*The Mathematical Education of Teachers* is the eleventh volume in the *Conference Board of the Mathematical Sciences: Issues in Mathematics Education Series*, written as a resource for those engaged in the education of teachers of mathematics. The volume is made up of two parts: the first six chapters, which provide mathematics faculty of two-year colleges, four-year colleges, and universities a summary of the curriculum and policy issues that affect present and future teachers of mathematics. The last three chapters are directed specifically at those faculty members in mathematics and mathematics education involved in teacher education.

Chapter one deals with the changing expectations for instruction and new realizations in the preparation of prospective teachers of mathematics. The mathematical knowledge needed for teaching in K-12 schools is documented, but, more importantly, the report stresses the need for a partnership between those teaching post secondary mathematics and mathematics education to enhance the education of prospective teachers. The report recognizes those math anxious students who gravitate to elementary teaching as being "...caught in a vicious 'cycle': poor K-12 mathematics instruction produces ill-prepared college students, and undergraduate education often does little to correct the problem..." (p.55). Hopefully, with the application of the ideas contained in this report, all prospective teachers will attain a deeper understanding of mathematics that will be passed on to their students. Questions surface for the reader at this point: Does teaching for understanding imply that students should not memorize their multiplication tables? Where does repetition or drill fit in? Should students use workbooks? Should there be more direction given on how students should use calculators, especially during K-8? These questions need to be addressed in this age of "constructing deep understanding" so that the interpretation of this report is what the authors intended it to be and to successfully break the 'cycle'.

Chapter two makes several recommendations regarding curriculum and instruction for prospective teachers of mathematics by mathematics departments, the need for cooperation between all the partners in the education of teachers, and the involvement of mathematicians in a national policy to improve mathematics teaching. The report recommends the following coursework:

- For prospective elementary teachers [grades 1-4], at least 9 semester-hours on fundamental ideas of elementary school mathematics.
- For prospective middle grades teachers of mathematics [grades 5-8], at least 21 semester-hours of mathematics, that includes at least 12 semester-hours on fundamental ideas of school mathematics suitable for middle grades teachers.
- For prospective high school teachers of mathematics [grades 9-12], to complete the equivalent of an undergraduate major in mathematics that includes a six-hour capstone course designed to connect their post-secondary mathematics courses with high school mathematics.

This capstone course is an interesting idea that would examine the conceptual difficulties, fundamental ideas and techniques in high school mathematics from an advanced perspective. This course should help reinforce the use of partnerships called for in this report.

The report stresses that these courses develop a deep understanding of the mathematics prospective teachers will teach as well as a mastery of the mathematics in the grades preceding and following what they teach. For existing and prospective elementary/middle grades teachers, this could provide some anxiety — but if the courses focus on gaining a deeper understanding of the mathematics being taught, their anxiety might be reduced. You never know, these teachers might even consider teaching mathematics in grades that are outside their comfort level.

It is encouraging to read that this report emphasizes the importance of mathematics specialists in the middle grades and furthering the professional development of all teachers of mathematics. Those currently teaching mathematics recognize that it would take a lot of cooperation by all the partners involved with teacher education to accomplish this task. Japanese lesson studies, which focus on teacher cooperation, could provide a possible vehicle for this professional development. Again, certain questions come to mind: How can we make everyone teaching mathematics a specialist? Why aren't our best high school mathematics teachers teaching grade nine?

The remaining chapters contain recommendations for teacher preparation and discussion regarding the preparation of teachers of mathematics. Many suggestions are given regarding the possible mathematical content necessary for teaching each level, but this is left for the reader. The mathematics taught to prospective teachers should not only be to improve their understanding and capability to teach mathematics, but also to enable them to understand the issues that occur in a mathematics class that focuses on students' mathematical thinking. Included are many examples and vignettes from lessons that illustrate actual children's mathematical thinking concerning what they can and cannot understand in K-5. These are followed by discussions of the mathematics needed by prospective teachers to follow and understand their students' thinking. In particular, I appreciated the discussion regarding getting students away from using surface feature cues or "key words" when solving problems. I hope this will get students to become more expert-like in their thinking and look for deeper relationships when solving problems.

However, if we are talking about students' problem solving, then prospective teachers should be taught a problem solving strategy in such a way that would help them and their future students gain control over their thinking in order to solve problems. To appreciate the impact of learning a problem solving strategy would require prospective teachers to wear the hats of being both a teacher and a student.

Representing the real number system by both the number line and a Venn diagram in middle grades is good advice for teachers of mathematics, but I am concerned how these teachers will incorporate the field axioms in their teaching. Will they need more guidance here since not every student will become a mathematician or will we return to the 'new math' of a quarter of a century ago?

Throughout this report there are examples of pattern recognition and inductive reasoning and the suggestion is made that students be given opportunities with or without technology to explore, conjecture, provide counterexamples and justify. In today's classroom there is not enough of this by students and it is no wonder that they have trouble with deductive reasoning well into post secondary education. The report encourages university mathematics instructors to "model for prospective teachers ways in which these aspects of mathematical learning can be commonplace in the mathematics classroom, and consciously make reasoning and understanding salient features of learning for their students" (p. 99).

The reader of this report would also have benefited from a more detailed explanation of the Van Hiele levels of geometric thought and the Action-Process-Object-Schema [APOS] theoretical perspective pertaining to functions.

The report underlines the importance of proportional reasoning so that a case could be made for this becoming a major section of the K-12 mathematics curriculum. It would be interesting to see how students would learn mathematics where the connections are more explicit. One thinking objective of this section could be to develop students' analogical reasoning ability. The lesson plans from such a section would demonstrate the importance of including thinking objectives, as well as learning objectives for all sections of the mathematics curriculum. The development of such lesson plans would hopefully transfer to prospective teachers' lesson plans after they graduate, possibly, breaking the 'cycle' of how they were taught.

When I read in the report a statement that says, "In traditional college preparatory curricula, the primary goal is preparing students for study of calculus" (p. 43), I think it is time for all the partners involved with teacher education to formally discuss why we teach mathematics in K-12.

The report contains many good references plus reports. I enjoyed reading this report, which I consider a launch pad for more discussion with all the players involved in improving the teaching of mathematics for our children now or in the future. At the University of Regina these discussions have just begun with mathematics education and mathematics instructors.

Rick Seaman is Assistant Professor of Mathematics Education at the University of Regina in Regina, SK, Canada.