Michael Starbird and Edward Burger are a great example of an academic duo. They have been collaborating on a range of projects and writing together for a while now. The book under review is the latest from this productive pair. And just like any literary historian would, a careful reader can enjoy tracing the evolution of this book by going over their joint bibliography. When viewed from such a perspective, this book can be seen as the culmination of several consecutive joint projects. I’d say, furthermore, that it is interesting not only as a standalone read but also as a capstone of a productive collaboration.
Let me tell you first what this book is not about. It is not, explicitly, about mathematics. In the broadest sense, this is a selfimprovement book, even a selfhelp book, carrying back cover blurbs from the cocreator of the bestselling Chicken Soup for the Soul series, the executive director of the Commission on Presidential Debates and the former president of Columbia Pictures. But it is definitely not a fluke when considered within the BurgerStarbird collaboration. The inside back cover flap boasts the mathematical and pedagogical achievements of its authors, and the book is in fact published by the Princeton University Press.
And when you read the book, you cannot but notice that these guys are mathematicians! Indeed many (perhaps the majority) of the examples are inspired by classroom experiences, though the authors have been careful not to limit themselves to the experiences of two mathematics professors. They try hard to provide examples from a variety of fields, aiming to engage with readers from diverse backgrounds who have a range of interests; exercises and case studies involve basketball, Picasso’s art, playing the cello, and others.
Let me now focus on the contents of the book. As its title suggests, Burger and Starbird distill their guidelines for effective thinking into five principles. I believe I will not be disclosing much if I list these five principles here; after all Starbird himself mentions some of these in his presentation titled “What Students Keep for Life: Elements of Effective Thinking” recorded at a recent MAAPREP workshop entitled Using Inquiry Based Learning in SecondYear Calculus and Courses for Prospective Teachers (UCSB, June 2012) which is available on YouTube.
So here, according to Burger and Starbird, are the five elements of effective thinking:

Understand deeply

Fail to succeed

Be your own Socrates

Look back, look forward

Transform yourself.
The titles Burger and Starbird give to their five principles are colorful and punchy, but a careful reader of one of their earlier books, namely The Heart of Mathematics, will notice that some of these were already a part of their “Lessons for Life” list, provided, among other places, on page 28. The original list contained ten items, but Burger and Starbird must have noticed that a shorter, distilled list would probably be easier to recall and more helpful for anyone who wishes to improve herself.
The book is short, a quick and easy read, and encourages the reader to stop and try the exercises. I also think the authors’ suggestion to read the book a couple of times, the first to get the point and the second to actually start applying the ideas and themes to one’s own thinking, is straight on. If you want to consciously improve yourself, to clarify your thinking, to encourage creativity and raised productivity, you might benefit from following the authors’ recommendation. Having read, admittedly, more than my share of books from this same genre, I must in fact underline this fact: The only way this book will be helpful to a reader is only if she follows this suggestion. Otherwise, a typical (nonmathematical) reader of selfhelp books will most likely move on and not gain much, possibly besides some sense that mathematicians can also have something of interest to say to her. Perhaps there will be some suggestion that will stick with her, like: “Fail to succeed,” or, in other words, “in order to get better, you will make mistakes. This is normal, and is no reason to give up”. Without picking the book up a second time to work out the exercises and fully engage with the proposed ideas, this reader will have lost an opportunity.
So this brings me to a question that some readers of this review might have about this book: Who will read it? The tone of the authors suggests that the book can be useful to anyone who is looking to improve his or her own thinking. However the fact that the publisher is a university press leads me to be more conservative and suggest that the main audience might be college students or other educated readers who look at university presses to create their seasonal reading lists. But will any of those people really want to improve their thinking? A saying in my native tongue, Turkish, tells of a fictional setup where all the minds were collected together and put on sale. Everyone then went ahead and bought their own brains all over again! We seem to be happy about the way our brains work, and most of the time are looking for shortcuts to improve our lot. Burger and Starbird’s suggestions are accurate, and would be helpful to anyone taking their suggestions seriously, but who will?
My cynicism about humanity aside, I thought this was an interesting book that I enjoyed reading. I liked the stories the authors put together, and would be proud to say that as a mathematician, I can see, at each step of the way, how a good mathematics education would help someone get better at effective thinking. So I agree with the premise of Burger and Starbird’s book. These five principles are important and would encourage the intellectual development of a young mind interested in improving herself. As a list of mathematical reasoning rules, the elements work well too. As a mathematics instructor I can easily see how we can use some of these principles to explicitly encourage our own students’ intellectual development in the classes we teach. I am not sure what fraction of the intended audience of this book will actually read it. Nonetheless, I admire Burger and Starbird for reaching out to nonmathematicians, and sharing what are basically mathematical insights and displaying their broad applicability.
Gizem Karaali is associate professor of mathematics at Pomona College and an editor of the Journal of Humanistic Mathematics.