Effective Teaching Techniques for Pre-Calculus and Trigonometry Success

(This is what is printed in the MAA program) Encouraging Effective Teaching Innovation Part A: Thursday, August 2, 9:00 11:55 AM, Governor s Square 12, Plaza Building Organizers: Susan Crook Loras College David Failing Lewis University Russ Goodman Central College Mami Wentworth Wentworth Institute of Technology This session will consist of presentations of demonstrably effective and innovative classroom techniques that address the reasoning behind, design, and implementation of resources or activities. This may include whole course techniques (not necessarily original to the presenter) or drop-in activities to bolster student learning and reflection in any course. Materials will be shared after the session at: http://mathfest2018.davidfailing.com. Keith Carlson University of Central Florida Exploration of Methods in the Teaching of Pre-Calculus There is a direct correlation of student competence in pre-calculus algebra and their success in calculus. Failure in calculus essentially prevents the completion of studies in a wide variety of subjects. At the University of Central Florida we have been concerned with the preparation of students for calculus. We have explored teaching that emphasizes those topics that frequently create difficulties for students learning calculus. In this talk we will outline the details of our work.

Exploration of Methods in the Teaching Exploration of Methods in the Teaching of Pre of Pre- -Calculus/Trigonometry Calculus/Trigonometry Keith Carlson Keith.Carlson@ucf.edu Department of Mathematics, University of Central Florida Orlando, Florida MathFest 2018 August 2, 2018

The MALL (Mathematics Assistance and Learning Lab) The MALL (Mathematics Assistance and Learning Lab) and the Emporium Model at UCF and the Emporium Model at UCF At UCF the Mathematics Assistance and Learning Lab (MALL) is a dedicated computer lab for students in intermediate algebra, college algebra, pre- calculus, and trigonometry When working in the MALL students have access to on-demand help from GTAs, undergraduate tutors, and instructors as well as the software system for their course The Emporium Model for teaching these courses has evolved over the last six years As originally conceived, the Emporium Model was implemented by requiring of all students at least three hours weekly time working in the MALL in addition to one lecture hour per week, except during test weeks

Trigonometry is Important for Calculus Trigonometry is Important for Calculus Students need to know basic properties of trig functions and graphs A variety of trig identities are useful: rational, reciprocal, complementary angle, Pythagorean as well as more complicated identities Calculus II integration techniques include trigonometric substitution Solid knowledge of trig functions and identities minimizes confusion with differentiation and integration rules The two common approaches to teaching trig are the right angle approach and the unit circle approach

Benefits to Right Triangle Approach Benefits to Right Triangle Approach Most students have encountered right triangles and acute angles before Many students are familiar with opposite, adjacent, and hypotenuse In the first quadrant all six trig functions are defined and are geometrically intuitive

Drawbacks to Right Triangle Approach Drawbacks to Right Triangle Approach For angles outside the first quadrant students may be confused that the acute angle of a reference triangle is not the original angle ? in question Some students are uncomfortable with the negative side lengths associated with reference triangles for finding trig functions of angles outside the first quadrant Without negative side lengths, another mnemonic like ASTC must be used for positive/negative function values

Benefits to Unit Circle Approach Benefits to Unit Circle Approach Handles any real number argument easily, which is why calculus instructors love it Makes visualizing radians easier Determining trig functions of ? based on values of ?, ?, and ? will automatically report positive, negative, and zero values, so that no mnemonic like ASTC is needed, though it provides a check

Drawbacks to Unit Circle Approach Drawbacks to Unit Circle Approach Abstraction difficult for many students Memorization of trig function values as ratios involving ?, ?, and ? (e.g., sin? =? ?) is necessary from the start, which some students may find less intuitive than a mnemonic like SohCahToa whose syllables describe ratios of sides of a right triangle

Arrangement of Topics to Decrease Number of DWFs Arrangement of Topics to Decrease Number of DWF s When we sought to decrease the number of D s, W s, and F s as semester grades, one thing we tried was rethinking time given to some of the topics One decision we made was to drop vectors, dot products, and cross products and leave these topics to physics This enabled us to increase the time given the material that seems to be most difficult for students, the material on identities through half- angle identities Increased time for this more abstract material was intended to allow more time for assimilation by students

Changes Made in the Emporium Model at UCF Changes Made in the Emporium Model at UCF Starting in the Fall 2014 semester the amount of time for identities was increased Starting in Fall 2016 two changes were made: The textbook was changed, from a text that used the unit circle approach, to a text that started with a right triangle approach and then soon presented the unit circle approach The number of 50 minute lectures per week was changed from one to two, while the number of required hours in the MALL per week was decreased from three to two (1 + 3 to 2 + 2)

Data of Numbers of DWFs by Fall/Spring Semesters Data of Numbers of DWF s by Fall/Spring Semesters Semester % Total DWF s Spring 2014 37.06 Fall 2014 48.09 Spring 2015 41.18 Fall 2015 47.33 Spring 2016 47.92 Fall 2016 22.47 Spring 2017 25.03 Fall 2017 19.44 Spring 2018 15.77

Impact of Changes in Teaching Trig Impact of Changes in Teaching Trig We believe that careful arrangement of the contents and an improved mixture of lecture and work at the MALL has helped students master the content more effectively We hope that the decline in DWF s is indicative of a greater number of students being prepared to take calculus and less student dissatisfaction We are collecting data over a period of three years to attempt to determine the impact of our trig on Calculus I performance

Further Changes in Teaching Trig Further Changes in Teaching Trig This coming year we are planning to adopt a new textbook and initiate a more adaptive approach to our instruction The adaptive approach most weeks will include an optional pre-test, a graded study plan, and a graded post-test taken in the MALL The pre-tests will provide students with an opportunity to prove that certain things they already know do not require additional work; the post- test will provide evidence of the effectiveness of work done in the study plan that week We are interested in how the adaptive approach might affect student preparation for Calculus I

Thank You for Your Thank You for Your A Attention ttention Comments or Questions? Visit us: Department of Mathematics University of Central Florida Orlando, Florida 32816