Wednesday, October 25, 2006

How much is a million?

It's been a long time since I posted about the Numeracy class, mainly because I am not teaching it this year, so it's no longer at the front of my mind. But, this year's Numeracy teachers are struggling with the same concepts that I did. One of these killer concepts is rounding. This does not seem like a very difficult idea, but for our students, learning how to round a number is a huge challenge. We've tried all sorts of scaffolding, conceptual development, practice with the algorithm, and some kids just can't get it.

Ultimately, we think it comes down to a continued lack of understanding of the base-10 system. These students missed out on some very important mathematics in their first few years of school, and this is making everything else inordinately difficult for them. This week, one of the classes is piloting an activity where the students have the goal of collecting one million pennies. Each day, students will have to count how many pennies there are so far (pennies will initially be collected in a big jar). The idea is that students will eventually lose patience with this, and propose the idea of some sort of stacking or grouping. The teacher will then magically produce a container that has slots to divide the pennies into groups! When groups of 10 are no longer enough, then bigger groups of 100 will be used, and so forth.

Aside from learning more about the base-10 system, and the relative sizes of the different places, we hope that this activity will help students understand more about large numbers. We do an activity in the class where the whiteboard is divided into categories (thousands, millions, billions, trillions), and teams are given strips of paper with a quantity and a number, anad they must stick it to the board in the right column. For example, a strip might read "Number of people in San Jose (1)", or "The average income of a family of four (50)". It should come as no surprise that students usually have no idea where to put the strips of paper. In any case, this year's students seem to really doubt that there are about a million people in San Jose - they think the number is bigger by orders of magnitude. With the pennies activity, students will hopefully see that reaching a million is a bit harder than they think. But if they do reach a million, they will definitely earn their pizza party!


Anonymous said...

I'm curious to learn how you figured out the connection between rounding and base-10. Do you think they need to understand the concept of grouping in that specific (and most common) use first, in order to go beyond grouping by 10s and, say, round to the nearest 2? Or were you thinking along the lines that rounding to the nearest whole number involves considering the tenths in order to decide if we should round up or down? Can the students add and subtract reliably?

As far as the large numbers, my guess is that most adults have not much sense about how much a million, billion, or trillion of anything really is. (Myself included.) But we can definitely tell you that they are an order of magnitude apart, and solve problems by using that fact. It always surprises me where the deficiencies are when you start digging... I would not have guessed rounding to be a difficult thing.

Dan Wekselgreene said...

The first unit that we do in Numeracy is all about base-10 concepts. Rounding (believe it or not) is one of the "capstone" concepts of the unit. We start off with developing the idea of exchange rate, and we use base-10 blocks to help students understand how to do multi-digit addition and subtraction. We have our 9th graders actually using these blocks to understand the idea of "borrowing" and "carrying" - we reframe them in terms of "exchanging". Most students can add reliably when they come to us, but most cannot subtract reliably.

We also work on learning the place value names and having students identify what place value a digit is in, what it is worth, and properly saying the name of "big" numbers (such as 1,254,302).

When we get to rounding after all of this, the hope is that students have internalized the idea of exchange, and what it means. Here are the typical mistakes students make in rounding:

Round 1842.32 to the tens place.

1) 4 (Students ignore the direction to round because they don't understand it, and instead write the number that is in the specified place value.)

2) 42.32 (Students start to get the idea of the concept, but cut off the rest of the number to the left of the rounded place value. I'm not quite sure what deficiency this indicates, although it's clear that there is a deep misunderstanding here.)

3) 1830 (Students see that "2 is less than 5" so they "round down" by subtracting 1 from 4. Is this progress? I'm not sure..)

4) 1840.00 (This is definitely progress, as it is correct. It then takes more effort to get students to understand where adding 0s matters and where it doesn't).

When students have to model rounding with the base-10 blocks, they usually get the problems right. But when they do it with only numbers, they make these mistakes, again and again. I'm not sure exactly what is going on, but it is clear that they are not yet fluent with understanding how digits in a number work.

Darren said...

Students will memorize anything put on the walls. They'll do this because there's not much else they can do without getting into trouble when they're bored in class.

DCP Students:
Number of people that can fit in the Shark Tank:
Number of people that will fit in the Oakland Coliseum (yeah, I know, it'll never be filled by the Raiders, but still!):
Number of people in (pick an easily identifiable locale):
Number of people in California:
Number of people in the US:
Number of people in the world:
US debt:

Most of those are an order of magnitude or so apart.

And while you're putting stuff on the board, identify the Greek metric prefixes for units greater than one: exa, peta, tera, giga, mega, kilo, hecto, deka. Then do the Latins for less than one: deci, centi, milli, micro, nano, pico, femto, atto.

Anonymous said...

Your post made me think of a website a love. It's called The MegaPenny Project, and it shows what a million pennies would look like. Here is the link:

Dan Wekselgreene said...

Thanks for the site, Mrs. Simpson. It's got some great visuals!