If you have an advanced degree in your field, I suspect you might experience an inverse relationship between your need for new knowledge and the availability of learning opportunities to gain knowledge in the areas outside your expertise. As I continue to deepen my knowledge of the teaching and learning of mathematics, I feel a need for broadening and deepening my knowledge of mathematics in general. When everyone turns to you with questions because you have a Ph.D, it becomes increasingly difficult to engage in serious learning as you used to when you were a graduate student hungry for knowledge. Zvi Artstein, the author of the book Mathematics and the Real World: The Remarkable Role of Evolution in the Making of Mathematics could serve as the teacher we all want in our fantasies — like the professor we all met in graduate school who has a quirky sense of humor, yet is profoundly insightful and knowledgeable. I felt I met that teacher again, this time not “in my face” but in the comfort of my living room with time and space for reflection.
Artstein covers 71 intriguing topics organized into 10 chapters, ranging from evolutionary roots of mathematics in humans to fundamental issues related to the teaching and learning of mathematics. Each chapter provides a list of critical questions that motivates the reader to keep reading. For examples, Why square the circle? What do genetics and mathematical calculations have in common? What does a mathematician do when he gets to the office in the morning?
If you are in a position in your career where you are invited for speaking at a variety of outreach events to introduce mathematics for the public, you may find his book an invaluable source of examples and ideas for illustrating the ways mathematics connects to our lives. As an educator of mathematics teachers, I found the last two chapters of his book most interesting: “The nature of research in mathematics” and “Why is teaching and learning mathematics so hard?”
If you are interested in the evolution of humans or the history of mathematics, Artstein’s account of human mathematical (in)ability as part of the evolutionary process or the history of mathematicians should be thought-provoking enough to attract you to grab this book. If you are interested in developing a freshman seminar course in which a primary goal is experiencing academic discourse and exploring quantitative thinking, Artstein’s book can be a course text to keep students on their toes.
Although it is evident Artstein worked hard making his materials accessible for general readers, the nature of the mathematics-oriented discussion in the book can still be difficult for novice in mathematics. For example, my undergraduate students who are currently taking an elementary abstract mathematics course were asked to read sections 59 (Numbers as Sets, Logic as Sets); 60 (A Major Crisis); and 61 (Another Major Crisis). Most students found the section 59 the most relevant text for reading, but I found the section 60 the most relevant.
Some students may benefit from select portions of the book rather than an indiscriminate assignment of reading. It helps that Artstein provides supplementary notes containing relatively more formal mathematics, separating them from the rest of the chapter by a different font couched between rules. Additionally, I was surprised with student responses such as “I actually understood some parts of the book,” “It was interesting to read about the Dedekind cut outside the classroom,” and “Wow, there are lots of mathematicians in history!” These statements imply that our students need to read about the success, struggles, and stories behind the theorems in texts with formal language of mathematics only.
I recommend that college mathematics instructors read books like this and consider ways to incorporate them into instruction. An unintended consequence can be that your students not only have opportunities to revisit the mathematics you taught but also learn to “talk” mathematics, since that is what Artstein is essentially doing in his book: talking about “all mathematics considered.”
Another benefit from reading this book could be the opportunity to pick the author’s brain to reflect on the critical questions in the teaching of mathematics. For example, I read his book when I taught a course in which students learned the basic language of set theory and logic. As I witnessed students struggling to make sense of the formal structure of mathematics with definitions and axioms in relation to their intuitive understanding of mathematics, one of his insights struck a chord with me: “Set theory was created simply to establish mathematics on solid fundamentals, and not so that we should better understand how to use mathematics, and certainly not in order for us to teach mathematics better (p. 396.)” In a similar vein, I had a kick out of reading an account of his experience with a mathematics teacher educator discussing whether it is appropriate for the teacher to say, “Benjamin is a member of the set of children in Class 2A” rather than “Benjamin is a student in Class 2A.” Artstein says, “This is a distortion and misrepresentation of the role of mathematics and is harmful indoctrination.” If you wonder which statement Artstein considers as inappropriate and why, I encourage you to get his book. In that sense, I recognize the potential of this book as a text for a faculty learning community to increase dialogue between mathematicians and mathematics educators.
My closing message for potential readers is that you will enjoy this long intimate conversation with a brilliant mathematician who is willing to speak in crisp, eloquent, and original language to discuss the manifold and intricate making of mathematics.
Woong Lim (firstname.lastname@example.org) is an Assistant Professor of Mathematics Education at Kennesaw State University, GA. His research interests include interrelations between language and mathematics, content knowledge for teaching, and social justice issues in mathematics education.