A lesson in percentages

I'm hardly the first writer to kvetch about how the dang kids these days -- or any day, really -- don't have any sort of number sense.  My kid is working on his summer math assignment, which includes a page of percentage problems.  The questions themselves are not just reasonable, but important.  "What is 31% of 75" or "28 is 25% of what number" are to mathematical literacy what the offside rule is to soccer -- not everyone understands, but you'd dang well better understand if you want to be considered fluent.

My complaint, therefore, is not that Milo's class is studying the wrong thing.  It's how they approach the problems.  He is required to do the problems the same way I was taught 30-odd years ago:  set up a proportion, translating English to mathematics.  In this parlance, "of" means to multiply, "is" is an equals sign, "percent" means to make a fraction over 100.  No calculator is allowed.  And thusly, Milo and his classmates usually get the right answer.  They often don't notice when they do a routine backwards and say that 31% of 75 is 220, but they usually get the right answer.

I've no doubt that there is some sort of validity to this pedagogy, especially if some sort of national exam is going to require precise answers to such questions with no calculator.  But consider: beyond the test, what do we really want functional high school students and adults to be able to do with percentages?  I personally would prefer my class to be skilled estimators.  What's 31% of 75?  It's about 25, or maybe 24, because 31% is just about a third.  And I would prefer that no one in my class or family* rejoin "well, actually, one-third is 33.3333 repeating percent, so you're wrong."

* For their own sake, so they don't get thrown in the scorpion pit

Me, I'd teach this topic like a video game.  

Start with obvious reference percentages: 50% is a half, 25% is a fourth, 33% is a third.  And use them intuitively to solve problems quickly.  For example, I'd set up a competition: everyone gets 30 seconds to do, say, five no-calculator problems with just these obvious percentages.  Score something like one point for getting "close" in a way defined by the teacher, and an additional point for being right-on.  Guessing is encouraged, and essentially required by the time limit.  Students are practicing making intelligent guesses, and refining their guesses.

Once the class is getting bored with the obviousness, do tricksier problems.  Now the additional point would be awarded to the student closest to the right answer.  Don't demand any formal work or method, but discuss and share methods.  After doing, say, "What is 66% of 210," one student might suggest they knew that the answer had to be more than 105, because 66% is more than half.  But perhaps someone else noticed that 66% is twice 33%, and so is two-thirds -- and perhaps someone else explains how they estimated 2/3 of 210 without painstakingly dividing by three and multiplying by two.  

What does this have to do with physics?  I use essentially this same method when teaching circuits to freshmen in conceptual physics.  They learn to estimate, not calculate, voltages across series resistors and currents through parallel resistors.  And, by unit's end, they have a better sense for the answers than do seniors who have been taught to calculate.

I understand math teachers' obsession with routine and algorithm.  When weak students -- students without any innate number sense, and without any serious interest in the subject -- simply need to get exact answers, well, algorithm can be a friend.  I'm telling you, though, an estimating approach can work wonders.  Even weak students can make progress by guessing and checking.  I've seen it happen.  If that culminating test is multiple choice, even the weak students will be able to pick out correct answers from a lineup.  

And, perhaps if a page of problems didn't represent a multi-hour sentence to proportions, cross-multiplication, and hand arithmetic, such students might develop an interest in the subject.  Or at least a competence with it.