Simple Solutions to Complex Problems

Felt it was time for another update. I've been busy with my own studies and a side project on Tumblr these past few weeks, so forgive me for going silent. Anyway, I've been cruising through the vast collections of educational resources on the Internet looking for something that I could use to further my own collective knowledge in the world of science and technology, and I came across something very interesting on the Google Educators website  that I think could help the masses better understand that black art we call mathematics. There is a section called Computational Thinking, which is something that we computer scientists and software engineers use to write programs (more on this later).

Now, to most of us, MATH is the most feared, and most frowned upon four letter word. I personally respect math, but many do not. We are taught from a very young age that math is something that we need to get through life; and this is true. But then why is it that the way by which we learn math relies on an archaic learning style where we see "ideal" examples that we can easily solve by hand if we know the trick of that one type of problem? This is an idea that has confounded me since I began my struggle with mastering math. And then it dawned upon me in my first year of college that the reason I struggled so much in the past, is because I realized that ideal examples completely break down when they are exposed to complexity of the real world. It was on that day that  I realized that I spent eighteen years of my life learning how to do math completely backwards.

What do I mean by this? Let's consider the pattern by which we learn how to do computational mathematics. We start our mathematical journey by learning to count things. And this is how we should begin. Counting works in the real world. And our brain can grasp this and acknowledge this implicitly. We start out by using counting agents like fingers and toes to keep count. Eventually we move to computing agents, but before we do that, we move to paper calculations. I think that this is the start of the problem.

Many math teachers will probably crucify me when I say this, but we don't need to learn how to do paper calculations anymore. We have computing agents to do these calculations for us. Now before you ready your pitchforks, hear me out for a second. If we already have the tools in place (ie counting agents, and computing agents), then why are we wasting so much time on fixed interval calculations, and not immediately graduating the learner into real world calculations? I am of course talking about Calculus, that dreaded math that the layman can never hope to understand. Or can he? This brings me back to concept of Computational Thinking.



There are many techniques our brain uses to process CT, but the ones that are outlined by Google's section are as follows:

1. Decomposition: This is probably the most fundamental form of CT. We use it all the time without even realizing it (which is key). Our brain is designed to output the information it takes in as complete objects, but it can also break those objects apart into their individual pieces. For example, you sit down to dinner, and you take that first bite of a pizza or what have you, and the first thing you taste is something that we associate with how pizza should taste. But what about after that? Suddenly, hints of basil and tomato begin to permeate your pallet, the spice of the pepperoni, the sweetness of the cheese, the earthiness of the yeast risen dough. It all breaks down into something that makes you eager to take another bite, and another. We can do the same thing in math. In other words, we can break a number into its individual pieces. Take for example, the number 20. There's several approaches to this, but let's say we want to just take little bites of the number 20 in breaking it apart. We know that 20 can be expressed as 5*4, or 10*2. So if we take just one more bite, we see that 20 can be expressed as 2*2*5. In algebra, we'd call these the "prime factors" of 20, but how much algebra did I actually use? None. My brain implicitly decomposed 20 into its prime factors.
2. Pattern Recognition: The brain loves to find trends in natures. Pattern recognition is simply the ability to note common similarities and differences. Once again, the human brain can do this implicitly. For example, when you were a child, you might've had the unfortunate experience of touching the stove when it was hot. And what was the result? Other than the searing pain, your brain quickly made the connection that fire is hot, and we don't touch fire because it will burn us. Continuing with the example, we see that you likely shrieked in pain, and your parents came running. What's the pattern here? If I hurt myself, and I cry, then mommy and daddy will come running to my aid. Now let's apply this to math. Why is the product of two negative numbers a positive number? This is something that I had the hardest time with in school. I eventually accepted it as law without fully understanding it. But let's examine this phenomenon. We know that:

• (-4)(4)=-16
• (-4)(2)=-8
• (-4)(1)=-4
• (-4)(0)=0
• (-4)(-1)=4

          Do you see the pattern? Of course you do. And that is because your brain is designed to see patterns. It isn't designed to necessarily understand the principle that two negative numbers when multiplied together is a positive number, but it will immediately grasp the fact that a pattern emerges when you keep a constant, and decrement be until b is less than 0.

There are two more principles of CT that Google goes over, but for the sake of the length of this post, I'll let you research those for yourself. My point is that our brain is a marvelous computing agent that is designed in such a way that is counter to the current way mathematics is taught in the public sector. I believe that if we instead teach students to decompose complex, real world mathematical problems, to identify patterns, abstract the critical steps, and then write the algorithm to solve the problem, the student will then have a full and comprehensive understanding of math, and maybe even come to love its subtle intricacies, and infinite puzzles. And how do we do this? We teach them how to write programs. Instead of teaching the student how to use a calculator, we teach them how to BUILD the calculator. We live in a modern world, and there is no place for archaic methodologies in a modern world. We sit on the forefront, the very pinnacle of a movement that could potentially change this world into a place worth living in. If we continue to use archaic methods of instruction, and (worse yet), continue to dumb down the material that students NEED to understand to inherit this world. We'll miss the boat on this movement, and the consequences will be dire.

I'll close with an amazing presentation by Conrad Wolfram who addresses this very problem, as well as possible solutions in great detail.

Enjoy!