Mathematics is as much a set of ways of thinking—an evolving set of methods and habits of mind—as it is a body of results that have, over the centuries, been derived from those ways of thinking. Students who wish to advance in science or mathematics need a strong background in many of the results—the facts and procedures and topics that often form the tables of contents of our textbooks. But all students, those who may someday choose advanced mathematics and also those who will not, need the habits of mind that the study of mathematics hones and refines. They need to understand how mathematical results are created, and they need to be able to create results of their own. These ways of thinking must therefore be made an explicit part of mathematics curriculum, just as science curricula teach ideas about scientific method alongside the facts that scientific methods have uncovered.

However, the methods of mathematics are many and varied, and they don’t have standard names in wide popular use. The choices we made for this site were intended to be practical. We attempted to create a list of the habits of mind that would be most helpful to teachers looking for good problem sets to use with their classes. We hope that our choices are clear, useful, and reasonably comprehensive.

Let’s search for problems that involve problem-solving strategies for working with graphs.