Blog Archives

A Possible Mathematical Theory Behind The Coming Cicada Infestation

The eastern United States is about to be overrun by billions of cicadas who will crawl out of the ground and create a deafening commotion. The interesting thing about their emergence is that they only come out every 17 years. Some scientists think that this is a coincidence, but the late Stephen Jay Gould, one of the major figures of evolutionary biology, postulated that the fact that this number is prime might not be an accident. He reasoned that if these periodical cicadas were to come out every, say, 12 years they would coincide with the emergence of predators whose life cycles are 1, 2, 3, 4, 6, and 12 years. Because their life cycle is 17 years, only predators with life cycles of 1 and 17 years coincide with the cicadas and it is easier for them to survive. In other words, periodical cicadas evolved to minimize their exposure to predators. You can learn more about this possible connection between number theory and biology in this Nature article and in a more detailed math paper [PDF] from the Courant Institute at New York University. Even if questions remain about the validity of this particular theory, it is an important reminder that purely mathematical ideas can provide fertile ground for scientific theories in any discipline.

A Comprehensive Guide To Teaching K-8 Mathematics

K-8 math terms

One of the effects of a highly decentralized education system in the US is the lack of a single guide to teaching any single subject. In mathematics, especially at the K-8 level, this has been an acute problem with no easy solution. Teachers have to do their own research, rely on the opinion of colleagues, and hope that their Web surfing or professional development classes lead them to good materials and guides. Unfortunately, even if they find useful bits of content scattered in online forums, websites, or books, how to bring all of it together into one cohesive mathematical narrative remains a mystery. Standard school textbooks, because of their low quality, are unfortunately not useful.

To address this problem, James Milgram, a Stanford mathematician and one of the top math education experts in the country, put together The Mathematics Pre-Service Teachers Need to Know [PDF], a 564 page guide to teaching K-8 mathematics. A few key facts about this monumental work stand out. First of all, unlike many good (but less comprehensive) mathematics books, Milgram’s work does not introduce some radical curriculum intended only for elite Chinese and Russian students toiling away in some underground olympiad training camps. The book was funded by the Department of Education and deals primarily with core parts of the K-8 math curriculum. Secondly, because James Milgram, and many of the people who contributed to the book, are serious research mathematicians and not simply educators chasing the latest education fad, the content in the book is grounded in solid mathematics. Thirdly, Milgram includes a large amount of material borrowed from foreign textbooks (from Russia and Singapore) to illustrate the best practices that have been proven effective in teaching various topics.

The Mathematics Pre-Service Teachers Need to Know [PDF] corrects one of the main flaws of the standard mathematics curriculum — that it is a mile wide and an inch deep — by providing in-depth coverage of all of the core topics and not introducing extraneous concepts that cannot be fully and rigorously developed. At the same time, the book does venture into a few extracurricular areas which are important for developing mathematical maturity. While it can certainly be a definitive guide to K-8 mathematics, Milgram’s work is not a textbook, but a teaching guide. Teachers will find a myriad of pedagogical tips, exercises, and problems, but they will still need to do some work in finding additional challenges for their students. These 12 problems are a good place to start.

Photo Credit: Enokson

Zeno’s Paradox: Is Movement Possible?

Sometimes when you think really hard about something, you can reach a conclusion so contradictory to everyday experience, that it forces you to reexamine fundamental scientific and mathematical truths. That is exactly the predicament that Zeno of Elea, a Greek philosopher, reached almost two and a half thousand years ago. His famous Dichotomy paradox proposes that getting from point A to point B is impossible because it involves an infinite number of steps, which must take an infinite amount of time. As it turns out, the modern mathematics of calculus and infinite series is required to rigorously resolve this dilemma, but an intuitive explanation that captures the essence of the solution is possible. In his excellent animation, Colm Kelleher illustrates both the paradox and its resolution, and you don’t even need to know any calculus to understand the solution. This video is a good place to start when introducing the topic of infinite series and is also one more way to make a discussion of infinity more concrete.

Cut-the-Knot: A Hidden Treasure Trove of Mathematical Miscellany

cut the knot logo

The Web may contain almost every possible problem, puzzle, and article imaginable, but it’s decentralized nature makes it’s hard to locate good content in a sea of endless tutorials, amusing pictures, and commercial promotions. If you’re trying to find extracurricular mathematical materials you need to know where to look, but more importantly, what to look for. Knowing an erudite guide makes life much easier. Alexander Bogomolny, a professional mathematician and curator of mathematical recreations and other topics, is that guide. His site Cut-the-Knot is an enormous collection of fascinating articles, illustrations, and animations covering a wide range of mostly non-advanced mathematics. One of the defining features of his articles are the interactive Java applets that illustrate a problem or principle. The site has been continuously updated since 1997, which makes it among the most comprehensive such repositories online. Unfortunately, because it was created more than fifteen years ago, its age shows in the design and technology used (Java applets are no longer the preferred delivery mechanism for interactive media). Although Cut-the-Knot has garnered over twenty awards, including one from Scientific American, it is not as well known as it should be. If you’re looking for a source of enrichment for regular math classes this is one of the best places to start.

A Miniature Introduction To Infinity

Infinity is a topic that has for ages caused a great deal of both fascination and confusion among students. It is a mathematical abstraction that unlike other abstractions seems hard to make concrete. The fact there is more than one type of infinity and that infinity is often treated like a number but is not an element of what we know as the real numbers adds to the confusion. The charming little video below from the Open University takes a sixty second look at Hilbert’s paradox of the Grand Hotel, a comic, yet mathematically serious example of how to think about infinity. The name is a bit of a misnomer as it’s not really a paradox, but simply a question with a somewhat counterintuitive answer. The animation does not explore all aspects of Hilbert’s thought experiment, but it is a good start that will pique anyone’s curiosity.