Learning Math

Math did not come easy in school. I remember being bored with long division. I found it difficult to concentrate on the processes taught to me to solve the problem. So many steps and so much room for error. I found myself getting frustrated. I could barely engage long enough to finish my worksheets. It did not take long for my 9 year old brain to check out of math altogether. My report card in 4th grade almost reflected an ‘F’ in math. I say almost because I performed many extra-credit activities. I boosted my ‘F’ to a tolerable ‘B-.’ My standardized test scores in math reflected the same; a student who could not grasp math. In fact my scores were low enough to afford me the pleasure of being in “remedial” math for all of middle school. Then something strange happened to me in Geometry. I set the highest score on each exam. I went on to do the same in Algebra II and Pre-Calculus. When I went to college, the university hired me to tutor Calculus and Physics. This was quite a switch and not the trajectory that my 4th grade teacher predicted for me.

So what changed? Why all of a sudden is a switch thrown in the brain that allows a student to go from almost no comprehension to mastery in mathematics. The answer I think is simple. It boils down to the difference between the abstract and concrete sides of math. We learn everything we know first from our own concrete experience before we can abstract. We learn to count, not by memorizing the numbers in order but by counting objects. We soon learn that, no matter what objects we are counting, the order of the numbers never change. This simple fact is our first experience in abstraction. We abstract the idea of the number 2 that exists independently of the things we are counting. For me geometry class made sense. There were shapes. The properties of shapes such as triangles and squares were very intuitive. However, In my 4th grade long division experience, all I knew was a process. The goal: master the processes. I didn’t really understand what it was I doing. The process of long division is uninteresting to me. Looking at shapes and understanding how these objects are related ignited my curiosity. I could now work hard in my math because I am solving problems.

Math is a series of abstractions. Each level of abstraction is simplifying and generalizing the concrete world. It is easy to lose sight of the concrete world in math and rely on memorization, math tricks, processes and shortcuts. Math can become an exercise in mastering processes and not understanding. Before the mind is ready to abstract effectively a concrete intuition must be built. Take division for instance. When I ask my 5 year old what 12 divided by 2 is, he has no idea. However, if I give him 12 jellybeans and ask him to share equally with his brother so that they each get the same number, he divides by 2 with no problem. He understood one question but not the other. One question is abstract, one question is concrete. He can consistently do this with any even number. He also realized that when he has to divide an odd number by 2 there is always one left over. (Dividing and having a remainder is pretty advanced for a 5 year old). With my 7 year old, I play with the number 12 a little bit more. I ask him how many jellybeans do 3 friends get if you have 12 jellybeans to share. I ask about 4 friends, then 6 friends. He learns that three groups of 4 make 12. He learns 2 groups of 6 make 12. In this game he is touching on several mathematical concepts. Division, multiplication, factoring and expression. This is now a game where there is a puzzle to figure out. The more a student interacts with the concrete world in this way, the stronger the concrete intuition is built.

Fireside Chat: Mathematics

Sarah Deconink, our head of the math department, laid out some pretty brilliant insight on how we try and teach math classically. I was amazed and excited about what she shared. We really only know what we can build from the concrete. From the concrete we then begin to abstract as we developmentally reach that age where we can do that. By continually going back to concrete things we know we never lose math to the very tempting abstract processes they can become. When math is simply a set of steps to follow as opposed to problems to solve we can lose the whole purpose to math: to understand creation and thereby our creator.

Do I use math in my everyday life?

Any math teacher will hear: "I will never use this in life!" The emphatic statement that really reveals our common understanding of math. Since the enlightenment, it has seemed that mathematics, in the way it is taught, has but one use: utility.  If it has no immediate use to my everyday life, what good or use is it really? My algebra students moan when we are mastering factoring. In their minds they can see no correlation to manipulating expressions and equations to careers and rigors of everyday life. Outside of balancing a checkbook and investing money, of what use is math? If I don't go into a field like Engineering or business finance, why do I need subjects like Algebra, Geometry and Calculus? These classes seem like really hard work for little to no return. This common misconception, I believe, comes from a long line of false understanding about what subjects like Algebra are and what they were historically used to teach. Algebra to most people is simply a set of tools used to solve interesting problems in life. Here is an example: If there are two tax rates, both state and federal that are interdependent of each other a straightforward calculation is impossible. However, a system of two equations and two unknowns can be expressed and therefore solved. This is useful and the idea is this is for what Algebra is. But, that is actually only a consequence of Algebra. Algebra was originally taught to teach people how to think, not just solve problems. Lets go back to balancing the check book. Theoretically it is easy to balance a checkbook. Most people know what they should do. They should save money and live below their means. Do people normally do that? No, most American's live on 110% of their income. This shows that even though we know what to do, that doesn't mean we do it. Why? Because we make emotional decisions. This is a problem. Algebra engages the mind on a level that forces it to think rationally. It forces one to think in terms of logic and does not allow answers that feel right, only ones that are logically provable. The field is based on axioms not statistical analysis. Algebra helps the mind make logical decisions as opposed to emotional ones. Algebra teaches someone how to think and that is something we use everyday of our life!

What do numbers tell us about God?

Does a number exist outside of 'stuff.' In other words, the number three represents the quantity of say 3 apples. However, if I had no apples, does the number 3 still exist? This may seem odd but the question is valid. We rely on numbers but what exactly are they? Are they merely constructs that we have come up with that help us understand the world around us or are they something more? Do they help see things about God? I think they do. As a math teacher I work with numbers and variables everyday. The numbers and variables help us make sense of the world around us. They help us solve problems. But it is interesting that when we do math it seems to always correspond to real world situations. For example, if I know a rate of speed and an amount of time, I can easily calculate the distance traversed = d = r*t. But why? Why does a math equation work? Could it be that numbers are exactly HOW God created the world. In John 1 we see Jesus equated with the word, 'Word.' The word 'Word' comes from the greek word 'Logos.' Logos means divine order. God uses words to create in Genesis 1. I think numbers are skeleton of the cosmos. I think pure math is the transcendent Word of God that holds everything together.

What exactly are you?

What are you of? What makes you, you. Weird question huh? But think about it. What "stuff" makes up your human body. Well, lets see there is the skeletal structure, muscles, nervous system, and somewhere in science I heard we were 70% water.

What's your point?

My Point comes from reading the The God Delusion. Dawkins quotes Steve Grand (pg. 416) who points out the paradox of memory. Think of something that happened in your childhood. How do you remember that, if you were never there?

Wait, what!? What do you mean I was never there? I mean, there is not a single atom in your being aside from bone molecules that were in you when you had that experience. The atoms making up who you are are completely different. The body continues to regenerate itself and dispose of dead cells.

This paradox exists because we tend to only view 'real' things in terms of physical things. Could non-physical things be just as real? Yes, the electromagnetic waves that we cannot see, taste, hear, touch or smell make our phones download data and carry data out. That is just as real even though we may not think of it as "stuff."  You could be said to be the same. The DNA in you replicates cells and your brain somehow stores memory and gives you personality and abilities.

So what makes you, you and what is consciousnesses?

What if I told you: The floor you're standing on is empty space.

What if I told you there was no such thing as a solid surface?  And no, I am not smoking pot, I am reading a book by a prominent evolutionist, Richard Dawkins.

Consider page 412 of The God Delusion:

Science has taught us, against all evolved intuition, that apparently sold things like crystals and rocks are really composed almost entirely of empty space.

Dawkins goes on to give this illustration. If you took the nucleus of an atom as a fly and put it in the center of the Chief's stadium in KC, the next atom next to it would have to be placed outside the stadium. That is a lot of empty space between the two. So how on earth does a rock feel solid it it is almost all empty space?

This is how it works, It is not solid surface that keeps you from going through walls or falling through the floor, its a force field that exists between the atoms that prevents us from going through it. Those force fields can be broken with enough force, thus why a sledge hammer is a good way of taking out dry wall.

Now your eye balls receive light within what is called the visible light spectrum. That spectrum of light is large enough to bounce off of the atoms and not penetrate through them. Other bands of light like X-rays can penetrate but what our eyes can see cannot. Therefore, the light that bounces off and comes into our eye balls our brain interprets as "Solid."

Kinda makes you wonder what is really "REAL".

Time for some Relativity

All right kids, ever wonder why people consider Einstein a genius? What were his contributions?What is this whole thing about 'Relativity' and that famous equation E=mc^2?

Allow me to brake it down...One Step at a time. First, what Einstein predicted and was later proved is that time does NOT tick the same for everyone. You may be sitting there thinking...but wait, I thought time does tick the same for everyone that is why we have time zones? I thought the same thing. I thought it was a bunch of psycho babel smart people used to confuse the rest of us so we keep giving them grant money because we are to self-conscious to admit we don't know what the heck they are talking about.

Actually, time ticks slower the faster you move. So, if you are standing still on the earth time ticks by you at a certain rate...but if you are on a plane traveling at 600 mph for example time ticks slower than the guy standing on the ground.

The idea is known as "Time Dilation" and here is how it works.

v = velocity of guy on plane

c = speed of light

t = how many seconds ticked by for the guy on the ground

t' = how many seconds would have ticked by for the guy in the plane

Any questions?