Blog Post #3 in the series "Finding Inspiration and Joy in the Words of Others"

The mathematics I do remember is the mathematics in which I understand how and why it works.—Sarah (2001)

 These words are pinned to the bulletin board in my office. The sentence was written several years ago by a preservice teacher in a reflection about her mathematical understanding and serves as a reminder of the contribution of how and why to one’s mathematical knowledge. Often, how and why are not always embraced as relevant understandings by those who want to get to an answer quickly or who simply want to use procedures and step-by-step processes.

Earlier this summer, I taught a graduate course on mathematical problem solving to a class of young teachers. We tackled a variety of problems from several branches of mathematics—contest problems, recreational mathematics problems, and open-ended problems. We expanded our repertoire of problem-solving tactics and strategies, and we developed perseverance in our efforts to find satisfying solutions. Through authentic engagement in mathematical problem solving, these teachers encountered and recognized rich connections within the mathematics they know and to the subjects they teach. Each teacher developed a plan for providing students with more problem-solving opportunities.

NCTM has long espoused the power of problem solving. Its Agenda for Action (1980) states, “True problem-solving power requires a wide repertoire of knowledge, not only of particular skills and concepts but also of the relationships among them and the fundamental principles that unify them.” Clearly, the relationships among skills and concepts, unifying principles, and mathematical processes are where the how and why of mathematical understanding reside.

In the foreword to Mathematical Mosaic: Patterns and Problem Solving, Ravi Vakil states, “Most ‘big ideas’ and recurring themes in mathematics come up in surprisingly simple problems or puzzles that are accessible with relatively little background” (p. 10). Each of us has favorite problems that we like to use in class—problems that connect to the real world, problems that generate a mathematical topic, problems that contain rich connections, and problems that yield surprising or unexpected results. Problem solving invites our students to encounter the big ideas of mathematics and to uncover the how and why of mathematical concepts.

With all its mathematical potential, problem solving can do more than unite mathematical concepts and processes. It has the power to create and nurture a community of learners, sharing and celebrating the journey toward deeper understanding.

References

National Council of Teachers of Mathematics (NCTM). 1980. Agenda for Action: Recommendations for School Mathematics of the 1980s. http://www.nctm.org/standards/content.aspx?id=17278

Vakil, Ravi. 1996. Mathematical Mosaic: Patterns and Problem Solving. Ontario: Brendan Kelly Publishing. 

Tom Evitts, TAEvit@ship.edu, is a mathematics teacher educator at Shippensburg University of Pennsylvania and is the current president of the Pennsylvania Association of Mathematics Teacher Educators (PAMTE). He is a frequent presenter at NCTM annual and regional meetings and enjoys helping others find, make, and strengthen mathematical connections.