Challenge
Professor Rolland Trapp wanted a strong visualization tool to help his students better engage with and understand key mathematical concepts.
Solution
Trapp adopted Maple, with its in-depth visualization tools, to provide his students with hands-on problem solving, allowing them to interact with formulas and develop a better understanding of how they work.
Result
Students have provided positive feedback about Maple, and using it has created a higher level of engagement with course materials. Trapp also used Maple to create an interactive multivariable calculus textbook using MathApps.
Providing students with a strong understanding of complex ideas and concepts is a critical component of teaching. Using visualization tools to help students envision how formulas are impacted by changing data and parameters can go a long way in improving their grasp of mathematical materials. This is especially critical in science, technology, engineering and mathematics (STEM) courses, where students regularly encounter difficult concepts. Rolland Trapp, a professor at California State University San Bernadino (CSUSB), relies on Maple’s visualization tools to provide his students with a deeper understanding of how formulas work.
Trapp was introduced to Maple while completing his post-doctorate at the University of Michigan. The school was piloting multivariable calculus modules using Maple. Trapp was immediately impressed by Maple’s visualization capabilities, and has used it ever since. “I liked that you could develop 3-D intuition relatively easily in Maple,” he said. “There are good computational aspects, but it’s the visualization that I’ve really enjoyed.”
Maple allows Trapp to better communicate his thought processes to his students by letting him provide them with an image of the concepts in his mind. “I have an image of what’s going on, and being able to communicate that with the graphic capabilities of Maple is very beneficial,” he said. “It’s a powerful tool, period. Pictures, words or static images, they just don’t accomplish the same thing. Maple gives the students a much better understanding of what’s going on.”
CSUSB has a long-standing Research Experiences for Undergraduates program, which is funded by the National Science Foundation. Trapp uses Maple with these undergraduate students to help them visualize concepts such as the translation from topology to geometry in hyperbolic link complements. “I’ve been using Maple for several years now to support these undergraduate research projects,” he said. “Maple has a lot of different components that are beneficial to the students’ learning.”
Trapp uses Maple to provide his students with hands-on, interactive experience that allows them to better process mathematical ideas. “When I teach a course like upper division geometry, I use Maple for visualizing figures in projective space,” he said. “When you use homogenous coordinates for the projective plane, it’s very strange for students to think of a line through the origin as a specific point in the projective plane. It’s abstract ideas like that which they find difficult to process. So using Maple to illustrate what a projective figure looks like, they really find that beneficial.”
Trapp says that his students are constantly asking questions about Maple and wanting to learn more about its capabilities, so they provide a lot of feedback on its efficacy. “Well-designed lessons can lead students to discover important concepts and, conversely, important concepts are illustrated by well-chosen examples,” he said. “I can spend an entire lesson discussing level curves of surfaces, and even have students do analytic examples, but three minutes interacting with a three-dimensional graph containing a surface intersected by a horizontal plane in Maple is worth twenty minutes of description and lecture about algebra.”
In addition to teaching, Trapp is authoring a multivariable calculus textbook that uses Maple, relying on Math Apps as an integral part of the exposition. He was motivated to write the book because he found that his students were having a hard time understanding formulas, and he wanted to provide them with greater knowledge and understanding of what each formula signified and how it worked. “I was talking with my son about what I needed to do to create a helpful text, and he said ‘you have to make it interactive,’” Trapp said. “That got me looking at Maple’s MathApps. I am creating a standard multivariable calculus book as far as content goes, but I create apps along the way that help illustrate different aspects of the theory and shed light on what the formulas really mean.”
“The goal of most math educators is to help student internalize mathematics,” he said. “MathApps in Maple provide a nice picture for students to have in their mind as they work through concepts. I’ve also had the opportunity to showcase my work in Maple in some professional presentations, which has garnered positive feedback from colleagues.”
Trapp has embedded hyperlinks in the textbook that students can click on, taking them to a MathApp that better explains the specific concept and allows the student to play around with different parameters and see how the formula is impacted. Maple is especially good at helping students make the connection between 2-D to 3-D, Trapp said. “Students have a hard time jumping from 2-D, which they’ve studied for years, to 3-D in a couple weeks at the beginning of a multivariable calculus course,” he said. “Maple helps them make that transition. I’ve tried other math software, but Maple’s images and visualization capabilities are so much better. It’s user-friendly and more aesthetically pleasing.”
Ultimately, the benefit of using technology in the classroom comes down to providing students with practical experience to help solidify their conceptual understanding of mathematics. Working with Maple provides students with strong visuals for concepts and ideas that can be difficult to grasp. The online textbook provides students with an interactive way to engage with the course materials. “It’s pretty standard material being presented, but I’ve taken advantage of the graphics in Maple to create material that make it more interactive for students,” he said. “Maple is an essential part of the book.”
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