Blog Archives

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.

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.

Startup Engineering: A Course for Advanced High School Students and Their Teachers


Successful startup founders seem to be starting their companies earlier than ever before. Whereas Bill Gates and Mark Zuckerburg dropped out of college to start their companies, David Karp whose company, Tumblr, was recently acquired by Yahoo, dropped out of high school to work on his own software projects. While leaving school is not a requirement for successful entrepreneurship, starting early can help.

High school students who have completed an AP computer science course, or who have had previous programming experience, and are interested in learning about the way software engineering is done at the top startups, can now take a startup engineering course that will not only expose them to the latest technology and methodologies, but will give them an opportunity to launch their own project. Stanford University is offering this free online course through Coursera, and although it is a massive open online course (MOOC) it will have a few unique features that set it apart from other MOOCs. First of all, students will be exposed not only to the technical tools of the trade, but also to the business side of starting a company. Secondly, and perhaps more importantly, students will complete a final project that will entail building their own web application, and the best projects will qualify for prizes from sponsoring startups. The course starts June 17 and should be an interesting learning experience for both students and their teachers.

Photo credit: hackNY

Mechanical MOOC: A New Type of Online Course That Teaches Basic Programming

mechanical mooc picture

Most massive open online courses (MOOCs) seem to follow a familiar pattern. Thousands of students sign up for free structured courses that require first watching videos of university professors lecturing and then doing homework and taking exams following a strict schedule. These MOOCs tempt students looking for convenient and free access to top universities, but unfortunately most of those who sign up drop out without completing the courses. One of the problems maybe that not everyone learns at the same pace and some students may simply not have enough time outside of work and school to put in the required hours every week. The Mechanical MOOC, a collaboration between MIT OpenCourseware, OpenStudy, Peer to Peer University, and Codecademy, is taking a stab at addressing some of the underlying problems plaguing most MOOCs. In their course A Gentle Introduction to Python, they are eliminating an instructor and a strict schedule and instead encouraging students to work in groups to learn at whatever pace works for them. There will be a mailing list that will coordinate all learning activities and direct students to the the appropriate resources, but beyond that there will be complete freedom for groups of students to work together and help each other while following their own schedule. Most of the material will come from MIT OpenCourseWare and OpenStudy will provide a question and answer forum to facilitate group discussions. The course begins in June and given it’s flexible nature is worth a try for those who have had a hard time committing to any of the other MOOCs. You can sign up here.

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.

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.

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.