Saturday, June 16, 2012

An Open Letter to Parents that Think they are "Bad At Math"

Dear Parent,

I may or may not know you, but if you are reading this it probably means you and I have one thing in common: we love our children desperately and ache to see them happy and successful. So, without further delay, I will jump to the point. Loving mathematics is a critical component in your child's development.

Now, assuming you really are a parent to whom this article is addressed, you believe that you are "bad at math."

PLEASE STOP BELIEVING THAT.

You honestly do not know if your bad at math because this country struggles to teach people what mathematics is. I have written about this at length here, here, and here, and a real mathematician wrote about it here. But the shortened summary is that learning "steps" to compute an answer is NOT mathematics. Pretty much everything they taught you before college (and much of what they taught you in college) is not mathematics.

Did you struggle in grade school with getting the multiplication quizzes done on time? It doesn't matter. THAT IS NOT MATHEMATICS.

Did you struggle in middle school with remembering how to solve the "two trains leaving a station" problems? It doesn't matter. THAT IS NOT MATHEMATICS.

Did you struggle in high school to remember which formula and/or "trick" to use to solve some problem? It doesn't matter. THAT IS NOT MATHEMATICS.

Different people call these examples different things; in our house we call them computation. Computation is important especially to subjects like science and engineering (and it is great for balancing a checkbook, etc), but truthfully it has very little to do with mathematics.

So, please, take a big deep breath and let go all of your "I'm bad at math" tension. I want to start over with you on a new clean slate.

Yes, with you. While we started this conversation about your child's education, please believe me when I say that it starts with you. If you can't believe that mathematics (real mathematics) can be fun and interesting, your child will struggle to believe it unless they have a natural affinity for it. And even if they have an affinity for it, wouldn't you want to understand and appreciate your child's gifts?

Again, please forget everything you think you know about the math you think you're bad at.

Let's start with a game. This game has a game board with exactly three squares and exactly three pieces. For convenience, let's use game pieces from a popular game like Monopoly. We'll use the car, the hat, and the iron. Here's a visualization.



There is only one other additional piece. The "room" piece. The room piece fits over either square 1 and 2 or square 2 and 3. Let's start with it around square 1 and 2 (the car and the hat):


The objective of this game is simple. You want the board to look like this:


Here are the rules:


  1. Any two adjacent pieces can switch places.
    1. NOTE: For this rule, the room and its contents are treated as a single piece (so the room and its two pieces could switch places with a piece adjacent to the room).
    2. NOTE: Pieces may not change places across room boundaries (a piece within the room cannot switch with a piece outside the room).
  2. The room can switched at any time from its current two squares to the other two squares (e.g., it can move from being around 1,2 to being around 2,3 or back).
If it isn't immediately obvious how to reach objective, it is probably because I described the rules poorly. Nevertheless, try the different rules on the pieces of the board and see what you can do with it.

Did you try it? If you didn't, please go back and do it at least in your head. Did you reach the objective?

Here is how I did it.

START:


STEP 1 (using rule 2, moving the room):









STEP 2 (using rule 1, swapping adjacent pieces):









FINAL STEP (using rule 2, moving the room):


As you were doing this game, you were experimenting with math. Did you know that? Did you know that you were messing around with some of the core math ideas that make the steps we use in our multi-digit multiplication work?

In mathematics, multiplication is said to have certain properties. These properties say how you can change a certain multiplication problem and still get the same answer.

  1. The commutative property of multiplication says that 3 x 4 is the same as 4 x 3. In our game, it is like swapping the two adjacent pieces. Note that the "room" piece is just parentheses. So, you can also say that (3 x 4) x 3 is the same as 3 x (3 x 4). Did you see the whole room move like a single piece?
  2. The associative property of multiplication says that (3 x 4) x 3 is the same as 3 x (4 x 3). In our game, it is moving the room from squares 1 and 2 to squares 2 and 3.
We use these two properties extensively and all the time without thinking. Here is an example.

What is 2000 x 3? You probably think that is pretty easy if you remember your grade school computation. I was taught you multiply the 2 and the 3, then add as many zeros as there were in both numbers. The answer is just 6000.

But why? Why does that work?

Please note that asking why is not to be able to remember the steps better. Asking "why" is not to make us better computers. Asking "why" IS MATHEMATICS.

It turns out that we can show why the way my teacher told me to solve 2000 x 3 is correct using the game we played earlier. It can be done with just the commutative and associative properties. Watch.

  1. 2000 x 3 = (2 x 1000) x 3
  2. 2 x (1000 x 3)      associative rule
  3. 2 x (3 x 1000)      commutative rule
  4. (2 x 3) x 1000      associative rule
  5. 6 x 1000
  6. 6000
Did you have any fun with the numbers? Any fun whatsoever? Or did you have any fun with the puzzle game?

Alternatively, did you think it was the least bit interesting?

If so, you have just had a spark of enjoying real mathematics. For fun, why don't you see how you need to extend the game to solve a problem like 2000 x 300 = (2x1000) x (3x100). This introduces a small additional wrench into the game and rules, but you still solve it using just the associative and commutative properties. If you still don't like numbers, can you just extend the puzzle game above and play with it? Notice that you will have "two rooms" now instead of one. How do the rules need to be augmented? What makes the most sense?

I have yet to work with a child that didn't think this was fun. My second child really struggles with computation, but he absolutely loved this exercise the other day. I didn't even have him "solve" the multiplication problem; I just wanted him to get some experience to using mathematical properties (i.e., game rules) to be able to move around the numbers (i.e., the game pieces). He was so excited about it he didn't want to stop.

And please don't ask me how this is "practical." Of course we want education to be useful and practical to students in modern society. But we also want our children to be come educated. And by educated, I mean capable of meaningful, critical thinking. The convenient side effect of learning mathematics is that eventually it can be really helpful in computation.

But far more importantly, mathematics, real mathematics, stretches the mind and opens up new avenues of exploration and adventure. 

So, in closing, please stop thinking you were bad at math. Give it another try. Look for some fun and adventure in the mathematical world. If you don't know where to start, ask someone who does. Find that little spark of joy at solving a puzzle or crafting something new.

I promise that your little spark can turn into something wonderful for you, and for your child. It can become a treasure you will carry with you all the days of your life.

Sincerely,

-- A guy who failed all his math classes in college and had to re-take them but still loves math and thinks it's really, really interesting.

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In case you didn't catch it, "type h personality" means "type homeschooled". As I explain in the original post, this title was originally meant to be derisive, but I liked it and have converted it, in my own mind, at least, into all the amazing things I see home schoolers do every day! Go Type-H'ers!!!!

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