by Glenda Lappan, NCTM President 1998-2000
NCTM News Bulletin, November 1999

Every day in every mathematics classrooms around the world, teachers of mathematics work hard to help students learn. Sometimes their learning excites us and sometimes their lack of it disappoints us. And sometimes we think the students have learned when in reality they have not.

Some of us are very attuned, on a daily basis, to what our students are learning, to what they are understanding, and to what they can do with their knowledge. Others of us are less so. What makes the difference? Our effectiveness, in large part, is enhanced or limited by the depth and breadth of our understanding of mathematics itself.

Our own content knowledge affects how we interpret the content goals we are expected to reach with our students. It affects the way we hear and respond to our students and their questions. It affects our ability to explain clearly and to ask good questions. It affects our ability to approach a mathematical idea flexibly with our students and to make connections. It affects our ability to push each student at that special moment when he or she is ready or curious. And it affects our ability to make those moments happen more often for our students.

We probably can all agree with the importance of content knowledge at some level. But we need to continue to hone our mathematics content knowledge throughout our teaching careers. It is easy to develop routines in our teaching so that we simply never get to the parts of the book we do not enjoy or do not feel comfortable teaching. Recently, an elementary school teacher told me that in the more than 20 years she has been teaching, she has never taught the geometry sections of her text because she simply does not know the geometry. That was a brave admission. And in fact, she was taking a geometry workshop because she had decided she owed it to her students to learn it.

This story easily could have been about a teacher at any other level. The environments, including colleges and universities, in which we are educated and work simply are not very conducive to the in-depth analysis of the foundational ideas in grades K–12 mathematics. For example, many of us were taught to use formulas for finding areas of certain geometric shapes, but we were never pushed to think about why those formulas make sense. Consequently, we teach our students to approach measurement at a very instrumental level--"plug and chug" as we laughingly say--rather than teaching with an emphasis on understanding what it means to measure or in ways that promote seeking relationships among figures and formulas. Our students become the ones who remember that 2 r and  r² are formulas for something about a circle but do not understand mathematics well enough to figure out which must be the formula for area and which for circumference.

Our own knowledge or lack thereof can be the stumbling block to our students' learning. So what can we do about this? There are many possible avenues for enhancing our knowledge of mathematics.

One way is to learn from your own text materials. Try starting a math study group around the text materials that you use. A group can be as small as two people--you and a friend. Meet on a regular basis and together examine the mathematics that you are teaching. Ask questions about how to teach each lesson more effectively. This leads to examining the mathematics in more in-depth ways. What questions will help students better understand this idea? How does this lesson fit with the next? Where is this idea going? To what is it connected? Why is it important? What examples or analogies will help students understand? This can be a safe place to even ask such questions as, "Where on earth did this come from?"

Examining students' work is another effective way to deepen your mathematical knowledge--especially if you do so with other colleagues. The conversations that focus on trying to figure out what a student really knows invariably lead to enhancing our own understandings of mathematics. One summer my colleague and I were working with high school teachers from 12 schools in our area. We grouped the 24 teachers in pairs across schools and gave each pair some examples of students' work to grade. Wow! On some of the papers, the range of scores led to great debates about what the student really knew. In order to have these conversations, the teachers found themselves carefully analyzing the mathematics of the problems. And there you have it--mathematics learning at its best.

Other avenues open to us all are more formal workshops or courses that focus on content we want to understand better. But even if we attend such offerings outside our schools, our knowledge still has to be translated in ways that help us teach the content of our specific curriculum. The bottom line is that we have to deeply understand the mathematics that we actually teach if our students are to have a chance to do the same.

Don't let this school year go by without adding to your own knowledge of mathematics. Get involved and get others involved in understanding this beautiful subject that we teach.