John A. Beachy

As a mathematician, I am motivated by the sheer joy of discovery and understanding that I find in my work. As a teacher, I am motivated by the desire to help others to find that same spark of joy in discovering and understanding new concepts.

The most rewarding part of my career as a teacher is to work with a student one-on-one and to witness the moment that he or she finally grasps the essence of a new idea. That moment can surprise and delight a student at any level, from beginning algebra to category theory. Sharing that moment, however brief, and no matter how deep the subject matter, always boosts my enthusiasm, and continues to provide the motivation for the necessary hours of preparation and grading.

I joined the NIU faculty in 1969, and in the subsequent years I have certainly found enjoyment in teaching at all levels. The classes nearest to my own work have been at the graduate level, where I have taught a number of students on the verge of completing their doctoral research. At the other end of the spectrum, I have found it rewarding to work with students just beginning to learn college algebra. Their problems are different, but their joy at finally understanding a concept that had seemed impossible to comprehend is just as visible, and just as exciting.

Much of my teaching effort has been spent on developing and teaching the first abstract algebra course, at the junior-senior level. Most of the students are taking their first axiomatic course, and have difficulty with the level of abstraction. The course is in many ways a writing course, since the students are just learning to construct and present proofs. I try to pose a series of problems that will gradually develop their powers of analysis and teach them some of the fundamental tools of the trade. A careful development is necessary; it is important to ask the students to stretch to reach the goals of the course, but they must also believe that the goals are within reach.

Writing a textbook for the abstract algebra course was a natural outgrowth of my teaching. I do not think that I could have stopped the process, even if I had been able to predict the amount of time it would consume. During the writing, the many helpful comments from students and the many debates with my co-author certainly sharpened my presentations in the classroom.

In lecturing, I confess that my personal goal is just simply to enjoy the hour. That is not to minimize the level of the work, since I also confess to enjoying the challenge of difficult tasks. The rewards for a student are also proportionally greater when hard work is involved. The lecture must be intriguing and challenging, but within reach of the students who are genuinely studying. I hope to be able to capture the interest of my students, and I know that I cannot do that without showing my own interest. I would hope that students feel that I challenge them to do their best. I would also hope that they see evidence that I have a great deal of empathy with their struggles to learn. That is one reason that teaching should be complemented by research, to keep a sense of humility in the face of ever expanding knowledge.

One of the major challenges in my teaching is the use of technology in mathematics and the use of that technology in teaching. I have experimented with various technologies, as they have become available. It is all too easy to be seduced by the promise that learning will be made easier. Too often the technology simply adds another layer of difficulty, and does not make a genuine contribution to teaching and learning. Our major challenge is to use the technology appropriately, and to make certain that it does contribute to learning.

Some of my most successful classes have used old rather than new technology. In small classes, I often have students work in groups at the blackboard. Chalk and slate is tried-and-true technology; it has the advantage of covering a large area, so all of the work is visible to everyone. This leads to a more cooperative setting than if one person is making a presentation to the entire class. This cooperative effort is important, but the individual effort is equally important. I believe that most learning in mathematics occurs in very personal, individual efforts to solve problems. The moment of understanding can be an intensely intimate one. Because learning in mathematics is so individualized, it is rare that an entire class of students is ready for a simultaneous quantum step.

In the entire process of teaching and learning, I believe that mutual respect plays a major role. This is one of the areas in which I have seen a great deal of change during the years I have been teaching. I do my best to respect my students as individuals, and to treat them fairly, with an open discussion of grading. It is important to have this basis for learning, since respect for others seems to have diminished in society at large.

One of my reasons for choosing a career in teaching was to be of service to others. I have been amply rewarded by my students, and I have never regretted the decision.