Monthly Archives: July 2013

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.

A Short Animated Introduction to Reading Music (And a Bit of Bach)

Although numerous studies have shown the importance of music education in schools, few students learn the basics of music theory as part of core curriculum requirements. Apart from the obvious reasons, this is unfortunate because music is so intertwined with math and science and has been a source of inspirations for many great mathematicians and scientists. In fact, Albert Einstein once said that “the theory of relativity occurred to me by intuition, and music is the driving force behind this intuition. My parents had me study the violin from the time I was six. My new discovery is the result of musical perception.”

If you have never learned how to read music, the following animation by Tim Hansen does a good job of conveying the essentials. After you watch it, listen to some Bach (i.e. mandatory music for mathematicians) and watch an incredible visual representation of music that is rich in mathematical structure. Enjoy!

An Online Problem Solving Course for Young Kids

logo for young mooc math class

We’ve mentioned the work of James Tanton, Maria Droujkova, and Yelena McManaman before, and now they have teamed up to offer a one month long online math problem solving course. mpsMOOC13: Problem Solving for the Young, the Very Young, and the Young at Heart revolves around a small set of accessible nonstandard math problems that kids and parents solve together. The solutions and discussions are recorded and reported on the course website resulting in a community-generated math education research project.

The course has already started, but you can still do all of the problems and follow the discussions. If you’re homeschooling this course will be especially useful, and you should stay tuned for similar future courses from this team.

A Video Game Company Recruits, Trains, and Reaches Out to High School Students

The desire to create their own video games is one of the top reasons why kids give programming a try. As a result, teaching programming through computer game creation has become increasingly popular. One can now easily find books and after school programs that follow this trend, but as tempting as this approach may be, it suffers from false expectations. A student who wants to learn how to make video games doesn’t realize that computer science educators are more interested in teaching him how to program than in helping him create something that resembles the slick professionally-produced games that he plays at home. Of course, the initial excitement of creating your own computer program may compensate for unmet expectations, but it’s not clear that this is a good trend.

What if, on the other hand, students could create real modern video games alongside industry professionals? That is the idea behind Pipeline, an outreach effort by Valve, a major video game company. They have recruited a group of high school students who are working alongside their much older colleagues on video game titles that will be sold to millions of players around the world. These students are not only exposed to the video game industry, but are sharing their experience with their peers through the Pipeline website. This may be the easiest way to learn about what it actually takes to make a professional video game. Learning to code is important no matter what your goals are, but if you’re interested in joining the video game industry, signing up for Pipeline will give you a realistic view of the work and knowledge required to do so.

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.

A Simple Animated Explanation of Free Falling and Zero Gravity

As children we first learn about the notion of weightlessness in outer space and the idea of zero gravity, but these concepts are actually a bit more nuanced than may appear. For example, why satellites orbit Earth instead of crashing into it because of Earth’s gravitational pull can be a mystery if you have never studied physics. The following TED Education animation does a good job of illustrating the basic principle behind orbiting objects without going into too many details.

If you’ve been exposed to high school physics, the Wikipedia article on weightlessness is uncharacteristically clear on some of the more advanced aspects of the subject. In particular, you may be surprised to learn that Einstein in developing his theory of relativity realized that gravitational interaction cannot be felt when all other forces are removed and this led him to consider the possibility that gravity is the result of the curvature of space. If you’re interested in the details of relativity and a modern interpretation of Newtonian mechanics, Leonard Susskind’s set of physics lectures is the best place to start.

An Animated Introduction to Ontology: Is a Copy the Same as the Original?

Questions of equality and equivalence are of fundamental importance in mathematics and computer science. In everyday use we are usually comfortable with a vague definition of equality, but in programming for example, two objects may be identical in one instance and different in another. This is usually a great source of confusion for inexperienced programmers. In mathematics, equality has multiple meanings and uses and even basic subjects like high school geometry introduce the notions of similarity and congruence that represent two different levels of equality.

Of course, equality and equivalence are also part of the branch of philosophy called ontology. In the following classic animation, John Weldon presents the topic as a fun thought experiment that asks the question: what does it mean to be? Watch it and be amazed by the philosophical nuances of existence.