Reading the Comics, February 2, 2016: Pre-Lottery Edition


So a couple weeks ago one of the multi-state lotteries in the United States reached a staggering jackpot of one and a half billion dollars. And it turns out that “a couple weeks” is about the lead time most syndicated comic strip artists maintain. So there’s a rash of lottery-themed comic strips. There’s enough of them that I’m going to push those off to the next Reading the Comics installment. I’ll make do here with what Comic Strip master Command sent us before thoughts of the lottery infiltrated folks’ heads.

Punkinhead: 'I was counting to five and couldn't remember what came after seven.' Tiger: 'If you're counting to five nothing comes after seven.' Punkinhead: 'I thought sure he would know.'
Bud Blake’s Tiger for the 28th of January, 2016. I do like Punkinhead’s look of dismay in the second panel that Tiger has failed him.

Bud Blake’s Tiger for the 28th of January (a rerun; Blake’s been dead a long while) is a cute one about kids not understanding numbers. And about expectations of those who know more than you, I suppose. I’d say this is my favorite of this essay’s strips. Part of that is that it reminds me of a bit in one of the lesser Wizard of Oz books. In it the characters have to count by twos to seventeen to make a successful wish. That’s the sort of problem you expect in fairy lands and quick gags.

Mort Walker’s Beetle Bailey (Vintage) from the 7th of July, 1959 (reprinted the 28th of January) also tickles me. It uses the understanding of mathematics as stand-in for the understanding of science. I imagine it’s also meant to stand in for intelligence. It’s also a good riff on the Sisyphean nature of teaching. The equations on the board at the end almost look meaningful. At least, I can see some resemblance between them and the equations describing orbital mechanics. Camp Swampy hasn’t got any obvious purpose or role today. But the vintage strips reveal it had some role in orbital rocket launches. This was in the late 50s, before orbital rockets worked.

General: 'How's your porject coming along to teach the men some science, Captain?' Captain: 'Wonderful, sir. Six months ago they didn't know what the square root of four was! Now they don't know what this [ blackboard full of symbols ] is!'
Mort Walker’s Beetle Bailey (Vintage) for the 7th of July, 1959. This is possibly the brightest I’ve ever seen Beetle, and he doesn’t know what he’s looking at.

Matt Lubchansky’s Please Listen To Me for the 28th of January is a riff on creationist “teach the controversy” nonsense. So we get some nonsense about a theological theory of numbers. Historically, especially in the western tradition, much great mathematics was done by theologians. Lazy histories of science make out religion as the relentless antagonist to scientific knowledge. It’s not so.

The equation from the last panel, F(x) = \mathcal{L}\left\{f(t)\right\} = \int_0^{\infty} e^{-st} f(t) dt , is a legitimate one. It describes the Laplace Transform of the function f(t). It’s named for Pierre-Simon Laplace. That name might be familiar from mathematical physics, astronomy, the “nebular” hypothesis of planet formation, probability, and so on. Laplace transforms have many uses. One is in solving differential equations. They can change a differential equation, hard to solve, to a polynomial, easy to solve. Then by inverting the Laplace transform you can solve the original, hard, differential equation.

Another major use that I’m familiar with is signal processing. Often we will have some data, a signal, that changes in time or in space. The Laplace transform lets us look at the frequency distribution. That is, what regularly rising and falling patterns go in to making up the signal (or could)? If you’ve taken a bit of differential equations this might sound like it’s just Fourier series. It’s related. (If you don’t know what a Fourier series might be, don’t worry. I bet we’ll come around to discussing it someday.) It might also remind readers here of the z-transform and yes, there’s a relationship.

The transform also shows itself in probability. We’re often interested in the probability distribution of a quantity. That’s what the possible values it might have are, and how likely each of those values is. The Laplace transform lets us switch between the probability distribution and a thing called the moment-generating function. I’m not sure of an efficient way of describing what good that is. If you do, please, leave a comment. But it lets you switch from one description of a thing to another. And your problem might be easier in the other description.

John McPherson’s Close To Home for the 30th of January uses mathematics as the sort of thing that can have an answer just, well, you see it. I suppose only geography would lend itself to a joke like this (“What state is Des Moines in?”)

Wally explains to the Pointy-Haired Boss that he's in the Zeno's Paradox phase of the project, in which 'every step we take gets us halfway closer to launch', a pace that he hopes 'it will look' like he's keeping up. First week in, he is.
Scott Adams’s Dilbert for the 31st of January. The link will probably expire around the end of February or start of March.

Scott Adams’s Dilbert for the 31st of January mentions Zeno’s Paradox, three thousand years old and still going strong. I haven’t heard the paradox used as an excuse to put off doing work. It does remind me of the old saw that half your time is spent on the first 90 percent of the project, and half your time on the remaining 10 percent. It’s absurd but truthful, as so many things are.

Samson’s Dark Side Of The Horse for the 2nd of February (I’m skipping some lottery strips to get here) plays on the merger of the ideas of “turn my life completely around” and “turn around 360 degrees”. A perfect 360 degree rotation would be an “identity tranformation”, leaving the thing it’s done to unchanged. But I understand why the terms merged. As with many English words or terms, “all the way around” can mean opposite things.

But anyone playing pinball or taking time-lapse photographs or just listening to Heraclitus can tell you. Turning all the way around does not leave you quite what you were before. People aren’t perfect at rotations, and even if they were, the act of breaking focus and coming back to it changes what one’s doing.

To Borrow A Page


If you’re very like me you wonder sometimes about subtraction, like, where it comes from and how people have thought about it over time. I’m particularly interested in different ideas of what negative numbers have meant, but my amateur standings in the history of mathematics keep me from easily finding what I want to know. I’m not seeking pity; I know many interesting things as it is.

Pat Ballew’s On This Day In Math Twitter recently posted the above. It links to an article about how subtraction has been represented in history, with a particular focus on ways borrowing has been taught.

I have a particular horror for the mathematics books quoted in it that demand people work out borrowing problems without the use of any extra marks. That is, if working out “5276 – 3739”, no fair writing the first number as “52716” along the way. I can accept that, to someone experienced with arithmetic, the writing out of borrowing steps is unnecessary. And the steps do make for a pretty cluttered page. But it seems to me that especially in the learning stage this sort of false work is essential. Any new skill is hard, and it’s worth making some mess to be sure nothing essential is left out.

Ballew also mentions a fascinating point. The ordinary homeworks and assignments and preparation papers for teachers may have found their way into public libraries. These could be great guides to the ways people actually did calculations, or learned how to do calculations, in past eras. I don’t know how much material there is, or how useful it is. I confess that while I love mathematical history, it is a remote love.

A Summer 2015 Mathematics A To Z: step


Step.

On occasion a friend or relative who’s got schoolkids asks me how horrified I am by some bit of Common Core mathematics. This is a good chance for me to disappoint the friend or relative. Usually I’m just sincerely not horrified. Much of what raises horror is students being asked to estimate and approximate answers. This is instead of calculating the answer directly. But I like estimation and approximation. If I want an exact answer I’ll do better to use a calculator. What I need is assurance the thing I’m calculating can sensibly be the thing I want to know. Nearly all my feats of mental arithmetic amount to making an estimate. If I must I improve it until someone’s impressed.

The other horror-raising examples I get amount to “look at how many steps it takes to do this simple problem!” The ones that cross my desk are usually subtraction problems. Someone’s offended the student is told to work out 107 minus 18 (say) by counting by ones from 18 up to 20, then by tens from 20 up to 100, and then by ones again up to 107. And this when they could just write one number above another and do some borrowing and get 89 right away, no steps needed. Assuring my acquaintance that the other method is really just the way you might count change, and that I do subtraction that way much of the time, doesn’t change minds. (More often I do that to double-check my answer. This raises the question of why I don’t do it that way the first time.) Though it does make the acquaintance conclude I’m some crazy person with no idea how to teach kids.

That’s probably fair. I’ve never taught elementary school students, and haven’t any training for it. I’ve only taught college students. For that my entire training consisted of a single one-credit course my first semester as a Teaching Assistant, plus whatever I happened to pick up while TAing for professors who wanted me to sit in on lecture. From the first I learned there is absolutely no point to saying anything while I face the chalkboard because it will be unheard except by the board, which has already been through this class forty times. From the second I learned to toss hard candies as reward to anyone who would say anything, anything, in class. Both are timeless pedagogical truths.

But the worry about the number of steps it takes to do some arithmetic calculation stays with me. After all, what is a step? How much work is it? How hard is a step?

I don’t think there is a concrete measure of hardness. I’m not sure there could be. If I needed to, I’d work out 107 minus 18 by noticing it’s just about 110 minus 20, so it’s got to be about 90, and a 7 minus 8 has to end in a 9 so the answer must be 89. How many steps was that? I guess there are maybe three thoughts involved there. But I don’t do that, at least not deliberately, when I look at the problem. 89 just appears, and if I stay interested in the question, the reasons why that’s right follow in short order. So how many steps did I take? Three? One?

On the other hand, I know that in elementary school I would have had to work it out by looking at 7 minus 8. And then I’d need to borrow from the tens column. And oh dear there’s a 0 to the left of the 7 so I have to borrow from the hundreds and … That’s the procedure as it was taught back then. Now, I liked that. I understood it. And I was taught with appeals to breaking dollars into dimes and pennies, which worked for my imagination. But it’s obviously a bunch of steps. How many? I’m not sure; probably around ten or so. And, if we’re being honest, borrowing from a zero in the tens column is a deeply weird thing to do. I can understand people freezing up rather than do that.

Similarly, I know that if I needed to differentiate the logarithm of the cosine of x, I would have the answer in a flash. It’d be at most one step. If I were still in high school, in my calculus class, I’d need longer. I’d struggle through the chain rule and some simplifications after that. Call it maybe four or five steps. If I were in elementary school I’d need infinitely many steps. I couldn’t even understand the problem except in the most vague, metaphoric way.

This leads me to my suggestion for what a “step” is, at least for problems you work out by hand. (Numerical computing has a more rigorous definition of a step; that’s when you do one of the numerical processing operations.) A step is “the most work you can do in your head without a significant chance of making a mistake”. I think that’s a definition that clarifies the problem of counting steps. It will be different for different people. It will be different for the same person, depending on how experienced she is. The steps a newcomer has to a subject are smaller than the ones an expert has. And it’s not just that newcomer takes more steps to get to the same conclusion than the expert does. The expert might imagine the problem breaks down into different steps from the ones a newcomer can do. Possibly the most important skill a teacher has is being able to work out what the steps the newcomer can take are. These will not always be what the expert thinks the smaller steps would be.

But what to do with problem-solving approaches that require lots of steps? And here I recommend one of the wisest pieces of advice I’ve ever run across. It’s from the 1954 Printer 1 & C United States Navy Training Course manual, NavPers 10458. I apologize if I’m citing it wrong, but I hope people can follow that to the exact document. I have it because I’m interested in Linotype operation is why. From page 308, the section “Don’t Overlook Instructions” in Chapter 7:

When starting on a new piece of copy, or “take” is it is called, be sure to read all instructions, such as the style and size of type, the measure to be set, whether it is to be leaded, indented, and so on.

Then go slowly. Try to develop even, rhythmic strokes, rather than quick, sporadic motions. Strive for accuracy rather than speed. Speed will come with practice.

As with Linotype operations, so it is with arithmetic. Be certain you are doing what you mean to do, and strive to do it accurately. I don’t know how many steps you need, but you probably won’t get a wrong answer if you take more than the minimum number of steps. If you take fewer steps than you need the results will be wretched. Speed will come with practice.

Reading the Comics, June 11, 2015: Bonus Education Edition


The coming US summer vacation suggests Comic Strip Master Command will slow down production of mathematics-themed comic strips. But they haven’t quite yet. And this week I also found a couple comics that, while not about mathematics, amused me enough that I want to include them anyway. So those bonus strips I’ll run at the end of my regular business here.

Bill Hinds’s Tank McNamara (June 6) does a pi pun. The pithon mathematical-snake idea is fun enough and I’d be interested in a character design. I think the strip’s unjustifiably snotty about tattoos. But comic strips have a strange tendency to get snotty about other forms of art.

A friend happened to mention one problem with tattoos that require straight lines or regular shapes is that human skin has a non-flat Gaussian curvature. Yes, that’s how the friend talks. Gaussian curvature is, well, a measure of how curved a surface is. That sounds obvious enough, but there are surprises: a circular cylinder, such as the label of a can, has the same curvature as a flat sheet of paper. You can see that by how easy it is to wrap a sheet of paper around a can. But a ball hasn’t, and you see that by how you can’t neatly wrap a sheet of paper around a ball without crumpling or tearing the paper. Human skin is kind of cylindrical in many places, but not perfectly so, and it changes as the body moves. So any design that looks good on paper requires some artistic imagination to adapt to the skin.

Bill Amend’s FoxTrot (June 7) sets Jason and Marcus working on their summer tans. It’s a good strip for adding to the cover of a trigonometry test as part of the cheat-sheet.

Dana Simpson’s Phoebe and her Unicorn (June 8) makes what I think is its first appearance in my Reading the Comics series. The strip, as a web comic, had been named Heavenly Nostrils. Then it got the vanishingly rare chance to run as a syndicated newspaper comic strip. And newspaper comics page editors don’t find the word “nostril” too inherently funny to pass up. Thus the more marketable name. After that interesting background I’m sad to say Simpson delivers a bog-standard “kids not understanding fractions” joke. I can’t say much about that.

Ruben Bolling’s Super Fun-Pak Comix (June 10, rerun) is an installment of everyone’s favorite literary device model of infinite probabilities. A Million Monkeys At A Million Typewriters subverts the model. A monkey thinking about the text destroys the randomness that it depends upon. This one’s my favorite of the mathematics strips this time around.

And Dan Thompson’s traditional Brevity appearance is the June 11th strip, an Anthropomorphic Numerals joke combining a traditional schoolyard gag with a pun I didn’t notice the first time I read the panel.


And now here’s a couple strips that aren’t mathematical but that I just liked too much to ignore. Also this lets Mark Anderson’s Andertoons get back on my page. The June 10th strip is a funny bit of grammar play.

Percy Crosby’s Skippy (June 6, rerun from sometime in 1928) tickles me for its point about what you get at the top and the bottom of the class. Although tutorials and office hours and extracurricular help, and automated teaching tools, do customize things a bit, teaching is ultimately a performance given to an audience. Some will be perfectly in tune with the performance, and some won’t. Audiences are like that.

Lewis Carroll and my Playing With Universes


I wanted to explain what’s going on that my little toy universes with three kinds of elements changing to one another keep settling down to steady and unchanging distributions of stuff. I can’t figure a way to do that other than to introduce some actual mathematics notation, and I’m aware that people often find that sort of thing off-putting, or terrifying, or at the very least unnerving.

There’s fair reason to: the entire point of notation is to write down a lot of information in a way that’s compact or easy to manipulate. Using it at all assumes that the writer, and the reader, are familiar with enough of the background that they don’t have to have it explained at each reference. To someone who isn’t familiar with the topic, then, the notation looks like symbols written down without context and without explanation. It’s much like wandering into an Internet forum where all the local acronyms are unfamiliar, the in-jokes are heavy on the ground, and for some reason nobody actually spells out Dave Barry’s name in full.

Let me start by looking at the descriptions of my toy universe: it’s made up of a certain amount of hydrogen, a certain amount of iron, and a certain amount of uranium. Since I’m not trying to describe, like, where these elements are or how they assemble into toy stars or anything like that, I can describe everything that I find interesting about this universe with three numbers. I had written those out as “40% hydrogen, 35% iron, 25% uranium”, for example, or “10% hydrogen, 60% iron, 30% uranium”, or whatever the combination happens to be. If I write the elements in the same order each time, though, I don’t really need to add “hydrogen” and “iron” and “uranium” after the numbers, and if I’m always looking at percentages I don’t even need to add the percent symbol. I can just list the numbers and let the “percent hydrogen” or “percent iron” or “percent uranium” be implicit: “40, 35, 25”, for one universe’s distribution, or “10, 60, 30” for another.

Letting the position of where a number is written carry information is a neat and easy way to save effort, and when you notice what’s happening you realize it’s done all the time: it’s how writing the date as “7/27/14” makes any sense, or how a sports scoreboard might compactly describe the course of the game:

0 1 0   1 2 0   0 0 4   8 13 1
2 0 0   4 0 0   0 0 1   7 15 0

To use the notation you need to understand how the position encodes information. “7/27/14” doesn’t make sense unless you know the first number is the month, the second the day within the month, and the third the year in the current century, and that there’s an equally strong convention putting the day within the month first and the month in the year second presents hazards when the information is ambiguous. Reading the box score requires knowing the top row reflects the performance of the visitor’s team, the bottom row the home team, and the first nine columns count the runs by each team in each inning, while the last three columns are the total count of runs, hits, and errors by that row’s team.

When you put together the numbers describing something into a rectangular grid, that’s termed a matrix of numbers. The box score for that imaginary baseball game is obviously one, but it’s also a matrix if I just write the numbers describing my toy universe in a row, or a column:

40
35
25

or

10
60
30

If a matrix has just the one column, it’s often called a vector. If a matrix has the same number of rows as it has columns, it’s called a square matrix. Matrices and vectors are also usually written with either straight brackets or curled parentheses around them, left and right, but that’s annoying to do in HTML so please just pretend.

The matrix as mathematicians know it today got put into a logically rigorous form around 1850 largely by the work of James Joseph Sylvester and Arthur Cayley, leading British mathematicians who also spent time teaching in the United States. Both are fascinating people, Sylvester for his love of poetry and language and for an alleged incident while briefly teaching at the University of Virginia which the MacTutor archive of mathematician biographies, citing L S Feuer, describes so: “A student who had been reading a newspaper in one of Sylvester’s lectures insulted him and Sylvester struck him with a sword stick. The student collapsed in shock and Sylvester believed (wrongly) that he had killed him. He fled to New York where one os his elder brothers was living.” MacTutor goes on to give reasons why this story may be somewhat distorted, although it does suggest one solution to the problem of students watching their phones in class.

Cayley, meanwhile, competes with Leonhard Euler for prolific range in a mathematician. MacTutor cites him as having at least nine hundred published papers, covering pretty much all of modern mathematics, including work that would underlie quantum mechanics and non-Euclidean geometry. He wrote about 250 papers in the fourteen years he was working as a lawyer, which would by itself have made him a prolific mathematician. If you need to bluff your way through a mathematical conversation, saying “Cayley” and following it with any random noun will probably allow you to pass.

MathWorld mentions, to my delight, that Lewis Carroll, in his secret guise as Charles Dodgson, came in to the world of matrices in 1867 with an objection to the very word. In writing about them, Dodgson said, “”I am aware that the word `Matrix’ is already in use to express the very meaning for which I use the word `Block’; but surely the former word means rather the mould, or form, into which algebraical quantities may be introduced, than an actual assemblage of such quantities”. He’s got a fair point, really, but there wasn’t much to be done in 1867 to change the word, and it’s only gotten more entrenched since then.

Reading the Comics, April 21, 2014: Bill Amend In Name Only Edition


Recently the National Council of Teachers of Mathematics met in New Orleans. Among the panelists was Bill Amend, the cartoonist for FoxTrot, who gave a talk about the writing of mathematics comic strips. Among the items he pointed out as challenges for mathematics comics — and partly applicable to any kind of teaching of mathematics — were:

  • Accessibility
  • Stereotypes
  • What is “easy” and “hard”?
  • I’m not exactly getting smarter as I age
  • Newspaper editors might not like them

Besides the talk (and I haven’t found a copy of the PowerPoint slides of his whole talk) he also offered a collection of FoxTrot comics with mathematical themes, good for download and use (with credit given) for people who need to stock up on them. The link might be expire at any point, note, so if you want them, go now.

While that makes a fine lead-in to a collection of mathematics-themed comic strips around here I have to admit the ones I’ve seen the last couple weeks haven’t been particularly inspiring, and none of them are by Bill Amend. They’ve covered a fair slate of the things you can write mathematics comics about — physics, astronomy, word problems, insult humor — but there’s still interesting things to talk about. For example:

Continue reading “Reading the Comics, April 21, 2014: Bill Amend In Name Only Edition”

The Mathematics Of A Pricing Game


There was a new pricing game that debuted on The Price Is Right for the start of its 42nd season, with a name that’s designed to get my attention: it’s called “Do The Math”. This seems like a dangerous thing to challenge contestants to do since the evidence is that pricing games which depend on doing some arithmetic tend to be challenging (“Grocery Game”, “Bullseye”), or confusing (“The Check Game”), or outright disasters (“Add Em Up”). This one looks likely to be more successful, though.

The setup is this: The contestant is shown two prizes. In the first (and, so far, only) playing of the game this was a 3-D HDTV and a motorcycle. The names of those prizes are put on either side of a monitor made up to look like a green chalkboard. The difference in prize values is shown; in this case, it was $1160, and that’s drawn in the middle of the monitor in Schoolboard Extra-Large font. The contestant has to answer whether the price of the prize listed on the left (here, the 3-D HDTV) plus the cash ($1160) is the price of the prize on the right (the motorcycle), or whether the price of the prize on the left minus the cash is the price of the prize on the right. The contestant makes her or his guess and, if right, wins both prizes and the money.

There’s not really much mathematics involved here. The game is really just a two-prize version of “Most Expensive” (in which the contestant has to say which of three prizes and then it’s right there on the label). I think there’s maybe a bit of educational value in it, though, in that by representing the prices of the two prizes — which are fixed quantities, at least for the duration of taping, and may or may not be known to the contestant — with abstractions it might make people more comfortable with the mathematical use of symbols. x and all the other letters of the English (and Greek) alphabets get called into place to represent quantities that might be fixed, or might not be; and that might be known, or might be unknown; and that we might actually wish to know or might not really care about but need to reference somehow.

That conceptual leap often confuses people, as see any joke about how high school algebra teachers can’t come up with a consistent answer about what x is. This pricing game is a bit away from mathematics classes, but it might yet be a way people could see that the abstraction idea is not as abstract or complicated as they fear.

I suspect, getting away from my flimsy mathematics link, that this should be a successful pricing game, since it looks to be quick and probably not too difficult for players to get. I’m sorry the producers went with a computer monitor for the game’s props, rather than — say — having a model actually write plus or minus, or some other physical prop. Computer screens are boring television; real objects that move are interesting. There are some engagingly apocalyptic reviews of the season premiere over at golden-road.net, a great fan site for The Price Is Right.

What Is Calculus I Like?


Although I haven’t got a mathematics class to teach this term, at least not right now, I have thought a bit about it and realized that I’ve surprisingly missed a nearly universal affair: I haven’t had a Calculus I course, the kind taught in big lecture halls capable of seating hundreds of students, literally several of whom are awake and alert and paying attention. The closest I’ve come is a history-of-computation course, with a nominal enrollment of about 130 students, and a similarly sized Introduction to C; but the big mathematics course college students are supposed to get through so they learn they really don’t like calculus, I haven’t done. While I was teaching assistant for some Calculus I courses, I never had professors who wanted me to attend lecture as a regular thing, and I just came in to do recitations.

More, I never had Calculus I as a student. I was in a magnet program in high school that got me enough advanced placement credit that I skipped pretty near the whole freshman year of the mathematics major sequence, and I could jump right into the courses with 30-to-40 student enrollments like Vector Calculus and Introduction to Differential Equations. That was great for me, but it’s finally struck me that I missed a pretty big, pretty common experience.

So I’m curious what it’s like: what the experience is, what students are expecting from their professors, what professors expect from students, how those expectations clash. I know the sorts of class methods I liked as a student and that I like as an instructor, but not how well that fits the attempt to teach a hundred-plus students who are just there because the school requires the passing of some mathematics courses and this is the one they offer 140 sections of.

Probability Spaces and Plain English


Lucas Wilkins over on the blog Jellymatter writes an article which starts from a grand old point: being annoyed by something on Wikipedia. In this case, it’s Wikipedia’s entry on the axioms of probability, which, like many Wikipedia entries on mathematical subjects is precise, correct, and useless.

Why useless? Because while the entry does draw from a nice, logically rigorous introduction to the way probability is defined, it’s done by way of measure theory, a mildly exotic field of mathematics — I didn’t get my toes wet in it until my senior year as a math major, and didn’t do any serious work with it until grad school — for a subject, probability, that an eight-year-old could reasonably be expected to study. (Measure theory gets called in for a number of tasks; in my grad school career, its biggest job was rebuilding integral calculus, compared to what I’d learned in high school and as an undergraduate, for greater analytic power. So, yes, calculus can be done harder.)

Wilkins goes on to explain the same topic but in plain English, to what seems to me great effect, including an introduction to measure theory that won’t make Wikipedia’s precise-but-curt definition make sense, but will leave someone better-prepared to read it.

Where Do Negative Numbers Come From?


Some time ago — and I forget when, I’m embarrassed to say, and can’t seem to find it because the search tool doesn’t work on comments — I was asked about how negative numbers got to be accepted. That’s a great question, particularly since while it seems like the idea of positive numbers is probably lost in prehistory, negative numbers definitely progressed in the past thousand years or so from something people might wildly speculate about to being a reasonably comfortable part of daily mathematics.

While searching for background information I ran across a doctoral thesis, Making Sense Of Negative Numbers, which is uncredited in the PDF I just linked to but appears to be by Dr Cecilia Kilhamn, of the University of Gothenburg, Sweden. Dr Kilhamn’s particular interest (here) is in how people learn to use negative numbers, so most of the thesis is about the conceptual difficulties people have when facing the minus sign (not least because it serves two roles, of marking a number as negative and of marking the subtraction operation), but the first chapters describe the historical process of developing the concept of negative numbers.

Continue reading “Where Do Negative Numbers Come From?”

How Long Is A Bad Ellipse Question?


Something came to mind while thinking about that failed grading scheme for multivariable calculus. I’d taught it two summers, and the first time around — when I didn’t try the alternate grading scheme — I made what everyone assured me was a common mistake.

One of the techniques taught in multivariable calculus is how to compute the length of a curve. There are a couple of ways of doing this, but you can think of them as variations on the same idea: imagine the curve as a track, and imagine that there’s a dot which moves along that track over some stretch of time. Then, if you know how quickly the dot is moving at each moment in time, you can figure out how long the track is, in much the same way you’d know that your parents’ place is 35 miles away if it takes you 35 minutes of travelling at 60 miles per hour to get there. There are details to be filled in here, which is why this is fit in an advanced calculus course.

Anyway, the introduction of this, and the homeworks, start out with pretty simple curves — straight lines, for example, or circles — because they’re easy to understand, and the student can tell offhand if the answer she got was right, and the calculus involved is easy. You can focus energy on learning the concept instead of integrating bizarre or unpleasant functions. But this also makes it harder to come up with a fresh problem for the exams: the student knowing how to find the length of a parabola segment or the circumference of a circle might reflect mastering the idea, or just that they remembered it from class.

So for the exam I assigned a simple variant, something we hadn’t done in class but was surely close enough that I didn’t need to work the problem out before printing up and handing out the exams. I’m sure it will shock you that an instructor might give out on an exam a problem he hasn’t actually solved already, but, I promise you, sometimes even teachers who aren’t grad students taking summer courses will do this. Usually it’s all right. Here’s where it wasn’t.

Continue reading “How Long Is A Bad Ellipse Question?”

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.

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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.

Mr. Shauver - Learner Educator

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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What is .19 of a bathroom?


I’ve had a little more time attempting to teach probability to my students and realized I had been overlooking something obvious in the communication of ideas such as the probability of events or the expectation value of a random variable. Students have a much easier time getting the abstract idea if the examples used for it are already ones they find interesting, and if the examples can avoid confusing interpretations. This is probably about 3,500 years behind the curve in educational discoveries, but at least I got there eventually.

A “random variable”, here, sounds a bit scary, but shouldn’t. It means that the variable, for which x is a popular name, is some quantity which might be any of a collection of possible values. We don’t know for any particular experiment what value it has, at least before the experiment is done, but we know how likely it is to be any of those. For example, the number of bathrooms in a house is going to be one of 1, 1.5, 2, 2.5, 3, 3.5, up to the limits of tolerance of the zoning committee.

The expectation value of a random variable is kind of the average value of that variable. You find it by taking the sum of each of the possible values of the random variable times the probability of the random variable having that value. This is at least for a discrete random variable, where the imaginable values are, er, discrete: there’s no continuous ranges of possible values. Number of bathrooms is clearly discrete; the number of seconds one spends in the bathroom is, at least in principle, continuous. For a continuous random variable you don’t take the sum, but instead take an integral, which is just a sum that handles the idea of infinitely many possible values quite well.

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