## Someone Else’s Homework: A Probability Question

My friend’s finished the last of the exams and been happy with the results. And I’m stuck thinking harder about a little thing that came across my Twitter feed last night. So let me share a different problem that we had discussed over the term.

It’s a probability question. Probability’s a great subject. So much of what people actually do involves estimating probabilities and making judgements based on them. In real life, yes, but also for fun. Like a lot of probability questions, this one is abstracted into a puzzle that’s nothing like anything anybody does for fun. But that makes it practical, anyway.

So. You have a bowl with fifteen balls inside. Five of the balls are labelled ‘1’. Five of the balls are labelled ‘2’. Five of the balls are labelled ‘3’. The balls are well-mixed, which is how mathematicians say that all of the balls are equally likely to be drawn out. Three balls are picked out, without being put back in. What’s the probability that the three balls have values which, together, add up to 6?

My friend’s instincts about this were right, knowing what things to calculate. There was part of actually doing one of these calculations that went wrong. And was complicated by my making a dumb mistake in my arithmetic. Fortunately my friend wasn’t shaken by my authority, and we got to what we’re pretty sure is the right answer.

## Someone Else’s Homework: A Postscript

My friend aced the mathematics final. Not due to my intervention, I’d say; my friend only remembered one question on the exam being much like anything we had discussed recently. Though it was very like one of those, a question about the probability of putting together a committee with none, one, two, or more than two members of particular subgroups. And that one we didn’t even work through; I just confirmed my friend’s guess about what calculation to do. Which is good since that particular calculation is a tedious one that I didn’t want to do. No, my friend aced it by working steadily through the whole term. And yes, asking me for tutoring a couple times, but that’s all right. Small, steady work adds up, in mathematics as with so much else.

Meanwhile may I draw your attention over to my humor blog where last night I posted a bit of silliness about number divisibility. Because I can’t help myself, it does include a “quick” test for whether a number could be divided by 21. It’s in the same spirit as tests for whether a number can be divided by 3 or 9 (add the digits add see whether that sum’s divisible by 3 or 9) or 11 (add or subtract digits, in alternate form, and see whether that sum is divisible by 11). The process I give is correct, which is not to say that anyone would ever use it. Even if they did they’d be better off testing for divisibility by both 3 and 7. And I don’t think I’d use an add-the-digits scheme for 7 either.

## Someone Else’s Homework: Was It Hard? An Umbrella Search

I wanted to follow up, at last, on this homework problem a friend had.

The question: suppose you have a function f. Its domain is the integers Z. Its rage range is also the integers Z. You know two things about the function. First, for any two integers ‘a’ and ‘b’, you know that f(a + b) equals f(a) + f(b). Second, you know there is some odd number ‘c’ for which f(c) is even. The challenge: prove that f is even for all the integers.

My friend asked, as we were working out the question, “Is this hard?” And I wasn’t sure what to say. I didn’t think it was hard, but I understand why someone would. If you’re used to mathematics problems like showing that all the roots of this polynomial are positive, then this stuff about f being even is weird. It’s a different way of thinking about problems. I’ve got experience in that thinking that my friend hasn’t.

All right, but then, what thinking? What did I see that my friend didn’t? And I’m not sure I can answer that perfectly. Part of gaining mastery of a subject is pattern recognition. Spotting how some things fit a form, while other stuff doesn’t, and some other bits yet are irrelevant. But also part of gaining that mastery is that it becomes hard to notice that’s what you’re doing.

But I can try to look with fresh eyes. There is a custom in writing this sort of problem, and that drove much of my thinking. The custom is that a mathematics problem, at this level, works by the rules of a Minute Mystery Puzzle. You are given in the setup everything that you need to solve the problem, yes. But you’re also not given stuff that you don’t need. If the detective mentions to the butler how dreary the rain is on arriving, you’re getting the tip to suspect the houseguest whose umbrella is unaccounted for.

(This format is almost unavoidable for teaching mathematics. At least it seems unavoidable given the number of problems that don’t avoid it. This can be treacherous. One of the hardest parts in stepping out to research on one’s own is that there’s nobody to tell you what the essential pieces are. Telling apart the necessary, the convenient, and the irrelevant requires expertise and I’m not sure that I know how to teach it.)

The first unaccounted-for umbrella in this problem is the function’s domain and range. They’re integers. Why wouldn’t the range, particularly, be all the real numbers? What things are true about the integers that aren’t true about the real numbers? There’s a bunch of things. The highest-level things are rooted in topology. There’s gaps between one integer and its nearest neighbor. Oh, and an integer has a nearest neighbor. A real number doesn’t. That matters for approximations and for sequences and series. Not likely to matter here. Look to more basic, obvious stuff: there’s even and odd numbers. And the problem talks about knowing something for an odd number in the domain. This is a signal to look at odds and evens for the answer.

The second unaccounted-for umbrella is the most specific thing we learn about the function. There is some odd number ‘c’, and the function matches that integer ‘c’ in the domain to some even number f(c) in the range. This makes me think: what do I know about ‘c’? Most basic thing about any odd number is it’s some even number plus one. And that made me think: can I conclude anything about f(1)? Can I conclude anything about f at the sum of two numbers?

Third unaccounted-for umbrella. The less-specific thing we learn about the function. That is that for any integers ‘a’ and ‘b’, f(a + b) is f(a) + f(b). So see how this interacts with the second umbrella. f(c) is f(some-even-number) + f(1). Do I know anything about f(some-even-number)?

Sure. If I know anything about even numbers, it’s that any even number equals two times some integer. Let me call that some-integer ‘k’. Since some-even-number equals 2*k, then, f(some-even-number) is f(2*k), which is f(k + k). And by the third umbrella, that’s f(k) + f(k). By the first umbrella, f(k) has to be an integer. So f(k) + f(k) has to be even.

So, f(c) is an even number. And it has to equal f(2*k) + f(1). f(2*k) is even; so, f(1) has to be even. These are the things that leapt out to me about the problem. This is why the problem looked, to me, easy.

Because I knew that f(1) was even, I knew that f(1 + 1), or f(2), was even. And so would be f(2 + 1), that is, f(3). And so on, for at least all the positive integers.

Now, after that, in my first version of this proof, I got hung up on what seems like a very fussy technical point. And that was, what about f(0)? What about the negative integers? f(0) is easy enough to show. It follows from one of those tricks mathematics majors are told about early. Somewhere in grad school they start to believe it. And that is: adding zero doesn’t change a number’s value, but it can give you a more useful way to express that number. Here’s how adding zero helps: we know c = c + 0. So f(c) = f(c) + f(0) and whether f(c) is even or odd, f(0) has to be even. Evens and odds don’t work any other way.

After that my proof got hung up on what may seem like a pretty fussy technical point. That amounted to whether f(-1) was even or odd. I discussed this with a couple people who could not see what my issue with this was. I admit I wasn’t sure myself. I think I’ve narrowed it down to this: my questioning whether it’s known that the number “negative one” is the same thing as what we get from the operation “zero minus one”. I mean, in general, this isn’t much questioned. Not for the last couple centuries.

You might be having trouble even figuring out why I might worry there could be a difference. In “0 – 1” the – sign there is a binary operation, meaning, “subtract the number on the right from the number on the left”. In “-1” the – sign there is a unary operation, meaning, “take the additive inverse of the number on the right”. These are two different – signs that look alike. One of them interacts with two numbers. One of them interacts with a single number. How can they mean the same thing?

With some ordinary assumptions about what we mean by “addition” and “subtraction” and “equals” and “zero” and “numbers” and stuff, the difference doesn’t matter much. We can swap between “-1” and “0 – 1” effortlessly. If we couldn’t, we probably wouldn’t use the same symbol for the two ideas. But in the context of this particular question, could we count on that?

My friend wasn’t confident in understanding what the heck I was getting on about. Fair enough. But some part of me felt like that needed to be shown. If it hadn’t been recently shown, or used, in class, then it had to go into this proof. And that’s why I went, in the first essay, into the bit about additive inverses.

This was me over-thinking the problem. I got to looking at umbrellas that likely were accounted for.

My second proof, the one thought up in the shower, uses the same unaccounted-for umbrellas. In the first proof, the second unaccounted-for umbrella seemed particularly important. Knowing that f(c) was odd, what else could I learn? In the second proof, it’s the third unaccounted-for umbrella that seemed key. Knowing that f(a + b) is f(a) + f(b), what could I learn? That right away tells me that for any even number ‘d’, f(d) must be even.

Call this the fourth unaccounted-for umbrella. Every integer is either even or odd. So right away I could prove this for what I really want to say is half of the integers. Don’t call it that. There’s not a coherent way to say the even integers are any fraction of all the integers. There’s exactly as many even integers as there are integers. But you know what I mean. (What I mean is, in any finite interval of consecutive integers, half are going to be even. Well, there’ll be at most two more odd integers than there are even integers. That’ll be close enough to half if the interval is long enough. And if we pretend we can make bigger and bigger intervals until all the integers are covered … yeah. Don’t poke at that and do not use it at your thesis defense because it doesn’t work. That’s what it feels like ought to work.)

But that I could cover the even integers in the domain with one quick sentence was a hint. The hint was, look for some thing similar that would cover the odd integers in the domain. And hey, that second unaccounted-for umbrella said something about one odd integer in the domain. Add to that one of those boring little things that a mathematician knows about odd numbers: the difference between any two odd numbers is an even number. ‘c’ is an odd number. So any odd number in the domain, let’s call it ‘d’, is equal to ‘c’ plus some even number. And f(some-even-number) has to be even and there we go.

So all this is what I see when I look at the question. And why I see those things, and why I say this is not a hard problem. It’s all in spotting these umbrellas.

## Someone Else’s Homework: Some More Thoughts

I wanted to get back to my friend’s homework problem. And a question my friend had about the question. It’s a question I figure is good for another essay.

But I also had second thoughts about the answer I gave. Not that it’s wrong, but that it could be better. Also that I’m not doing as well in spelling “range” as I had always assumed I would. This is what happens when I don’t run an essay through Hemmingway App to check whether my sentences are too convoluted. I also catch smaller word glitches.

Let me re-state the problem: Suppose you have a function f, with domain of the integers Z and rage of the integers Z. And also you know that f has the property that for any two integers ‘a’ and ‘b’, f(a + b) equals f(a) + f(b). And finally, suppose that for some odd number ‘c’, you know that f(c) is even. The challenge: prove that f is even for all the integers.

Like I say, the answer I gave on Tuesday is right. That’s fine. I just thought of a better answer. This often happens. There are very few interesting mathematical truths that only have a single proof. The ones that have only a single proof are on the cutting edge, new mathematics in a context we don’t understand well enough yet. (Yes, I am overlooking the obvious exception of ______ .) But a question so well-chewed-over that it’s fit for undergraduate homework? There’s probably dozens of ways to attack that problem.

And yes, you might only see one proof of something. Sometimes there’s an approach that works so well it’s silly to consider alternatives. Or the problem isn’t big enough to need several different proofs. There’s something to regret in that. Re-thinking an argument can make it better. As instructors we might recommend rewriting an assignment before turning it in. But I’m not sure that encourages re-thinking the assignment. It’s too easy to just copy-edit and catch obvious mistakes. Which is valuable, yes. But it’s good for communication, not for the mathematics itself.

So here’s my revised argument. It’s much cleaner, as I realized it while showering Wednesday morning.

Give me an integer. Let’s call it m. Well, m has to be either an even or an odd number. I’m supposing nothing about whether it’s positive or negative, by the way. This means what I show will work whether m is greater than, less than, or equal to zero.

Suppose that m is an even number. Then m has to equal 2*k for some integer k. (And yeah, k might be positive, might be negative, might be zero. Don’t know. Don’t care.) That is, m has to equal k + k. So f(m) = f(k) + f(k). That’s one of the two things we know about the function f. And f(k) + f(k) is is 2 * f(k). And f(k) is an integer: the integers are the function’s rage range). So 2 * f(k) is an even integer. So if m is an even number then f(m) has to be even.

All right. Suppose that m isn’t an even integer. Then it’s got to be an odd integer. So this means m has to be equal to c plus some even number, which I’m going ahead and calling 2*k. Remember c? We were given information about f for that element c in the domain. And again, k might be positive. Might be negative. Might be zero. Don’t know, and don’t need to know. So since m = c + 2*k, we know that f(m) = f(c) + f(2*k). And the other thing we know about f is that f(c) is even. f(2*k) is also even. f(c), which is even, plus f(2*k), which is even, has to be even. So if m is an odd number, then f(m) has to be even.

And so, as long as m is an integer, f(m) is even.

You see why I like that argument better. It’s shorter. It breaks things up into fewer cases. None of those cases have to worry about whether m is positive or negative or zero. Each of the cases is short, and moves straight to its goal. This is the proof I’d be happy submitting. Today, anyway. No telling what tomorrow will make me think.

## Someone Else’s Homework: A Solution

I have a friend who’s been taking mathematical logic. While talking over the past week’s work they mentioned a problem that had stumped them. But they’d figured it out — at least the critical part — about a half-hour after turning it in. And I had fun going over it. Since the assignment’s already turned in and I don’t even know which class it was, I’d like to share it with you.

So here’s the problem. Suppose you have a function f, with domain of the integers Z and rage of the integers Z. And also you know that f has the property that for any two integers ‘a’ and ‘b’, f(a + b) equals f(a) + f(b). And finally, suppose that for some odd number ‘c’, you know that f(c) is even. The challenge: prove that f is even for all the integers.

If you want to take a moment to think about that, please do.

First thing I want to do is show that f(1) is an even number. How? Well, if ‘c’ is an odd number, then ‘c’ has to equal ‘2*k + 1’ for some integer ‘k’. So f(c) = f(2*k + 1). And therefore f(c) = f(2*k) + f(1). And, since 2*k is equal to k + k, then f(2*k) has to equal f(k) + f(k). Therefore f(c) = 2*f(k) + f(1). Whatever f(k) is, 2*f(k) has to be an even number. And we’re given f(c) is even. Therefore f(1) has to be even.

Now I can prove that if ‘k’ is any positive integer, then f(k) has to be even. Why? Because ‘k’ is equal to 1 + 1 + 1 + … + 1. And so f(k) has to equal f(1) + f(1) + f(1) + … + f(1). That is, it’s k * f(1). And if f(1) is even then so is k * f(1). So that covers the positive integers.

How about zero? Can I show that f(0) is even? Oh, sure, easy. Start with ‘c’. ‘c’ equals ‘c + 0’. So f(c) = f(c) + f(0). The only way that’s going to be true is if f(0) is equal to zero, which is an even number.

By the way, here’s an alternate way of arguing this: 0 = 0 + 0. So f(0) = f(0) + f(0). And therefore f(0) = 2 * f(0) and that’s an even number. Incidentally also zero. Submit the proof you like.

What’s not covered yet? Negative integers. It’s hard not to figure, well, we know f(1) is even, we know f(a + b) if f(a) + f(b). Shouldn’t, like, f(-2) just be -2 * f(1)? Oh, it so should. I don’t feel like we have that already proven, though. So let me nail that down. I’m going to use what we know about f(k) for positive ‘k’, and the fact that f(0) is 0.

So give me any negative integer; I’m going call it ‘-k’. Its additive inverse is ‘k’, which is a positive number. -k + k = 0. And so f(-k + k) = f(-k) + f(k) = f(0). So, f(-k) + f(k) = 0, and f(-k) = -f(k). If f(k) is even — and it is — then f(-k) is also even.

So there we go: whether ‘k’ is a positive, zero, or negative integer, f(k) is even. All the integers are either positive, zero, or negative. So f is even for any integer.

## Reading the Comics, July 30, 2016: Learning Tools Edition

I thank Comic Strip Master Command for the steady pace of mathematically-themed comics this past week. It fit quite nicely with my schedule, which you might get hints about in weeks to come. Depends what I remember to write about. I did have to search a while for any unifying motif of this set. The idea of stuff you use to help learn turned up several times over, and that will do.

Steve Breen and Mike Thompson’s Grand Avenue threatened on the 24th to resume my least-liked part of reading comics for mathematics themes. This would be Grandma’s habit of forcing the kids to spend their last month of summer vacation doing arithmetic drills. I won’t say that computing numbers isn’t fun because I know what it’s like to work out how many seconds are in 50 years in your head. But that’s never what this sort of drill is about. The strip’s diverted from that subject, but it might come back to spoil the end of summer vacation. (I’m not positive what my least-liked part of the comics overall is. I suspect it might be the weird anti-participation-trophy bias comic strip writers have.)

Ryan North’s Dinosaur Comics reprint for the 25th is about the end of the universe. We’ve got several competing theories about how the universe is likely to turn out, several trillion years down the road. The difference between them is in the shape of space and how that shape is changing. I’ve mentioned sometimes the wonder of being able to tell something about a whole shape from local information, things we can tell without being far from a single point. The fate of the universe must be the greatest example of this. Considering how large the universe is and how little of it we will ever be able to send an instrument to, we measure the shape of space from a single point. And we can realistically project what will happen in unimaginably distant times. Admittedly, if we get it wrong, we’ll never know, which takes off some of the edge.

Dinosaur Comics reappears the 28th with some talk about number bases. It’s all fine and accurate enough, except for the suggestion that anyone would use base five for something other than explaining how bases work. I like learning about bases. When I was a kid this concept explained much to me about how our symbols for numbers work. It also helped appreciate that symbols are not these fixed or universal things. They’re our creations and ours to adapt for whatever reason we find convenient. In the past we’ve found bases as high as sixty to be convenient. (The division of angles into 360 degrees each of 60 minutes, each of those of 60 seconds, is an echo of that.) But when I was a kid doing alternate-base problems nobody knew what I was doing or why, except the mathematics teacher who said I might like the optional sections in the book. We only really need base ten, base two, and base sixteen, which might as well be base two written more compactly. The rest are toys, good for instruction and for fun. Sorry, base seven.

Scott Meyer’s Basic Instructions rerun for the 27th is about everyone’s favorite bit of intransitivity. Rock-Paper-Scissors and its related games are all about systems in which any two results can be decisive but any three might not be. This prospect turns up whenever there are three or more possible outcomes. And it doesn’t require a system to be irrational or random. Chaos and counterintuitive results just happen when there’s three of a thing.

I remember, and possibly you remember too, learning of a computer system that can consistently beat humans at Rock-Paper-Scissors. It manages to do that by the oldest of game theory exploits, cheating. Its sensors look for the twitches suggesting what a person is going to throw and then it changes its throw to beat that. I don’t know what that’s supposed to prove since anyone who’s played a Sid Meier’s Civilization game knows that computers already know how to cheat.

Thom Bluemel’s Birdbrains, yes, you can be in my Reading The Comics post this week too. Don’t beg.

Bill Schorr’s The Grizzwells for the 28th is a resisted word problem joke. It doesn’t use the classic railroad or airplane forms, but it’s the same joke anyway.

Benita Epstein’s Six Chix for the 29th is probably familiar to the folks taking electronics. The chart is a compact map used as a mnemonic for the different relationships between the current (I), the voltage (V), the resistance (R), and the power (P) in a circuit. When I was a student we got this as two separate circles, one for current-voltage-resistance and one for power-current-voltage. Each was laid out like the T-and-O maps which pre-Renaissance Western Europe used to diagram the world. While I now see that as a convenient and useful tool, as a student, I was skeptical that it was any easier to use the mnemonic aid than it was to just remember “voltage equals current times resistance” and “power equals voltage times current”. I’ve always had an irrational suspicion of mnemonic devices. I’m trying to do better.

Brian Boychuk and Ron Boychuk’s Chuckle Brothers for the 30th is a return of the whiteboard full of symbols to represent deep thinking. The symbols don’t mean anything as equations, though that might be my limited perspective. And that also might represent the sketchy, shorthand way serious work is done. As an idea is sketched out weird bundles of symbols that don’t literally parse do appear. In a publishable paper this is all turned into neatly formatted and standard stuff. Or we introduce symbols with clear explanations of what they mean so that others can learn to read what we write. But for ourselves, in the heat of work, we’ll produce what looks like gibberish to others and that’s all right as long as we don’t forget what the gibberish means. Sometimes we do, but the gibberish typically helps us recapture a lost idea. (I offer the tale of a mathematician with pages of notes for a brilliant insight which she has to reconstruct from a lost memory to would-be short story writers looking for a Romantic hook.)

## Reading the Comics, July 23, 2016: Familiar Friends Week Edition

This past week was refreshing. The mathematics comics appeared at a regular, none-too-excessive pace. And some old familiar friends reappeared. Some were comic strips that haven’t been around in a while. Some were jokes that haven’t been. Enjoy.

Bill Whitehead’s Free Range for the 17th is the first use of the meth/math lab pun to appear in the comics since September 2014 by my reckoning. And only the second in my Reading the Comics series. I’m surprised too. For all this goes around Twitter and other social media I’d imagine it to make the comics more.

Scott Hilburn’s The Argyle Sweater gets back in my review here for the first time since April, to my amazement. Used to be you couldn’t go two weeks without Hilburn looking for my attention. And here’s the first Roman Numerals joke since … I don’t quite feel up to checking just now. I’m going to go ahead and suppose it’s the first one since the last time Samson’s Dark Side Of The Horse was here.

It’s anachronistic to speak of Ancient Roman students getting ‘C’ grades. Of course it is; it could hardly be otherwise. It’s a joke; how much is that to be worried about? But if I haven’t been mislead the use of letters, A through E-or-F, in student evaluations is an American innovation of the late 19th century. It developed over the 20th century and took over at least American education, in conjunction with the 100-through-0 points evaluation scale. And in parallel to the Grade Point Average, typically with 4.0 as its highest score.

Samson’s Dark Side Of The Horse makes a comfortable visit back here on the 20th. It’s another counting-sheep and number-representation gag. I love the third panel’s artwork.

Mark Anderson’s Andertoons for the 22nd is a joke about motivating mathematics study. I believe I’ve mentioned this before, but there was a lovely bit on The Mary Tyler Moore Show along these lines once. Fantastically stupid newsman Ted Baxter was struggling to do some arithmetic until Murray Slaughter gave him the advice: “put a dollar sign in front of it”. Then he had the answer instantly.

Nat Fakes’s Break of Day for the 22nd brings back mathematics as signifier of the hardest homework a kid can have, or the toughest thing someone can have to think about. Fine enough stuff, although it isn’t really that stunning to think a parent might not understand what the kid’s homework is about. Often the point of an assignment is not to learn how to do something, but to encourage thinking about ways one could do something. That’s a hard assignment to create, and a harder one to do, and a very hard one to help with. As adults we get used to looking at problems as calculations to identify and do as swiftly as possible. That there is value in wandering around the slow routes needs remembering.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 22nd riffs on … I’m not sure exactly. The idea that the sort of meaningless nonsense that makes for good late-night dorm conversations when you’re 20 comes back around to being the cutting edge of theoretical physics, I suppose. It’s funny enough. A complaint often brought against the most cutting edge of theoretical physics is that it’s so abstract that there aren’t any conceivable tests that would say whether a calculation is right or not. In that condition mathematics and theoretical physics merge back into a thread of philosophy and its question of how can we know what it is for something to be true. Once we have a way of discerning whether an idea might happen to be true we’re ejected again from philosophy and into a science. And then the scientist makes a smug, snarky comment about the impossibility of testing philosophical conclusions.

Since the late 19th century much cutting-edge physics has involved counter-intuitive results. Often they have premises that strain intuition, as see relativity, or that seem to violate it altogether, as see quantum mechanics. But they turn out so very right so very often it’s hard not to feel excited and encouraged by this. Who wouldn’t look for a surprising and counter-intuitive explanation for the world as thrilling and maybe even right idea? I don’t blame anyone for looking to a wild idea like “what if the universe is made of math”. I don’t know what that would mean exactly unless we suppose we do live in a universe of Platonic Forms, in which case perfection runs counter-intuitively to me. I do understand being excited by the question. But the answers probably won’t be that much fun.

## Reading the Comics, June 29, 2016: Math Is Just This Hard Stuff, Right? Edition

We’ve got into that stretch of the year when (United States) schools are out of session. Comic Strip Master Command seems to have thus ordered everyone to clean out their mathematics gags, even if they didn’t have any particularly strong ones. There were enough the past week I’m breaking this collection into two segments, though. And the first segment, I admit, is mostly the same joke repeated.

Russell Myers’s Broom Hilda for the 27th is the type case for my “Math Is Just This Hard Stuff, Right?” name here. In fairness to Broom Hilda, mathematics is a lot harder now than it was 1,500 years ago. It’s fair not being able to keep up. There was a time that finding roots of third-degree polynomials was the stuff of experts. Today it’s within the powers of any Boring Algebra student, although she’ll have to look up the formula for it.

John McPherson’s Close To Home for the 27th is a bunch of trigonometry-cheat tattoos. I’m sure some folks have gotten mathematics tattoos that include … probably not these formulas. They’re not beautiful enough. Maybe some diagrams of triangles and the like, though. The proof of the Pythagoran Theorem in Euclid’s Elements, for example, includes this intricate figure I would expect captures imaginations and could be appreciated as a beautiful drawing.

Missy Meyer’s Holiday Doodles observed that the 28th was “Tau Day”, which takes everything I find dubious about “Pi Day” and matches it to the silly idea that we would somehow make life better by replacing π with a symbol for 2π.

Hilary Price’s Rhymes With Orange for the 29th uses mathematics as the way to sort out nerds. I can’t say that’s necessarily wrong. It’s interesting to me that geometry and algebra communicate “nerdy” in a shorthand way that, say, an obsession with poetry or history or other interests wouldn’t. It wouldn’t fit the needs of this particular strip, but I imagine that a well-diagrammed sentence would be as good as a page full of equations for expressing nerdiness. The title card’s promise of doing quadratic equations would have worked on me as a kid, but I thought they sounded neat and exotic and something to discover because they sounded hard. When I took Boring High School Algebra that charm wore off.

Aaron McGruder’s The Boondocks rerun for the 29th starts a sequence of Riley doubting the use of parts of mathematics. The parts about making numbers smaller. It’s a better-than-normal treatment of the problem of getting a student motivated. The strip originally ran the 18th of April, 2001, and the story continued the several days after that.

Bill Whitehead’s Free Range for the 29th uses Boring Algebra as an example of the stuff kinds have to do for homework.

## Reading the Comics, February 23, 2016: No Students Resist Word Problems Edition

This week Comic Strip Master Command ordered the mention of some of the more familiar bits of mathematical-premise stock that aren’t students resisting word problems. This happens sometimes.

Rick Stromoski’s Soup to Nutz for the 18th of February finds a fresh joke in the infinite-monkeys problem. Well, it uses a thousand monkeys here, but that hardly matters. If you had one long-enough-lived monkey at the typewriter, in principle, we could expect them to type the works of Shakespeare. It’s how long it takes that changes. In practice, it’s going to be too long to wait for anyway. I wonder if the monkeys will ever get computers to replace their typewriters.

Anyway, the point of the paradoxes is not something as trite as “silly Ancient Greeks didn’t understand calculus”. They had an awfully good understanding of what makes calculus work. The point is that either space and time are infinitely divisible or else they aren’t. Either possibility has consequences that challenge our intuitions of how space and time should work.

Dave Blazek’s Loose Parts for the 19th uses scientific notation. It’s a popular way to represent large (and small) numbers. It’s built on the idea that there are two interesting parts to a number: about how big it is, and what its leading values are. We use some base, nearly always 10, raised to a power to represent how big the number is. And we use the rest, a number between 1 and whatever the base is, to represent the leading values. Blazek’s channel 3 x 103 is just channel 3000, though. My satellite TV package has channels numbering from 6 up through 9999, although not all of them. Many are empty. Still, it would be a more excessive number of options if he were on channel 3 x 106, or 3,000,000.

Russell Myers’s Broom Hilda for the 22nd shows Nerwin trying to learn addition by using a real-world model. I tend to be willing to let people use whatever tool they find works to learn something. But any learning aid has its limits, and trying to get around them can be challenging, or just creepy.

Dave Whamond’s Reality Check for the 22nd is another version of that rounding-up joke that’s gone around Comic Strip Master Command, and your friends’ Facebook timelines, several times now. Well, I enjoy how suspicious the sheep up front are.

Rick Kirkman and Jerry Scott’s Baby Blues for the 23rd I include mostly because I wanted some pictures to include here. But mathematics is always a reliable choice when one needs scary school work to do. And I grant that fraction are particularly unsettling. There is something exotic in being told 1/2 is much bigger than 1/6, when one knows that 2 is so much smaller than 6. And just when one’s gotten comfortable with that, someone has you subtract one fraction from another.

In the olden days of sailors and shipping, the pay for a ship’s crew would be in shares of the take of the whole venture. The story I have read, but which I am not experienced enough to verify, depends on not understanding fractions. Naive sailors would demand rather than the offered 96th (or whatever) share of the revenues a 100th or 150th or even bigger numbers. Paymasters would pretend to struggle with before assenting to. Perhaps it’s so. Not understanding finance is as old as finance. But it does also feel like a legend designed to answer the question of when will someone need to know mathematics anyway.

David L Hoyt and Jeff Knurek’s Jumble for the 24th is not necessarily a mathematics comic. It could be philosophy or theology or possibly some other fields. Still, I imagine you can have fun working this out even if the final surprise-answer jumped out at me before I looked at the other words.

## Reading the Comics, August 3, 2015: Things That Make Me Cranky Edition

My edition name sounds cranky and I’m sorry for that. But the fact is a couple of the comics in this roundup did things that irritated me. I hope you don’t think worse of me when you’ve heard why they made me cranky.

Patrick Roberts’s Todd the Dinosaur (July 31) is a riff on the infinite-monkey problem, often discussed in the comics. Todd isn’t quite into the perfect randomness that the thought experiment wants. The strip does make me wonder if there have been any variations on the infinite monkey problem in which, instead of a series of randomly typed characters, random words are picked instead. On the one hand, there are many more possible words than there are letters every time something is to be typed. On the other hand, obvious nonsense like ‘gazurlnikov’ won’t turn up. But it’s easier to imagine a keyboard than it is a random pick of all the words in the language.

Lorie Ransom’s Daily Drawing (July 31) is some compass and protractor wordplay. Protractors aren’t part of the classical set of tools used for geometric proofs — compasses and straightedges alone do it — although they are convenient things to have. And they can be used to confirm hunches or refute possibilities, in much the same way trying out a specific case of a problem can guide one to solving a general problem.

Tony Carillo’s F Minus (August 1) makes a joke that I admit I don’t quite get. I think it’s trying to say that you get better pay with more mathematics training. That ought to be a nice affirmation of my chosen field’s value, although it comes across to me as snotty. For one, typically, more training in any field correlates with higher salaries. It’s not some magic that only mathematics has. For another, salary is not the measure of the worth of something, nor should it be.

Tom Thaves’s Frank and Ernest (August 1) tries to use up the mathematics puns for this installment.

Steve Breen and Mike Thompson’s Grand Avenue (August 2) is another entry in the “kids refusing mathematics for summer break” string of comics. And, of course, the kids display an ironic understanding of probability while trying to avoid Grandma’s mathematics workbooks. I’m on the kids’ side here, by the way. Previous summer installments have shown Grandma making the kids do tedious, boring, repetitive calculations that make me, now, not want to do mathematics. It’s in the name of getting them back in practice before the school year starts, but as depicted, it’s an attempt to crush all the joy of mathematics. At least working out best ways to hide is a use of probability that has some clear purpose and some fun to it. The daily strips for this week seem to be going in a different direction.

Dave Coverly’s Speed Bump (August 2) makes me wonder something I never thought about before. Would Romans see the symbols I, V, X, and so on as “one” and “five” and “ten” and so on? I mean, certainly they would in contexts where a number was expected. But if they just encountered the symbol without context, would they read it as the letter or as the number? I rate this as my favorite of this set of strips because it has given me something so fresh to ponder.

## Reading the Comics, July 7, 2015: Carrying On The Streak Edition

I admit I’ve been a little unnerved lately. Between the A To Z project and the flood of mathematics-themed jokes from Comic Strip Master Command — and miscellaneous follies like my WordPress statistics-reading issues — I’ve had a post a day for several weeks now. The streak has to end sometime, surely, right? So it must, but not today. I admit the bunch of comics mentioning mathematical topics the past couple days was more one of continuing well-explored jokes rather than breaking new territory. But every comic strip is somebody’s first, isn’t it? (That’s an intimidating thought.)

Disney’s Mickey Mouse (June 6, rerun from who knows when) is another example of the word problem that even adults can’t do. I think it’s an interesting one for being also a tongue-twister. I tend to think of this sort of problem as a calculus question, but that’s surely just that I spend more time with calculus than with algebra or simpler arithmetic.

And then Disney’s Donald Duck (June 6 also, but probably a rerun from some other date) is a joke built on counting sheep. Might help someone practice their four-times table, too. I like the internal logic of this one. Maybe I just like sheep in comic strips.

Eric Teitelbaum and Bill Teitelbaum’s Bottomliners (June 6) is a bit of wordplay based on the idiom that figures will “add up” if they’re correct. There are so many things one can do with figures, though, aren’t there? Surely something will be right.

Justin Thompson’s Mythtickle (June 6, again a rerun) is about the curious way that objects are mostly empty space. The first panel shows on the alien’s chalkboard legitimate equations from quantum mechanics. The first line describes (in part) a function called psi that describes where a particle is likely to be found over time. The second and third lines describe how the probability distribution — where a particle is likely to be found — changes over time.

Doug Bratton’s Pop Culture Shock Therapy (July 7) just name-drops mathematics as something a kid will do badly in. In this case the kid is Calvin, from Calvin and Hobbes. While it’s true he did badly in mathematics I suspect that’s because it’s so easy to fit an elementary-school arithmetic question and a wrong answer in a single panel.

The idea of mathematics as a way to bludgeon people into accepting your arguments must have caught someone’s imagination over at the Parker studios. Jeff Parker’s The Wizard of Id for July 7 uses this joke, just as Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. did back on June 19th. (Both comic strips were created by the prolific Johnny Hart. I was surprised to learn they’re not still drawn and written by the same teams.) As I mentioned at the time, smothering people beneath mathematical symbols is logically fallacious. This is not to say it doesn’t work.

## Reading the Comics, May 4, 2015: Hatless Aliens Edition

I have to make two confessions for this round of mathematics comic strips. First is that I was busy for like two days and missed about a jillion comic strips. So this is the first part of some catching-up to do. The second is that I don’t have a favorite of this bunch. The most interesting, I suppose, is the Mr Boffo, because it lets me get into a little trivia about Albert Einstein. But there’s not any in this bunch that made me smile much or that gave me a juicy topic to discuss. Maybe tomorrow.

Steve Breen and Mike Thompson’s Grand Avenue ran a week of snarky-answers-to-word-problems strips. April 28th, April 30th, and May 2nd featured mathematics questions. This must reflect how easy it is to undermine the logic of a mathematics question. The April 27th strip is about using Roman numerals, which I suppose is arithmetic. I’m not sure there’s much point to learning Roman numerals. We don’t do any calculations using the Roman numeral scheme except to show why Arabic numerals are better. All you get from Roman numerals is an ability to read building cornerstones and movie copyright dates. At least learning cursive handwriting provides the learner with a way to make illegible notes.

## Reading the Comics, February 4, 2015: Neutral Edition

Several of the comic strips that’ve been sent my way the past couple days touch on cultural neutrality in mathematics problems. People like to think of mathematics as a universal language, which makes me think of, for example, the quipu — twisted woolen cords with smaller cords tied to the main one — that Incans used to represent numbers. Even knowing the number one is supposed to represent doesn’t help me work out how to read the thing, and that’s not even doing calculations, just representing a number.

Darby Conley’s Get Fuzzy (February 1) uses several mathematics questions as part of a “general knowledge” quiz. Mathematics questions, particularly reasoning questions, are held up a good bit as examples of general knowledge since we’ve always cherished reasoning as a particularly precious sort of thinking, and because it’s easy to convince oneself that arithmetic and logic problems are culturally neutral. They’re not, but I would agree that “one times four” or the candy-counting problem are more culture-neutral than naming places with “-ham” or (to invent something not in the strip) identifying prime ministers of Canada would be. Really intriguing to me, though, is that Conley has Bucky Katt mention the Times as a newspaper without comics and the Daily News as one with: I had believed the strip to be set in or around Boston in the past, while this is pretty soundly a New York reference. Perhaps Conley’s let his daily comics lapse into reruns because he’s been moving, very slowly, across Connecticut?

Mac and Bill King’s Magic in a Minute (February 1) isn’t really a mathematics puzzle, but it does employ mathematical symbols in a way that I remember fondly from a bunch of “stories with holes” — superficially nonsensical problems which have logical resolutions if you can avoid being hobbled by implicit assumptions — so it’s really well-fitted for kids of the right age.

## Reading the Comics, January 24, 2015: Many, But Not Complicated Edition

I’m sorry to have fallen behind on my mathematics-comics posts, but I’ve been very busy wielding a cudgel at Microsoft IIS all week in the service of my day job. And since I telecommute it’s quite hard to convincingly threaten the server, however much it deserves it. Sorry. Comic Strip Master Command decided to send me three hundred billion gazillion strips, too, so this is going to be a bit of a long post.

Jenny Campbell’s Flo and Friends (January 19) is almost a perfect example of the use of calculus as a signifier of “something really intelligent people think of”. Which is flattening to mathematicians, certainly, although I worry that attitude does make people freeze up in panic when they hear that they have to take calculus.

The Amazing Yet Tautological feature of Ruben Bolling’s Super-Fun-Pak Comix (January 19) lives up to its title, at least provided we are all in agreement about what “average” means. From context this seems to be the arithmetic mean — that’s usually what people, mathematicians included mean by “average” if they don’t specify otherwise — although you can produce logical mischief by slipping in an alternate average, such as the “median” — the amount that half the results are less than and half are greater than — or the “mode” — the most common result. There are other averages too, but they’re not so often useful. On the 21st Super-Fun-Pak Comix returned with another installation of Chaos Butterfly, by the way.

## Reading The Comics, November 4, 2014: Will Pictures Ever Reappear Edition

I had assumed that at some point the good folks at Comics Kingdom would let any of their cartoonists do a panel that’s got mathematical content relevant enough for me to chat about, but apparently that’s just not happening. So for a third time in a row here’s a set of Gocomics-only comic strips, with reasonably stable links and images I don’t feel the need to include. Enjoy, please.

Fred Wagner’s Animal Crackers (October 26) presents an old joke — counting the number of animals by counting the number of legs and dividing by four — although it’s only silly because it’s hard to imagine a case where it’s easier to count the legs on a bunch of animals than it is to count the animals themselves. But if it’s the case that every animal has exactly four legs, then, there’s what’s called a one-to-one relationship between the set of animals and the set of animal legs: if you have some number of animals you have exactly four times that number of animal legs, and if you have some number of animal legs you have exactly one-fourth that number of animals, and you can count whatever’s the more convenient for you and use that to get what you’re really interested in. Showing such a one-to-one relationship exists between two interesting things can often be a start to doing more interesting problems, especially if you can show that the relationship also preserves some interesting interactions; if you have two ways to work out a problem, you can do the easier one.

Mark Anderson’s Andertoons (October 27) riffs on the place value for numbers written in the familiar Arabic style. As befitting a really great innovation, place value becomes invisible when you’re familiar with it; it takes a little sympathy and imagination to remember the alienness of the idea that a “2” means different things based on how many digits are to the right (or, if it’s a decimal, to the left) of it.

Anthony Blades’s charming Bewley (October 27) has one of the kids insisting that instinct alone is enough to do maths problems. The work comes out disastrously bad, of course, or there’d not be a comic strip. However, my understanding is that people do have some instinctive understanding even of problems that would seem to have little survival application. One test I’ve seen demonstrating this asks people to give, without thinking, their answer to whether a multiplication problem might be right or wrong. It’s pretty quick for most people to say that “7 times 9 equals 12” has to be wrong; to say that “7 times 9 equals 59” is wrong takes longer, and that seems to reflect an idea that 59 is, if not the right answer, at least pretty close to it. There’s an instinctive plausibility at work there and it’s amazing to think people should have that. Zach Weinersmith’s Saturday Morning Breakfast Cereal for October 31 circles around this idea, with one person having little idea what 1,892,491,287 times 7,798,721,415 divided by 82,493,726,631 might be, but being pretty sure that “4” isn’t it.

Saturday Morning Breakfast Cereal (October 30) also contains a mention of “cross products”, which are an interesting thing people learning vectors trip over. A cross product is defined for a pair of three-dimensional vectors, and the interesting thing is it’s a new vector that’s perpendicular to the two vectors multiplied together. The length of the cross product vector depends on the lengths of the two vectors multiplied together and the angle they make; the closer the two vectors multiplied together are, the smaller the cross product is, to the point that the cross product of two parallel vectors has length zero. The closer the two vectors multiplied together are to perpendicular the longer the cross product vector is.

More mysterious: if you swap the first vector and the second vector being cross-multiplied together, you get a cross product that’s the same size but pointing the opposite direction, pointing (say) down instead of up. Cross products have some areas where they’re particularly useful, especially in describing the movement of charged particles in magnetic fields.

(There’s something that looks a lot like the cross product which exists for seven-dimensional vectors, but I’ve never even heard of anyone who had a use for it, so, you don’t need to do anything about it.)

Eric the Circle (November 2), this one by “dDave”, presents the idea that that the points on a line might themselves be miniature Erics the Circle. What a line is made of is again one of those problems that straddles the lines between mathematics and philosophy. It seems to be one of the problems of infinity that Zeno’s Paradoxes outlined so perfectly thousands of years ago. To shorten it to the point it becomes misleading, is a line made up of things that have some width? If they’re infinitesimals, things with no width, then, how can an aggregate of things with no width come to have some width? But if they’re made up of things which have some width, how can there be infinitely many of them fitting into a finite space?

We can form good logical arguments about the convergence of infinite series — lining up, essentially, circles of ever-dwindling but ever-positive sizes so that the pile has a finite length — but that seems to suggest that space has to be made up of intervals of different widths, which seems silly; why couldn’t all the miniature circles be the same? In short, space is either infinitely divisible into identical things, or it is not, and neither one is completely satisfying.

Guy Gilchrist’s Nancy (November 2) uses math homework appearing in the clouds, although that’s surely because it’s easier to draw a division problem than it is to depict an assignment for social studies or English.

Todd Clark’s Lola (November 4) uses an insult-the-minor-characters variant of what seems to be the standard way of explaining fractions to kids, that of dividing a whole thing into smaller pieces and counting the number of smaller pieces. As physical interpretations of mathematical concepts goes I suppose that’s hard to beat.

## Reading the Comics, October 25, 2014: No Images Again Edition

I had assumed it was a freak event last time that there weren’t any Comics Kingdom strips with mathematical topics to discuss, and which comics I include as pictures here because I don’t know that the links made to them will work for everyone arbitrarily far in the future. Apparently they’re just not in a very mathematical mood this month, though. Such happens; I’m sure they’ll reappear soon enough.

John Zakour and Scott Roberts’ Working Daze (October 22, a “best of” rerun) brings up one of my very many peeves-regarding-pedantry, the notion that you “can’t give more than 100 percent”. It depends on what 100 percent means. The metaphor of “giving 110 percent” is based on the one-would-think-obvious point that there is a standard quantity of effort, which is the 100 percent, and to give 110 percent is to give measurably more than the standard effort. The English language has enough illogical phrases in it; we don’t need to attack ones that are only senseless if you go out to pick a fight with them.

Mark Anderson’s Andertoons (October 23) shows a student attacking a problem with appreciable persistence. As the teacher says, though, there’s no way the student’s attempts at making 2 plus 2 equal 5 is ever not going to be wrong, at least unless we have different ideas about what is meant by 2, plus, equals, and 5. It’s easy to get from this point to some pretty heady territory: since it’s true that two plus two can’t equal five (using the ordinary definitions of these words), then this statement is true not just everywhere in this universe but in all possible universes. This — indeed, all — arithmetic would even be true if there were no universe. But if something can be true regardless of what the universe is like, or even if there is no universe, then how can it tell us anything about the specific universe that actually exists? And yet it seems to do so, quite well.

Tim Lachowski’s Get A Life (October 23) is really an accounting joke, or really more a “taxes they so mean” joke, but I thought it worth mentioning that, really, the majority of the mathematics the world has done have got to have been for the purposes of bookkeeping and accounting. I’m sorry that I’m not better-informed about this so as to better appreciate what is, in some ways, the dark matter of mathematical history.

Keith Tutt and Daniel Saunders’s chipper Lard’s World Peace Tips (October 23) recommends “be a genius” as one of the ways to bring about world peace, and uses mathematics as the comic shorthand for “genius activity”, not to mention sudoku as the comic shorthand for “mathematics”. People have tried to gripe that sudoku isn’t really mathematics; while it’s not arithmetic, though — you could replace the numerals with letters or with arbitrary symbols not to be repeated in one line, column, or subsquare and not change the problem at all — it’s certainly logic.

John Graziano’s Ripley’s Believe It or Not (October 23) besides giving me a spot of dizziness with that attribution line makes the claim that “elephants have been found to be better at some numerical tasks than chimps or even humans”. I can believe that, more or less, though I notice it doesn’t say exactly what tasks elephants are so good (or chimps and humans so bad) at. Counting and addition or subtraction seem most likely, though, because those are processes it seems possible to create tests for. At some stages in human and animal development the animals have a clear edge in speed or accuracy. I don’t remember reading evidence of elephant skills before but I can accept that they surely have some.

Zach Weinersmith’s Saturday Morning Breakfast Cereal (October 24) applies the tools of infinite series — adding up infinitely many of a sequence of terms, often to a finite total — to parenting and the problem of one kid hitting another. This is held up as Real Analysis — – the field in which you learn why Calculus works — and it is, yeah, although this is the part of Real Analysis you can do in high school.

John Zakour and Scott Roberts’s Maria’s Day (October 25) picks up on the Math Wiz Monster in Maria’s closet mentioned last time I did one of these roundups. And it includes an attack on the “Common Core” standards, understandably: it’s unreasonable to today’s generation of parents that mathematics should be taught any differently from how it was taught to them, when they didn’t understand the mathematics they were being taught. Innovation in teaching never has a chance.

Dave Whamond’s Reality Check (October 25) reminds us that just because stock framing can be used to turn a subtraction problem into a word problem doesn’t mean that it can’t jump all the way out of mathematics into another field.

I haven’t included any comics from today — the 26th of October — in my reading yet but really, what are the odds there’s like a half-dozen comics of obvious relevance with nice, juicy topics to discuss?

## Reading the Comics, December 3, 2013

It’s been long enough for a couple more mathematics-themed comics to gather, so, let me share them with you. The comics easily available to me may be increasing, too, as dailyink.com has indicated they’re looking to make it easier for people who aren’t subscribers to their service to look at the daily strips. I’d be glad to include them back in my roundup of mathematics strips, at least when I see them making mathematics jokes; there’ve been surprisingly few of them lately. Maybe the King Features Syndicate artists know it’s generally too much effort for me to feature them for a joke about how silly word problems are and have been saving us both the trouble.

Frank Page’s Bob the Squirrel began a sequence November 20 with the kid Lauren doing her math homework and Bob the Squirrel, one of multiple imaginary squirrels which I follow on Twitter, helping. It starts with percentages, a concept I admit that other people find harder than I ever did, probably because the “per cent” just made it clear to me at a young age what the whole thing was about. On the 21st Bob claims to have known a squirrel named Algebra, which wouldn’t be the strangest name for a squirrel. “Algebra”, the word, isn’t drawn from anyone’s name; it’s instead drawn from the title of the book Hisab al-jabr w’al-muqabala, Kitab al-Jabr wa-l-Muqabala (“The Compendious Book On Calculation By Completion and Balancing”), written by Muḥammad ibn Mūsā al-Khwārizmī, whose name did give us the word “algorithm”, which is the kind of successful word-generating power that you usually expect only from obscure Swedish towns. Bob closes things off with your standard breaking-the-word-problem sort of joke.

## Reading the Comics, September 21, 2013

It must have been the summer vacation making comic strip artists take time off from mathematics-themed jokes: there’s a fresh batch of them a mere ten days after my last roundup.

John Zakour and Scott Roberts’s Maria’s Day (September 12) tells the basic “not understanding fractions” joke. I suspect that Zakour and Roberts — who’re pretty well-steeped in nerd culture, as their panel strip Working Daze shows — were summoning one of those warmly familiar old jokes. Well, Sydney Harris got away with the same punch line; why not them?

Brett Koth’s Diamond Lil (September 14) also mentions fractions, but as an example of one of those inexplicably complicated mathematics things that’ll haunt you rather than be useful or interesting or even understandable. I choose not to be offended by this insult of my preferred profession and won’t even point out that Koth totally redrew the panel three times over so it’s not a static shot of immobile talking heads.

## Reading the Comics, August 18, 2013

I’m sorry to have fallen silent so long; I was away from home and thought I’d be able to put up a couple of short pieces along the way, and turned out to be rather busy doing other things instead. It’s given me at least one nice problem with dramatic photographs to use in a near-future entry, though, so not all is lost (although I’m trying to think of a way to re-do the work in it that doesn’t involve quite so much algebra; I’m afraid of losing my readers and worse of making a hash of the LaTeX involved). Meanwhile, it’s been surprisingly close to a month since the last summary of comic strips with mathematical themes — I imagine the cartoonists are taking a break on Students In Classroom setups what with it being summer vacation across so much of the United States — so let me return to that.

## Reading the Comics, July 5, 2013

I’m surprised to discover it’s been over a month since I had a roster of mathematics-themed comic strips to share, but that’s how things happen to happen. It’s also been a month with repeated references to “finding square roots”, I suppose because that sounds like a really math-y thing to do. It’s certainly computationally challenging; the task of finding such is even a (very minor) moment in Isaac Asimov’s magnificent short story about arithmetic, “The Feeling Of Power”. I remember reading the procedure for finding them when I was a kid, and finding that with considerable effort, I was able to, though I’d probably refuse to do more than give a rough estimate of such a root nowadays.

Bill Watterson’s Calvin and Hobbes (June 4, rerun) is another entry in the long string of jokes about “why bother studying mathematics”, but Watterson’s craft lifts it above average. Admire that fourth panel: that’s every resistant student in one pose.

## Complex Experiments with Grading Mathematics

While I’ve never managed to attempt an experimental grading system as the one I enjoyed in Real Analysis, I have tried a few more modest experiments. The one chance I’ve had to really go wild and do something I’d never seen before, sadly, failed, but let me resurrect it enough to leave someone else, I hope, better-informed.

The setting was a summer course, which the department routinely gave to graduate students as a way of keeping them in the luxurious lifestyle to which grad students become accustomed. For five weeks and a couple days I’d spend several hours explaining the elements of vector calculus to students who either didn’t get it the first time around or who wanted to not have to deal with it during the normal term. (It’s the expansion of calculus to deal with integrals and differentials along curves, and across surfaces, and through solid bodies, and remarkably is not as impossibly complicated as this sounds. It’s probably easier to learn, once you know normal calculus, than it is to learn calculus to start. It’s essential, among other things, for working out physics problems in space, since it gives you the mathematical background to handle things like electric fields or the flow of fluids.)

What I thought was: the goal of the class is to get students to be proficient in a variety of techniques — that they could recognize what they were supposed to do, set up a problem to use whatever technique was needed, and could carry out the technique successfully. So why not divide the course up into all the things that I thought were different techniques, and challenge students to demonstrate proficiency in each of them? With experience behind me I understand at least one major objection to this, but if the forthcoming objection were to be dealt with, I’d still have blown it in the implementation.

## Real Experiments with Grading Mathematics

[ On an unrelated note I see someone’s been going through and grading my essays. I thank you, whoever you are; I’ll take any stars I can get. And I’m also delighted to be near to my 9,500th page view; I’ll try to find something neat to do for either 9,999 or 10,000, whichever feels like the better number. ]

As a math major I staggered through a yearlong course in Real Analysis. My impression is this is the reaction most math majors have to it, as it’s the course in which you study why it is that Calculus works, so it’s everything that’s baffling about Calculus only moreso. I’d be interested to know what courses math majors consider their most crushingly difficult; I’d think only Abstract Algebra could rival Real Analysis for the position.

While I didn’t fail, I did have to re-take Real Analysis in graduate school, since you can’t go on to many other important courses without mastering it. Remarkably, courses that sound like they should be harder — Complex Analysis, Functional Analysis and their like — often feel easier. Possibly this is because the most important tricks to studying these fields are all introduced in Real Analysis so that by the fourth semester around the techniques are comfortably familiar. Or Functional Analysis really is easier than Real Analysis.

The second time around went quite well, possibly because a class really is easier the second time around (I don’t have the experience in re-taking classes to compare it to) or possibly because I clicked better with the professor, Dr Harry McLaughlin at Rensselaer Polytechnic Institute. Besides giving what I think might be the best homework assignment I ever received, he also used a grading scheme that I really responded to well, and that I’m sorry I haven’t been able to effectively employ when I’ve taught courses.

His concept — I believe he used it for all his classes, but certainly he put it to use in Real Analysis — came from as I remember it his being bored with the routine of grading weekly homeworks and monthly exams and a big final. Instead, students could put together a portfolio, showing their mastery of different parts of the course’s topics. The grade for the course was what he judged your mastery of the subject was, based on the breadth and depth of your portfolio work.

Any slightly different way of running class is a source of anxiety, and he did some steps to keep it from being too terrifying a departure. First is that you could turn in a portfolio for a review as you liked mid-course and he’d say what he felt was missing or inadequate or which needed reworking. I believe his official policy was that you could turn it in as often as you liked for review, though I wonder what he would do for the most grade-grabby students, the ones who wrestle obsessively for every half-point on every assignment, and who might turn in portfolio revisions on an hourly basis. Maybe he had a rule about doing at most one review a week per student or something like that.

The other is that he still gave out homework assignments and offered exams, and if you wanted you could have them graded as in a normal course, with the portfolio grade being what the traditional course grade would be. So if you were just too afraid to try this portfolio scheme you could just pretend the whole thing was one of those odd jokes professors will offer and not worry.

I really liked this system and was sorry I didn’t have the chance to take more courses from him. The course work felt easier, no doubt partly because there was no particular need to do homework at the last minute or cram for an exam, and if you just couldn’t get around to one assignment you didn’t need to fear a specific and immediate grade penalty. Or at least the penalty as you estimated it was something you could make up by thinking about the material and working on a similar breadth of work to the assignments and exams offered.

I regret that I haven’t had the courage to try this system on a course I was teaching, although I have tried a couple of non-traditional grading schemes. I’m always interested in hearing of more, though, in case I do get back into teaching and feel secure enough to try something odd.

## Reblog: Mathematical Creativity: Multiple Solutions to the Pencil Sharpener Problem

TheGeometryTeacher has here the four kinds of results gotten from a class given a word problem (about the time needed for a certain event to occur). I like not just the original problem but the different approaches taken to the answer. It seems to me often lost to students, or at least poorly communicated to them, that nearly any interesting problem can be solved several ways over. Probably that’s a reflection of wanting to teach the most efficient way to do any particular problem, so showing more than one approach is judged a waste of time unless the alternate approach is feeding some other class objective.

Given the problem myself, I’d be inclined toward what’s here labelled as the “guessing and checking” approach, as I find a little experimentation like that helps me get to understand the workings of the problem pretty well. If the problem is small enough this might be all that I need to get to the answer. If it’s not, then the experience I get from a couple guesses and seeing why they don’t work would guide me to a more rigorous answer and one that looks more like the graph depicted.

Guessing and checking gets little respect, probably because when you’re trying to train the ability to calculate like “what is eight times seven” it’s hard to distinguish informed guessing from a complete failure to try. (The correct answer is, of course, “nobody knows”; the sevens and eights times tables are beyond human understanding.)

But when you’re venturing into original work for which there may be no guidance what a correct answer is (or whether there is one), or when you’re trying to do something for fun like figure out “What are the odds my roller coaster car will get stuck at the top of a ride like Top Thrill Dragster?” guessing and correcting from that original guess are often effective starting points.

I enjoy watching students exploring a problem that forces them to come up with their own structure for solving it. Today, a group got a chance to mess around with The Pencil Sharpener Problem, which is a problem I posted a month or so ago. (I’ll leave you to read it if you are curious what the problem is.)

From my perspective, what makes this problem interesting for the students is the ease with which it is communicated and the complexity with which is it solved. It seems quite easy. The answer is fairly predictable, but the students quickly found out that if they were going to solve this problem accurately, they were going to need two things:

1. A way to organize their thoughts and,

2. a way to verify their answer.

As long as the solution process included those two things, the students ended up fairly successful…

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## How Big Was West Jersey?

A book I’d read about the history of New Jersey mentioned something usable for a real-world-based problem in fraction manipulation, for a class which was trying to get students back up to speed on arithmetic on their way into algebra. It required some setup to be usable, though. The point is a property sale from the 17th century, from George Hutcheson to Anthony Woodhouse, transferring “1/32 of 3/90 of 90/100 shares” of land in the province of West Jersey. There were a hundred shares in the province, so, the natural question to build is: how much land was transferred?

The obvious question, to people who failed to pay attention to John T Cunningham’s This Is New Jersey in fourth grade, or who spent fourth grade not in New Jersey, or who didn’t encounter that one Isaac Asimov puzzle mystery (I won’t say which lest it spoil you), is: what’s West Jersey? That takes some historical context.

## Reading the Comics, November 11, 2012

Since Scott Adams’s Dilbert hasn’t done anything to deserve my scrutiny let me carry on my quest to identify all the comic strips that mention some mathematical thing. I’m leaving a couple out; for example, today (the 11th) Rob Harrell’s Adam @ Home and Bill Amend’s FoxTrot mentioned the alignment of digits in the date’s representation in numerals, but that seems too marginal, and yet here I am talking about it. I can’t be bothered coming up with rules I can follow for my own amusement here, can I?

## Reading the Comics, October 25, 2012

As before, this is going to be the comics other than those run through King Features Syndicate, since I haven’t found a solution I like for presenting their mathematics-themed comic strips for discussion. But there haven’t been many this month that I’ve seen either, so I can stick with gocomics.com strips for today at least. (I’m also a little irked that Comics Kingdom’s archives are being shut down — it’s their right, of course, but I don’t like having so many dead links in my old articles.) But on with the strips I have got.

## Reading the Comics, October 13, 2012

I suppose it’s been long enough to resume the review of math-themed comic strips. I admit there are weeks I don’t have much chance to write regular articles and then I feel embarrassed that I post only comic strips links, but I do enjoy the comics and the comic strip reviews. This one gets slightly truncated because King Features Syndicate has indeed locked down their Comics Kingdom archives of its strips, making it blasted inconvenient to read and nearly impossible to link to them in any practical, archivable way. They do offer a service, DailyInk.com, with their comic strips, but I can hardly expect every reader of mine to pay up over there just for the odd day when Mandrake the Magician mentions something I can build a math problem from. Until I work out an acceptable-to-me resolution, then, I’ll be dropping to gocomics.com and a few oddball strips that the Houston Chronicle carries.

## Reading the Comics, September 26, 2012

I haven’t time to write a short piece today so let me go through a fresh batch of math-themed comic strips instead. There might be a change coming to these features soon, both in the strips I read and in how I present them, since Comics Kingdom, which provides the King Features Syndicate comic strips, has shown signs that they’re tightening up web access to their strips.

I can’t blame them for wanting to make sure people go through paths they control — and, pay for, at least in advertising clicks — but I can fault them for doing a rotten job of it. They’re just not very good web masters, and end up serving strips — you may have seen them if you’ve gone to the comics page of your local newspaper — that are tiny, which kills plot-heavy features like The Phantom or fine-print heavy features like Slylock Fox Sunday pages, and loaded with referrer-based and cookie-based nonsense that makes it too easy to fail to show a comic altogether or to screw up hopelessly loading up several web browser tabs with different comics in them.

For now that hasn’t happened, at least, but I’m warning that if it does, I might not necessarily read all the King Features strips — their advertising claims they have the best strips in the world, but then, they also run The Katzenjammer Kids which, believe it or not, still exists — and might not be able to comment on them. We’ll see. On to the strips for the middle of September, though:

## Bad Luck on Deal Or No Deal

Mathstina, in a post from August 25, put put a video from the Australian version of Deal Or No Deal which showed a spectacularly unlucky contestant, a contestant unlucky enough to inspire word problems. I quite like game shows, partly because I was a kid in an era — the late 70s and early 80s — when the American daytime game show was at a creative and commercial peak, when one could reasonably expect to see novel shows on two or three networks from 9 am until 1 or 2 pm, and partly because they give many wonderful, easy-to-understand mathematics problems. Here’s one I based on the show and used as an exam problem.