Concrete learning

[velvetpage]

Part of my job involves making sure kids can draw or stack blocks to represent the number 100 000. That's as high as grade fives in Ontario are expected to be able to understand.

Well, here's a much bigger number, represented in stacks of $100 bills.

Probably the most difficult thing about math, either scientific math or advanced economics, is the understanding of quantity that comes with it. One trillion dollars is really hard to visualize. Nobody will ever hold that much cash in their hands, or even in their warehouses. It's 10 to the power of 12, and it's just at the limit of how big something can be and still be relatively simple to represent visually (as this website did.) Once you get past that, it becomes a lot easier to compare it to something else: like, one in ten trillion is one drop of water in an Olympic-size swimming pool, for example. (Note: I don't know if that's true.) I think a big part of the reason most kids have trouble with math in the higher grades is quite simply that nobody takes the time or the energy to draw those comparisons for them in such a way that they understand the concept of "orders of magnitude."

Math gets hard at exactly the rate that it gets abstract. Make it less abstract, and most people can do a lot more of it. The problem with the traditional methods of teaching math has always been that it left math as an abstract concept long, long, long before most people were ready to think, "I can manipulate these numbers in certain ways without thinking about what the numbers actually mean at every point along the way."

Anyhow. An economic visualization and a little plug for concrete methods of teaching mathematics, at the same time.

Comments

[daednu.livejournal.com]

I don't know if it's just my brain, or the way I was taught math. All math is abstract to me. It's something I just cannot feel my way around. The fastest way to make me feel stupid is to throw math at me. Particularly surprise math. Like when I'm working a cash and someone gives me a twenty to pay for a $1.79 coffee (no problem!) but then changes his mind 4 different times about what money his paying with.

[velvetpage.livejournal.com]

It's probably the way you were taught - as though numbers had no meaning off the page. Your kids are probably going to grow up with a better understanding than that because their teachers are expected to teach using manipulatives, so they're likely to get to high school without ever being asked to learn a math concept without representing it visually in some way - with blocks or at least with drawings and most often with both.

When you're never asked to think about what the numbers mean but only about how to make them move around to become more abstract answers, you never learn the key thing about math: it's a language used to describe the world. Absolutely anything can be described with numbers, and every math question is describing something that exists in the real world.

[merlyn4401.livejournal.com]

That's scary.

[velvetpage.livejournal.com]

I'm not so sure. I think the piece that you and I are missing, now that we have a visual, is the piece to tell us where that money comes from. I know just enough about economics to know that money is created by other money. The government is shoring up businesses with money that exists because the government says it exists, or will exist by the time they pay it. It's all an exercise in fantasy-world generation, where the world they're creating is one that has this trillion bucks in it. The world didn't have that money until the American government decided to plop it into the financial industry.

You read in the paper that world markets have lost 20% of their value, or some such, since September. But what does that actually mean? Did 20% of the food that was growing suddenly stop growing? Did 20% of the machinery that was being used suddenly stop working? The fact is, that wealth was paper wealth to begin with (or computer wealth.) It had whatever value the bankers and money-market people said it had, because it didn't represent anything concrete.

Now of course, sooner or later it gets tied into something that feels concrete - like a mortgage on a brick-and-mortar house, or a new car coming off the line at GM. Still, those things are worth what the market says they're worth, no more and no less - so the ability to pay for them is determined by the same folks who are telling us that 20% of the world's wealth has disappeared in the last six months.

We're not being given the visuals to represent what's really going on, in part because there are no visuals. It's like the cheshire cat - "When I use a word, it means exactly what I mean it to mean - neither more nor less." Part of the way to solve the crisis may be to actually link money to concrete wealth again. I don't know. But I do know I'm not willing to let myself be drawn into how scary it is without a better understanding of exactly what it means.

[merlyn4401.livejournal.com]

I think whether it is tied to concrete wealth or paper wealth, it's still scary. That's an obscene amount of money.

[sassy-fae.livejournal.com]

0_0

*nodnod*

velvetpage, I think I'm going to repost that link on my LJ. (Though I don't think I'll use this icon, as you've ended up using it more than me, and everyone will think it's you posting! *giggles*)

[catarzyna.livejournal.com]

I've long felt I have a learning disability when it comes to math. I think I am right in my line of thinking but inevitably I am usually wrong. It could be how I was taught or it could be a learning disability. I am really good at problem solving and analysis when math is not attached to the issue. I've thought about taking a math class or two once I am finished with my degrees but I'd much rather take a pottery class. :-)

[stress-kitten.livejournal.com]

I think you'd be surprised at how much better you understand math with an adult brain.

I took Math for Elementary School Teachers and we went through the math curriculum from basic numbering systems (I can add, subtract and multiply in binary, yo!) to trigonometry and quadratic equations... and realised I was actually pretty good at it. And understood it a hell of a lot better using an adult brain to learn it.

[catarzyna.livejournal.com]

I'm not so sure because I was in a nursing program after I finished my BA degree. I struggled with the math; I was passing but I opted to not risk a med error. Thanks for the perspective though.

[zorinlynx.livejournal.com]

What's scary is just one of those double-stacks at the corner of the sea of pallets is enough for a person to live a decadent lifestyle for the REST OF THEIR LIFE, and leave a nice chunk of cash to their children too.

When you try to imagine the quantity of money being spent, it is absolutely mind blowing, especially when you think of it in individual terms, in your own finances.

I'm predicting inflation. It's like the US government just got the Midas touch; everything they touch turns to gold. But everyone knows that if that happened, gold would quickly become worthless.

[velvetpage.livejournal.com]

Coupled with the deflation that comes about because there are no assets backing up the value of the original losses, and I think you're right - the end result will be inflation even greater than what it looks like it's worth.

[amarafox.livejournal.com]

Holy CRAP that's a lot of money

[mar2nee.livejournal.com]

This post makes more clear to me what you meant when I was talking about learning math, and how it always frustrated me, trying to tell friends that they didn't NEED to understand it, they just needed to follow the steps.

Difficult combination of problems - because math is abstract, by it's nature, even the concept of numbers, using numbers to represent something, then using those representations to make manipulations...

Cool picture.

[velvetpage.livejournal.com]

Yeah, the kids who got enough of a glimmer of understanding that they were content to follow the steps after that were the ones who did well in math. But it can be argued (and has been!) that even the good math students mostly didn't understand the concepts when it was taught that way - they just understood the steps. Or sometimes, they understood the concepts but couldn't figure out how the steps were derived from the concepts, like me with division. I never understood how long division worked or why it worked, even though I could do it and I did understand the concept of division itself.

[mar2nee.livejournal.com]

If you understand the concept, and can do the procedure- does it matter if you understand why it works?

[velvetpage.livejournal.com]

It matters to motivation for a lot of people, and it matters when you take the step from that procedure to the next one up. If you don't understand why the procedure works, then when you go into an exam and can't remember the formula, you have no skills to fall back on to figure out how to get to the answer.

You can't build on knowledge you don't have, and memory is very closely tied to how well we understand the connections between concepts. If you don't understand how to connect math concepts to each other or to the real world, then it doesn't matter how well you can manipulate them - you're not going to be able to apply them outside of contrived situations and you're going to have trouble remembering them after the impetus of the exam is gone.