Holl Mathematical Discovery Workshop for Teachers 2014
Leslie Hogben, Dio Lewis Holl Chair in Applied Mathematics, Iowa State University
Ulrica Wilson, Associate Professor, Morehouse College



Eight high school teachers and one community college teacher were invited to the Iowa State University campus for the Holl Mathematical Discovery for Teachers Workshop July 14 – 18, 2014.  Participating teachers included
The workshop focused on a mathematical discovery process, including numerical experiments to gather data, observing patterns in the data, making conjectures, and proving (or disproving) the conjectures.  Implementation of these ideas with students was also discussed.  The participants learned to use the free mathematical software Sage and used it to gather data.

The workshop was structured around two mathematical problems, presented by Dr. Leslie Hogben, Dio Lewis Holl Chair in Applied Mathematics and Professor of Mathematics at Iowa State University, and Dr. Ulrica Wilson, Associate Professor of Mathematics at Morehouse College.  The workshop divided into two research groups to explore the problems.

One problem involved mathematical analysis of the Lights Out! game, which was sold as a handheld electronic game in the 1990s and can be played online, e.g., http://www.neok12.com/games/lights-out/lights-out.htm.  There is a 5 x 5 grid of lights, each of which can be either on or off.  Each light is also a button, and when you press a button you change (turn on to off, and off to on) that light and each of its neighbors (above, below, left, right) that exist. The goal is to turn all the lights off.   This game can be analyzed using matrices, which are rectangular arrays of numbers, and arithmetic mod 2 (in mod 2, 1+1= 0, as well as the more usual 0 + 1 = 1 = 1+0 and 0 + 0 = 0).  Variations including different size grids were explored. 
LightsOut! group

The other problem involved iterating (repeatedly applying) a particular function to see if the number does not change (i.e., is a fixed point), goes to a fixed point, or some other behavior.  The particular function looked at initially was to square each digit and then take the sum.  For example, 1 is a fixed point (because 12 = 1) and 13 goes to 1 (because 12 + 32 = 10, and 12 + 02 = 1).  Variations including summing cubes and other powers of the digits were explored. 
Iteration group

On Friday each group presented its results to the other group. The iteration group found a bound for each power (such as square or cube), so that if you check all numbers below the bound you know you have found all fixed points for that power, proved the bound worked, and found all fixed points for several powers. A paper has been written.  The Lights Out! group studied additional configurations of lights and found several patterns for which starting arrangemts of lights could be tunred off.  In addition to mathematical results, the Lights Out! group developed a model for the lights/buttons using two-colored disks; this made it possible to experiment concretely with various arrangements of lights and made the problem more accessible to students.  

Dr. Heather Bolles of Iowa State University helped with the development of the workshop and also participated in some sessions.  Dr. Travis Peters of Culver-Stockton College, MO, who teaches future teachers, also participated.

About Dio Lewis Holl (from LAS News Release):

"Dio Lewis Holl received his AB in 1917 from Manchester College, his AM in 1920 from Ohio State University, and his PhD in 1925 from the University of Chicago with a thesis entitled “Viscous Fluid Motion in Eccentric Cylinders”. He joined the faculty at Iowa State in 1925, and he served as Head of the Department of Mathematics from 1945 until his death in 1954.

"Dio Holl was well known for his excellent teaching skills and his interest in high quality education. He supervised 12 PhD students in areas such as elasticity, fluid flows, and stress analysis of materials. He envisioned the Department of Mathematics at Iowa State College as one of applied mathematics, i.e. focusing on areas such as dynamics, elasticity and plasticity. He did, however, hire more ‘core’ mathematicians later in his term as Head, including some algebraists, topologists, and analysts.

"The Department of Mathematics expresses its gratitude to the donors of the Dio Lewis Holl Chair in Applied Mathematics for their generous gift. The donors have expressed ‘their desire for the professor selected to be passionate about teaching and educating students who are going to be math teachers, to better reflect the qualities of their father.” One of the donors, Elizabeth Holl Bierbaum, a daughter of Dio Holl, serves on the Advisory Board of the Department of Mathematics where her insight and passion for mathematics are much appreciated."

About Leslie Hogben
Leslie Hogben is the Dio Lewis Holl Chair in Applied Mathematics, and this workshop is supported by Holl Chair funds.   Dr. Hogben has focused on a variety of different aspects of mathematical education and research.  She has worked with middle and high school teachers in the Masters of School Mathematics (MSM) program.  She is the author of more than eighty research papers, and particularly enjoys introducing people to mathematical research and discovery.  She has been a leader in the use of research groups in mathematics, bringing together undergraduates, graduate students, postdoctoral associates, and faculty at undergraduate colleges to successfully solve problems.  She says, “Teachers can convey the excitement of discovery to their students.  This was certainly true for me -- I was fortunate to be taught by a truly inspiring math teacher when I was in middle and high school, who had us exploring open questions in graph theory.”