Teaching Math via Computers and Computer Programming: What Do You Think?

Conrad Wolfram has a presentation about what he feels is a weak, antiquated way of teaching math in school. Instead of all hand-written work on paper, use computers to get students thinking about everyday problems. He argues that problems are dumbed-down in school, and that real-world calculations are not done that would better engage students, as well as lead to better math skills that are necessary in today's world. Because math is done on computers in research and the workplace, this would allow students to build the knowledge, tools and skills that are relevant in today's world, rather than the knowledge, tools and skills that were necessary 50 years ago in an age of agricultural and manufacturing jobs.

Personally I think he has a good point. However, I am convinced that doing just about anything one-way is not a good idea. Variety is necessary. There is something to be said for doing things by hand to learn process and the nuts and bolts of a computation. But I do think technology can be and should be used more frequently than is presently done, as this is a student's future. Also, not everyone will likely learn more if done on a computer. Some students do in fact enjoy pencil and paper problems, and can learn a great deal with this technique. I also think that many learn, or at least gain greater insights, interest and relevance of math through applications in something like physics. I know I finally got a grip on what calculus was all about after using it in physics, and many students have told me the same thing.

I am interested in your take on this as students...what do you think?

Very Good video about gravity and Einstein's Idea

This is the snippet from Brian Greene's Elegant Universe. It deals with Einstein's general theory of relativity, and the notion of warped space-time leading to what we call gravity. There are some very good graphics which help us get some pictures in our heads as to how this all works. I encourage you to watch it, and also the 'how to' video below that has to do with relativity and the principle of equivalence, to see how Einstein reasoned light is bent by gravity.

Some Advice - To Embrace Complex Networks to Find Simple Solutions

Scientist Eric Berlow shows a few examples of how one can think about complex systems and networks in order to find simpler structures and solutions. In networks where there are hubs (i.e. agents of the network which have many connections compared to most agents), one can look and focus on the first few orders of connectivity to begin looking at the key components and eliminate 'noise.' Rather than be freaked by a complex problem, step back and look at the overall picture to pick out the key pieces of the problem. Sound familiar? This is the approach we take for something like those systems with tension. We only look at the forces that may affect the motion and don't worry about the others. So we continuously try to simplify the complexity into simpler pieces. Or in circuit analysis, we isolate smaller networks of resistors, and simplify those to single resistors, until a complex circuit is redrawn as a series circuit. This is the idea Berlow is promoting. Check it out, and let me know what you think!

Do keep in mind, though, that this is not foolproof. Some times this approach makes a problem more manageable and it can lead to some sort of solution, or at least some sort of approximation, but other problems have so many intricacies that this approach leads to nowhere. It is a strategy you may try to see where it takes you.

The Basic Principle of General Relativity - Principle of Equivalence

Einstein published General Relativity in 1915. Similar to special relativity, where he only considered frames of reference moving with constant relative velocity, general relativity is based on simple principles that lead to fantastic consequences. In this case, Einstein reasoned that the Principle of Equivalence was the fundamental building block concept for gravity. It states that the effects of acceleration on a system are indistinguishable from the effects of gravity on that system. This video shows an example of this principle in action, and shows how you can quickly figure out gravity should bend the path of light, even though light has no mass (so Newton would say this cannot happen)! This principle also explains why inertial mass (the mass in F = ma) is equivalent to gravitational mass (the mass in F = GMm/r^2), which is why all objects fall at the same rate in a gravitational field. Note that general relativity leads to the Big Bang, black holes, stellar evolution, gravitational red shifts, precession of planetary orbits, effects on time (which are needed for GPS to be so accurate), frame dragging as planets and objects move through space-time, and so on. All are confirmed through observation and experiments. Check it out!

How to do Inelastic Collisions where bullets stick in blocks

Here is a case where bullets are fired into blocks, and by measuring what happens afterwards allows us to figure out the speed of the bullet before the collision. A classic example is a ballistic pendulum, and this is compared to a similar spring problem. Because the collisions are INelastic, kinetic energy before and after is not the same (KEo > KEf). But, conservation of momentum is what connects the before and after pictures of such collisions. Check it out.

Dark Matter - Interactive Forum from Scientific American

Check out the interactive discussion about dark matter, a favorite topic in modern physics, particularly in the astrophysics and particle physics fields. Dark matter is a generic term for anything that cannot be seen in the electromagnetic spectrum, but affects other objects gravitationally. This could be known matter such as neutrinos, neutron stars, brown dwarfs, or just unseen stars and galaxies. Or, it could be new forms of matter outside of the standard model in particle physics. There are names like WIMPS, MACHOS, axions, and others. Always a good time checking into cutting-edge ideas!

Spring Problems and the use of Energy

Springs provide a nice example of NON-constant force. We know Hooke's law, F = -kx, where x is the displacement of the spring from its equilibrium position. Keep in mind the - sign is for direction: the force always tries to bring a spring back to equilibrium, which is in the opposite direction relative to the direction of the displacement (a point of stable equilibrium). Here are examples of using energy conservation to solve spring problems.