This project builds on a very successful program of summer institutes for mathematics teachers, graduate students and university faculty at the Park City Mathematics Institute (PCMI) run by the Institute for Advanced Study and supported in part by Boston University and EDC. In this project they will produce, publish and disseminate a strong mathematics content curriculum for in-service secondary mathematics teachers and prepare a group of specialized teacher-leaders to deliver this curriculum across the country. The program works on the theory that an immersion experience moves teachers to a higher order of understanding; formality and precision are crucial in mathematics; and teachers thrive in a mathematical community. Important components of this project are that expert teachers will ensure that the mathematics is relevant to the professional lives of secondary teachers and mathematicians will be core members of the development and review team. The materials produced for these courses contain ingredients for a problem-driven curriculum connecting advanced mathematical knowledge with the mathematics used by secondary teachers, in and out of their classrooms. Mathematics courses that connect upper-division university mathematics to secondary mathematics are not typically available to teachers. This program uses very well respected mathematics and mathematics educators to prepare content rich materials that will be tested at a prominent facility with teachers. They will also be working with presenters at this time to prepare facilitators guides for the materials. The developers currently have five three-week courses. The proposal will allow them to 1) add exposition and detailed commentary for facilitators; 2) add courses where there may be gaps in the mathematical topics and 3) prepare teachers who will work with mathematicians to develop and teach the materials to their colleagues using the PCMI professional development approach.
The Mathematics for Teaching problem-solving course materials, which this project supported, were designed and implemented at the Institute for Advanced Study Park City Mathematics Institute’s Secondary School Teachers Program (SSTP), which served as the lab for the project. The 55 to 60 teachers in the SSTP meet each weekday for three weeks during the summer with two hours of mathematical problem solving, followed by one hour of Reflecting on Practice. Afternoons are devoted to Working Groups, which are small groups whose goal is to produce activities that can serve as resources for other teachers (for eventual online publication) and other participant-organized activities plus a significant amount of interaction with the other programs of the PCMI Summer Session (Research mathematicians, graduate students, undergraduate students, undergraduate faculty, and mathematics education researchers). The materials for the morning mathematical problem-solving course were designed through a process that began with a team from Education Development Center and Boston University that developed the framework for the morning mathematical problem-solving course materials and broadly designed the mathematics problems. The team consists of Al Cuoco, Glenn Stevens, and a small group of secondary teachers and educators that changes from year to year. The instructors, who were on the original design team, designed the actual course while at PCMI. They revised the problem sets every night to accommodate what happened in class that day, and, by the end of the three weeks of PCMI, the draft of each course is the result of careful thought in Boston and some extremely thorough rewriting based on real live field tests in Park City. It is this collection of courses that Mathematics for Teaching polished into a published set of professional development materials. The American Mathematics Society is the publisher of the volumes in the Mathematics for Teaching series, which consists of: (1) Volume I, Famous Functions in Number Theory. This course develops the idea of arithmetic (multiplicative) functions like the function that counts the number of integer factors of a positive integer. In very informal settings, it introduces ideas of Dirichlet series as combinatorial tools, investigating ideas like: the number of ways an integer can be written as the sum of two squares, the number of lattice points inside a circle as a function of its radius ("Gauss’ circle theorem"), classical functions like the Euler -function, the sum of divisors function, and ways to create new multiplicative functions from old via Dirichlet convolution. (2) Volume 2, Probability through Algebra. The course is designed to show how middle and high school mathematics can be applied to statistical analyses of phenomena, both from within and from outside mathematics: an analysis of randomness in sequences of integers, the use of polynomial algebra and the idea of generating functions to investigate distributions, the use of expectation to analyze popular games, and an investigation of phenomena associated with the integers: the distribution of primes, the probability that a random integer is square-free, and so on. (3) Volume 3, Applications of Algebra and Geometry to the Craft of Teaching. One theme in the course is to use algebra (Gaussian and "Eisenstein" integers) and geometry (conics) to design problems that "come out nice" such as Pythagorean triples, integer sided scalene triangles with a 60? angle, cubic polynomials with lattice point zeros and extrema, triangles in the plane with lattice point vertices and integer side length.
