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.

The Scale of the Universe and How We Measure It

Students often ask about the existence of the largest, smallest, or most distance objects that exist. These questions undeniably provide intellectual entertainment, especially when we can visualize the answers with ease. Take a look at this beautiful interactive animation created by Cary and Michael Huang to get a sense of the the kinds of distances and sizes that exist in the universe, and then look at the Royal Museums Greenwich animation that introduces the physics of measuring distances to macroscopic objects in the Universe. The video skips the details of how some of the distances are calculated but is a good starting point for further geometrical explorations that are not beyond the school curriculum.

12 Elementary Math Problems that Capture the Essence of Mathematical Thinking

girl solving problem

One of the most abused terms in mathematics education is problem solving. The term has been hijacked to mean anything from plugging numbers into the quadratic formula to repeating the same steps over and over again when calculating a derivative in calculus class. Neither of these activities could be further from the work of real mathematics, but what kind of problem solving constitutes true mathematical thinking? Alexandre Borovik and Tony Gardiner, both practicing mathematicians, provide a compelling answer in their paper: A Dozen Problems. These twelve problems are accessible even to elementary school students, yet they convey the archetypal paradigms of genuine mathematical thinking. The problems don’t require much mathematical background, certainly nothing beyond the regular school curriculum, but some of them require a good deal of mathematical sophistication. Most of these problems are part of the classical canon of math problems in Russian math literature and have been used in thousands of extracurricular math programs in Russia and the former Soviet Union. This paper is a good starting point if you’re interested in expanding your mathematical horizons beyond the regular school curriculum but are intimidated by difficult olympiad problems that require extensive extracurricular math knowledge.

(Photo credit: Kathy Cassidy)

Art of Problem Solving Classes

aops_logoWe prefer to review high quality online courses that are free, but for the Art of Problem Solving (AOPS) classes we need to make an exception. The Internet is flooded with free courses taught by first-rate instructors, but with scale comes an often overlooked problem. A popular free online course that has an enrollment in the thousands cannot provide the kind of student-teacher interaction that is vital for learning. Art of Problem Solving math classes are relatively inexpensive, but because of limited enrollment, students can communicate with their instructors in real-time. Unlike massive open online courses, Art of Problem Solving classes are built on fairly basic technology that does not include audio or video, but this does not take away from the learning experience because students receive individual attention and are required to do in-class work that is immediately available for their instructors to review. The other defining aspect of these classes is the quality of the mathematical content. Unlike various other online resources, AOPS focuses on sometimes difficult yet engaging problems (after all, problem solving is in their name) instead of simple textbook exercises. For advanced or motivated students in grades 5-12 who do not have access to local high quality math instruction and for homeschoolers, these classes are worth looking into.

Mindstorms: Children, Computers, And Powerful Ideas

mind storms book

Some of the most crucial steps in mental growth are based not simply on acquiring new skills, but on acquiring new administrative ways to use what one already knows.

If you’re only going to read one book on learning Mindstorms: Children, Computers, and Powerful Ideas has to be it. On the surface, this may seem like an outdated book about computer science education, but it is really a profound study of how people learn anything from math and physics to juggling and skiing. Written by Seymour Papert, one of the pioneers of artificial intelligence and the creator of what would become the MIT Media Lab, Mindstorms illustrates the idea that children can use their own “objects-to-think-with,” intellectual structures of their own making, to acquire, and more importantly, to work with increasingly complex and abstract knowledge. The book is filled with concrete examples of students learning something completely new or even fear-inducing (like math) using knowledge and intuition that they already have. Logo, the programming language that Papert co-created, serves as the primary example of a tool that helps students reason about new ideas (not just in math) in a perfectly rigorous yet comfortably intuitive way. Whether you’re interested in math and computer science education at the K-12 level or want a deeper understanding of how people learn without all the education jargon than this book will be indispensable.

2013 Cambridge (MA) Science Festival April 12 – April 21

If you’re anywhere near the Boston area in the second half of April don’t miss the 7th annual Cambridge Science Festival featuring lectures from some of the world’s top researchers, hands on activities, theatrical performances, a robot zoo, and much more. One of the outstanding features of the Cambridge Science Festival, unlike other science events, is that everyone from elementary school kids to adults with science backgrounds will find something interesting to see or do there. If you haven’t been to Cambridge Science Festival before you can see videos of some past activities on their Youtube channel.

The Story of Martin Gardner and Mathematics as Magic

Martin Gardner, one of the greatest recreational mathematicians of all time, is responsible not only for helping popularize mathematics as an art form and as a form of recreation but for inspiring a generation of future mathematicians to pursue it as a profession. He wrote the Mathematical Games column for Scientific American for a quarter of a century, and his mathematical and scientific gems have found their way into dozens of foreign publications as well as numerous research papers. The Nature of Things documentary gives the viewer an up close look at Martin Gardner’s work and the work of other people with whom he collaborated. If you’re looking for some mathematical entertainment with serious mathematical substance, or simply want a glimpse into the playful nature of mathematics and mathematicians, this is a highly recommended film.

A Mathematician’s Lament


There is such breathtaking depth and heartbreaking beauty in this ancient art form. How ironic that people dismiss mathematics as the antithesis of creativity. They are missing out on an art form older than any book, more profound than any poem, and more abstract than any abstract

Much ink has been spilled on the subject of why K-12 mathematics education needs improving, but rarely has anyone made the point in such an eloquent manner as to make it mandatory reading for teachers, parents, and students of mathematics education. This is exactly what former research mathematician and current math teacher Paul Lockhart has done in his impassioned essay “A Mathematician’s Lament” [PDF].

This is not another dry statistics-filled analysis comparing competing education reforms, but a powerful cry for teaching thinking over mindless procedure following. As an added bonus, Lockhart includes two beautiful geometry problems that capture the essence of mathematical reasoning and that you can start playing with right away. Yes, playing — we need more of that in our math classes.

(Photo by Mikey Angels)