Coherence and enrichment across the middle and secondary levels: Four mathematically authentic learning experiences

Abstract

(ProQuest: ... denotes formulae omitted.)INTRODUCTIONThe Oliver Heaviside function. Fermat's Last Theorem. Modular arithmetic. Dihedral groups. These areas are commonly studied in college, perhaps in an undergraduate course in engineering or a graduate-level seminar in mathematics or computer science. Is there a place for these advanced topics in the high school curriculum? Can teachers of 9-12 connect content to these areas? Most importantly, can students benefit by engaging in problem settings where these ideas are introduced? In this article, we argue that the answer to each of these questions is yes.Teachers of mathematics undoubtedly see important connections in their subject area but these connections may not be evident to students. By blurring the boundary between the high school and college curriculum, we offer a number of benefits to the high school experience. First, exploring challenging content can whet students' appetites to the investigative nature of mathematics. Even if these investigations prove difficult, students confront the limits of their mathematical knowledge and are encouraged to learn more; this is a valuable lesson. Second, there are beneficial ties to areas such as engineering, physics, biology, or art. In very simple cases, innocent questions can carve a pathway to research-oriented mathematics. Third, students may come to realize that pure mathematics can be interesting and engaging-lest we say this is often shocking to students! Finally, these experiences may provide some much needed enrichment to those who need it. Interesting problems that use novel tools can leave a lasting impression on students.This article is principally about introducing young students to mathematical ideas and practices not typically explored at the high school level. We discuss four problems with appropriate origins in high school settings. The problems then flourish, extend, and enlighten by using sophisticated tools typically reserved for college mathematics. In The Parking Garage, we introduce a modeling problem suitable for a standard algebra course (Grade 8-10) but we use the Heaviside function as the centerpiece. Students learn about piecewise and discontinuous functions-important objectives in the high school curriculum-but via a tool typically used by engineers. In Clock Arithmetic, students operate in the familiar modulo 12 as this builds upon previously learned elapsed time problems that surface in CCSSM in Grade 4 (CCSSI, 2010). Then through student-created informal definitions and generalizations, discussions in modular arithmetic lead to non-trivial extensions in number theory and randomness. In Pythagoras meets Fermat and Newton, students begin with a ubiquitous mathematical theorem (Grade 8) and engage in the interplay of informal reasoning (conjecturing, experimenting) and formal reasoning (generalizing, extrapolating). This experience mirrors that of professional mathematicians and finishes with the ultimate prize: a generalized mathematical proof. In Star Polygons, we present the definition of a star polygon, a construction that naturally lends itself to line and rotational symmetry explorations, recurrent topics in the K-12 curriculum (Grades 4, 9-10). However, further investigation uncovers rich connections to group theory, usually reserved for undergraduate mathematics students but accessible to high school students given this context.In each of these cases, we import tools and practices more prevalent to college mathematics into the high school experience. Doing so supports mathematics as a unified body of knowledge with clear connections from grade to grade (CCSSI, 2010). As NCTM (2000) reminds us, Mathematics is an integrated field of study.. ..The notion that mathematical ideas are connected should permeate the school mathematics experience at all levels (p. 64).THE PROBLEMSThe Parking Garage. One of the most important concepts in all of mathematics is that of function. …