

When Wrong is All Right
by Gautam Borooah, Mathematics
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 problemsolving
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 shortterm goal of correcting any errors they see rather than the
longterm goal of fostering ideas and criticalthinking 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 appproach
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 lessthanperfectlyefficient method was a closer reflection of students'
thought processes and gave them a more realistic expectation of what they
should expect problemsolving 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 problemsolvers. 