Georgetown College. Georgetown University.

By LiAnna Davis

Problem solving is a recurring theme in all of Dr. Jim Sandefur’s classes. Certainly he wants his students to learn to solve the mathematical problems in front of them. Yet even more than this, Dr. Sandefur hopes his students can learn something about how they work best to solve a problem, reflecting on what the best approach to a problem is rather than just finding the quickest way to the correct answer. Dr. Sandefur, a professor in Georgetown’s Department of Mathematics, researches in the growing field of mathematics education, or scholarly teaching—a field that directly impacts the pedagogy in his courses.

Collaborating with Dr. Manya Raman of Sweden’s Umea University, Dr. Sandefur is developing a series of video tools that help teach undergraduates how to write mathematical proofs. Dr. Raman focuses on the psychology of mathematical thinking, while Dr. Sandefur focuses on the mathematical content. Such tools can be integrated in courses Dr. Sandefur teaches for Mathematics majors and non-majors alike, including an Ignatius Seminar course on the mathematics of games and puzzles.

“I find it a real challenge to both make mathematics interesting to students and to make them realize that it is interesting and valuable to them,” he says. “Really, in games, to win it’s a problem solving situation. You’re trying to figure out the best approach. It uses the same thinking process in solving proofs and mathematical problems.”

Constructing proofs is a key component of Georgetown’s Foundations of Mathematics course, a gateway class to the major. When Dr. Sandefur started teaching the class, he quickly realized that students tended to memorize proofs the night before a test and otherwise simply ignore them. But proof construction is a technique for problem solving that plays an important role in students’ future academic success, making it a critical skill to teach in such a course. In conceptualizing and developing his syllabus, Dr. Sandefur realized he needed to determine how students learned problem solving before he could adequately teach it.

Dr. Sandefur set aside his research on differential equations and focused his attention on math education. Math education research addresses how students learn the skills necessary to succeed in the discipline and provides curriculum development tools for professors. As a mathematician, Dr. Sandefur knew the material he was teaching; he just needed to discover the best way of presenting it to his students.

Professors and graduate students in the discipline generally have enough experience in mathematics to determine the best problem-solving route before the proof begins, making it seem like a daunting task to undergraduates who often try and fail on several methods. So Dr. Sandefur decided to look to upper-level students as his problem solving examples.

“I videotaped juniors and seniors while they were doing their homework problems,” he explains. “On the tapes, you see these students get stuck in very similar ways to the students taking the Foundations course. They make several false starts and may go through three different approaches until something works.”

Dr. Sandefur has experimented videotaping students both individually and in groups. He finds the videos of the group work to be most useful because the presence of multiple people presents a natural dialogue in which the participants explain their thought process to each other. This discussion about why they choose certain methods demonstrates the problem-solving skills students need to learn in the Foundations course; the videos thus present a case study of these skills in action. In watching these videos, students new to the mathematics major can learn how to make progress on their own (see related video).

Dr. Sandefur credits the contributions of his excellent group of Georgetown student volunteers for the program’s success. All the students who are videotaped in the modeling sessions are willing to have their successes and failures documented for new students to see. While this could be embarrassing for some, Dr. Sandefur’s student volunteers are all nevertheless eager and willing to help further the learning experience and improve the teaching methods in the discipline.

Although the research is still in the preliminary stages, Dr. Sandefur has had some success in his courses using the tapes. Now he is trying to determine what forms the videos should take to make them successful learning tools for students. He is considering either placing the videos online for students to watch individually or dedicating time in class for everyone to watch together. He and Dr. Raman hope to create a collection of videos and a set of papers on their effects in mathematics classrooms. The positive reinforcement Dr. Sandefur has experienced from his students encouraged him to try to expand the program to other schools.

“I enjoy the student interaction,” he says. “It’s nice helping them learn how to learn. The amount of growth I see in students is incredible. My job is always interesting and exciting.”

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