For a long time now, I’ve been writing journal articles and blog posts about large number innumeracy. A basic take-away from my research has been that untrained members of the general population have a wide range of skills for dealing with large numbers. Basically, people are very competent at arithmetic up to about 1,000; they can order the basic scale words thousand, million, billion, and trillion, and can generally even write them correctly. What people struggle to do is to relate across orders of magnitude. In one study, for instance, 65 of 67 college undergraduates represented one thousand, one million, and one billion correctly as numerals, even as one third of them made huge errors estimating their relative magnitudes. This week, a popular internet meme has made my point beautifully.
As Snopes notes, 1.3 Billion is 13×10^8, while 300 million is 3×10^8, so this is $4.33 per person, not $4.33 million (shameless plug: you can see this nicely rendered using our Graspable Math app). As you can see, people who buy this are doing the basic arithmetic just fine, but are off by a factor of 1 million–they just don’t know how to relate the scale words, even though they know how many zeros. Of course, people may really not believe it–but our data is consistent with the idea that many people would.
Apparently, this has come up before. Again from Snopes:
With the lotto meme, the format of the numbers helped make it difficult to accurately assess the math. I saw it the other day and immediately went to check the math, at which time I typed the following (in IPython):
The first two lines were to make sure I wasn’t being dumb about the scientific notation (i.e., I counted zeros to make sure I had the right numbers). Then I divided the one by the other and got 4.33.
At which time I said to myself, "self, that math works out." I mean, the meme says "1.3 billion", "300 million", and "4.33 mil." Hence, it all makes sense.
All of which is to say two things. First, your notation doesn’t make the dumb mistake I made, i.e., 13e8 and 3e8 make the comparison obvious, whereas my 1.3e9 and 300e6 make it less obvious. Second, my answer – 4.33 – matched the answer on the meme, just without that pesky "mil" dangling there.
For you of all people to (a) forget that you’re way better at dealing with these numbers than just about anyone else, and (b) ignore the hybrid number nonsense… why, it’s borderline disgusting!
Okay, i totally agree with (b). I should have more explicitly mentioned that problem–the hybrid notation makes this much more misleading, and the switch from million to mil makes it even harder. As for (a), I don’t think I am better! I caught this one, it’s true, but for most problems that involve estimating across orders of magnitude I’m as bad as any other mathy person–which is to say, really really bad.
Much as with the defense budget post awhile back, the Powerball math error requires that folks not do a common sense check. The average American didn’t spend $4+ million on lottery tickets. There can’t be a bigger jackpot than what folks spend in tickets (the state makes rather than loses money on lotteries). When the math outcome just doesn’t seem plausible ….
Perhaps I’m inclined to do this because I was apt to make silly errors in math problems as a kid, yet good enough at math to usually understand it. I find the not doing common sense checks to protect against these sorts of innumeracy problems quite interesting. (I also have a tendency to do what I’ve noticed a lot of Asian students do … claim and really believe I’m not good at math when a number of objective standards would suggest I’m fine if certainly not excellent at it. I blame David).
Two ideas on this. One, perhaps folks just don’t have any ideas about what is plausible, though when it comes to how much people spend on the lottery they should. Two, it may be the fact the math intimidates people and they just want THE answer and don’t want to think about it. I’ve had trouble getting college students to do what is basically 4th grade division. They COULD do it … they had the skills. They just found it repulsive or painful somehow and did NOT WANT to do it.
Yes, I agree so completely. There isn’t a huge research emphasis, nor is it part of the common core, but this idea of finding a way to check your answer is such a key part of mathematical learning and thinking. It’s not surprising for even bright, math-sophisticated people to make (or believe) this error. the math-sophisticated people are just more likely to say "could that be true?" and work out some way of checking it.
I am not sure (and again, would love to know more research regarding) why people find this process so aversive. I suspect that part of it is that if you’re getting answers wrong all the time, it can feel like you don’t understand what’s happening. If no one explains to you that math JUST IS that field where we all get the answers wrong most of the time (and then check them), it can amplify the natural frustration, maybe?
One natural experiment you might do. The British call what we call a "billion" a "thousand million". (a billion is a million million to them). It might be clearer that 1,000 million divided by 300 million is NOT 4.33 million. Of course, the Obamacare example suggests that apparently even if the wording is the same the math goes haywire. At any rate, I wonder if the different language in the US and UK makes any difference.
Thanks for the suggestion!! My group is doing some work right now, looking at alternative number systems to see how confusions carry over into them. I’ll have an update on this, I hope, very soon!