### by Gautam Borooah, Mathematics

#### Teaching Effectiveness Award Essay, 2004

Math 113, Abstract Algebra, gives many students both their first view of the elegance and rigor of mathematical thought and the first test of their creative problem-solving skills. But the polish of textbook mathematics often intimidates students and inhibits their creativity. Since mathematics in books is (almost) always correct and students’ work is often wrong, they think that they cannot produce “real” mathematics. They are so afraid of coming up with a wrong idea that they do not articulate any ideas at all: they are too afraid to try. This is compounded by instructors who focus on the short-term goal of correcting any errors they see rather than the long-term goal of fostering ideas and critical-thinking skills. The problem I identified was how to unlock students’ creative potential. I needed not only to eliminate their fear of making mistakes, but also to show them that making mistakes is an essential part of the mathematical creative process.

I began by setting the students problems that encouraged an experimental, questioning approach. For example, I would ask “Is statement X true or false?” (rather than “Prove that statement X is true”) to give students a wrong path to follow (they might try proving that statement X is false) from which they could still gain useful insights. In group work I emphasized a democratic approach in which students worked on problems together and were encouraged to follow through the ideas of all the group members regardless of where they led. This gave a better understanding of the problem than simply doing it the fastest way. When solving a problem in lecture, I first solicited a wide range of responses from students; specifically asking for intuitive guesses, approaches that might not work and potential misreadings. These were represented as useful ways of building understanding and necessary steps on the path to the right solution rather than as mistakes to dismissed. By not privileging the right answers, I was able to build a more inclusive class dialogue in which everyone’s ideas were valuable. As a result, previously silent students became more confident about speaking out in class and confident students learned to listen. Then, rather than using the fastest method, I would begin on a “wrong” approach, and demonstrate information and intuition that could be gleaned from this. Solving the problem by a less-than-perfectly-efficient method was a closer reflection of students’ thought processes and gave them a more realistic expectation of what they should expect problem-solving to feel like. Lastly, when grading students’ work, I was careful to give credit and praise for creative ideas even when they did not directly help to solve the question and to indicate how and where such ideas could be used. My only stipulation was that students engage with the problem in some way.

Throughout the semester, in conversation, and in formal and informal evaluations, students told me that they were becoming more comfortable expressing their ideas in stark contrast to many previous classes. And beyond merely expressing themselves, the students used their critical skills to evaluate their ideas and choose the most appropriate for the task at hand. Over the course of the semester, the typical response to a hard problem changed from “I have no idea how to start this” (Translation: “Any ideas I have may be wrong so I’m going to keep quiet”) to “I don’t know if this will work, but let’s try this” to “I had a few ideas and one of them worked.” This freedom of expression produced better written homework and exams and lively class discussion: all filled with more ideas and more correct ideas than anything I had seen before. The students (even those who had been most nervous about getting things wrong) had become creative and competent problem-solvers.