MathEd Forum

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 math-the 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 Science-Tech 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 tic-tac-toe 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 non-trivial 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), 519-524.

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 18-20, Vancouver, BC, 111-124. http://www.complexityandeducation.ualberta.ca/conferences/2007/Documents/CSER07_Gerofsky_Goble.pdf

Kimberly White-Fredette 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 real-world 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?" (White-Fredette, 2009/2010, p.26).
Reference: White-Fredette, K. (2009/2010). Why not philosophy? Problematizing the philosophy of mathematics in a time of curriculum reform. The Mathematics Educator, 19(2), 21-31.

2:00 p.m. Adjournment