Stop Mindlessly Memorizing the Order of Operations

Breaking news: the order of operations that elementary schools teach students is not a fundamental law of nature but a convention to make our lives easier. Unfortunately, many students add PEMDAS (as the order of operations is commonly called in the US) to the list of mystical yet unquestionable truths to be memorized and feared. Everyone’s life might be a bit easier if we realized that mathematical expressions are written in a special mathematical language, and that like any language it has its own rules. The English language, for example, has spelling rules that dictate how to spell the word “bite” in the sense of eating and the word “byte” in the sense of data stored in a computer. If it wasn’t for those rules, there would be a great deal more confusion, and different people would read the same sentence in multiple ways. The same is true in mathematics. The notation and rules that we learn in school have developed over centuries to make reading and writing mathematical expressions an unambiguous activity. In the short video below, Henry Reich explores the conventions we use today and reminds us that thinking deeply about even the most basic ideas is more important than memorizing them. If you’re interested in the history of modern mathematical notation Ask Dr. Math has a bit more information.

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.

Quadratics Done Simply and Properly


A common middle school algebra topic like quadratics is all too often spread across hundreds of confusing textbook pages and several years of tedious instruction. It’s as if the subject is a mix of rocket science and neurosurgery. In reality, if you set aside some of the more beautiful geometric examples, the basic algebra of quadratic equations should not pose much of a problem for a typical math student. James Tanton confirms this with his Guide to Everything Quadratic [PDF], a short 64 page booklet on the fundamentals of quadratic equations. The guide presents the algebra of quadratics in an intuitive and mathematically sound way with plenty of examples. In addition to the algebra section, there is a section on graphing quadratics, and a section on fitting quadratics to data. As an added bonus, Tanton demonstrates a quick way to graph parabolas and includes a set of exercises that walk the reader through the derivation of the cubic formula, something that almost never appears in the standard school curriculum. If you want a more geometric perspective we recommend the book Lines and Curves. For those approaching quadratics for the first time, Tanton’s guide can replace or at least supplement most of the coverage of quadratics in a standard algebra textbook.

Dimensions: A Beautiful Excursion Through Geography, Geometry, and Topology

Unfortunately, some of the most beautiful mathematics is hidden from most people because it is so difficult to visualize. A good explanation has limited reach when the discussion at hand is about geometry, especially when it spans more than two dimensions. We may have an abundance of technology to help illustrate the subject, but someone still needs to spend an enormous of time and energy creating the kind of visualizations that are mathematically accurate, yet breathtaking. Fortunately, a group of French engineers, mathematicians, and education enthusiasts have done some of this hard work and produced Dimensions, an incredible nine part animated film that is nothing short of a visual feast featuring some of the most important and beautiful ancient and modern mathematics

The first chapters of the film introduce geography and the geometry of the sphere. Later chapters extend our intuition about two and three dimensions to four dimensions. The final chapters are more advanced but present a fairly elementary treatment of complex numbers and some topology. Every new idea is presented by an important mathematical personality, putting the whole narrative into a historical context. Although you can watch all nine chapters in one sitting, they are not all connected and it might be easier to watch them separately. The film website has a useful guide to help you choose what to watch, and we can’t recommend watching it enough.

Lively Chemistry Crash Course for Those in a Hurry

Although chemical experiments can yield exciting results, the theoretical part of chemistry may appear overly dry to students who are not already interested in it. Those who are studying the subject and need to review it may be overwhelmed by the sheer volume of details that they need to memorize. In both of these cases, it is helpful to have a highly condensed and lively summary of the key concepts. That is exactly what Hank Green accomplishes in his video series, Chemistry Crash Course. The videos in this series are short and entertaining, but they still highlight fundamental concepts. You can think of the collection as an extended trailer for the much broader and deeper subject or as a fun way to review for a chemistry test. The videos do not replace a textbook or a good teacher but they pack enough content into a few minutes that we recommend pausing them to process all of the information. If you have encountered any chemistry at all, these videos will be a bit more useful than if you have never heard about the existence of atoms.

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.

Plus Magazine: A Collaboration Between Mathematicians and Educators

plus magazine

Unfortunately, too many of the English language math textbooks that students see on a daily basis are written exclusively by professional educators without any serious input from mathematicians. As a result, these books are too much about teaching procedures and not enough about inspiring future mathematicians and scientists. At the other extreme, textbooks (usually at the college level) written by mathematicians tend to be dry and extremely dense. They may present all of the necessary definitions, lemmas, and theorems, but there is not enough room left for applications.

Plus Magazine, a University of Cambridge project is an attempt to correct this situation. It is an online publication that features articles, podcasts, book reviews, and news stories that makes mathematics relevant to those who are don’t grasp its importance and it is a collaboration between full-time educators and full-time researchers and practitioners of mathematics. You can find an article on why the violin is so hard to play and learn about the research of a recent Abel Prize winner (one of the top awards for mathematical research). The site also includes a collection of interesting nonstandard math problems and quite a bit of the content is related to physics. This should be a useful resource both for teachers and high school students.

The Theoretical Minimum: An Introduction to Modern Physics for the Curious Amateur

photo of Susskind

After Walter Lewin wows you with his theater of physics and you become intrigued by the possibility of parallel universes, you may be interested in some of the details behind modern physics. Unfortunately, at that point, you will most likely run into a serious roadblock. Contemporary theoretical physics is steeped in advanced theoretical mathematics, and most textbooks are geared towards future researchers, not intellectually curious individuals with limited backgrounds in either subject.

Luckily, Leonard Susskind, a Stanford Physicist and one of the fathers of string theory, comes to the rescue with The Theoretical Minimum, his unique series of courses on modern physics. The outstanding feature of Susskind’s lectures is that they do not shy away from mathematical derivations; the concepts are introduced in a completely rigorous way, yet they are made accessible to people who have never studied much math or science beyond advanced high school courses. In effect, these lectures offer both a physics and mathematics education for the price of one (figuratively speaking — the courses are free). Susskind develops the material from first principles and introduces all of the math that the physics requires. His target audience is adult continuous learners who want more detail than can be found in popular lectures, but bright high school students will benefit from seeing what life as a physics major entails. It doesn’t look too scary at all.

Jetpacks, Rocket Science, and Basic Physics for Beginners

Rocket science is usually a term associated with something too complicated for mere mortals to comprehend. In reality, the basic principles are fairly simple and involve basic middle school or high school physics. In this video, Derek Muller illustrates Newton’s laws of physics as they apply to rockets and jetpacks and mentions a few other interesting facts along the way. This is an attention-grabbing introduction to the foundations of classical mechanics and as usual with such videos, motivates a further more detail-focused exploration of the subject.

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