How many stars are in this picture? In the universe?
I’m currently part of a professional study group of Montessorians, and we are reading Tracy Zager’s excellent book Becoming The Math Teacher You Wish You’d Had. In Chapter 6 (page 128), she shares a spectacularly unfortunate problem (as an exercise in improving such tasks). The problem itself says something like this:
The average American drinks about 61 cups of soda per month. About how many cups of soda is that per year?
Not such a bad problem right there. But before asking the actual question, the creator of this problem “scaffolds” the problem-solving (or in this case, obfuscates the problem solving) by giving several steps. First, the children are supposed to estimate an answer and identify the order of magnitude of the result (using some very confusing instructions about circling numbers on a grid). Then, the author asks the child to calculate the exact answer.
This is absurd! Why in the world should we pretend that it’s meaningful to calculate an exact answer to a question for which we have only inexact information? Given the errors inherent in working out something like the average amount of soda an American drinks (Who are you surveying? If you’re basing this on sales, how do we know the soda was actually drunk? Does it vary by region? Age? Economic status?), why should we pretend 732 (61 x 12) is a significantly better answer than 720 (60 x 12), which is a sensible pair of numbers to use for estimating an answer? If we want children to calculate precise answers to multiplication questions, let’s give them questions where precise answers matter.
This approach also reduces the chance that you’ll have an interesting discussion about what counts as a good answer to a question like this. In what contexts might you care about those extra 12 cups of soda? Those extra 12 cups aren’t going to make much of a difference if you’re calculating how much sugar an average American gets with their soda in a year. But, it might make a big difference if you’re a soda manufacturer trying to decide how much soda to make next year. Even more importantly, we need to dig into the question of what “about 61 cups” even means. How much do individuals vary? Do some drink no soda at all? Others drink hundreds of cups?
This brings us to the real problem here: estimation is an underrated skill. This problem reinforces the idea that estimations are not as good as “precise” answers, or even that estimates are just bad precise answers. But this simply isn’t true.
Overly precise answers are stupid. The constant, subtle message that the point of math is precise calculation leads to absurdities like saying we should put 3.66666667 people in each of our cars. If we want children to practice calculating precise answers to multiplication problems, we should ask them questions where precise answers are meaningful. And when precise answers are ridiculous, we should ask children to think about what answer is meaningful.
Frequently, as in the problem above, we don’t have precise information. In this instance, pretending to have an exact answer is dishonest. It’s not a big rhetorical leap from “Americans drink around 732 cups of soda per year,” to “Every American drinks 732 cups of soda per year,” but it’s a big conceptual leap.
It’s the mathematical equivalent of prejudice, and in fact, it’s just this sort of leap that can create all sorts of bias. Machine learning algorithms (which are created by humans!) can easily be given the information that “a higher than average number people in this zip code have faced more structural discrimination and therefore may have less financial stability,” to denying people in that zip code mortgages (which, in the US, typically means penalizing people for their skin color.)
More precise answers are often not very useful. When I’m trying to calculate how much money to budget for groceries each month, there’s no benefit to trying to work it out to the penny. Prices change, our needs change, some months we eat over at friends’ houses more often and don’t buy as many groceries, and so on.
Precise answers are often impractical or impossible to find. No one knows how many stars are in the visible universe, but using star surveys of sectors of sky, and what we know or assume about the way matter is distributed in space, we can estimate that there are between $10^{22}$ and $10^{24}$ stars in the universe. That’s a huge range, but it’s much narrower than shrugging our shoulders and saying “we can’t count them all, so there’s no way to know anything.”
Mental estimation is good for you, and for your children. Even when you are trying to find a precise calculation, estimation can help keep us grounded in reality. If I’m trying to solve a problem like 3,997 x 41, a good place to start is to say to myself, this is pretty close to 4,000 x 40, which is 160,000, so the answer should be in the ballpark of 160,000. Now I know that if my final answer is 25,462, it’s got to be wrong.
Estimation requires very different skills than precise calculation, and that means it’s an opportunity for different children to succeed. Not every child learns their math facts and remembers procedures easily, and this frequently means they struggle to learn precise calculation. But these same students may be excellent at making visual estimates, noticing when an answer doesn’t make sense, or remembering to check precise calculations against a prior estimate.
Good estimates frequently require more mathematical knowledge than following an algorithm. In my example above about ballparking 3,997 x 41, there is a huge amount of mathematical subtext going on. First, I need to think about these numbers as actual quantities, so that I can estimate similar quantities. This requires recognizing that the left-most digits affect the quantities most, but also that I can’t ignore the other digits (3,000 wouldn’t be nearly as good an estimate as 4,000). Then, I need enough knowledge of place value and multiplication to mentally compute 4,000 x 40. And finally, I need to understand what information is important in that result. It’s not the 16 that matters, so much as the order of magnitude. A final calculation close to 16,000 would not be very accurate, despite having a leading 16.
Compare this to the potentially mindless magic that goes on when following an algorithm, and you’ll see why the estimate was actually much more sophisticated.
I hope this gave you some food for thought. Coming up soon: ways to incorporate estimation into a Montessori environment.