Reflection, the ruminating power which is so strongly developed in children and is somehow lost with much besides of the precious cargo they bring with them into the world. There is nothing sadder than the way we allow intellectual impressions to pass over the surface of our minds, without any effort to retain or assimilate. ~ Charlotte Mason
Mathematics requires a wide variety of intellectual habits. Attention, “the power of turning the whole force of the mind upon the subject brought before it,” is the most important. Math will never make sense if one doesn't pay attention to the big picture and the details. To illustrate this process, I'll guide you through another interesting aspect of Pascal's triangle.
How does one place full attention onto a topic? The same things you do to help you focus while reading with an eye toward narrating also apply in learning math. Eliminate distractions. Read when the mind is most alert. Stop after each main point to make sure you understand before moving on. Do something else when the brain gets fuzzy. Any break in a chain of mathematical logic may lead to confusion.
If you added each row in Pascal's triangle, you might have noticed a pattern — if you were paying attention. Here is the number sequence produced: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1,024, 2,048, 4,096, 8,192, 16,384, 32,768, 65,536, and 131,072.
You can apply the same operation to one number in the sequence to produce the next. Do you see what it is?
Don't spoil the fun by reading further until you've thought about this for awhile!
Does the relationship between a sum and the sum of the next row make sense? Because math is often presented as a series of tricks and cute mnemonics, we're too satisfied with incomplete understanding. Suppose we're told that any number raised to the zeroeth power is one. Oh, wait! I know, “Zero is my hero with a superpower of one.” Things like that make me uneasy because simply applying reason reveals the sense of it.
Math should make sense if one starts with the most basic understandable thought and then adds to it in a beautiful chain of reason. What do I do when something doesn't come easy? I hit the search engine until I find an explanation that makes sense. I leave a comment on someone's blog. I ask my math savvy friends. Cultivating thoroughness, “the habit of dissatisfaction with a slipshod, imperfect grasp of a subject,” leads to greater understanding.
Take that a step further. If repeated addition is multiplication, what is repeated multiplication?
Imagine writing “2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2.” Saying, “Two times two times two times two times two times two times two times two times two times two times two times two times two times two times two times two times two times two.”
How tedious is that?
The shorthand way is, “Two raised to the eighteenth power,” written in the format below.
So, the tenth row in which you add numbers yields 1,024, which is 2 to the tenth power.
And, the seventeenth row in which you add numbers yields 131,072 is 2 to the seventeenth power.
You may not have noticed but we are missing the very top number. Why? The row at the very top has only one number, which means you don't add anything.
If the first row of adding yields 2, then the row before that is row zero. So, applying what we have said before, row zero (the row before the first row in which you add numbers) yields 1, which is 2 to the zeroeth power. (Yes, zeroeth is a word - I was thorough and looked it up.)