I’m reading a great book right now called “Is God A Mathematician?” by Mario Livio. It’s a fascinating book about whether mathematics is discovered or invented. It’s a really interesting book (so far) and has really gotten me excited about math, again.
It got me thinking about how I was taught math and why I disliked math so much. I remembered some of the things I learned and, for whatever reason, focused on the formula for the area of a circle: (A = ?r2).
What I asked myself was, “why is that the formula of a circle? Why, if you multiply the radius times itself and then times pi, some constant, do you get the area of the circle?” After digging into this, I realized why I didn’t like math in school.
Pi is a constant, meaning the number never changes. But, as I learned reading this book, it represents something. Pi is equal to the ratio of the circumference (C) of a circle to its diameter (D). No matter what the size of the circle, this ratio is always the same, 3.14159…it goes on, literally, forever.
I never knew that! Well, to be fair, maybe I did and just forgot. But I certainly didn’t remember it. That’s really interesting. But what does that do for our formula?
Maybe we can substitute (C/D) for pi into our formula to help it make more sense?
Area of a Circle
Now the formula for the area of a circle is A = (C/D) r2. Well, the diameter of a circle is twice the radius, right? If we substitute 2r for D, we get A = (C/2r) r2. We can cancel out the r in 2r because of r2 on top, and we’re left with A = (C/2)r.
The area of a circle is equal to its circumference times its radius, divided by 2.
But, does this make sense? Kind of. Here’s how I understand it. Take a circle, like this one:
Now, let’s show the radius:
The radius has a length of r (of course) and a width of 1 unit. If we show the radius x times, it might look like this:
The formula, again, says we should show the radius C/2 times, though, right? (C/2) * r means show r (C/2) times:
Now, here’s where it breaks down for me. Doesn’t that just give you half the area? Why divide by 2? Isn’t this what we’re looking for?
Anyway, the point is this: this formula (C/2)r makes so much more intuitive sense to me. To understand the area, just draw the radius as many times as the circumference is long. Pretty soon, you’ll fill up the circle. Count the number of radii you had to draw to get there, multiply by the length of the radius and you’ve got the area. In that sense, it’s no different from the area of a square.
Back to Teaching
Which brings me to my point. When we teach math, why aren’t we teaching this way? The way I learned math (rote memorization and application) only taught me to memorize. When it came to word problems, I was abysmal because I didn’t know why I should be using which formula.
I suspect that same problem is found in every subject. My goal is to help my kids learn to find out why what they’re doing works. I want them to understand the mechanics and inner workings of everything they do so they can manipulate it, change it and use it anyway they need to, in order to solve whatever problem they’re having.
What do you think? Is this a worthwhile endeavor or just unnecessarily confusing?