
THE
FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

FIELDS
MATHED FORUM MEETING AGENDA
Theme: THE
HEART OF MATHEMATICS
November
26, 2011
10AM  2PM
Fields Institute, 222 College Street, Toronto



AGENDA
10:00 a.m.  10:10 a.m. Reports: OAME, CMS, CMESG, OCMA, OMCA,
etc.
10:10 a.m.  11:10 a.m. William Byers (Professor emeritus of mathematics
and statistics at Concordia University in Montreal)
11:10 a.m. 12:00 p.m. Discussants: John Mighton and Judy Mendaglio
12:00 p.m. 1:00 p.m. Lunch Break (Light refreshments provided)
1:00 p.m. 2:00 p.m. General Discussion: What is at the heart
of mathematics and how relevant is it to your teaching goals?
10:10AM  11:10 a.m. The Heart of Mathematics
Presenter: William Byers (Professor emeritus of mathematics
and statistics at Concordia University in Montreal)
Abstract: What motivates people to get into mathematics? Why
do we love math? What is the ingredient that brings it to life?
In an attempt to get at this magical ingredient let us distinguish
as philosophers do between mathematical process and content. Content
is what many of us think of as "real" math. I will take
the position that mathematics is process, in other words that
the creative acts of doing and understanding mathematics are what
is real and what we call content is only a snapshot of our understanding
at a given moment in time.
Everyone has a philosophy of math but for most of us it is implicit
not explicit. How does questioning these assumptions that we bring
to the table change us as teachers and communicators of math?
I believe that the shift to process that I propose is radical
and, once made, we will come to see mathematics in a new light.
For example, some think that the essence of math is its logical
structure but I claim that ambiguity is present everywhere in
math, that ambiguity is, in fact a key to getting a deeper appreciation
for math. Second, if math (and science) is something that you
do then it is dynamic not static as is our understanding. It follows
that you can never say, "I understand continuity or randomness
(definitively)." There are inevitable blind spots in our
knowledge and our understanding. The problematic and the uncertain
are essential aspects of mathematics that need to be acknowledged
and addressed.
This talk will draw on my experience as a university teacher of
maththe things I tried to do, my successes and my frustrations.
My eventual conclusion that something is wrong; something basic
is missing. In an attempt to isolate this missing element I had
to go back to the beginning. I feel that such a reexamination
is important not only for teachers and students of math but for
all people from the Prime Minister and CEO's to university administrators
who use mathematics to help them define and solve the problems
that they face in their personal and professional lives.
Short Bio: Bill Byers recently retired from Concordia's Department
of Mathematics and Statistics in order to try doing something
new. So far this has resulted in two books published by Princeton
University Press: How Mathematicians Think: Using Ambiguity, Contradiction,
and Paradox to Create Mathematics (2007) and The Blind Spot: Science
and the Crisis of Uncertainty (May, 2011). The former was the
winner of the 2007 Library Journal Best ScienceTech book in mathematics
and was one of Choice's 2007 outstanding academic titles. It has
been positively reviewed in such journals as Nature, New Scientist,
Times Higher Education Supplement, Notices of the AMS, and many
others (http://press.princeton.edu/titles/8386.html). The Blind
Spot is new but has generated interest from various circles including
senior economists and the Harvard Business Review (http://press.princeton.edu/titles/9406.html).
11:10 a.m.  12:00 p.m. Discussants: John Mighton and Judy Mendaglio
Short Bio: John Mighton is a mathematician, author, playwright,
and the founder of JUMP Math. He tirelessly volunteers his time
and expertise at JUMP as the lead curriculum developer for the
JUMP Math Student Workbooks and Teacher's Manuals. He also donates
all proceeds from publications to JUMP. Dr. Mighton completed
a Ph.D. in mathematics at the University of Toronto and was awarded
an NSERC fellowship for postdoctoral research in knot and graph
theory. He is currently a Fellow of the Fields Institute for Research
in Mathematical Sciences and has also taught mathematics at the
University of Toronto. Dr. Mighton also lectured in philosophy
at McMaster University, where he received a Masters in philosophy.
Short Bio:Judy Mendaglio is the Curricular Head, Mathematics,
at a high school in Peel District, an executive member of CHAMP,
on the Board of Directors of OAME, and a member of the Steering
Committee of the Fields MathEd Forum. Teaching Mathematics in
high school is her third career.
Judy added, "I am completely, totally in love with Mathematics
and teaching Mathematics. I have been a student of mathematics
for too many decades to reveal. I was not one of those students
who was always 'really good' at Math  I was especially good at
languages, actually (which some would think qualifies me for math).
I studied it [Mathematics] because I needed to know more. Since
leaving graduate school all those many decades ago, I have continued
my studies informally (by reading books such as Prof Byers'),
largely to gain deeper understanding of procedures that I had
already learned and to build connections that I never knew existed.
(My high school math education did not include anything as radical
as graphing! All algebra and geometry, all the time.)"
12:00 p.m.  1:00 p.m. Lunch Break (Light refreshments provided)
1:00 p.m. 2:00 p.m. General Discussion: What is at the heart
of mathematics and how relevant is it to our teaching goals?
Citations to guide the discussion:
Paul Halmos wrote:
"What does mathematics really consist of? Axioms (such as
the parallel postulate)? Theorems (such as the fundamental theorem
of algebra)? Proofs (such as Godel's proof of undecidability)?
Concepts (such as sets and classes)? Definitions (such as the
Menger definition of dimension)? Theories (such as category theory)?
Formulas (such as Cauchy's integral formula)? Methods (such as
the method of successive approximations)? Mathematics could surely
not exist without these ingredients; they are all essential. It
is nevertheless a tenable point of view that none of them is at
the heart of the subject, that the mathematician's main reason
for existence is to solve problems, and that, therefore, what
mathematics really consists of is problems and solutions. 'Theorem'
is a respected word in the vocabulary of most mathematicians,
but 'problem' is not always so. 'Problems,' as the professionals
sometimes use the word, are lowly exercises that are assigned
to students who will later learn how to prove theorems. These
emotional overtones are, however, not always the right ones. The
commutativity of addition for natural numbers and the solvability
of polynomial equations over the complex field are both theorems,
but one of them is regarded as trivial (near the basic definitions,
easy to understand, easy to prove), and the other as deep (the
statement is not obvious, the proof comes via seemingly distant
concepts, the result has many surprising applications). To find
an unbeatable strategy for tictactoe and to locate all the zeroes
of the Riemann zeta function are both problems, but one of them
is trivial (anybody who can understand the definitions can find
the answer quickly, with almost no intellectual effort and no
feeling of accomplishment, and the answer has no consequences
of interest), and the other is deep (no one has found the answer
although many have sought it, the known partial solutions require
great effort and provide great insight, and an affirmative answer
would imply many nontrivial corollaries). Moral: theorems can
be trivial and problems can be profound. Those who believe that
the heart of mathematics consists of problems are not necessarily
wrong." (Halmos, 1980, p. 519)
Reference: Halmos, P.R. (1980). The heart of mathematics. The
American Mathematical Monthly, 87(7), 519524.
Susan Gerofsky and J. Scott Goble wrote:
"Davis and Hersch (1981), in their book The Mathematical
Experience, interviewed dozens of research mathematicians and
found that 'the typical working mathematician is a Platonist on
weekdays and a formalist on Sundays. That is, when he [sic] is
doing mathematics, he is convinced that he is dealing with an
objective reality whose properties he is attempting to determine.
But then, when challenged to give a philosophical account of this
reality, he finds it easiest to pretend that he does not believe
in it after all' (p. 321)." (Gerofsky & Goble, 2007,
p.111)
Reference: Gerofsky, S., & Goble, J.S. (2007). A conversation
on embodiment at the heart of abstraction in mathematics education
and music education. Complexity Science and Educational Research
Conference Feb 1820, Vancouver, BC, 111124. http://www.complexityandeducation.ualberta.ca/conferences/2007/Documents/CSER07_Gerofsky_Goble.pdf
Kimberly WhiteFredette wrote:
"Mathematics teachers are seldom asked to explore philosophy
beyond an introductory Philosophy of Education course. If one
is going to teach mathematics, one should ask "Why?"
What is the purpose of teaching mathematics? What is the purpose
of mathematics in society at large? Should not mathematics' purpose
be tied to how we then teach it? These questions come back to
teachers' perception of mathematics, and more specifically, their
philosophies of mathematics. (…) I end this article by revisiting
a definition of philosophy of mathematics: 'The philosophy of
mathematics is basically concerned with systematic reflection
about the nature of mathematics, its methodological problems,
its relations to reality, and its applicability' (Rav, 1993, p.
81). If our goal in mathematics education reform is to make mathematics
more accessible and more applicable to realworld learning, we
should then help guide today's teachers of mathematics to delve
into this realm of systematic reflection and to ask themselves,
'What is mathematics?" (WhiteFredette, 2009/2010, p.26).
Reference: WhiteFredette, K. (2009/2010). Why not philosophy?
Problematizing the philosophy of mathematics in a time of curriculum
reform. The Mathematics Educator, 19(2), 2131.
2:00 p.m. Adjournment

