About Matthew

I'm a pre-service teacher based out of southwest Michigan. I am certified to teach both mathematics and physics at the secondary level.

Sweden, LittleBigPlanet, etc!

Wow, this blog is dead.  Like seriously dead.

Not that it matters all that much, given that I’m not sure anyone actually follows it!  I have been getting a lot of spam comments, though… so at least the bots are reading…

My life has been a bit crazy over the past 3 months.  I landed a job in the games industry, working for Tarsier Studios as they design LittleBigPlanet for the Vita.  For the job, I’ve  moved to Sweden.  Malmo, Sweden, to be precise.  It’s a nice little city; clean, dense, and surprisingly quiet.  The details of the work I’m doing have to remain a bit secretive at this time, but I can divulge that I’ve been doing level design and some basic art passing on story levels.  I’ve even worked on a minigame or two!

The job so far has been fantastic.  I really wasn’t sure what my role was going to be, what with being a community member and all.  I am allowed a surprising amount of freedom, which when coupled with the fact that I’m working with so many other passionate people, has led to some awesome creations.  I have two roommates who are in the exact same position as myself – community creators working as level designers, which has worked out really well.  For those of you familiar with LittleBigPlanet, you might recognize their PSN handles, Slaeden-Bob and Lockstitch.  Interestingly enough, we are all teachers (or on the track to become teachers).  Crazy world.

At any rate, I’m home for the Holidays.  I get two weeks to relax (and take a break from all things LittleBigPlanet), then I fly back to Sweden.  There’s lots to do yet, and it will be a mad dash to the finish!


Portal 2: Modern physics in popular culture? Nice.

So, I’m playing through the new game Portal 2, and about an hour in, I stumble across this:

Ah, the memories...

(Click to see large view)

Man, this picture is just chock full of physics goodness.

On the bottom right, we have the Maxwell Equations, with which I always struggled to remember when first learning about electricity and magnetism. Directly beneath “UNREASON” is the famous Heisenberg Uncertainty Principal which is a mind-bending property of quantum mechanics, wherein the accuracy of position and momentum are inversely related by the constant ħ (“h-bar”). In layman’s terms, the more we know about an object’s momentum, the less we know about its position (and visa-versa). The famous thought experiment that is associated with the uncertainty principal, Schrödinger’s cat, is the main focus of the scene. (Wikipedia: Schrödinger’s cat)

The idea is that due to the uncertainty principle, a cat in a closed box with enough poison to kill it is at some time both alive and dead. When the box is opened, however, the cat is either alive or dead. It is said that by observing an event, the outcome is inherently changed. Confusing stuff, right? It’s very interesting, but one has to play quite the game of mental gymnastics to understand the arguments at work.

The equations to the left all appear to be related to special relativity. Relativity applies when objects are moving very fast with respect to one another (think fractions of the speed of the light) and thus time is dilated and length is contracted, according to observers. The Greek symbol γ (“gamma”) represents the Lorentz Factor which is used in Lorentz Transformations (the transformations necessary to explain the measurements of things that move really fast).

It’s been a while since I’ve studied these things, but it’s awesome to see them popping up in popular culture.  Most people will never study the realm of quantum mechanics or relativity, which is too bad.  For those who do, we are privy to some excellent jokes that reference these ideas.

To quote the animated show, Futurama, when the cast was watching a close finish at a horse race:

Professor Farnsworth: “No fair! You changed the outcome by measuring it!”

Arguing Semantics: the obelus, or division symbol: ÷

We’ve all seen it, and we all know what it means… right?  Commonly referred to as the division symbol, I’m talking about the obelus, ÷.

Now we’re used to seeing the obelus in elementary mathematics problems of the form:

No surprises thus far, I hope.  We use this symbol almost interchangeably with the idea of division and it even exists as the symbol on our calculators for the division operation.  When we punch the button on the calculator, though, it will generally display as a solidus (or slash, “/”).  The interesting thing to note here is that we treat these as interchangeable symbols, and there should be no reason to think otherwise.

Recently, however, a discussion came on a forum that I frequent focusing on the potential ambiguity of the obelus.  More specifically, a problem was posed.  To what does the following expression reduce?

Too simple, right?  We just use our order of operations, PEMDAS (or BEMDAS), so that it reduces as follows:

Well, here’s the interesting part.  We are making the assumption that the obelus is used exactly like the solidus and that it is representing the basic division operation.  To look at why this assumption might not be founded in truth, let’s consider the origins of the obelus.

A friend of mine named Adrian, after doing some reaching, found an old German book called Teutsche Algebra by Johann H. Rahn, dated 1659.  It’s entirely in German, but the beauty of mathematics is that it’s a universal language.  Without being able to read any of the words, I was able to decipher many of the symbols.  It starts with an elementary introduction to the notation to be used in the book, which includes addition, subtraction, multiplication, exponentiation, and of course division.  Interestingly the symbol for exponentiation was a swirl:
At any rate, the obelus is introduced to represent division on page 8.  Previously, the obelus had been used to mark ancient manuscripts that were believed to be corrupt.  As it is used in Teutsche Algebra, however, was the first time it is used in a more modern sense.  There are pages of sample uses for division and all of them are analogous to how it used today.  This is the first known use of the obelus to represent division.
Things get interesting on page 76, however, when the following assertion is made:
For clarity, and since this is provided without context, this can be likened to the following, which is more relate-able:
Now, this clearly doesn’t obey the aforementioned order of operation rules, which suggests something about the nature of the obelus as a symbol for division.  As presented here, it is a line operation that divides the expression entirely (without need for parenthesis or bracketing), pun intended.
When presented with  this information, my first instinct was to defend the original answer to the problem above as a more modern interpretation of the obelus symbol.  My thought was that the use of symbols and their inherent interpretations evolve over time.  I considered that parenthesis were not used as a form of grouping, so the inherent system is slightly devolved from what it is today.  Hell, as I mentioned, this text even uses a swirl to indicate exponentiation, and we clearly don’t use a swirl anymore.
I was content with this way of looking at the situation until that same friend located a more recent book called A First Book in Algebra, by Wallace C. Boyden, dated 1895.  Adrian’s Google-fu is very strong, apparently.  In this text, the following is presented:
Which is clearly aligned with the usage in Teutsche Algebra.  Now, I don’t know where this wild-goose-chase will take me next, but it seems that when the obelus is used as a line operation, it splits the line evenly into numerator and denominator, which is removed from my previous assumptions.  Returning to the original problem, then, yields:
So, which is it?  288 or 2?
My TI-84 thinks it’s 288, as does Wolfram Alpha.  Now, before you start arguing that Wolfram is the infallible machine that we often assume, check out this result.  My intuition even says that it has to be 288, but my (and mostly Adrian’s) research leaves me wondering.  What do you guys think?
I know one thing is for certain… for my own sanity, I will not be using the obelus in my future classrooms.  There’s just one thing that bothers me, though:
Um…what does that button do?
Bonus Points: There is a calculation error on page 15 of Teutsche Algebra.  Can you find it?

New home!

My blog has officially been relocated to a more appropriate and official address.  If you’re reading this, you’ve found it.  This blog will also serve as my professional website and electronic portfolio.

Poke around as you’d like.  I’m just trying to get my footing with all the tools and options.