**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.