Something that I encouraged by modeling as a teacher was that many new discoveries could be found if one is willing to take risks. Since deciding to become a teacher when I was 6 years old, I remember how fascinated I was with mathematics. Even though mathematics was very structured, I soon learned that within that structure were numerous opportunities to discover new ways of solving problems. Several years later when I became aware of Number Sense, I had an opportunity to make discoveries of how to solve mathematics problems faster and more efficiently.
Prior to becoming a teacher, most of the risk taking that I did was not seen by anyone. I would study problems from Number Sense tests and I would try to develop shortcuts for how to solve them. The joy that I got from those discoveries was immense. They inspired me to devote my life to searching for better ways to solve problems from basic mathematics to calculus. Early in my career I shared those discoveries with my peers. When I became a mathematics coach I began teaching them to members of my team. Soon I was sharing those discoveries at workshops and with instructional mathematics workbooks that I was writing.
In the classroom, I constantly had students that would ask me if there was another way of solving certain problems. In general, I would first explain how to solve a problem in the traditional way, then I would invite them as I attempted to find a non-traditional way of solving the same problem. In numerous cases I made incredible discoveries. The students joined me as I celebrated a new discovery. During these situations I emphasized that the discovery was made possible because I was willing to take risks. I encouraged them to do the same.
A couple of days ago, I was teaching my students how to solve a problem using synthetic division. After doing several examples, a student asked me how I would do a problem where the divisor was 3x + 2. I told them that I had never known of a way to solve it. I then told them, if they were willing, to work with me in searching for a way to solve a problem where the x term had a coefficient other than one. After several attempts we were successful. I informed my students that I had been asked a similar question each year for 44 years and I had never known how to solve that type of problem. After the successful solution, I informed them that I would begin sharing this new discovery with other students that I would be tutoring in the future. I know that there are many other careers where risk taking has had a major impact in the lives of millions of people, and it is incredible how risk taking can make you a better teacher.