Blog Archives

Moebius Noodles: A Mathematical Playground for Young and Old

Moebius Noodles book

Contrary to popular belief, mathematics is not an activity that requires textbooks, calculators, and years of training. Because it consists of such fundamental notions as symmetry, classification, counting, and geometric transformations — all concepts that come naturally to even the youngest children — mathematics can truly be studied at any age. If you have picked up a copy of Math From Three to Seven and are wondering whether there is something similar for kids that are younger still, you should take a look at Moebius Noodles.

This book, the work of Yelena McManaman, Maria Droujkova, and Ever Salazar, is a beautifully illustrated collection of activities that engages young kids (even toddlers) in discovering fundamental mathematical principles and abstractions. For example, why wait until middle school or high school to learn about functions when you can think about them in any almost any context? For instance, Moebius Noodles proposes an activity where a child is given the name of a baby animal (like “kitten”) and must identify the corresponding adult animal name (in this case “cat”). The child has just created a baby-to-mother function and there are endless other possible activities that reinforce this idea of mappings between sets. The book covers basic ideas involving numbers, symmetry, functions, and even a little bit of calculus. If you’re a parent or preschool teacher interested in fun activities that involve both playing with and internalizing fundamental mathematical concepts, then Moebius Noodles is worth your time.

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

Curriculum Notes: Teaching Logarithms and Their History

John Napier

Most textbooks present logarithms as just one more set of mechanical procedures to be memorized and repeatedly applied. Their history of how and why they were invented, however, is rarely presented. In addition, all too often the common sense proofs of the basic properties of logarithms are not emphasized at all. James Tanton, a research mathematician-turned-teacher, presents a quick, rigorous, yet interesting way of teaching logarithms in his take on logarithms essay [PDF]. Like many of his essays, this one contains links to videos on the subject for those who prefer to watch rather than read. If you’re a teacher this will make your lesson preparation easier.

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)

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.

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)