Traditional expectations for students studying geometry frequently take forms such as: "What is the perimeter?"; "Which angles are congruent?"; "Find the volume." While proficiency with procedural skills retains its importance in the Common Core standards for mathematics, understanding of procedures involved in calculations has acquired increased importance. In the introduction to the Common Core State Standards for Mathematics we read:
Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness.
One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from.
Although the new emphasis on understanding is evident throughout the Common Core standards, in some cases it is particularly prominent. We find good examples in two related high school geometry standards G-GMD.1 and G-GMD.2. These standards direct students to an understanding of the volume formulas that they have learned to recite, write, and apply. Mastery of these standards requires students to be familiar with Cavalieri's principle.
Mathematicians have used Cavalieri’s principle (not necessarily under this name) for thousands of years. It finds its way to school as an essential element of high school geometry only now. ATI believes that Instructional Dialogs can aid both students and teachers (and even parents) in working with Cavalieri’s Principle. To this end, we have developed two dialogs devoted to this fundamental concept. One is a teacher version, which contains supplemental notes and insights to prepare teachers for content delivery. The other is a teacher-student version, containing the same instructional content but designed for use with groups of (or individual) students. These dialogs were carefully crafted to build upon geometric concepts students are already expected to understand, and the questions posed are articulated to assess understanding of Cavalieri’s principle as well as the ability to apply it.
These published dialogs are introductory in nature. In the near future, ATI plans to publish related dialogs to address the role of Cavalieri’s principle in explaining or proving volume formulas.
We are pleased to present these tools, and trust they will prove useful in teaching a potentially-challenging topic.