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A Free Stanford Online Course to Develop Your Mathematical Thinking

Most math classes try to teach computation skills and not much more. This is true not only in school where students memorize mechanical procedures and how to plug numbers into formulas, but also in university courses targeting scientists and engineers. Computational fluency is important but it is only a stepping stone to mathematical maturity.

Higher level mathematics requires the ability to prove mathematical statements, which in turn, requires the ability to think logically and create convincing and rigorous arguments. If this sounds more like something taught in law school, it’s because much of math education has been divorced from actual math. We’ve addressed the topic of mathematical thinking before, but now there is an online course that teaches the most foundational concepts in non-computational math.

That course, Introduction to Mathematical Thinking is offered by Stanford University via Coursera and is taught by Keith Devlin, who is a well known and charismatic math popularizer, educator, and researcher. The main purpose of this course is to serve as a transition between computation-fixated school math classes and undergraduate math major courses. In some ways, this is a traditional course that many university math departments require, but as a MOOC it is now accessible to anyone from high school students to math teachers. As Professor Devlin says in the introductory video (below), the course does not teach students new mathematics; instead it teaches them how to think mathematically and work with the standard mathematical language that involves notions like equivalence relations and logical quantifiers.

The course has been offered before with tens of thousands of students and has received excellent reviews. If you have never been exposed to anything beyond plug and chug math, this course is for you. Once you acquaint yourself with the basics of mathematical thinking, you will gain a deeper appreciation of some important topics that are usually left out of the regular school math curriculum.

Euclidean Geometry As a Game In Your Browser

Euclidean Geometry Game

Euclidean geometry is one of the most beautiful math topics that is part of the standard school curriculum, yet it is consistently butchered by standard teaching practices. Even more surprising is that straight edge and compass constructions are usually excluded from geometry classes instead of being used as a way to lure students in. Using a compass and straightedge to construct a geometric figure feels like playing a game and has the added benefit of being a hands-on activity.

Using paper and pencil has been the traditional way to do constructions but software makes it a great deal easier, especially in the age of sloppy handwriting. Although a number of geometry software packages exist, none is as simple or as addictive as this web app. No download is required and you can immediately start constructing the shapes that are offered as challenges on the right side of the app page. The challenges are ordered by difficulty and you gain points when solving them. Although the app lacks instructions, it shouldn’t take more than a few minutes to get a hang of how it works. If you teach geometry, do your students and yourself a favor — let them try out a few challenges and they’re likely to get hooked not just on the game, but on geometry as well.

Better Explained: A Site That Develops Your Intuition

better explained

When learning mathematics, you need a concrete mental representation of the often extremely abstract ideas that you need to internalize. Unfortunately, most math books, even good ones, don’t provide the needed intuition. Fortunately, Kalid Azad has come to the rescue with Better Explained, a site dedicated to helping anyone learning a new subject develop a way to think about it in intuitive and concrete terms.

The math topics covered include basic arithmetic, probability and statistics, exponents, complex numbers, and even advanced topics like calculus and the Fourier Transform. Better Explained is not comprehensive, nor is it rigorous; its goal is to give its readers the kinds of insights that will enable them to jump start further in-depth learning on their own. Khalid’s tone is that of a student who at one point also struggled with the given concepts and is a welcome change from the often austere tone found in textbooks and lectures. The site features excellent articles on programming and a few other topics, but it certainly stands out for its clear and concise math articles.

A Comprehensive Introduction to Information Theory for Complete Beginners

The term ‘information age’ is a modern cliche, yet few realize that the word information has a precise mathematical meaning with far-reaching consequences. Information theory is one of the great developments of the twentieth century that spans multiple disciplines including mathematics, computer science, electrical engineering, and biology. Unfortunately, although some of the fundamental ideas of this subject are easy to convey to even the youngest students, it is completely absent from the school curriculum.

Luckily, the filmmaker, Brit Cruise has created “The Language of Coins,” a series of videos about information theory that is accessible to a general audience. The series begins with a close look at the way we communicate and continues on to more advanced topics like Markov chains, which is an important modern tool of applied mathematics. In all, there are sixteen videos; twelve are already available online and the remaining ones will be posted soon (the complete playlist is available on Youtube). You need to know about information theory and if you don’t, you should start with this excellent series.

Richard Feynman on the Differences between Mathematics and Physics

As previously discussed, mathematics is not a science in the same way as physics, chemistry, and biology, yet because it is treated as a scientific discipline in school, students rarely understand it’s role. If you’re a math teacher, ask your students the following question: “Biology is the study of living organisms, physics is the study of matter, motion, energy, and forces, but what does math study?” You are certain to elicit a great deal of confusion.

In the following video, Richard Feynman, one of the great physicists of the twentieth century, attempts to answer the question by differentiating between the mental models of mathematicians and those of physicists (and by extension other scientists). Feynman, whose mathematical abilities stood out even among other elite physicists, was supremely qualified to compare the different approaches and to elucidate the peculiar nature of mathematical research. His lecture was recorded almost half a century ago, and lacks the polish of more modern science productions, but it more than makes up for it in both substance and Feynman’s impassioned lecture style. This is a must-see lecture for high school students who have an interest in a math or science career.

Is Mathematics Real? A Thought-Provoking Discussion for Any Math Class

Mathematics is so frequently put into the same category as the sciences that students often assume that it is one of the many scientific disciplines, just like physics, chemistry and biology. This can become a problem when students try to understand the reason for studying mathematics. Most students can immediately see that biology is the study of living organisms and the immediacy of that subject makes it both instantly appealing and comprehensible. In mathematics, however, as soon as the studied objects become sufficiently abstract and far removed from everyday experience, students fail to see their significance. As layers of abstraction are added, visualizing mathematics becomes even harder than picturing microscopic cells.

When confusion arises about the nature of mathematics, it can be helpful to introduce a few ideas from the philosophy of mathematics. That is exactly, what the following PBS Idea Channel video does. The question of whether mathematics is a science that studies objects that exist in this universe or is a mental construct that is aesthetically elegant and just happens to be the best language we know for describing nature, remains unanswered, but the discussion is important. Without it, students may never suspect that mathematics plays a unique role in human history and that it spans almost all disciplines. For those who want a deeper take on the nature of mathematics, Eugene Wigner’s classic paper on The Unreasonable Effectiveness of Mathematics in the Natural Sciences will provide much more food for thought.

NRICH: An Organized Collection of Math Enrichment Problems and Activities


If you liked the expository writing in Plus Magazine, you may enjoy Nrich, a sister project from Cambridge University. The site features hundreds of math problems and activities organized by grade and ability level, as well as by topic. Although Nrich has a section for students, teachers who need to prepare lesson plans may find it more useful. The site content is closely aligned with US and British math curriculum standards, which should make it particularly appealing to educators.

An outstanding feature of the site is it’s emphasis on math enrichment topics that are usually outside a standard school curriculum, yet close enough to it to be relevant in a regular math class that needs to follow strict education guidelines. Another welcome aspect of Nrich is that professional mathematicians, not just math educators oversee the project, making sure that it is both mathematically sound and relevant. There is even a forum for those who need math help. Nrich may not be the easiest site to navigate, but it does contain a convenient topic directory that organizes all of the content. This project is worth exploring and should contain something useful for anyone teaching or learning math.

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.

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.