Monthly Archives: June 2013

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.

10 Science Tricks for Entertainment and Further Exploration

In the era of iPhones and iPads, magic tricks involving everyday items may not be as exciting as they once were, but in a classroom they are still an effective teaching tool. Richard Wiseman has created a video featuring ten science-based stunts which are an excellent demonstrations of various principles of physics. Some of these tricks were once popularized by the great Martin Gardner and you can learn more about the science behind them in his books. Our favorite stunt, which is more mathematical than the others, involves cutting a hole in a small postcard so that a person can climb through it. Watch the video and start your next science conversation or class with one of the tricks in it. Any science knowledge gained from it is sure to be more memorable than a boring science textbook.

The Stupidity of the “Math Wars”

we love math tshirt

Unless you’re a professor of education, you may not have noticed that for the last several decades a war has been raging between math education reformers and those from the traditional camp. Their disagreement boils down to one simple question: should students learn traditional computational algorithms (like the division algorithm for two numbers)? Reformers believe that achieving computational fluency is secondary to developing thinking, while traditionalists argue that not only are the basic arithmetic algorithms a fundamental part of mathematics, but that their mastery enables students to progress in the subject.

The historical roots of this conflict are irrelevant to us because like many social policy debates, it is heavily politicized. A more pertinent issue is how does all of this affect parents, teachers, and students. In a recent opinion piece in the New York Times, Alice Crary and W. Stephen Wilson offer an interesting analysis of the false dichotomy created by this ideological battle and come to an obvious conclusion: facility with mechanical mathematical procedures is inseparable from mathematical thinking. A student can’t develop abstract reasoning abilities without being able to easily manipulate the numbers and symbols that represent that reasoning.

Crary and Wilson point out that studying mathematics while de-emphasizing computation is like studying history without the actual historical facts. Even if historians have developed a way of thinking about their subject, they cannot do so without reference to specific facts. Similarly, both professional mathematicians and students of mathematics need to be comfortable with a certain set of basic computation techniques and a body of fundamental facts, before they can either prove new theorems or learn new material.

If you’re a parent or teacher, you need to give your kids an opportunity to practice algebraic and arithmetic computation and at the same time, let them work on challenging non-standard problems that develop their thinking. For a thorough treatment of the computational side of K-8 mathematics we suggest starting with James Milgram’s The Mathematics Pre-Service Teachers Need to Know, and if you need a textbook right away, Singapore Math is a solid choice. For developing problem solving skills and mathematical maturity, we’ve already mentioned a few excellent places to start. The most important takeaway from the “math wars”, is to stay away from standard US math textbooks and to steer clear of anything that has the word reform in it or looks like it was written without any input from well-known professional mathematicians.

Photo Credit: _Untitled-1

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.

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

hackathon

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

DIY.org: A Social Network for Young Engineers and Inventors

Do-it-yourself projects are vital components of a good STEM education, but until recently there have not been good online DIY resources targeting kids. DIY.org, a social network designed for children to share their creations with each other, seeks to change that. The site lets kids learn from each other and from a growing collection of tutorials. DIY.org is not restricted to purely engineering creations; users can share anything they make, from baked bread to artwork.

Perhaps, the greatest benefit of a site like this is that it allows kids to create a portfolio of their creations. Far too often, the talents and abilities of kids are hidden behind letter grades, numerical scores, and short teacher assessments. We are living, however, in a time when showcasing your work is becoming increasingly more important than bragging about your grades. Instead of aiming solely for perfect test scores, it may make more sense to enjoy the process of working with your hands and your creativity while acquiring useful knowledge and building up your resume. If you’re interested in a more advanced community, Instructables is worth exploring.

Vacuum Cleaners, Cannons, and the Quantum Mechanics of Empty Space

The force exerted by air molecules is something that we take for granted every day, but it is a surprisingly powerful force with equally surprising applications. For example, a vacuum cleaner “sucks” dirt in by creating a partial vacuum that allows the air outside the vacuum cleaner to push dirt into it. Similar reasoning can be used to create a vacuum cannon that relies on air pressure to eject a projectile at great (even supersonic) velocity. The following video from the Sixty Symbols Youtube channel illustrates this and provides further technical details.

If you’re interested in learning more about vacuums, Sixty Symbols has a follow up video that reveals their surprising quantum mechanical nature. You may be surprised to learn that a complete vacuum cannot really exist and that empty space actually contains energy. The conversational nature of the video doesn’t allow for a rigorous treatment of the subject, but instead offers an enticing glimpse into the exciting world that research scientists get to explore.

Wolfram Alpha: A Computational Knowledge Engine with Educational Applications

wolfram alpha logo

Sometimes one discovers a tool that is incredibly useful, yet surprisingly not as widely known as it should be. Wolfram Alpha, an intelligent search engine that responds to queries with answers as opposed to a list of links, has been around since 2009, but many teachers and students still haven’t heard of it.

This is surprising because there might not be another online tool that is as applicable in as many academic subjects as Wolfram Alpha. Because it is based on Mathematica, one of the top computational software packages used by scientists, mathematicians, and engineers all over the world, it is clearly extremely good at answering math questions. Beyond complex numeric computations, it can do symbolic computation (like factoring polynomials) that is much harder than simply crunching numbers. This computational power applies to mathematics and all of the sciences (is math a science?) but Wolfram Alpha is more than just a fancy online graphing calculator.

It is actually an intelligent system that taps into a myriad of online data sources that enable it to answer questions in almost any field. For example, suppose you wanted to know the identities of the characters in Shakespeare’s A Midsummer Night’s Dream. A simple query would yield the answer and provide additional details. To give a sense of the scope of it’s knowledge engine, Wolfram has conveniently created a page of sample uses that covers a wide range of human activities. If you have never heard of Wolfram Alpha, you will be surprised when you first use it.

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.