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The Creation of Math

Seminar Leader: Jim Dudley, Department of Mathematics

Seminar Description:
From earliest times, humans have not only used mathematics but have built upon the math they have inherited or acquired from others.  This seminar focused on the growth and development of mathematics over time by focusing on the people, ideas and controversies that have contributed to the creation of math.  Four major areas in the advancement of mathematics guided the seminar: 

1. The math from cultures all around the world that survives in the archaeological record but that existed mostly before the time of “history” or “biography.”  This includes African and Native American counting systems and designs, and the astronomically based math of ancient civilizations in Mexico and Babylonia. 
2. The Greek mathematicians Pythagoras and Euclid who survive not only in the Pythagorean theorem and Euclidian geometry but also in ways of thinking about mathematics still relevant (and debatable) today. 
3. The period of the 17^th and 18^th centuries in Europe whose mathematicians (Descartes, Newton, Euler, and Gauss, among many others) developed most of the mathematics of the traditional school math curriculum (analytic geometry, analysis of functions, logarithms, trigonometry, calculus, and the mathematics of sequences and series, of combinations and permutations, and of graph theory). 
4. The 20^th century that contributed an intense and well-documented period of doubt about the nature of math itself, the “crisis in foundations.”  The century just past also witnessed an amazing explosion of mathematics, creating the new disciplines accessible to today’s students in statistics, matrix theory, computing, and fractal geometry.  

The seminar relied upon activities at all grade levels that illuminate the history and development of mathematics.  For example, crossing from one continent to another and passing back and forth through time is the spiral.  It is perhaps the ideal mathematical concept for uniting the diverse strands of our research.  Embedded within it are the right triangles of the Egyptians, the wandering-from-home petroglyphs of North America and design motifs from around the world, the Golden Section of the Greeks, Cartesian coordinates, logarithms and trigonometry from the Europe of the Enlightenment, and the ecological awareness of the 20^th century.  Throughout the seminar, teachers will be asked to investigate how people think and have thought about mathematics, as well as how they (and their students) DO math.  While we explore and develop the activities that relate to the four main areas of focus, our discussions should be enlivened by controversies old and new such as the existence of zero and infinity; whether math is created or discovered; whether math is more a thing of beauty or of utility; on the connection of math to religious truth, logical certainty, and political beliefs; and the (mis)identification of math with “dead White males”.

Seminar Readings:
Texts that guided in this research include Hersh and Davis’s The Mathematical Experience and Descartes’ Dream, both taking a refreshingly skeptical look at how mathematics was viewed and practiced towards the end of the 20^th century.  Reuben  Hersh’s What Is Mathematics, Really? examines the philosophies behind math that eventually led to the clash between Platonism and formalism during the “crisis” period and espouses a “humanist” philosophy to replace both.  A Mathematician’s Apology by G.H. Hardy questions the usefulness of math and argues that beauty is the better judge of a mathematical theorem.   The Apology also reveals where, by whom, and under what circumstances mathematics was practiced in the first third of the 20^th century.  And it tells the story of the brief and unconventional career of the brilliant Indian mathematician Srinivasa Ramanujan.  Finally, Harold Jacob’s Mathematics, a Human Endeavor traces topics from the past through investigations about number, shape, and pattern.

About the Seminar Leader:
With Masters degrees both in history and mathematics education, the history of mathematics has been a natural and long-time interest of mine.  During my 24 years as a high school teacher, and in numerous summer middle school programs, I worked in as many readings and activities related to the history of math as I thought the students could stand.  As an ATI Fellow in 1999, I selected a topic relating the histories of astronomy and trigonometry.  Now as a Lecturer with the UNM Mathematics and Statistics Department, I’m pleased to be able to expose students in the pre-calculus curriculum, as well as prospective educators in math-for-teachers courses, to many of the same kinds of activities and ideas.