Opening Session  Glenda Lappan and Alan Tucker

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The remarks by Roger Howe, Liping Ma, and H.H. Wu address different aspects of the challenges to improve the mathematical education of teachers. They are notable in the common assumptions and goals that underlie their specific concerns. Indeed, one of the most successful aspects of the Mathematical Education of Teachers report was the process by which constructive input to early drafts was obtained from mathematicians across a broad spectrum. While there were places where opinions differed, there was much that all responses had in common. This is especially true of the eleven recommendations in Chapter 2 of the MET report.
In different ways, Howe, Ma and Wu were all making the point that mathematics can only be learned with deep understanding when it is learned in a rich array of
contexts.
Howe spoke about the substantive mathematical underpinnings of basic arithmetic procedures and how teachers must learn this mathematical context of arithmetic. Chinese school textbooks start building one type of contentof arithmetic in the first grade with, for example, a picture of some ducks around a river: some in the water, some on the shore and some in the air. Chinese firstgraders make up Arithmetic expressions corresponding to different combinations of ducks or to removing some group from the total.
Wu's discussion of division of fractions makes it clear why it is unrealistic to expect a generalist U.S. elementary school teacher (teaching all subjects) to be able to provide high quality instruction in fraction and decimal arithmetic. Judy Sowder’s middle grade chapters in the MET report left me feeling somewhat overwhelmed by the complexity of the understanding needed to master division in all its various contextsn/p could mean splitting n cookies into p equal parts of size n/p; or n/p could be the number of batches of cookies can you make with n cups of sugar if each batch requires p of sugar.
On the other hand, I assume that everyone in this room learned fractions, including division by fractions, with little difficulty. This brings up a major challenge that I like to share with new graduate TAs in my department at Stony Brook. If I were free to choose instructors for beginning college mathematics courses, I would not consider mathematics graduate students. Rather I would look for people who had struggled with the beginning college mathematics but gone on to do well in the course (and would have taken additional college mathematics courses). These people are likely to be much better able to anticipate what is hard to the typical student in the class about the course material and appreciate why it is hard. Graduate students in mathematics, and faculty, are likely to see concepts as 'obvious' and have no idea why students cannot understand them.
Faculty teaching a course about school mathematics to future teachers, especially elementary grade teachers, are likely to be in the same situation. It is very challenging to connect with the way these students think. Moreover, there is often a good amount deprogramming to dohelping students identify and abandon flawed ways of thinking about mathematics before good reasoning habits can start to be developed.
Mathematics faculty are used to explaining how to do problems and giving tests to see how well the students succeed at this task. The situation with future teachers needs to be different. The students who earn C's as well as the students who earn A's are both going into the classrooms. Faculty now have the responsibility to try to bring all students up to an A or B level of understanding. They cannot sit back and judge with a final exam or endofchapter test who learned the material well. They have to try to continually monitor students' progress and step in early to help students develop the key steps in mathematical reasoning and understand key concepts. I see this as akin to the way physicists break down atoms into more basic elementary particles that are broken down into more fundamental ' 'building blocks' of matter. We have to identify corresponding fundamental building blocks involved in acquiring mathematical reasoning and knowledge. This is both a daunting and an exciting assignment.