Have you wondered what you could do to be a better teacher? I have recently become interested in this question. And, being trained as a research mathematician, I am interested in what research has already been done in this area to help me. I have many questions: How do I find and judge umdergraduate mathematics education research? If I collect data on my classes, how can I get valid interpretations of my results? Where do I go for help in interpreting my data? How do I even get started? My impression of mathematics education research, at least at the undergraduate level, has been that there are conflicting results about whether to teach reform or traditional calculus, whether technology helps or has little effect on student performance, and similar confusing results. There does not seem to be much out there to address my questions, so I jumped at the opportunity to review this book.

In a nutshell, this book is a guide for reading and evaluating mathematics education research, especially undergraduate mathematics education research. Its purpose seems to be to encourage research mathematicians, who do much of the teaching of undergraduate mathematics, to use education research to actually inform their teaching. In order to use this research, though, mathematicians need to know how to find and evaluate the research already done, and this is the focus of this book.

From the back jacket of the book we find these statements.

"Mathematics education research, carefully conducted, is something far more fundamental and widely useful than might be implied by its use by the advocates of innovation in undergraduate mathematics education. Most simply, mathematics education research is inquiry by carefully developed research methods aimed at providing evidence about the nature and relationships of many mathematics learning and teaching phenomena. It seeks to clarify the phenomena, illuminate them, explain how they are related to other phenomena, and explain how this may be related to undergraduate mathematics course organization and teaching.

This book — the collaborative effort of a research mathematician, mathematics education researchers who work in a research mathematics department and a professional librarian — introduces research mathematicians to education research. The work presents a non-jargon introduction for educational research, surveys the more commonly used research methods, along with their rationales and assumptions, and provides back ground and careful discussions to help research mathematicians read or listen to education research more critically."

So, let's look at what the book contains. There are 12 chapters in all. The first two chapters provide an overview to the subject. The titles are "Evidence-based Pedagogy" and "Recognizing Research", respectively. Subtitles in each chapter bring out the flavor of each one. There are sections labeled, "Mathematicians and Proof", "Mathematics Education Research and Mathematics Education Reform", "Mathematicians and Mathematics Education Research", "Evidence", "Bias" and "Theory" in the first chapter. The discussion revolves around trying to convince a mathematician grounded in deductive argument and formal proof to accept mathematics education research whose arguments are based on evidence instead of formal proof. The second chapter includes only two sections, "Criteria of mathematics education research" and "Qualitative and Quantitative Methods". Here, there are several criteria to look for in mathematics education research: clear research questions, adequate description of the context, statement of methods, analysis procedures, appropriate generality in results and separating conclusion from conjecture. In talking with others involved in this type of research, before reading this book, I had found these to be the required pieces in this type of research.

The meat of the book is in the next two sections, each section having three chapters. The two sections deal with quantitative research and qualitative research respectively. The chapters are:

**Part I. Quantitative Research** |
**Part II. Qualitative Research** |

Ch. III. Critiquing Quantitative Research |
Ch. VI. Critiquing Qualitative Research |

Ch. IV. Reliability and Validitiy in Quantitative Research |
Ch. VII. Reliability and Validity in Qualitative Research |

Ch. V. A Survey of Statistical Methods |
Ch. VIII. A Survey of Qualitative Methods |

These chapters give several ideas of what to look for and how to judge the research presented. The quantitative research details should be familiar to anyone who has taught an elementary statistics course. The qualitative research details were generally new to me, and quite interesting, although this may be the place many research mathematicians begin to say that education research is fuzzy and does not feel like research.

The book ends with four chapters under the section heading of "Using Mathematics Education Research Appropriately". It includes chapters titled: "Teaching Experiments, Quasi-experimental Research, and Threats to Validity", "Evaluation, Assessment, and Research", "Finding Research: The Literature Search", and "From Consumer to Producer". These chapters discuss:

- The problems with teaching experiments — they are not really experiments in the research sense.
- The use of databases to find what research has already been done.
- The suggestion that research mathematicians interested in this type of research should find mathematics education researchers and work together to become producers of this type of research.

This book was easy to read, yet not simplistic. There was some jargon, but it should be understandable by a mathematically trained audience, the intended audience for the book. I particularly liked Chapter 9, "Teaching Experiments, Quasi-experimental Research, and Threats to Validity". This chapter pointed out the problems with many of the education research projects, particularly teaching experiments, which have been used to justify calculus reform and other similar current topics.

There were many proofreading errors. There were some in formulas and several TEXing errors. Others may not see or be bothered by these, but I find them quite annoying.

As I was reading the book, one article was praised as being an example of good research. I actually checked out the article and found it did satisfy the criteria in the book, but I found the article to be dealing with a minute portion of teaching, and not what I had expected when I requested the article. I found that I kept wishing for some source of expository/summary articles on what is believed so far. I did not find this. Maybe I was expecting too much. Is this book enough to start a research mathematician reading and critically evaluating education research? Maybe. At least when I read an article after reading this book, I knew what to look for in it. And I think I could do a moderately successful search of the literature now. Could I perform good research? Probably not, but in cooperation with or guidance from a mathematics education researcher I would now consider being involved in undergraduate mathematics education research.

Who would be able to use this book? The publishers/authors suggest university-based research mathematicians, graduate students in mathematics who are planning academic careers, and mathematics department chairs and their deans. I would add to this college faculty at institutions where teaching is the primary duty. Each of us who teaches undergraduate mathematics can benefit by even the exercise of looking at our teaching and looking at the research and judging whether the research presented could help us improve our teaching. This book can help you get started.

Mary Shepherd, (shephemd@potsdam.edu) is Assistant Professor at SUNY College at Potsdam in Potsdam, New York. Her special interests are differential geometry and music.