Professor Ilias Kotsireas wanted a solution to help his students better adapt to his Experimental Mathematics approach; to help them better visualize problems, work through solutions in a less time-consuming manner, and assist them in completing an in-depth portfolio project.
Kotsireas implemented Maple into his courses, getting students to solve algorithmic problems and work through difficult conjectures as a more effective method of learning key concepts.
The students and the University responded positively, citing Maple’s ease of use, functionality, notation and visualization characteristics. Students required less assistance after Maple was introduced, and a higher percentage completed their course work.
With an enrollment of over 20,000 students, Wilfrid Laurier University is a rapidly-expanding institution that is highly regarded in the academic community. Ilias Kotsireas, a professor in the Department of Physics and Computer Science, and Director of the CARGO Lab, has been teaching at the university since 2001. In his mathematics courses he has found that Maple improves his students’ ability to learn the materials and understand difficult concepts.
Kotsireas began using Maple in the late 1990’s, when he was completing his Masters and his Ph.D. in France with renowned French mathematician and computer scientist Daniel Lazard. Lazard is a professor emeritus at Université Pierre et Marie Curie in Paris and is regarded as one of the most influential pioneers in the field of computer algebra. His algorithms were introduced into the earliest versions of Maple. Lazard introduced Kotsireas to Maple and he was amazed by its capabilities. “I couldn’t believe what he was showing me in Maple was possible,” Kotsireas said. “The functionality was way beyond other tools I had known; I was very impressed.”
When Kotsireas started teaching, he introduced Maple into his classroom. In teaching math courses with a computer science focus, he had students use Maple programs to solve algorithmic problems. Kotsireas used to assign individual assignments and group activities to his students, but found that those assignments led to some topics being skipped over or not fully investigated. He switched to an individual approach where, throughout the course, students develop a full portfolio using Maple, which forces them to work through all key concepts in the course. Kotsireas received positive feedback from students for this approach. “To be honest, I expected some resistance,” he said. “I expected the students to say ‘it’s too hard, it’s too much’. But I’ve received only positive feedback.” He said the University is also pleased with the new approach as it creates sustainable engagement with students.
A key aspect of Kotsireas’ teaching approach is what he refers to as Experimental Mathematics, a research area with important pedagogical ramifications, largely developed by the late Jonathan M. Borwein. He encourages his students to develop a conjecture and solve it step by step. This cultivates a “learn by doing” approach that promotes active learning and helps students better learn and understand key mathematical concepts. Due to the difficulty of some of these problems, being able to visualize each step can make experimentation easier and provide greater insights. Maple is a key tool in this approach, as it allows students to engage and interact with the problems in ways not possible by other means. “Maple helps make abstract concepts more tangible and down to earth,” Kotsireas said. “You can visualize examples and show students how to develop intuition for creating formulas, which is what experimental mathematics is all about, and a very effective way for students to learn.”
He kicks off each course with a three-hour tutorial introducing students to Maple. Since instituting the tutorial, Kotsireas has noticed that students are able to learn and work through the course material more independently and more students are completing their course work. The course is designed as a platform for students to gain an understanding of symbolic and numerical computations. Maple is an ideal tool to help them achieve that understanding, Kotsireas said. “Maple helps tremendously in instigating a culture of experimentation. Students learn the concepts better by working with Maple; it’s a good way to engage them and get them to learn on their own.”
Kotsireas said he chose Maple over other mathematics software for its ease of use. “Maple allows the students to see math differently, and that makes the subject exciting,” he said. “Maple is more intuitive, less effort-intensive and more natural than other options on the market.”
Apart from his teaching, Kotsireas also uses Maple heavily in his research projects. Combinatorial Design Theory has been his area of research for over 13 years. He finds Maple to be very effective in symbolic computing and parallel programming and heavily relies on it as a tool for metaprogramming. He used Maple to introduce new symbolic computation techniques in his research and this was very well accepted within the group of established researchers in this area.
In his 20-plus years of working with Maple, Kotsireas has seen Maple evolve, and feels that the changes over the years have been impressive. He encourages the University to adopt Maple in more courses to enhance the efficiency of its educational offerings and to provide a valuable tool for other instructors to use in their teaching.
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