Monday, August 09, 2010

Our Plan for the Year: A Mathematical Rabbit Trail Part III

Even though math can help inspire works of beauty as described in my last post, some readers may still be math atheists or agnostics, especially about high school math. Somehow we have lost touch with the great discoveries and how they were made in an effort to cram mechanics into young skulls. What gets transmitted as dull, dry mechanics in high school came out of the understanding of patterns.

Try a coin tossing problem. Take a fair coin and toss it. How many ways can you end up with tails? You have two possible outcomes, tails or heads. Here is a summary:


Toss two fair coins. How many ways can you end up with a head and a tail? You have four possible outcomes: both tails (TT), both heads (HH), and a head and a tail (TH or HT). Here is a summary:


Toss three fair coins. How many ways can you end up with two heads and a tail? Now, it is getting complicated! Ug! You have eight possible outcomes: both tails (TT), both heads (HH), two heads and a tail (HHT, HTH, or THH), or two tails and a head (TTH, THT, or HTT). Here is a summary:


Try four fair coins. What are the odds of getting two tails and two heads? What? Come on!!! That is too much work!! A mathematician would look for a pattern to make the calculations more convenient. What you end up building here is Pascal's triangle:



Would you rather mechanically crank out the different outcomes or figure out an elegant pattern that would help you answer the question with ease? Spend some time puzzling out the next row and check here to see if you got it right. To prove to you that Pascal's triangle yields the correct answer, I worked out answers to two more problems:


Here is another neat thing about this triangle. To get the total number of outcomes, you have to add up all the numbers in a row. Being able to see the pattern in the answers is another time saver. Study the answer for a moment and think about the relationship between the number of coins and the total number of outcomes.

Instead of adding up a whole bunch of numbers, possibly making errors, take 2 to the power of the number of coins. Seeing the pattern enables you to reduce the number of keystrokes on the calculator to a handful, regardless of the number of coins.

I have not forgotten our friend Fibonacci. The Fibonacci numbers are also in Pascal's triangle: 1, 1, 2, 3, 5, 8, 13, 21, . . . By adding the numbers in diagonals, you get Fibonacci numbers! SWEET!


Let me take you on a trip back to high school math. I don't expect you to remember this, but, if you took Algebra II, you probably had problems like this:



Click the pictures and look at the coefficents (the numbers in red). Do you see Pascal's triangle?

In this series of posts, I am trying to make a point. How differently would people view math if it had been taught as a way to understand a complex world and as a search for meaning, pattern, and beauty? Charlotte Mason wrote, "Nothing can be more delightful than the careful analysis of numbers and the beautiful graduation of the work, 'only one difficulty at a time being presented to the mind'" (page 263). This little exercise shows how careful analysis and slow unfolding of new ideas heightens interest and creates joy at a new discovery.

While Blaise Pascal was not the first human to discover this triangle, he was the first to make it known to the Western world. A Christian philosopher, he wrote, "We arrive at truth, not by reason only, but also by the heart." I see mathematics as a way to baptize the imagination for absolute truth. We live in a world where values are relative (what is true for me may not be true for you). In public schools, teachers are not able to share their faith in the classroom. However, math teachers can share the concept of absolute truth in a way that no other teacher can. When the time is right, God will take what the mind knows about truth from a mathematical point of view and start melting the heart toward truth from a spiritual point of view.

Tomorrow, I will share our plan for math!

4 comments:

Stephanie said...

I want you to be my math tutor! ;-) Sign me up for Glasers RDI for Math Atheists and Agnostics program. *LOL*
In just a few posts, you've made a lot of things come together for me.

poohder said...

I can clearly see God's hand at work here and of course it's amazing. But I've gotta be honest. I don't FEEL the beauty you do. As you have succinctly said in your blog, math was ruined for me a long time ago by poor teachers and by tackling parts of math when I didn't even have the earlier stuff. The curriculum moved TOO fast for me. I have tried not to give my children the "fear" of math, but I'm afraid my "brick and mortar school" already has it.

The Glasers said...

Yeah!!!!! I guess that is why I never seem to be happy with math curricula. My next post will explain why.

The Glasers said...

Poohder, it just goes to show the power of nothing: going slowly at a person's pace, focusing on a shared undertsanding, etc. It also shows how long negative episodic memories last!