John Allen Paulos, who has accomplished the unheard-of double of writing best-sellers about mathematics and inserting a word (‘innumeracy’) into the language, has attempted another ambitious feat – bringing mathematics to bear on one of the few subjects it has yet to examine: biography and autobiography. A Numerate Life (Prometheus Books, 2015) is simultaneously a charming memoir and a highly entertaining venture into mathematics, literature, and philosophy. This is one of those rare books that, when you have finished a section, you are torn between going on to the next section and re-reading the last section to make sure that you got everything out of it. The subtitle of the book is “A Mathematician Explores the Vagaries of Life, His Own and Probably Yours”. You’ll be a little skeptical about that subtitle before you read the book, but when you finish it, you’ll realize the subtitle nails it — he’s talking, not just about himself, but about you. fascinating read.

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Today we’ll be talking about The Magic of Math (Basic Books, 2015)by Arthur Benjamin. This is a book that has the gee-whiz feeling you got when you first encountered George Gamow’s classic One, Two, Three … Infinity, but brought up to date and with much more in the way of solid mathematics that students can actually use. What makes this book especially appealing is that Benjamin emphasizes the magic that can be found in the mathematics that students study as they proceed through elementary and secondary schools. Students study arithmetic, algebra, geometry, and trigonometry, but too often the way these courses are taught resembles hospital food – nourishing but tasteless. Every teacher should have a copy of The Magic of Math to help combat the blandness and boredom that students too often experience in the classroom.

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Almost 400 years ago, Galileo wrote that the book of nature is written in the language of mathematics. Today, mathematics is integral to physics and chemistry, and is becoming so in biology, economics, and other sciences, although amid great controversy. The messy reality of biological creatures and their social relations cannot be captured in mathematical models or computer simulations, it is argued. But what is the relation between mathematics and physical reality? Do highly abstract mathematical formalisms and computer simulations yield empirical knowledge? If so, when, and how? In Reconstructing Reality: Models, Mathematics and Simulations (Oxford University Press, 2015), Margaret Morrison, Professor of philosophy at the University of Toronto, considers the epistemological status of the results of modeling and simulation as compared, and typically contrasted with, the results of experiment. She argues that no sharp distinction between simulating the world and measuring the world can be drawn in modern science, and that there is no justification for epistemically privileging the results of experiments over the new knowledge we derive from idealizations, abstractions, and fictional models.

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Christopher J. Phillips‘ new book is a political history of the “New Math,” a collection of curriculum reform projects in the 1950s & 1960s that were partially sponsored by the NSF and involved hundreds of mathematicians, teachers, professors, administrators, parents, and students. The New Math: A Political History (University of Chicago Press, 2015) explores the formation of an idea of the “American subject” in an environment where math was considered to be a component of intelligent citizenship. As classrooms became sites shaped by Cold War politics, efforts to reform mathematics curricula were bound up in ideas of subjectivity and discipline. Phillips pays special attention to the work of the School Mathematics Study Group (SMSG) in this context, looking closely at the textbooks that the SMSG produced for children studying at a range of levels. Importantly, The New Math explores not just the production of these textbooks but also what happened when they were actually brought into American classrooms and engaged by teachers, students, and parents. As a result, in addition to being a fascinating political history it’s also a model of how we can treat the archaeology of the classroom as a way to approach the history of science.

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The book discussed in this interview is Zombies and Calculus (Princeton University Press, 2014) by Colin Adams.   This is a truly unique book; a novel written in the first-person by the survivor of a zombie apocalypse who has managed to make it that far thanks to his knowledge of calculus.  The author starts his narrative by warning the reader that the book is not for the squeamish, but you shouldn’t be deterred by that, as I found the zombies to be more comical than horrific.  The book is especially worthwhile for the way it introduces some of the really intriguing applications of calculus that are not typically found in the standard three-semester calculus sequence, and the author has done a good job of making those applications relevant to his tale.  In an era when education competes with entertainment for attention – and generally fights a losing battle – books such as Zombies and Calculus are to be applauded.  They may prevent some of those taking math courses from becoming classroom zombies.

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The book discussed in this interview is How Not To Be Wrong: The Power of Mathematical Thinking (Penguin Press, 2014), by Jordan Ellenberg.  This is one of those rare books that belong on the reading list of every educated person, especially those who love mathematics, but more importantly, those who hate it.  Ellenberg succeeds in explaining the value of mathematical reasoning without ever needing to go into technical detail, which makes the book ideal for those who want to learn why mathematics is so important.  What makes the book doubly delightful is Ellenberg’s writing style; he intersperses the math with amusing anecdotes, dispensed with a sense of humor rarely found in books such as this.  The book is chock-full of OMG moments; the introductory anecdote about Abraham Wald and the missing bullet holes absolutely whets the appetite for more and Ellenberg never fails to deliver.

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[Re-published with permission from Inspired by Math] The MAA (Mathematical Association of America) sent me a review copy of their new book Learning Modern Algebra: From Early Attempts to Prove Fermat’s Last Theorem. I don’t typically review textbooks but the title and then the contents of the book convinced me that I needed to interview the authors. Joe Rotman wasn’t available but I was able to chat with the other co-author, Al Cuoco. I was really struck with Al’s passion about teaching the teachers as well as the students. Al shared some great insights about the ingredients that I think should go into every math textbook to help teachers and students to develop the right habits of mind to succeed. Here are some of the questions we discussed.

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[Re-posted with permission from Sol Lederman’s Wild About Math] I love novel ways of looking at arithmetic. I’m fascinated with how computers compute in binary, with tricks for simplifying calculations and with how Vedic mathematicians handle difficult arithmetic efficiently. So, when Princeton University Press sent me a review copy of their new book Count Like an Egyptian: A Hands-on Introduction to Ancient Mathematics (Princeton University Press, 2014), I immediately fell in love with it. I was delighted to learn even more techniques and the ideas behind them to deepen my appreciation of the beauty of what most consider to be mundane arithmetic.

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A conceptual space sounds like a rather nebulous thing, and basing a semantics on conceptual spaces sounds similarly nebulous. In The Geometry of Meaning: Semantics Based on Conceptual Spaces (MIT Press, 2014), Peter Gardenfors demonstrates that this need not be the case. Indeed, his research is directed towards establishing a formal, mathematically-grounded account of semantics, an account which – as expounded here – is nevertheless accessible. In this interview we discuss the essence of this proposal, focusing in particular on its implications for linguistic analysis, but also touching upon its relation to cognitive science and other related fields. The proposal makes testable predictions about the organization of individual linguistic systems, as well as their acquisition (and potentially their evolution over time). Notably, the “single domain constraint” posits that individual lexical items refer to convex regions of single domains. We discuss the significance of this idea as a bridge between linguistics and cognitive science, what would constitute its falsification, and how it can usefully be investigated from a linguistic standpoint.

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