Course Description Calculus I (MATH 153) is the first course in a sequence of calculus courses required for students in various majors including Computer Science, Engineering, Mathematics, Science, and Technology. The main objective of the course is building the essential skills, mastery, and understanding of the applications of several topics including analytic geometry of plane functions, limits, continuity, derivatives of functions and applications; exponential, logarithmic, trigonometric and inverse functions; indefinite and definite integrals; and the fundamental theorem of calculus. The prerequisite of this course is the completion of MATH 121, a trigonometric functions course, or an appropriate placement test score. The class size has increased over the last few semesters and averages about 40-45 students per class during the fall semester for MATH 153. During the spring semester when Calculus I is off sequence, the class size decreases. All the students in Calculus I have laptops and graphing calculators, although these technologies are not crucial to this lesson. The classroom setting varies among the sections of the course from long tables to individual desks, making it difficult to generalize student interactions in the lesson plan. The lesson follows content on finding critical numbers of a function and locating relative extrema on a closed interval. It includes two major theorems about continuity and derivatives of functions, which also have several applications, and the lesson usually takes approximately two days to complete. The students are split depending on their major into the two versions of the course. Calculus and Analytic Geometry I (MATH 156) contains mainly students majoring in Computer Science and Mathematics, and MATH 153 contains all other majors. The two versions are almost identical with the version for the Computer Science and Mathematics majors requiring more proofs and theory. Although this lesson was tested in a MATH 153 classroom, it could easily be used for MATH 156 or any calculus course including the topics of Rolle`s Theorem and the Mean Value Theorem.

Executive Summary The topic of the lesson is Rolle`s Theorem and the Mean Value Theorem. Learning Goals. 1. Students will understand the meaning of Rolle`s Theorem and the Mean Value Theorem, including why each hypothesis is necessary. 2. Students will complete problems and applications using Rolle`s Theorem and the Mean Value Theorem. 3. Students will appreciate the discovery process of developing mathematics and have a better understanding of the construction and proof of mathematical theorems. Lesson Design. The lesson was designed in order to emphasize the discovery process and the role of proof in mathematics. The first major piece of the lesson is an activity that asks students, in several steps, to draw graphs of functions satisfy various hypotheses. The last graph that students were asked to draw is impossible to draw, because any graph satisfying all of the required conditions would violate Rolle`s Theorem. Rolle`s Theorem is introduced in this way. A second activity involving graphs related to the Mean Value Theorem is used to introduce or study the Mean Value Theorem. These graphing exercises are intended to help students discover for themselves the two theorems and help them to appreciate the discovery process in mathematics. The second major part of the lesson is to work problems involving the theorems to better understand how the theorems are used and apply in practice. The variety of problems is intended to emphasize different aspects of the theorems, including why the hypotheses are necessary and how to apply the theorems to modeling applications and more abstract settings. The final part of the lesson is to prove the Mean Value Theorem assuming Rolle`s Theorem. This portion of the lesson is expected to be difficult for students, so ample time should be allotted for question and discussion. Major Findings. During the first round of the lesson, we learned that students seem to catch on quickly that the second graphing exercise is almost identical to the first and that therefore the last graph is impossible to draw. This seemed to cause a significant reduction in their engagement with the lesson. However, when this activity was changed for the second round, the decrease in performance on certain quiz and homework problems suggests that the repetition may actually have served its purpose of emphasizing the hypotheses present in the two theorems.

Printer Friendly Version of Complete Report The Final Report includes a detailed section on how to teach the lesson, which can be used in conjunction with the Lesson Outline and Lesson Materials below. The Outline and Materials are included as appendices to the Final Report. Also included in the Final Report are data, results and discussion of the findings of the two iterations of the lesson. Final Report

The Lesson Below are links to the lesson outline and the materials required for the lesson. Included with the materials are all of the in-class examples used as well as quiz, homework and exam problems. Lesson Outline Lesson Materials

The Study Full results of our study are linked below under "The Study." For convenience, the data tables and figures containing data are collected additionally under "Data." The forms used by observers during the lessons as well as the post-lesson student surveys are also linked below. The Study (with data) Data Observation Form Post-lesson Student Survey