Change. At the beginning, that's what I thought this book was about. Chris was going to try to convince me that teachers need to change the way they teach geometry. The idea of change is not totally reprehensible to me. In fact, I really like to try new ideas, both personally and professionally. However, teachers in my school system have been asked constantly for the last ten years to change something. Sometimes it is a complete change of curriculum structure. Sometimes it is a change of textbooks. Sometimes it is a change of technology. Sometimes it is even a complete change of county philosophy. So the idea of change, while nothing new, causes me to let out a rather deep sigh and to raise my eyes toward the sky with, "What now?"

Being a "seasoned" teacher who has taught every level of student in almost every course invented by the system, I found myself with two major concerns. First, I was very skeptical about the possibility of all students receiving all of the geometry they needed in a classroom comprised exclusively of self-constructed knowledge. There is no way that all students in all groups could be totally involved every day. There is no way that all the "needed" geometry could possibly be discovered with the amount of guidance that I perceived to be present.

This led to my second concern that, given the freedom to construct any knowledge they chose (be it right or wrong), the teacher seemed to be taking no responsibility for their learning. Isn't it my role as a teacher to at least guide students down the paths of knowledge even if I am not going to disseminate that knowledge by "traditional" teaching techniques? In fact, so many wonderful computer programs and manipulatives are available today, almost all geometry can be "discovered" by students in a more structured, controlled environment where the teacher could guarantee student exposure to all the "right" geometry. Sometimes I questioned Chris' motives in creating and continuing this class.

I have changed my views about my role as a teacher many times over the past twenty-four years. Much of this change of personal teaching philosophy has come about as a result of increased education and "mellowing" through the years. I certainly understand that I am neither the keeper nor the giver of all mathematical knowledge. I have taken my share of risks during this term of service. I have felt the "thrill of victory and the agony of defeat". I can truthfully say that the thrill has come more from the association with my students as people rather than their successes at the chalkboard. Therefore, the agony has come in the same way.

By the end of the book, I realized that this book WAS about change, but not curricular change. Rather, it was about changing young people in intangible ways. Significant lifelong changes come when young people learn to work together, respect each other's opinions, and think creatively. I believe that self-esteem is increased when a student succeeds at a difficult task. Self-confidence is not built through success with easy tasks, but rather through ones than are sometimes painfully attained.

The book was also about each reader developing an "effective style of relating to students and helping them learn." This is what I tell my student teachers. "Do not copy how I teach. Develop your own style which suits your personality."

I have come to believe, as a result of reading and contemplating this book, that I need to allow students to construct more of their own knowledge. There is a strong case for the intangible benefits to students through use of this technique. I know that my personality style desires too much control to ever be very successful by completely letting go of my class. However, I have begun to consider topics in which I could allow more room for student-constructed knowledge. I believe I could start with a general problem and allow students to relate the new statement or idea to previously learned ideas and construct new knowledge. This could be done, for example, when studying combinatorics and developing the formulas for combinations and permutations.

As for my two concerns in the beginning, I found at the end of the book that Chris directed the class much more than I was first led to believe and was indeed an essential element to the classroom. Secondly, what is "needed" geometry? How many geometry theorems constitute "all the geometry you will ever need"? Isn't it true that all geometry is derived from all the previous. If a student has the capacity to reason, then couldn't s/he reason all the necessary geometry at the point of necessity? The students in the BAB class ended up five percentage points behind the other geometry class in geometry concepts, but five hundred percentage points ahead of the other class in the intangibles.

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