I have always believed in allowing my students the opportunity to discover and construct their own knowledge of mathematics but I never considered such a radical move as asking the students to construct all of their knowledge for a course. "Build-a-Book" has helped raise several important questions about the aspects of teaching mathematics. The book has helped me reevaluate my pedagogical styles, my assessment, and my role as a teacher but I also believe that any good teacher should constantly reevaluate these aspects of teaching as we can always find ways to improve. The book also evoked many thoughts of how students gain meaning of mathematics.

First, I do believe students become more mathematically empowered by investigating and constructing their own knowledge of mathematics and the students understanding of the subject would be much deeper and profound through this style of instruction. However, through my own teaching experiences I have found that allowing students to construct their own knowledge of mathematics requires a more considerable amount of time than traditional teaching styles. As a teacher, my school system has expressed a concern that I make an effort to at least expose the students to all of the required curriculum. I personally would like my students to show mastery of the skills and knowledge that they do possess. This method of teaching would carry great risks. As a teacher, it would be difficult to direct the students toward construction of the required curriculum necessary to perform adequately on the department's exams. I wouldn't have thought such an extreme would be possible. I realize that Mr. Healy was still able to direct the students investigations by assigning the topics each group was to investigate but this still would not ensure that the class would cover the mandatory curriculum. My only rationale for the reason his students tested well on the departmental exams was that his students would probably have a better understanding of the material that they did cover and his class definitely had a better chance to develop their reasoning skills which would have aided them on the test.

I strongly believe the one area that Mr. Healy showed a major deficiency in was his assessments of the class. His assessment was not balanced with the curriculum and the instruction that was implied by "Build-a-Book". His class always worked in groups yet they were always assessed individually. The students' were usually tested about their knowledge of their book rather than their understanding, their ability to communicate, or their ability to reason and analyze problems. Explicit knowledge of their book was important but that was only one aspect among many in the class. I would have liked to have seen Mr. Healy give them a problem that required students to prove something and then asked them to write a paragraph or two to someone in their group explaining why their proof is correct or an open ended problem would have been nice.

The book really forces a teacher to reconsider their role. I was impressed with what Mr. Healy class accomplished with a teacher taking on such a passive role. I know I would have difficulty waiting (continuing to be passive) as long as Mr. Healy did on occasions when students became stagnant. This book truly shows the benefits of allowing the students to take an active role in their education but I believe when the students are struggling with the role then it is the teachers responsibility to step in and offer assistance and if absolutely necessary be a source of knowledge at times. I know it was important that the students didn't view Mr. Healy as an authority of the knowledge so that the students would question and analyze everyone's input as well as Mr. Healy's input but I think this attitude might still be possible even if Mr. Healy had to step in and occasionally and offer assistance to those students who had become dormant. I would have to promote some type of learning if I was the teacher of a class that had become stagnant for more than a week.

Through all of the criticism, it is still very tempting as a teacher to try implementing such an exciting idea. I have always tried to promote understanding through student discovery but at times, as students find a path that diverges from the expected goal I led them back to the expected outcome. I really think this book has helped me see that the cost in time spent on those divergent paths or unexpected learning outcomes can be very fruitful and are well worth the time spent. These paths can help promote students' mathematical self-esteem, they can enrich the students knowledge of mathematics in general as well as the intended goal, they can help develop reasoning skills, and the list goes on.

I think I enjoyed reading the book because it was thought provoking as well as entertaining similar to Mr. Healy's "No Book" class. This radical of a movement can be very exciting when it is met with such success which provided me with motivation to try and consider every aspect of this method of teaching.

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