Last week we started working with infinite geometric series, a topic I personally love.  First of all, it’s one of the few places in a high school curriculum where deep, genuine philosophical questions bubble all the way up to the surface of a mathematical discussion.  Second, it marks the place in my own academic life where I experienced a religious conversion to Orthodox Mathematicism:

In the beginning there was a single term.  And to that term the Teacher did add another of smaller magnitude.  Then a third term, smaller still, appeared upon the right hand side of the chalkboard, and it was revealed to me that the terms did decrease exponentially.  My heart saw that this shrinking and adding proceedeth forever and ever, terms without end, Amen.  And lo, when I beheld the sum, it was finite, and I knew that it was Good.

If my introduction to convergent series was a baptism, then using one to demonstrate that .999… = 1 was my confirmation.  Now, having done the same thing with my students, I think it might be even more interesting from this side of the desk.  In particular, two of their questions/comments highlight two very different understandings of infinity and the real numbers.

First, the ingredients of a metaphor.  If you’ve ever been a runner, this is easy.  If not, I’m going to need you to go on a quick jog before you read any farther so you can appreciate the rest of this carefully crafted rhetorical device.  I’ll wait…

When you drive the same stretch of road over and over again, you tend to experience it dynamically.  You pass a landmark, anticipate a curve, accelerate over a little rise.  The road changes in front of your eyes.  You see the road as a process.  But when you run along the same route, it looks completely different.  There is just this monolithic expanse of concrete laid out over the landscape.  You can creep around and explore its different features, but you experience the road essentially as a static object.  In other words, you experience the road as it actually is.  Keep this in mind as you read the following two questions from my actual students.

# D: “But Mr. Lusto, if .999… is exactly 1, then .999… plus .999… should equal exactly 2, but it doesn’t.  It’s 1.999…8.”

What a freaking fantastic argument!  Here’s a student who has accepted my proof, interpreted it, thought about it critically, and deduced a logical contradiction.  My heart swelled a little bit.  Unfortunately, the flaw in his reasoning highlights a fundamental misconception.  D is viewing .999… like a driver.  He sees it as a dynamic process, repeatedly appending a 9 to an ever-expanding sequence of 9s.  He might even accept that this can theoretically go on forever, but his point-of-view still gets him into some trouble.  When D mentally sums .999… and .999…, he’s suggesting that there are two “last 9s” that, when added, produce a trailing 8.  But of course there are no “last 9s.”  He’s implicitly terminated the process prematurely (which is to say, at all).  Hence his objection, though thoroughly beautiful, is ultimately illusory.

# J: “But Mr. Lusto, if .999… equals 1, then doesn’t 1.999… equal 2?  Then can’t we write every number in two different ways?”

This student views .999… like a runner.  The reason that .999… and 1 can be meaningfully thought of as equal is because they represent the same static value.  They’re just two different names for the same object.  Here’s a student who sees .999… as it actually is.  And now, because of that, his concern is genuine.  The fact that many real numbers have two decimal representations (one with infinite trailing 0s, one with infinite trailing 9s) is a true mathematical/philosophical problem.  In fact, it’s an important result: those sorts of numbers turn out to be dense in the reals (in the topological sense).  J may never care about, or even get enough math under his belt to understand, that statement,  but his view of the nature of infinity is already more nuanced than D’s.

Something to think about next time you’re driving.  Better yet, next time you’re running.

# Pretty Little Lies

I find myself lying to my students.  A lot.  I suppose it doesn’t much bother me on a moral level.  For one thing, my conscience is perhaps less muscular than it ought to be.  For another, I’m generally pretty open with my kids.  They know, for instance, that I’m divorced.  That I’m quitting smoking for my 30th birthday.  That for several years I was professionally violent.  I go out of my way to let them know that, within reason, I won’t shy away from their curiosity.  Still, I lie.

# Money for Nothing

In the conversation following a recent Mathalicious blog post, one commenter said:

“If we have evidence that current teachers are ineffective (and don’t international math test scores provide this evidence?), then why not let the non-educators take a shot?”

In reading through the comments, this particular point focused my attention because I suspect its primary sentiment runs deep within the larger discussion about the state of U.S. math education.  I.e., our international math ranking is poor; teachers must not be doing such a good job; so why not let somebody else take the wheel?

On the surface, this doesn’t seem a particularly unreasonable argument.  It does, however, make some tacit assumptions that are questionable, and even downright strange.  Namely:

# Teacher training is a negative-value-added process.

Any meaningful discourse about education has to address teacher preparation programs.  If teachers are truly failing en masse, then clearly something important and fundamental is lacking in their initial training.  Now of course these programs aren’t perfect.  They might not even be world-class.  But to suggest that non-educators would be better classroom teachers is to imply that I’m somehow a worse math teacher after year-long stint in grad school and months of practicum work and student teaching than I was the day I left the Marine Corps in search of a new career.  Which is ridiculous.  How can exposure to current research in pedagogical content knowledge, educational psychology, and legal/policy decisions make me worse?  How can active observation time in great, good, bad, and awful classrooms make me worse?  Hands-on experience with real students?  Regardless of the level of helpfulness of any of those things, I know that it’s strictly non-negative.  And please let me know if you can prove otherwise, because I’d like to sue for my \$25,000 back.

# The countries with better outcomes are being taught by non-educators.

If international math test scores suffice as proof of U.S. teachers’ ineffectiveness, then those countries with higher scores, one supposes, must provide some sort of evidence about effective practices.  Is Finland spanking us because it got its teachers out of the picture somehow?  Not quite.  Over there “teacher” routinely rates as the most admired profession among high school graduates; after a rigorous screening process, the most qualified teaching candidates are educated at government expense; all teachers are required to hold, at a minimum, a master’s degree (source for all the above).  Finland isn’t great in spite of its teachers; it just does a much better job of screening and training them.  Which all sounds easy and practical, but in order to replicate that in the U.S., our society’s view of the teaching profession would have to change dramatically.  You must believe your tax dollars are well spent paying pre-service teachers’ tuition.  You must believe teaching is an important and prestigious position.  You must believe that teachers shouldn’t have to take a vow of poverty to educate our children.  In short, the answer seems not to be getting non-educators into the game, but forcing über-educators into the game.

# People who devote all their professional time, energy, and resources to teaching can be called “non-educators.”

Like any other profession, education is vulnerable to a certain level of entrenchment.  Change can be slow and difficult, and new blood can’t hurt.  But if someone leaves a career in, say, mechanical engineering to spend all her working (and a whole lot of her non-working) life thinking, studying, worrying about, and practicing teaching math to high school students, then she has officially become an educator.  Because that’s what educators do.  But again, if the profession hopes to attract any of the best and brightest from other fields, then there is going to have to be a societal sea change that makes teaching a viable option for people who have gone to a whole lot of intellectual and financial trouble to become the best and brightest in the first place.  Those of us who have made those sacrifices in spite of the trouble/prestige ratio welcome you.

# War Games

Back in my previous existence as an artillery officer, I participated in the war for a little while.  Our main job—my Marines and I—was to provide counter-fire support for units in and around the city of Fallujah, Iraq.  Basically, whenever our guys started taking rocket and/or mortar fire, radar would track the source of those rounds and send us their point of origin as a target.  Then we would shoot at it.  Simple.  Kind of.

By the time I got to Fallujah, all the dumb bad guys had been selected out of the gene pool; the ones who were left knew that what they were doing was extremely risky, and they took steps to minimize that risk.  They tried their best to make every opportunity count, and our goal was to make it just as costly as possible for them to shoot at us.  It was a deadly serious game-theoretical problem for both sides.  A game measured in seconds.

# The (square) Root of Love

All right, fellas, huddle up.  We’re going to talk about the best way to find true love.  I mean, you can’t just go running around all willy-nilly hoping to bump into somebody great.  The world is a big place.  You need a strategy, man.  A dating plan of attack.

First, some ground rules, some general observations about romantic life, and a few restrictions in the interest of mathematical well-behavedness:

1. You are only going to meet a finite number of datable women over the course of your lifetime.  It will be a depressingly low number.
2. You are going to be an upstanding citizen and date only one woman at a time.
3. You will date a woman for some finite period of time, at which point you’ll make a decision either to pull the trigger and propose, or cut her loose.  Or, more likely, she’ll dump you first.
4. Once you propose, no takesies-backsies.  And once you cut a woman loose, you can’t ever reconsider her for marriage; she will hate you forever.
5. You are able to perfectly rank the women you have dated according to a strict, unambiguous order of preference.  Tie goes to the blonde.
6. You will encounter these women in random order.  That is, you are completely ignorant of where the next potential wife will stand in the overall rankings.
7. You will date a certain number of women without really considering any of them for a proposal.  In other words, you’ll take some time getting a feel for who’s out there.  Setting the bar.

In the world of mathematics, this is what’s known as an optimal stopping problem.  You’re going to date, and date, and date…, and stop.  Hopefully on the woman of your dreams (hence the optimal part).  In fact, this is one of those problems that’s so famous it goes by several (mildly sexist) names: the secretary problem, the sultan’s dowry problem, the fussy suitor problem.  Because it’s Valentine’s Day, we’ll call it the marriage problem.

# Smashmouth Mathematics

If it were physically possible to fold a piece of paper in half 50 times (it’s not), how thick would the resulting origami sculpture be?  Quick!  No fair calculating.  What does your gut say?

If you have absolutely no idea, I’ll tell you that a standard piece of printer paper, folded six times by high a school student with very little concern for symmetry or crease definition, has an average thickness somewhere between six and eight millimeters.  How much will that increase over the next 44 folds?  Any ideas?

# Computational Crisis

Let’s be clear: American software engineering is in crisis.  Thirty years ago our computer programs were the best in the world; now they routinely lag behind those from South Korea, Finland, China, and even…*gulp*…Canada.  In fact, a 2009 assessment found that U.S. reading software ranked 17th among the 34 OECD countries, math software a dismal 25th.  In the absence of radical reform, our code will cease to be competitive in an increasingly global economy, and we risk losing our preeminent place on the world stage.

In addition to poor test scores, American software has been suffering from increased feelings of alienation and disengagement.  In a survey from last year, 61% of programs said that they “strongly disliked” or “hated” compiling, and more than half said they would rather digitize Wuthering Heights than debug.  There’s no doubt the situation is dire.

But there is a solution.

# Greek to Me

I recently had a chance to do one of my favorite (and my students’ least favorite) things: talk about words in math class.  Math words.  I also had the opportunity to use one of my favorite math-teacher-type resources: a dictionary.  I don’t mean the glossary out of a math book, or a page from Wolfram MathWorld, or any one of the approximately 10.5 million web results (as of this writing) that the Google spits out when prompted with “math” + “dictionary.”  I’m talking about a good, old fashioned English dictionary, one of three left in my room by the previous English-teaching occupant: Webster’s Ninth New Collegiate, circa 1989.