This professional development activity consisted of five day-long workshops, which took place approximately every other month over a one-year period. Approximately 25 high school and middle school mathematics teachers attended each meeting—all had previously completed a six-week immersion experience in mathematics.
The workshops were designed to support a culture of exploration and problem solving among teachers. A key goal was to develop advanced mathematical ideas and show how they are related to teacher work in the classroom. The workshops provided numerous examples of activities that can be used at many different levels in secondary school classrooms. The activities were a mix of problems for teachers to attempt and discussions about how such problems might be used in the classroom. These were not "make and take" activities for teachers to bring back to their classrooms; rather, they were investigations of mathematical topics that underlie the school curriculum and that connect in some way to advanced mathematics—either in content or technique of exploration. Examples ranged from investigations involving repeating decimal expansions for rational numbers to elementary dynamical systems.
For example, at one workshop the activity theme was fitting functions to data. The question for the day was "How do you find a polynomial of minimal degree that agrees with an input-output table?" This activity started with a discussion of sequences and patterns, specifically the idea of finding the next term in a sequence, and contrasting that idea with the problem of giving an explicit formula for the nth term in a sequence. Several sample problems of increasing difficulty were posed. Two methods—Newton's difference formula and Lagrange interpolation—were developed.
The teachers tended to be quite good at recognizing simple patterns. They naturally compared consecutive terms in a sequence and quickly developed the habit of forming the sequence of differences between terms from a given sequence of terms. As the problems became more difficult, they began calculating higher order differences of the terms in the sequence and looking for patterns in those. The teachers experimented with families of sequences including geometric, polynomial, and Fibonacci. One teacher noticed that the sequence of successive differences of a geometric sequence is again geometric and explained the connection of this characteristic with the binomial theorem. Other teachers recognized that the successive differences of a polynomial sequence are eventually zero and one teacher even developed a method of working backwards to find the polynomial. One teacher explained how Lagrange interpolation was connected to the Chinese remainder theorem. Although the teachers would not be expected to engage their students in discussions of these advanced topics, they helped teachers understand the connections of topics in the K–12 curriculum to more advanced mathematics.
A state university offers semester-long, on-line science courses for elementary, middle grades, and high school science teachers. All science courses are taught by university STEM faculty, often co-facilitated by a graduate student and/or K–12 teacher. The courses include both independent work and required participation in on-line asynchronous discussions. In all courses, the primary emphasis is on disciplinary content. Emphasis on classroom applications varies by course and by instructor.
One of the offerings was a course for high school physics teachers on special relativity. This topic generally appears at the end of high school physics texts, if at all, and is rarely taught at the high school level. Special relativity is, however, critical for deep understanding of many topics that are taught in high school physics classes. Newton's Laws of Motion, for instance, are a staple of all such classes. Teachers who understand special relativity recognize that Newton's Laws work well for most situations students encounter, but are not appropriate in situations where an object's speed approaches the speed of light. The course focused on developing teachers' understanding of special relativity not so that they could teach it to their students, but primarily so that they would understand limitations of topics they do teach and avoid fostering misconceptions.
The course was structured in 13 week-long sessions. Participants spent an average of 10 hours per week on the course, including readings, homework assignments, and participation in the on-line discussion. Each week, the instructor (in this case a physicist) posed one or two rich questions for the class to consider during the week. For instance, one discussion focused on the twin paradox, a classic problem in special relativity. The instructor explained that one twin traveled for several years on a special spacecraft that flew constantly at speeds approaching the speed of light. When the twin returned home, he had aged, but not as much as his brother. How can special relativity explain this phenomenon?
For each discussion, one participant was assigned to be the "first responder" and post a comment to initiate the on-line conversation. Other participants joined the discussion, with the instructor occasionally asking a clarifying question, re-directing the conversation, or providing direct instruction in the rare case when the group could not make progress on its own.