Published: Feb. 5, 2014

Silva Chang projects her lessons from an iPad onto the board in Calculus classes.  She works out problems in different colors, asks students questions to predict next steps, and lets them volunteer answers.  Chang believes that seeing problems unfold live helps students better understand proper solving procedures.

### Writing Code to Illustrate Calculus Concepts

Chang writes code in Wolfram Mathematica to demonstrate more complicated surface area concepts in Calculus courses, including the use of rectangles to approximate area under a curve (Riemann Sum).  As illustrated in these images above that Chang coded in Mathematica, the user can move the cursor from left to right to increase the number of rectangles that could fit under the curve.  According to the Riemann Sum concept, the more rectangles that fit under the curve, the more accurately one can estimate the area under the curve.  Instantaneously maneuvering illustrations in Mathematica can much more effectively demonstrate a concept than would be taking the time to manually draw and erase images on a dry erase or chalkboard.

In addition to incorporating iPad apps and Mathematica images into lessons, Chang also encourages her students to use technology in their homework.  Specifically, Chang assigns her students Webassign textbooks.  These books have associated online homework problems that are slightly different for each student.  Webassign grades homework problems immediately so that Chang can more quickly see her students' results.  Additionally, Chang teaches students how to write a function in Microsoft Excel so that they can use spreadsheets to plug in different variables to a more complicated formula like Newton's Method.  She also encourages students to use the Desmos free graphing calculator website.

At the end of the week’s lessons, Chang may invite the students to to use their clickers to answer a sample question to assess their understanding of what they have learned.  This way, she can immediately anonymously display the students' results to a question, and everyone in the class can see how many students understood the associated concept.  If a significant number of students answered a question incorrectly, then Chang knows she should spend more time explaining the correct method.

Chang says that she hopes to one day become completely wireless so that she could actually walk around the room while teaching and even sit amongst students as she projects the problems that she is working out on her iPad.  Chang finds inspiration in the Khan Academy's methods of using videos with different colors for different steps to explain math problem solving methods.  Eventually, depending on the results of ongoing trials at other schools, such as at Harvey Mudd College, she may consider using the teaching concept of a flipped classroom, in which students would spend their homework time watching videos that demonstrate new methods.  Then, in class, students would practice working out problems themselves, this time with the professor present.

Ultimately, Chang strives to teach her students to understand larger mathematical concepts, more than just finding the correct numerical answer for a test question.

Written by: Moira McCormick