Hello, and thank you for visiting Math Rewind. I created this blog to convey to visitors a sense of how modern mathematics connects to its historical development as a way to help others understand mathematics more deeply, perhaps appreciate it more, and to use the concepts and ideas here to provide a richer learning experience for students. This site is primarily for math educators, but students and anyone in the general public is welcome to visit as well. Just understand that my focus is not on teaching mathematics in detail, but to describe an approach to teaching mathematics that I believe is better than the rather abstract way schools tend to teach it.
A little background is in order. I completed a bachelor of science degree in mathematics in the year 2000. With a degree under my belt, I considered myself knowledgeable and competent in the field. Over the years, I stayed current with modern mathematical concepts – using spare time to do problems in calculus, differential equations, complex analysis, abstract algebra, and more. My confidence in my mathematical abilities was very high. I later completed a master’s degree in education and another master’s degree in operations research. But for all of my competency in mathematics and in using sound educational principles to convey it to students, I came to the realization that I could do mathematics, but I could not think mathematically in the truest sense. That all changed after I decided to look into a book written by a Greek mathematician who lived over 2,300 years ago.
Euclid lived in Alexandria around 320 BCE, during the reign of Ptolemy I. He was a master geometer, regarded even today as one of the greatest mathematicians of all time. And the work for which he is most remembered, The Elements, was the main set of books (13 books in all) for teaching mathematics for quite a while and in particular the primary work for teaching plane geometry for over 2,200 years. It was the 13 books of Elements that led me to my realization that I was unprepared to think mathematically and what sparked my investigations into the historical development of mathematics.
Of course, I took a class in plane geometry in the ninth grade, and later, I took a course in non-Euclidean geometry in college. But when I opened Elements, I was amazed at what Euclid was able to accomplish over two millennia ago. I was astounded by two things. The first was that I did not know that a work of such far-reaching insight could have been written in those ancient times. Yes, I’d heard of Elements and the superlatives many have used to describe it and its author, Euclid of Alexandria, but to see it for myself was something entirely different. Second, it struck me that my high school geometry course used a book that was entirely derived from Elements. Why, I thought, did the school not simply use Elements as the textbook for teaching geometry? After considering my high school experience with geometry and working through Elements so many years later, I decided that my understanding of geometry and my ability to think mathematically about geometrical problems would both have been enriched if my school opted to use a translation of the ancient Greek text as the course book rather than the modern book we used instead.
Then, I got to wondering about what other early mathematical texts hold treasures that would be of use to students today. I obtained a copy of La Geometrie by French mathematician and philosopher, René Descartes. Admittedly, the only two things I knew about Descartes before working through that book was that the Cartesian coordinate system is named in his honor and that he is responsible for the philosophical assertion, Cogito, ergo sum (I think, therefore I am.). But La Geometrie is masterful, and it set the standard for mathematical writing. In Unknown Quantity: A Real and Imaginary History of Algebra, John Derbyshire (2006) said about La Geometrie that reading it, “you feel that you are looking at a modern mathematical text. It is the earliest book for which this is true, I think” (p. 93). Revisiting this early text is sure to reveal insights into the development of modern mathematical methods.
Later, I picked up a copy of Isaac Newton’s 1687 treatise, Philosophiæ Naturalis Principia Mathematica, or, Mathematical Principles of Natural Philosophy (often simply called the Principia) and Leonardo Pisano’s (better known at Fibonacci) 1202 work Liber Abaci, or the Book of Calculation, which popularized the Hindu-Arabic number system we use today and stood as the primary text on algebra for centuries after it was introduced. On and on, I acquired more and more works that are little (if at all) discussed in school with students who are studying mathematics and trying to grasp concepts for the first time. I also read many books about the history of mathematics and about the mathematicians who made history by introducing new concepts and techniques. Through all of this study, I realized that people who lived hundreds or even thousands of years ago had as much intelligence as any person living today. Further, the methods they developed to solve problems did not come easily and did not develop overnight as a modern math textbook might lead one to believe. In fact, modern mathematics curricula do a disservice to students by ignoring primary sources, focusing instead on current abstractions divorced from any treatment of the historical development of those abstractions.
So, now, I pose some important questions. Why does the current approach taken in mathematics curricula and textbooks do a disservice to students – what benefit does a historical development of concepts provide? What exactly does it mean to “think mathematically?” How would studying early (even ancient) mathematical literature result in a greater ability to think mathematically and assimilate new mathematical information more readily and intuitively? How can “forgotten” texts be used to enrich mathematics curricula for students today? These and other questions are what I will address using blog postings. Hopefully, as you read through the content here, you will see what I mean, and you will be able to teach mathematics in a way that may make all the difference with your students’ comprehension and retention of, and appreciation for, mathematics.
Derbyshire, J. (2006). Unknown quantity: A real and imaginary history of algebra. New York, NY: Penguin Group.