Breaking the Speed of Light

The e-mail arrived today. I am going to be invited to be a research assistant at MTSU this summer. The goal: break the speed of light through sound pulses. As long as the Einstein police don't pull me over, this will be great.

Hopefully, my research won't interfere with the classes I am planning to enroll in this semester. Expect more info as this story develops, stay tuned!

How is Mathematics Learned?

Practice vs. understanding: which is more important? Should students seek understanding early in the education game, or is it better to apply mathematical tools under the assumption that understanding will come with experience? According to Keith Devlin, NPR's The Math Guy, simple methodical practice is best. In a recent article published on the Mathematical Association of America's website, he suggests that practice is the natural method by which we learn any skill:

"When we learn a new skill, initially we simply follow the rules in a mechanical fashion. Then, with practice, we gradually become better, and as our performance improves, our understanding grows. Think, for example, of the progression involved in learning to play chess, to play tennis, to ski, to drive a car, to play a musical instrument, to play a video game, etc. We start by following rules in a fairly mechanical fashion. Then, after a while, we are able to follow the rules proficiently. Then, some time later, we are able to apply the rules automatically and fluently. And eventually we achieve mastery and understanding. The same progression works for mathematics, only in this case, as mathematics is constructed and carried out using our language capacity, the initial rule-following stuff is primarily cognitive-linguistic."

This approach is contradictory to my current method for studying mathematics (or any skill). Instead of seeking to fully understand a topic, perhaps it is more efficient to work problems in a mechanical fashion, slowly building understanding with time and practice.

Links:
Keith Devlin's article, "How do we learn math?"

Fibonacci Abounds in Tool's Lateralus

Based on the findings of a Kentucky blogger, the Fibonacci sequence manifests itself in Tool's current album, Lateralus. In the first instance, Maynard James Keenan sings the title track by keeping each line's syllables in step with the sequence. For example:

black [1]
then [1]
white are [2]
all I see [3]
in my infancy [5]
red and yellow then came to be [8]
reaching out to me [5]
lets me see [3]
there is [2]
so [1]
much [1]
more and [2]
beckons me [3]
to look through to these [5]
infinite possibilities [8]
as below so above and beyond I imagine [13]
drawn outside the lines of reason [8]
push the envelope [5]
watch it bend [3]

Maynard's mathematical musings don't end there. Drummer Danny Carey also plays the sequence ranging from one to thirteen throughout the song.

The article also suggests that there is some alternate order to the songs on the album, based on clues offered by Fibonacci and Maynard's lyrics.

Battlestar Galactica Season Finale Tonight!

Season two of Battlestar Galactica comes to its conclusion tonight. Lay Down Your Burdens Part 2 airs March 10 at 10:00E/9:00C.

From SciFi.com:

When Baltar wins the vote, Roslin considers stealing the election because she believes that he is a cylon collaborator.

Patrolling Season 2 Episode 4 Released

After a long wait, Rantmedia has released season II episode IV of Patrolling with Sean Kennedy.

Patrolling with Sean Kennedy introduces the viewer to the function over form world of military gear. Kennedy offers advice and insight to the question of survival and preparedness in his unique and sometimes tongue-in-cheek way.

Those new to the series may wish to view the first episode via streaming video here.

Links:
Rantmedia
Patrolling with Sean Kennedy
Download Episode IV

Proof Techniques: Forward-Backward Method

We all encounter it some time or another. Show that if statement A is true, then statement D is true. Upon diagnosing the patient, one may choose to make a direct proof. That is, assume A and work towards B. In symbols:

A→D
Let A be true. (lots of work). D. Therefore, A→D.

Unfortunately, D can sometimes be very hard to see. It is for this reason that the forward backward method is useful. Before assuming the left hand side (A), one should analyze D. Is there some statement C such that C→D? Then is there some statement B such that B→C? Now we need only show A→B, then by default A→D since D→C→B.

One should first work backwords towards A, then assume A to be true and prove the last statement from the backwards method.

Consider the following example:
Show that If right triangle XYZ with sides of length x, y, and hypotenuse z has an area of z^2/4, then triangle XYZ is isosceles.

It is very tempting to just assume the left hand side, but lets try working backwards first. What does it mean for a triangle to be isosceles? Ah, thats right, it means that the triangle has two sides of equal length. So if we were to show that x=y, then triangle XYZ would be isosceles.
Now lets try to step backwards again. what does it mean for two numbers to be equal? Not much. As you can see, things start getting trivial. Lets stop here and start working forwards to show x=y.

Assume the left side. That is assume the triangle has area z^2/4. Well, we know that the area of a triangle is A=(1/2)base*height. This yields (1/2)x*y = z^2/4, since A=z^2/4 by the problem definition. Remember, we want to show that x=y, but we have a nasty z in the way. Since this is a right triangle, we can get rid of it with the Pythagorean Theorem. since x^2+y^2=z^2, we can replace z^2. Now, we have (1/2)xy=(x^2+y^2)/4. Algebra shows that this is really a simple quadratic: x^2 + 2xy + y^2 = 0. Factoring, (x-y)^2=0, and so x-y=0. Subtracting the y reveals that x=y, which is the last step in our backwards process, meaning that we are done.

Remember, try to work backwards a bit before you assume the left hand side and work forwards in a direct proof. Sometimes the backwards steps will act as a blueprint to the direct proof.