“Instead of thinking about the monkeys I think about all the kids in the world, how they’re not TV they’re real, they eat and sleep and pee and poo like me. If I had something sharp and pricked them they’d bleed, if I tickled them they’d laugh. I’d like to see them but it makes me dizzy that there’s so many and I’m only one.”
--from Room by Emma Donoghue
In my five years of working in the education field, I have learned that most mathematical concepts can be placed on a continuum from easy to teach and learn, to tremendously difficult to teach and learn. The basics of three-dimensional objects, for example, are pretty easy since there are so many good models of three-dimensional solids in the real world. A shoebox is a model of a rectangular prism, a soup can is a model of a cylinder, and so on.
Two-dimensional geometry moves a little way down the continuum toward hard. There are many real-world models of two-dimensional figures, but they all have small problems with them. A piece of notebook paper is a pretty good model of rectangular region, but the paper does have some thickness, whereas a true rectangular region does not. The rails of train tracks are a decent model of parallel lines, but they are never perfectly straight and don’t continue forever. Plus, train tracks appear to get closer together as they move off into the distance, and explaining perspective to a child is not an easy thing to do.
There are ways to overcome these minor difficulties, though. For one thing, children don’t necessarily need to understand all these fine distinctions right away. The real thorns in my side are the concepts that are all the way at the hard end of continuum. One of these concepts is large numbers.
Learning to read and write large numbers is one thing. Once you get past the teens, our numeration system has a predictable pattern. But really understanding what the large numbers mean is much harder.
Instincts about large numbers are not something we are born with. When I was six or seven, I had a Dr. Seuss book called My Book About Me. Each page had blanks to fill in. My name is _______. I am _____ years old. I am _____ feet _____ inches tall. I happily filled in all these blanks, and eventually came to a page that asked me to count the number of steps from my room to the kitchen, from the kitchen to the door, from the door to the mailbox, and from the mailbox to the store. I counted the steps in the first three trips, then stopped. I'm not sure if I just didn't want to walk all the way to the store (which was about a mile down the road) or I thought my mother wouldn't let me, but I didn’t go. Instead, I simply wrote 100 in the blank, assuming that it couldn’t possibly be more than 100 steps. After all, 100 was a huge number.
I eventually learned a better sense of 100. There are many ways to visually compare 1 with 100. There are 100 pennies in two rolls, 100 squares in a 10-by-10 grid, and 100 paper clips in a box. If kids are exposed to comparisons like this (one penny versus two rolls of pennies), they will eventually gain a sense of how much 100 is, and come to understand that maybe 100 is not such a huge number after all. Lord knows that, as an adult, I have learned that $100 is not the fortune it seemed like when I was 7.
There are similar ways to help children gain a sense of 1,000. But what about 10,000? What about 100,000? What about a million? When is the last time you’ve seen that many of anything? I think part of the problem is that even when you do see collections that big, our brains have a hard time processing the individual pieces. There are photographs of crowds of 100,000 people, but when I look at such photos, I don’t see individual people. I see one mass of people. I don’t know if that is true of everyone, but I imagine that it is a common difficulty.
I openly admit that despite the fact that I am 28 years old and I have degrees in mathematics and learning sciences, I still struggle to really understand large numbers. My undergraduate alma mater recently received a $100 million gift. Many alumni were very excited about this, but I have to admit that my response was a blank stare. I don’t really understand what this means for the university. I know that they are going to use the money to open a medical school, but will the $100 cover the whole start-up cost? Will there have to me more fundraisers just to erect the buildings? Will it only take $50 million to first open it, so the rest can be used to sustain the school until it becomes profitable? I haven’t the faintest idea what $100 million will buy.
I also don’t have a clue about the population of the United States or the planet. I have heard the numbers before, of course, but they never stick because I really have no idea what they mean. Before I google it, here are my wild stabs at these populations: There are 3 million people in the United States and 2 billion in the world.
Here are the true populations, according to the census bureau: There are 311 million people in the United States and almost 7 billion on the planet. I guarantee that I will forget these numbers tomorrow, because at the moment my brain is struggling to really understand the difference between my guesses and the truth.
While I understand that my own personal anecdotes are not proof of anything, I relate them to illustrate the kinds of struggles that kids (and the adults they become) will face if they don’t develop a good sense of large numbers. This is, in fact, the kind of math that is used in everyday life, and I wish I had learned it better. And that returns me to my original point: This is a hard thing to teach.
There are some curricula and books out there that make a very good start, but I’m always on the lookout for inspiration for ways we could teach a better sense of large numbers. I often count my steps while I’m running, so maybe I will see how long it takes me to run a million steps (not all in one day, of course). I’m not particularly interested in finding out how long it takes me, but I am interested in finding out if it helps me understand just how much 1,000,000 is.
And if any of you would like to donate $100 million to the cause so that I can see what it takes to spend that kind of money, that would be great too. Thanks.
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