Must be the start of school or something. In today’s roundup of mathematically-themed comics there are a couple of strips that I think touch on the question of defining just what the problem is: what are you trying to measure, what are you trying to calculate, what are the rules of this sort of calculation? That’s a lot of what’s really interesting about mathematics, which is how I’m able to say something about a rerun Archie comic. It’s not easy work but that’s why I get that big math-blogger paycheck.
John Hambrock’s The Brilliant Mind of Edison Lee (September 2) talks about the shape of the universe. Measuring the world, or the universe, is certainly one of the older influences on mathematical thought. From a handful of observations and some careful reasoning, for example, one can understand how large the Earth is, and how far away the Moon and the Sun must be, without going past the kinds of reasoning or calculations that a middle school student would probably be able to follow.
There is something deeper to consider about the shape of space, though: the geometry of the universe affects what things can happen in them, and can even be seen in the kinds of physics that happen. A famous, and astounding, result by the mathematical physicist Emmy Noether shows that symmetries in space correspond to conservation laws. That the universe is, apparently, rotationally symmetric — everything would look the same if the whole universe were picked up and rotated (say) 80 degrees along one axis — means that there is such a thing as the conservation of angular momentum. That the universe is time-symmetric — the universe would look the same if it had got started five hours later (please pretend that’s a statement that can have any coherent meaning) — means that energy is conserved. And so on. It may seem, superficially, like a cosmologist is engaged in some almost ancient-Greek-style abstract reasoning to wonder what shapes the universe could have and what it does, but (putting aside that it gets hard to divide mathematics, physics, and philosophy in this kind of field) we can imagine observable, testable consequences of the answer.
Zach Weinersmith’s Saturday Morning Breakfast Cereal (September 5) tells a joke starting with “two perfectly rational perfectly informed individuals walk into a bar”, along the way to a joke about economists. The idea of “perfectly rational perfectly informed” people is part of the mathematical modeling that’s become a popular strain of economic thought in recent decades. It’s a model, and like many models, is properly speaking wrong, but it allows one to describe interesting behavior — in this case, how people will make decisions — without complications you either can’t handle or aren’t interested in. The joke goes on to the idea that one can assign costs and benefits to continuing in the joke. The idea that one can quantify preferences and pleasures and happiness I think of as being made concrete by Jeremy Bentham and the utilitarian philosophers, although trying to find ways to measure things has been a streak in Western thought for close to a thousand years now, and rather fruitfully so. But I wouldn’t have much to do with protagonists who can’t stay around through the whole joke either.
Marc Anderson’s Andertoons (September 6) was probably composed in the spirit of joking, but it does hit something that I understand baffles kids learning it every year: that subtracting a negative number does the same thing as adding a positive number. To be fair to kids who need a couple months to feel quite confident in what they’re doing, mathematicians needed a couple generations to get the hang of it too. We have now a pretty sound set of rules for how to work with negative numbers, that’s nice and logically tested and very successful at representing things we want to know, but there seems to be a strong intuition that says “subtracting a negative three” and “adding a positive three” might just be different somehow, and we won’t really know negative numbers until that sense of something being awry is resolved.
Andertoons pops up again the next day (September 7) with a completely different drawing of a chalkboard and this time a scientist and a rabbit standing in front of it. The rabbit’s shown to be able to do more than multiply and, indeed, the mathematics is correct. Cosines and sines have a rather famous link to exponentiation and to imaginary- and complex-valued numbers, and it can be useful to change an ordinary cosine or sine into this exponentiation of a complex-valued number. Why? Mostly, because exponentiation tends to be pretty nice, analytically: you can multiply and divide terms pretty easily, you can take derivatives and integrals almost effortlessly, and then if you need a cosine or a sine you can get that out at the end again. It’s a good trick to know how to do.
Jeff Harris’s Shortcuts children’s activity panel (September 9) is a page of stuff about “Geometry”, and it’s got some nice facts (some mathematical, some historical), and a fair bunch of puzzles about the field.
Morrie Turner’s Wee Pals (September 7, perhaps a rerun; Turner died several months ago, though I don’t know how far ahead of publication he was working) features a word problem in terms of jellybeans that underlines the danger of unwarranted assumptions in this sort of problem-phrasing.
Craig Boldman and Henry Scarpelli’s Archie (September 8, rerun) goes back to one of arithmetic’s traditional comic strip applications, that of working out the tip. Poor Moose is driving himself crazy trying to work out 15 percent of $8.95, probably from a quiz-inspired fear that if he doesn’t get it correct to the penny he’s completely wrong. Being able to do a calculation precisely is useful, certainly, but he’s forgetting that in tis real-world application he gets some flexibility in what has to be calculated. He’d save some effort if he realized the tip for $8.95 is probably close enough to the tip for $9.00 that he could afford the difference, most obviously, and (if his budget allows) that he could just as well work out one-sixth the bill instead of fifteen percent, and give up that workload in exchange for sixteen cents.
Mark Parisi’s Off The Mark (September 8) is another entry into the world of anthropomorphized numbers, so you can probably imagine just what π has to say here.
nebusresearch 
If teachers understood that when you bring in negative numbers you don’t just stick them to the left of the numbers you had before, you have actually created a new number system, for a new purpose, and made two copies of the original numbers (the natural numbers), stuck the copies together at zero, with one lot going to the left and one lot going to the right. It is completely arbitrary whether the ones to the right are called the positive numbers or the negative numbers. But of course that is far too mathematical for most people.
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You are right, that the introduction of negative numbers as this thing sort of slapped onto the left end of the number line sets up for a lot of trouble, particularly in subtraction and probably also in multiplication and even greater-than and less-than comparisons. But I also see why it’s so attractive to introduce it that way; it feels natural, or at least it look that way.
I wonder if there’s a way to introduce the subject more rigorously but still at a level that elementary school students, who generally aren’t very strong on abstract reasoning, will still find comfortable, and that won’t cause their parents to get upset that they’re being taught something too weirdly different from what they kind of remember from school themselves.
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In the case of Moose there, I’m tempted to say that, at least in part, he’s the victim of his education. If I had my way, students at some level would have a math class all about quickly winging it in real life situations. Shortcuts, etc. That would have been very helpful for me. It wasn’t until college physics when I realized you can start fudging numbers until you got “close enough.”
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He is, yes, certainly a victim there. The training to work out the exact problem specified is a good one, but what’s missed is knowledge that he actually gets to pick the exact problem. That’s a real shame since so much mathematics is a matter of picking the exact problem you want to solve, and how perfectly you have to solve it.
(Admittedly, Moose, by the definition of his character, would struggle with a problem simpler to calculate too.)
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That’s a great way of putting it. I feel like a lot of my mathematics woes would have been avoided if I’d learned fact that the “right” answer is the one that’s as accurate as you need it to be.
Ugh, I think back to college physics and wasting time trying to juggle so many decimals just because I thought the most accurate answer was the most desirable one, when I could have done the question in a quarter the time by saying “okay pi is 3, g is 10, and air resistance is a figment of an overactive imagination.”
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Decimals have this strange hypnotic effect on the human psyche. I don’t know what TV Tropes would call it but you see a piece of this in a science fiction show where a character is shown to be very smart by reeling out far more digits than could be meaningful or even useful, like the time Commander Data gave the travel time between galaxies down to fractions of a second.
It’s kind of a shame that fractions are kind of clumsy to work with, since the description of something as, say, one-quarter can avoid the trap of thinking that you have more precision than 0.25 really entitles you to. (And, yes, you can add a note about your margin for error, but I’m not convinced that people really internalize that, not without a lot of practice.)
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The joke involving those perfectly rational perfectly informed beings is my favorite – as it has several ‘levels’…
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Yeah, it has at that, hasn’t it?
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