Holl Mathematical Discovery Workshop for Teachers 2016
Leslie Hogben, Dio Lewis Holl Chair in Applied Mathematics, Iowa State University
Alejandro Andreotti, Director of Graduate Education, Masters of School Mathematics (MSM), Iowa State University

Four high school teachers came to the Iowa State University campus for the Holl Mathematical Discovery for Teachers Workshop June 13-17, 2016.  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,  They also shared several resources with each other, leading to new plans for their fall teaching.  The workshop began by sampling an assortment of mathematical problems.  These included:

Happy Numbers This problem involves 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 1^2 = 1) and 13 goes to 1 (because 1^2 + 3^2 = 10, and 1^2 + 0^2 = 1).  A number is called "happy" if it goes to 1.   For the sum of squares of digits, 1 is the only fixed point, there is one cycle, and every number is one of these or goes to one of these.  Variations include summing cubes and other powers of the digits.

Function Tables  A function table lists values of n and f(n) for a function f.   We looked at patterns and the use of repeated differences to recover a polynomial function through  interpolation.  

Crossing Numbers A graph is a set of vertices and edges between some of the vertices.  A planar graph can be drawn in two dimensions without any crossing so edges (except where they meet at vertices).  Not all graphs are planar, and the crossing number of a non-planar graph is the minimum number of crossings.  Crossing numbers were first contemplated by Paul Turan in a forced labor camp during World War II and continue to have practical applications.
 
After preliminary investigations of each problem over the first two days, the group settled on Happy Numbers for the last three days.  We investigated the density of happy numbers (what fraction of numbers at most n are happy, as n goes to infinity).  This amazing problem continues to produce new open questions.  We focused on fixed points and the density of numbers going to fixed points.



Annie Sullivan, Laura Scott, Trevor Barrett, and Diane Nead discuss mathematics.
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.”