in mind.           phenomena as self-organization and
                                                                                        pattern formation, to quantify the frac-
                                                                —Robert Hermann         tality, and so on. Why should this area
                                                                   Brookline, MA        not be developed by mathematicians?

                                                          (Received November 22,                       —Alexander Loskutov
                                                                          1999)                               Serge Rybalko
                                                                                                      Moscow State University

                                                        Chaos Theory: Present                   (Received December 7, 1999)
                                                        and Future
                                                         We were quite surprised by
                                                         the opinion column of          FYI Item “BMS Report on
                                                        Steven Krantz (Notices,         Institutes”
                                            October 1998), “See No Evil, Hear No        As a member of the panel commis-
                                            Evil, Speak No Evil”, in which Krantz in    sioned by the Board on Mathematical
                                            particular claimed that “there is not       Sciences of the National Research
Letters to the Editor                       one example of any scientific problem       Council, I write to correct several
                                            that has been solved (not just described)   inaccuracies in the “For your informa-
                                            using [chaos theory].” Indeed, there are    tion” note by Allyn Jackson entitled
Interdisciplinary Mathematics               plenty of popular “near-science” pub-       “BMS Report on Institutes” in the
I hope that the December 1999 issue         lications where authors sometimes           December 1999 issue of the Notices.
of the Notices is evidence that the math-   adduce highly dubious and speculative       First of all, the panel was told from the
ematical world is at long last seriously    conclusions that seemingly follow from      start that its task was to be totally
recognizing the importance of research      the mathematical theory of chaotic phe-     independent of the re-competition
and development of “interdisciplinary       nomena. In the history of science this      process for existing NSF-supported
mathematics”! I have been involved          frequently happens: One can mention         mathematical institutes. Second, I
with this since my experience working       cybernetics, singularity theory, and        recall no support whatsoever from
at Lincoln Laboratory in 1959 and have      synergetics, where similar inferences       panel members for an Oberwolfach-
seen at least some of my work become        have been posted. See also the letter by    style U.S. conference center. Third, the
of utility in control theory, mechan-       David Ruelle entitled “Achievements         panel strongly urged the preservation
ics, and physics. For the past five years   of Chaos Theory”, Notices, March 1999.      of funds for principal investigator
I have been trying to integrate some of        In our opinion, the main problem         grants. Finally, the panel’s recommen-
the mathematics I know with devel-          raised by Krantz concerns the social at-    dation of consideration of two new
opments in computer science.                titudes in a mathematical society. The      sorts of institutes was made with the
   I would like to make a modest sug-       example with a theater is very beautiful,   explicit caveat that such institutes
gestion. In my interactions over the        but it is necessary to note that among      should be funded with new funds and
years with scientists and engineers, I      dramatic and theater critics there are      not drain scarce resources from exist-
have noticed their difficulty in obtain-    authorities whose opinions are much         ing NSF research programs. Indeed, the
ing information about the frontiers of      more important than those of an ordi-       report states in its Executive Summary
mathematical research in a condensed        nary audience. Moreover, the theater        and repeats in its Recap and Closing
and intuitive form which might be use-      directors sometimes direct their per-       Comments the following statement,
ful in their work. Might something of       formances despite the critics: they are     italicized both times for additional em-
this nature be made available on a Web      artists and have their own attitudes.       phasis: “[T]he committee strongly
site?                                          How many mathematicians have pro-        believes that it would not be in the best
   One way to do this that has occurred     claimed a new paradigm and a new            interest of either the mathematical
to me might be to piggy-back on the         world picture? At the same time, math-      sciences community or society as a
grant applications which most of us         ematicians who deal with physical prob-     whole to transfer funding from exist-
write throughout our career. As part of     lems know quite well that methods of        ing mathematical sciences individual
this algorithm, one usually writes a de-    chaos theory have been successfully         (principal investigator) research grant
scription of past work and how it re-       applied in many fields of physics. In or-   programs to funding for existing or


438                                                  NOTICES   OF THE   AMS                              VOLUME 47, NUMBER 4
                                                                                                       Letters to the Editor


additional mathematical sciences            constant dollars both accounts have        Rota Students
institutes.”                                had healthy growth during the past         In regard to the two unknown Ph.D.
               —Eric M. Friedlander         two decades. In fact, in constant dol-     students of Rota mentioned on page
             Northwestern University        lars each account was at an all-time       205 of the February Notices, I can give
       (Received December 10, 1999)         high in 1998, the last year the tables     the following information about one
                                            provided by the OMB are not qualified      of them, Robert L. McCabe.
                                            with the word “estimate”. Those who           He received his degree in mathe-
Federal Funding for Basic                   wish to check the work should begin        matics from Boston University in 1971
Research                                    their navigations at http://w3.            with a dissertation entitled “On Dis-
In her opinion column “Opinion and          access.gpo.gov/usbudget/ and at            tensions of Positive Operators”. That
Responsibility” (Notices, December          http://www.bea.doc.gov/bea/dn/             has the sound of Rota. However, in the
1999), Mary Beth Ruskai raises an           gdplev.htm.                                abstract of his dissertation a different
important point. It appears that my            In all, the assertion in my essay of    adviser is listed. No doubt Rota helped
essay “The Immigration Law of 1990          August 1999 holds, though for reasons      him, but since Rota was not on the
and Its Effects” (Notices, August 1999)     not properly presented there. The ev-      faculty of Boston Unuversity, another
did not adequately address whether          idence is little short of overwhelming     acted in this role. I believe that McCabe
overall federal funding for basic           that federal funding for basic research    now teaches at the University of
research was or was not declining           was not declining in the 1990s. The de-    Massachusetts at Dartmouth, MA.
during the 1990s. The substance of this     clines happened twenty years earlier.         I cannot help with O. Murru.
question is too important to be left
hanging: news of healthy or of declin-                              —Damon Scott
                                                                                                             —W. A. Beyer
ing budgets can color perceptions on                                Rochester, NY
                                                                                            Los Alamos National Laboratory
a wide range of issues and should not              (Received December 15, 1999)
be made in error. Accordingly, I have                                                          (Received February 22, 2000)
researched the question in more detail
                                            Location of Joint Meetings
than I was able to do during the prepa-
ration of the original essay.               The selection of Washington, DC, for
   The Office of Management and             the winter meeting of the American
Budget publishes historical tables of       Mathematical Society was ridiculous.
funding by agency and funding by            Throughout the meeting there was
function and subfunction. Of the agen-      ice on the streets, and the temperature
cies, funding to the NSF has been the       was always below the freezing point.
statistic habitually employed in the        For those people in the Omni Shore-
mathematical community to gauge             ham Hotel, the walk to the Marriott              The Notices invites letters
federal commitment to our discipline.       Hotel was a real hazard. The map that         from readers about mathemat-
Of the functions and subfunctions of        showed the Marriott Hotel across from         ics and mathematics-related
federal outlays, Subfunction 251,           the Omni Shoreham was in error.               topics. Electronic submissions
“General science and basic research”,          There is no reason to select a north-      are best. Acceptable letters are
most closely aligns with basic research     ern or borderline site for the winter         usually limited to something
as Ruskai defines it. Funding to the NSF    meeting when some southern city               under one printed page, and
occupies the lion’s share of the ap-        would be better. To refresh your mem-         shorter letters are preferred.
propriations gathered into this             ory, here are a few locations of other        Accepted letters undergo light
subfunction; funding for the Space          winter meetings: 1987 in San Antonio,         copyediting before publication.
Program, NIH, and many other federal        Texas; 1988 in Atlanta, Georgia; 1989         See the masthead for electronic
technical programs do not appear            in Phoenix, Arizona; 1993 in San An-          and postal addresses for sub-
there at all. In addition, the portion of   tonio, Texas; 1996 in Orlando, Florida;       missions.
NSF funding that is devoted to edu-         1997 in San Diego, California; 1999 in
cation does not appear in Function          San Antonio, Texas.
250 at all, but rather in Function 500.        Other southern cities are possible:
   An examination of NSF funding and        namely, Miami, Florida; New Orleans,
Subfunction 251 funding, both in con-       Louisiana; and Houston, Texas.
stant dollars and as a fraction of the         Please schedule future winter meet-
Gross Domestic Product, reveals             ings of the AMS in the Deep South. If
surges in the 1960s and declines in the     one wishes to visit Washington, DC, it
1970s. Both accounts, as a fraction of      should be in the spring or fall.
GDP, have been holding their own
through the 1980s and ’90s: that for                             —A. W. Goodman
the NSF holding at about 75% of its                               Tampa, Florida
post-Sputnik levels, and that for Sub-
function 251 holding at about half. In              (Received February 10, 2000)


APRIL 2000                                           NOTICES   OF THE   AMS                                                439

				
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