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Math through the Ages: A Gentle History for Teachers and Others, Expanded Edition

Publisher: 
Mathematical Association of America/Oxton House
Number of Pages: 
276
Price: 
45.50
ISBN: 
978-0-88385-736-6

When I was asked to teach a summer course in the history of math, I gave the matter a bit of thought. First: did I want to teach in the summer? (Overnight I went from “Definitely not” to “Well, for this fun course, sure.”) Second: Did I know the history of math? Well, yes I did, but I was also modest. “I don’t have a timeline in my mind,” I qualified. That was okay. But I resolved to make a timeline for my students, a physical timeline, and towards giving them a mental timeline, I decided that, if possible, I’d use a text which, in some way, gave an overview — one which didn’t get too bogged down in the many exciting details but provided some way for the students, and for me, to grasp this never-ending story, in…

Well, “in a large nutshell”.

I located several candidates, and then I came across a review of this book. It begins, so I learned, with a sixty-page Part I titled “The History of Mathematics in a Large Nutshell”. “Yes!” I called out. Part II consists of twenty-three short sketches on selected topics, which are discussed in more detail (but not, for my taste, too much detail — four to six pages per sketch).

I was doubly thrilled to see that one of the authors is our own Fernando Gouvêa. I thus trusted that the book would be extremely well written, partly because I was familiar with some of his writing, both his published writings and his emails to me. In particular, I knew about his concern for clarity in textbook writing.

I was not disappointed. It felt like what they call “summer reading”, and not only because it was summer! I looked forward to my time with it the way one looks forward to time with a good novel. I almost wondered why, with such a book, a student would even need to take a course; the book practically reads itself! If I read it as I would a novel, students could, too.

“No one quite knows when and how mathematics began,” we read on page 6. “What we do know is that in every civilization that developed writing we also find evidence for some level of mathematical knowledge…” And thus we are drawn into the story, deeper and deeper. I already knew much of it, but there were holes in my knowledge, which were quickly and painlessly filled in. I had known a lot about the Greeks, in particular Euclid, but I hadn’t known much detail about Mesopotamia, and I hadn’t known that the ancient Chinese accepted zero, negatives, and the Pythagorean Theorem long before those to whom these ideas are credited. I had known about Fermat’s Last Theorem, and his other theorems, but I hadn’t known that he called them “negative propositions”, nor had I known that the very notions for which he is now famous were not as interesting to his friends and colleagues as his work on geometry.

The book is intended, largely, for math education students, which was another plus for my purposes, since the course I’d be teaching would be in the graduate school of education. It is to the authors’ credit that the book was not written for math majors; very little knowledge of math itself is assumed, though a math-hater would probably not enjoy the book. One of the book’s offerings that I enjoyed were the explanations given, in such a way that it’s informative for those who don’t know, and enlightening (or amusing…) in some way for those who do: p. 51:

The work of Gauss… Bolyai… Lobachevsky… and Riemann… finally settled the issue, leading to the discovery of the non-Euclidean geometries. Once again, the move was toward abstraction and rigor; rather than attempting to figure out what was “the geometry of the real world”, these mathematicians showed that there were several alternative ways to do geometry, each consistent within itself, each interesting, each correct. It must have seemed, at the time, to be a useless but beautiful dream, a move away from applicability. But it didn’t turn out that way.

This particular summer course would meet for five weeks, two days per week, three and a half hours per day. There are usually only a few students taking the course, and this summer there were only four. They were all going for their Masters in education, three in math education, a fourth in English.

The format of the book made it easy for me to decide on the syllabus. I’d provide the above-mentioned timeline, something which I myself wanted to own. I’d tell the students that this timeline was optional, for them to make use of as they saw fit; perhaps they could have it in front of them as I lectured and as we discussed. (My style of teaching is lecture/discussion.) I’d then go over some of the reasons that teachers of math need to know math history; there were several given and expanded on in one of the introductions to the book. Page 1: “Each contributor to that development was (or is) a person with a past and a point of view. How and why they thought about what they did is often a critical ingredient in understanding their contribution”. Page 2: “Telling such a story [like the story of how the child Gauss summed up the positive integers up to 100] achieves some useful things. It is, after all, an interesting story in which a student is the hero and outwits his teacher… the story can lead the class towards discovering the formula for themselves. {after all, if a ten-year-old could do it…)” Page 3: “People do things for a reason and their work typically builds on previous work in a vast cross-generational collaboration. Historical information often allows us to share this ‘big picture’ with students. For example, Sketch 17, on complex numbers, explains why mathematicians were led to invent this new kind of number that initially seems so strange to students.” And “Knowing the history of an idea can often lead to deeper understanding.” And finally, page 4: “they [past mathematicians] had trouble with the concept [for example, of negative numbers] itself… Understanding this helps us understand (and empathize with) the difficulties students might have.”

After going over these ideas (and after I gave a few ideas of my own, such as that exploring math history helped me to explore questions like, “If I had not been taught differential equations, would I have begun with solving the same equations that standard courses in diff eq begin with?”), we’d spend the first week (a total of seven hours) on “the large nutshell”. Then, for each of the following weeks, we’d do five or six of the sketches, which we wound up dubbing “small nutshells”. That would entail covering the entire 190-page book. The syllabus also included a three-or-four-page paper each week, plus an oral presentation on that paper, one long paper due the last week, and a book or article critique (also, something I thought of at the last minute, a paper of length as short as they wanted on “your math history; after all, in some respects, an individual’s life and work is a microcosm of the human race’s”).

After a while I decided that I didn’t want to be locked into any promises so I wrote on the syllabus, “Subject to changes and flexibilities on the part of the instructor and the students”. Indeed, that was the beauty of the book’s format. Within reason, I could take as long, or as short, with “the large nutshell” as we wanted or needed, and then do as many “small nutshells” (meaning sketches) as we had time left for — and whichever ones the class decided on. As teacher, I could thus relax, knowing that “the large nutshell” would certainly get done. We could then proceed to enjoy the short sketches to our hearts’ content. I want to emphasize that, to my knowledge, no other book affords this advantage.

This worked beautifully, though in ways that made me very glad that I had included in the syllabus the “subject to changes” disclaimer. I had indeed realized, while first reading through the text and beginning the process of lecture preparation, that, as teachers usually do and should, I’d be supplementing the text — through what I already knew, through things gleaned through other texts, and through the vaster and vaster material available online — and, while presenting the large nutshell, I soon also realized that I’d be borrowing from the smaller nutshells. I wouldn’t be able to resist this premature sharing. I wanted to give the students the same feel for math, and math history and culture, that I have, and perhaps then some. (Indeed, that “feel” was rapidly increasing.)

The book was written in such a way that, on the one hand, I noticed that it omitted, necessarily, some things that I did not want to omit (such as the Fibonacci series, and more about ancient Chinese math — in general, ethnomathematics [That’s my only criticism of the book.]), and on the other hand, it mentioned things that I had not known or thought much about. Concerning both, my curiosity was piqued. I spent whole mornings googling “Egyptian fractions”, “Babylonian geometry”, “mathematical games”, etc., and then becoming hungry for more details about things I already knew quite a bit about. In particular, upon googling Pythagorean Theorem, I came across the statement, “All proofs… depend upon Euclid’s Fifth Postulate. And in fact, in the presence of the first four postulates, the Pythagorean Theorem is equivalent to the Fifth Postulate.” That’s all that was said about that, and I thought, hmm. I decided that I just had to prove that the Pythagorean Theorem implied that famous Fifth. I spent half an hour thinking about it, then wrote down a proof to share.

All that found its way into my lectures, and into the students’ papers, presentations, and questions. Indeed, we were all hooked on the history of math, and Math through the Ages was the hook. As we approached the end of the large nutshell, I saw that I wanted to spend more time describing contemporary math and mathematicians. And the class as a whole (unanimous) wanted to play mathematical games, in particular ethnic ones like mancala and nim. The “large nutshell” got larger and larger; it wound up filling up more than two weeks, then more than three weeks, then closer to four. By the time we were ready to start the sketches, we had about a week and a half left, a total of ten and a half hours.

But nobody worried. In the process of covering “the large nutshell”, a huge portion of the “small nutshells” had also gotten covered — and then some. Perhaps this backfired on the idea of that “timeline feeling”, but that was compensated for by the natural-ness and spontaneity of our pace (and the “hardcopy” timeline handout). The point is that the format of this book made it possible for the class to be conducted in this flexible manner, and for the five weeks to constitute, as one of the students said, “a fun course”.

Shortly before completing the large nutshell I asked the students, as part of their homework, to think about which of the small nutshells they’d like us to do. By the next session, they had, among them, come up with ten, ranging from the history of zero and of negative and complex numbers, more details on the Pythagorean Theorem and FLT, to projective geometry, probability, and a little Statistics. The time remaining was just right to satisfy the students’ requests, with time left over for watching the dvd, The Great Pi/E Debate, and for having our own naturally-developing debates like “The Great Fractions/Decimals Debate” (We all voted for fractions.) and “The Great Should-Multiplication-Be-Introduced-in-Elementary-Schools-As Repeated Addition Debate” (sparked, as some of us might have read, by Keith Devlin’s posts on this site).

In short, the format of this book was perfect for the format of our course.

There are two versions of this book. The original edition is the one used in our course. The expanded edition came out in 2004, and includes problems and projects at the end of each of the short sketches (and only projects at the end of the large nutshell). The problems are very well chosen — not too easy, not too difficult, and helpfully worded. For example, p. 243 (in that edition), on set theory and infinities: “How much less than 1 is each term in this [convergent] sequence…” and “Even a toddler can see if she got her fair share of gum drops by matching them up with her brother’s share.” Also, the last problem in most of the sections strives to give readers a sense of the history of it all (“that timeline feeling”). For example, p. 244: “Georg Cantor lived and worked during a time of political and social turmoil in central Europe, much of it involving Germany. (a) What was the foremost political event in Germany in 1871… (b) What famous German composer…?” Etc. This device not only gives “that timeline feeling”, but also connects math history events with other history events. Thus readers learn history, period, not only math history.

As for the projects, pretty much ditto. For example, continuing with p. 244, the first project, “Write an essay comparing and contrasting the mathematical concept of infinity with the idea of infinity as it occurs in some other subject area…” That’s just plain fun! When I teach any math subject, history or actual math, I always try to emphasize how math terms are so often derived from “regular life terms”, so that students can appreciate the appropriateness, and in some cases the poetic-ness, of the math terms. The second project (proving things like “the set of rational numbers is the same size as the set of natural numbers”) is fun mathematically, and a bit challenging for those who don’t know Cantor’s work. And the third project, like the third problem, helps give readers a sense of history, and perspective, mathematically and otherwise.

I do have second thoughts about the phenomenon of problems and projects in textbooks. (Keep in mind that this is “just me…”) I realize that most textbook publishing companies want them, and so do most educational establishments. But I speculate that there might be a tradeoff to the advantages of the phenomenon. Without the problems and projects, our class did just fine. As described earlier in this review, we explored, we conversed, we played math games, and through these mostly spontaneous activities some of the very problems and projects from the new edition came up. Of course, there were problems and projects that the new edition contains that we did not think of, and also vice versa. But the process was more natural the way we did it, using the old edition, and if I were to teach the course again, I’d want to continue in this vein.

I think that, for some students, the presence of problems and/or projects (even if they’re not specifically assigned in the course) serves as reminder that this is, indeed, a textbook — something which students (in particular, math-anxious students) often have negative associations about. Thus, for some, there could be something a little (or a lot) intimidating, perhaps joy-squelching, about problems and projects presented so “officially”, “in writing”. (As we know, I did assign problems and projects, or the students chose them, but they weren’t so ubiquitous.)

I’ve said, and will say, that both my students and I felt that this book read like a “regular” book, a novel, something to relax with. Perhaps the inclusion of problems and projects takes that away, at least for some. Given these last three paragraphs, I’m glad that there weren’t too many problems and projects in the second edition (only two pages per short sketch (more for the large nutshell)), and that the original edition remains available.

One more point about the first edition vs. the second: It’s less expensive.

On the first day of the second week I had asked the students to write down how they felt about the book. Here is a medley of their comments: “So far I like the text very much. I find it easy to read and interesting. At times it can be a little too detailed. The ‘large nutshell’ is a helpful approach… I am really enjoying the text. [This from our English major] I find it easy to read and easy to understand. It almost reads more like a history book rather than a math book, and the authors don’t just expect you to know about the people and the math they discuss… I love the big intro which prepares you for their sketches. The text is more interesting than I would have predicted, but I find that other references are needed to provide more detail… I am definitely learning from the content of the text and plan on using the background historical information in my lessons for teaching.”

At the very end of the course, I asked again what they thought of the book, now that they’d read and/or dealt with some of the short sketches. The cry was unanimous. “Oh, I’m definitely planning to read through the entire book [now that the course was over and they had more time, in particular no more papers to write?]. It’s going to be bedtime reading!”

Indeed.


Marion Cohen teaches math at Arcadia University in Glenside, PA. She is the author of Crossing the Equal Sign, a poetry book about the experience of math. A new poetry book is forthcoming, Chronic Progressive, which is not about the experience of math — except (ahem) for the many “math poems” in it.

Date Received: 
Sunday, June 6, 2004
Reviewable: 
Yes
Include In BLL Rating: 
No
William P. Berlinghoff and Fernando Q. Gouvêa
Publication Date: 
2004
Format: 
Hardcover
Audience: 
Category: 
Textbook
Marion Cohen
09/2/2008

Preface

History in the Mathematics Classroom

The History of Mathematics in a Large Nutshell

  • Beginnings
  • Greek Mathematics
  • Meanwhile, in India
  • Arabic Mathematics
  • Medieval Europe
  • The 15th and 16th Centuries
  • Algebra Comes of Age
  • Calculus and Applied Mathematics
  • Rigor and Professionalism
  • Abstraction, Computers, and New Applications
  • Mathematics Today

Sketches

  1. Keeping Count
    Writing Whole Numbers
  2. Reading and Writing Arithmetic
    Where the Symbols Came From
  3. Nothing Becomes a Number
    The Story of Zero
  4. Broken Numbers
    Writing Fractions
  5. Something Less Than Nothing?
    Negative Numbers
  6. By Tens and Tenths
    Metric Measurement
  7. Measuring the Circle
    The Story of
  8. The Cossic Art
    Writing Algebra with Symbols
  9. Linear Thinking
    Solving First Degree Equations
  10. A Square and Things
    Quadratic Equations
  11. Intrigue in Renaissance Italy
    Solving Cubic Equations
  12. A Cheerful Fact
    The Pythagorean Theorem
  13. A Marvelous Proof
    Fermat's Last Theorem
  14. On Beauty Bare
    Euclid's Plane Geometry
  15. In Perfect Shape
    The Platonic Solids
  16. Shapes by the Numbers
    Coordinate Geometry
  17. Impossible, Imaginary, Useful
    Complex Numbers
  18. Half Is Better
    Sine and Cosine
  19. Strange New Worlds
    The Non-Euclidean Geometries
  20. In the Eye of the Beholder
    Projective Geometry
  21. What's in a Game?
    The Start of Probability Theory
  22. Making Sense of Data
    Statistics Becomes a Science
  23. Machines that Think?
    Electronic Computers
  24. The Arithmetic of Reasoning
    Logic and Boolean Algebra
  25. Beyond Counting
    Infinity and the Theory of Sets

What to Read Next

  • The Reference Shelf
  • Fifteen Historical Books You Ought to Read
  • The Internet and Other Media

Bibliography

Index

Publish Book: 
Modify Date: 
Wednesday, September 3, 2008

Dummy View - NOT TO BE DELETED