The importance of "definition" is obvious to mathematicians, but perhaps it is not always clear to them that it is not so obvious to teachers. The leaders of one MSP working primarily with underprepared middle and high school teachers have designed experiences to communicate to teachers the centrality of definitions in mathematics.
What often happens with this topic is that there is a focus on providing specific definitions and having teachers memorize and apply them. While doing so is certainly important in mathematics, teachers (and their students) may come to see definitions as statements given to them as if "dropped magically from the sky" to be discovered and memorized. This MSP is working to ensure that its courses provide opportunities for teachers to discuss the role of definitions in mathematics in general, and in their classrooms. The courses are also designed to provide teachers opportunities to examine the interplay between concept development and using the precise language of mathematics. The approach is to describe that concept thoroughly but without redundancy, so that teachers come to understand the value of the process in removing ambiguity, and in clarifying their own understandings.
In all courses offered by this MSP, teachers encounter mathematical definitions. For example, in the semester-long course on logic and reasoning, the topic of mathematical definitions often begins by asking teachers to "define" a familiar concept (e.g., a rectangle or prime number)—one in which they could recognize the "object" if they saw it. Teachers are then asked to create examples of objects that satisfy their "definition" and yet clearly violate their conceptual understanding of what would be acceptable. Teachers then consider notions of language precision, equivalent definitions, where definitions come from, and the importance of definitions in reasoning and justification.
An MSP offers numerous courses in science content, starting with introductory level content and moving up through advanced topics. One advanced biology course deals with genetics and cellular biology, including topics such as the sodium-potassium pump. This course includes both lectures and hands-on laboratory work. The class meets for two weeks, five days a week, eight hours per day.
One week of the course is devoted to genetics. It is taught by a STEM faculty member who is a genetics researcher, with deep understanding of both the content and processes involved with the study of genetics.
The instructor's understanding of research in genetics, and the ways in which it has been communicated with teachers, have played a crucial role in the learning experiences for participating teachers. For example, the assignments and assessments that the teachers engage with during the "genetics week" typically ask teachers to provide evidence of their understanding of both "content" and "understanding of the discipline." Teachers' assignments often involve them not only in explaining their understanding of a concept in genetics, but also in designing an experiment that could be used to test that concept.