Book Reviews - January 2008 - Volume 108 (1)

The Prime Number Theorem

Author: Graham J.O. Jameson
Cambridge University Press, 40 West 20th Street, New York, NY 10011-4211
2003; 252 pages, Hardback $75.00, Paperback $25.00

Reviewer: Medhat H. Rahim
Lakehead University, Thunder Bay, Ontario, Canada, P7B 1E5

This text intends to give a simple and basic exposition of the prime theorem and its proof, treating it as a subject in its own right rather that a peripheral topic of a wider subject. The topic is certainly suitable for the final year of an undergraduate program in mathematics. The proof of the theorem presented in the text takes students on a panorama. The students will encounter, probably at the first time, such unexpected charms as the Euler product, the various Dirichlet series derived from it, the corresponding convolutions, Moebius inversion, the inversion of Dirichlet series by suitable integrals, and the extension of the zeta function by a clever rewriting of the defining formula. The students will also meet the Riemann hypothesis, commonly regarded as the most important unsolved problem in mathematics. The main prerequisites, therefore, lie in analysis rather than number theory. That is, the text presupposes standard undergraduate courses in both real and complex analysis including, for example, Riemann integration, uniform convergence and Cauchy's integral theorem.

The text consists of six chapters where the basic account of the prime number theorem is contained in the first three chapters. The remaining three chapters continue the study further in three mutually independent directions. Chapter 4 is concerned with Dirichlet's theorem on the equal distribution of prime numbers among residue classes. This chapter requires a minimal knowledge of group theory. Chapter 5 addresses the problem of error estimates in the prime number theorem and similar results. The text shows the Mellin inversion method adapts in a straightforward way to provide the classical error estimate suggesting that Newman's proof is less suitable for this purpose. Finally, Chapter 6 contains an account of one version of the “elementary” proof of the prime number theorem by essentially number-theoretic methods, avoiding complex analysis.

Finally, the prime number theorem gives an asymptotic expression for the number of primes less than a given number - one of the great theorems in mathematics. As such, this text aims to give a simple and clear explanation of the theorem and its proof. For reader who earned an undergraduate degree and with modest background in real analysis and a minimal knowledge of group theory, this text would be a suitable addition to their mathematical resources.

Creating a Classroom of Young Scientists (Second Edition)

Author: Jeffrey W. Bloom
Routledge, Taylor & Francis Group, 270 Madison Ave., New York, NY 10016
2006; 463 pages, Paperback $20.99

Reviewer: John Eichinger
California State University, Los Angeles, Los Angeles, CA 90032

Jeffrey W. Bloom has written an outstanding elementary methods text that respects and utilizes children's unique and significant abilities to wonder, explore, and learn, as it encourages teacher innovation. The book emphasizes inquiry, curiosity, integration of disciplines, and conceptual understanding within a constructivist framework. Bloom endorses exploration, discussion, metaphor, and authentic engagement of learners, and encourages the building of inclusive classroom communities via active learning, reflection, negotiation of meaning by the group, and thoughtful assessment. A fundamental theme in the book is that of helping elementary students to develop a first hand understanding of the nature of science (NOS). Not surprisingly, his writing style is innovative and quite readable, as it strongly models the sort of creative, empirical, and investigatory thinking that he recommends.

Creating a Classroom of Young Scientists is broken into three parts. The first part, Getting Started, contains significant introductory material and an inspiring theoretical framework (discussed more extensively below) that provide a valuable context for the remainder of the book. The second part, “Teaching Science for Children's Meaningful Learning,” contains chapters entitled “The Nature of Science,” “Children's Learning and Sense Making,” “Teaching and Learning Through Inquiry,” “Assessing Children's Thinking, Learning, and Talk,” and a particularly thoughtful chapter entitled “Children's Talk” which offers a practical interpretation of social constructivism as it relates to student discussions. The book's comprehensive third section is composed of twelve appendices, including the following topics: safety, a summary of the U.S. National Science Standards, an overview of the Canadian science framework, learning for complexity, planning, field studies, a sampler of activities, data collection and analysis, fundamental science techniques, technology, science supplies/equipment and where to find them, and professional resources.

As noted above, Bloom's “Philosophical and Theoretical Framework” is a breath of professional fresh air in our current national atmosphere of instructional uniformity, suppression, and apprehension. For example, as he discusses the importance of pattern recognition, Bloom writes that “...we might consider the notion of a teacher's role as inviting chaos and trusting complexity.” (p.7, italics are Bloom's). What an invigorating and contemporary perspective, illuminating for neophytes and veterans alike! I only wish that he had shared an inventory of references that influenced his thinking in this area (maybe Professor Bloom can be coaxed to include such a list in future editions).

Creating a Classroom of Young Scientists compares very favorably with other elementary science education texts I have seen. Bloom's classroom activities are high quality, and his associated questioning strategies are outstanding. I would have recommended incorporating more activities directly into the chapter narrative, however, rather than sequestering most of them in the supplementary appendix. Presenting the activities in context would have more clearly demonstrated Bloom's ideas regarding exemplary classroom practice, especially for new teachers. Additionally, I would have very much appreciated an author index and/or full list of all references cited or recommended.

Overall, Bloom's Creating a Classroom of Young Scientists is a well written, thorough, and practical text for new teachers interested in incorporating critical thinking and NOS studies into all aspects of the elementary curriculum. The book will also serve as an enlightening and comprehensive reference for more experienced practitioners. The appendices alone are worth the price. Professor Bloom delivers on his title's promise to generate a collaborative and investigatory classroom atmosphere for elementary students and teachers.