One of the aspects of mathematics that students often do not see in a course is the evolution of a proof from scribblings on scrap to a finished product. Usually the text provides a highly polished proof, and quite often the instructor will either reproduce that polished solution or the ideas behind the proof. Since one of my goals is to have students complete the journey from one extreme to the other, it makes sense to have them both witness and practice all aspects of this process.
I first tried this in my intermediate real analysis class a few years ago. After the students and I read the statement of a theorem in the book, I would tell them to close their books. Using the blackboard as a big piece of scrap paper, we would try to figure out how to prove the theorem. This usually entailed writing down a column with what we know and one with what we want and trying to build a bridge between the two extremes. After this free-wheeling activity produced such a bridge, we would go back and look at the solution in the book. We would see if our scrapwork could be seen in the finished product or if our ideas produced an alternate proof. If our proof differed, I would often type our finished product and hand it out at the next class. While reading this solution, students would often find that I had streamlined arguments or had changed the argument to make the idea more clear. I would ask for their input as to the style and clarity of my finished product.
I soon realized that the students needed to be more involved in the process than simply observing/critiquing my work. If I knew that my approach to a topic or theorem would be different than the book's approach, I would tell the students, and we would do the scrapwork on the board. I started assigning the polishing to them. The grading was based not only on correctness but also clarity and to some extent style. Overall I was pleased with the development of their ability to translate scrap into a polished solution and it afforded me the opportunity to ask them more involved homework questions, either from the exercises in the book or from items that came up during class discussions. Examples would be to show that the Nested Interval Principle is equivalent to the Least Upper Bound Property or to prove the Increasing Function Theorem without using the Mean Value Theorem.
This process became more refined when I started using it in our discrete mathematics course. The students in this introduction-to-proofs course are less experienced than in the analysis class and thus need more guidance through the process. Also the problems are typically shorter, which gives me the time to write both the scrapwork and the finished solution in class. Again, I start a problem by writing SCRAP on the board and putting down a KNOW column and a WANT column. In looking for a bridge between the two columns we draw arrows from facts in the KNOW column to consequences of those facts that seemed to be closer to the statements in the WANT column. We also draw arrows in the WANT column from results that are sufficient to obtain what we want to our final goal. Once we have worked these two columns to points where there seems to be a connection, we start making that connection explicit.
In some circles, there seems to be a debate over whether instructors should emphasize "paragraph" proofs while deemphasizing "two column" proofs. In this class, the two column proof provides a nice intermediate step between the scrapwork (which tends to have arrows all over the place) and the finished product. It seems that both should be utilized, just as an outline provides a framework for a term paper.
While writing the final version, I am able to ask the class whether or not certain details should be put in or left up to the reader. This has led to some interesting discussions and to a better understanding of both the writer's and reader's roles in the process. I have even given problems on tests where I provide the scrapwork and ask students to write a finished product.
My perception is that by seeing and experiencing the whole process, students are writing better proofs and are starting to understand the importance of communication in mathematics. While this technique is certainly no panacea for a traditionally difficult topic, I feel that by involving students in the entire process, they are developing into better communicators of mathematics.
Robert Rogers (email@example.com) received his BS in Mathematics with certification in Secondary Education from SUNY College at Buffalo. his MS in Mathematics from Syracuse University, and his PhD in Mathematics from SUNY Buffalo. He is currently Associate Professor of Mathematics at SUNY College at Fredonia. His current interests are in analysis, the history of mathematics (as it pertains to teaching preservice teachers), the history of calculus and analysis, and applications of mathematicas to political science. He is particularly interested in the communication of mathematics and tries to convey this in his classes. He is a former recipient of the SUNY Fredonia President's Award for Excellence in Teaching and the MAA Seaway Section's Distinguished Teaching Award.