Author Archives: Chris

Reality Check

In accordance with the 7th Commandment of Blogging (If thy comment exceedeth two cubits in length, thou shalt write thine own damn post.), here is my personal response to Dan's latest question.  To be clear: even though the question was put to Mathalicious generally, and even though I bodily occupy a nontrivial fraction of that particular organization, it would be presumptuous of me to write anything approaching an official opinion, especially given the humbling brains attached to the people I spend my days with.  But I have thoughts.

We do thump the real-world drum pretty steadily around the office.  We have, as it's known in the biz, a niche.  But what exactly constitutes 'real-world' is an interesting question.  I don't think it's a particularly important question, but it's interesting insofar as it informs the practical decisions I make about what kinds of tasks I try to author, and what kinds of tasks I leave to other smart people, in other well-appointed niches.  And insofar as that term appears with some regularity in the CCSS.

A Line in the Sand

There's a philosophically defensible sense in which nothing we'd call a mathematical object is real (or, for that matter, an object).  They're abstract and causally independent and yadda yadda yadda.  There's another philosophically defensible sense in which everything we'd call a mathematical object is real.  Actually, a couple of senses, with varying definitions of reality.  And of course there are positions in between.  You could spend lifetime trying to untangle all the competing ideas of what (or whether) a number is.  And basically who cares.  In my mind, there's a simple way to draw a (note: not the) line on the curricular map between That Which Is Real and That Which Is Not:

Is this question self-referential?

In other words, are we using math to examine itself, or are we using it to inspect something outside its own borders?  So my working definition of 'real-world' math is a mathematical task that is not self-referential.  Which means, and I suppose you saw this coming, that my answer to Dan's question is none of the above.  None of those problems is real-world in any appreciable sense.  They are all questions about a circle, a square, and their respective areas: math looking at math.  A and B are obvious.  C and D are just promising hypothetical candy (which is the absolute worst kind of candy) for solving A or B.  I suppose E and F are both dipping their toes into the real-world, but self-consciously.

Okay, so those are some counterexamples.  Maybe a countercounterexample will help shed light on my litmus test.

For Instance...

Here are two problems:

1.  Is it true in general that P(A|B) = P(B|A)?  If not, can you express P(A|B) in terms of P(B|A)?

2.  Should innocent people be worried about the NSA's PRISM program?  How much should they be worried?

By my definition, the second question is 'real-world' and the first one is not, because the first question is mathematically self-referential (it's a question about a mathematical relationship phrased in mathematical language) and the second one is not (it's a question about personal liberty and national security phrased in natural language).  Of course they are the same question, and they are both excellent.  But from my chair, the very fact that those questions are the same is so non-obvious that connecting them requires a profound act of mathematical thinking.  Also, you get to do some really good math qua math.  I find that both professionally compelling and pedagogically useful.  That's why I do what I do.  Without speaking too forcefully for the rest of the team, I think that's part of the reason we do what we do.

Unpacking Circles

As I understand Dan's position (or at  least one particular aspect of Dan's position), there's no reason to create a distinction between, e.g., a circle's reality and the reality of health insurance.  In fact, for kids, a circle may be real (Platonist objections notwithstanding) in a much clearer and more visceral way.  And from there it's not a particularly ambitious leap to extend this reasoning such that all of mathematics can be considered practically real to human beings living in a world that includes mathematics.  I think that's about right.

But I also think it's valuable to make just this sort of distinction from time to time.  Learning mathematics (or maybe just learning) has a lot to do with forming connections.  You can know something about addition.  You can know something about subtraction.  But when you --- a much younger you --- begin to wrap your head around the connection between the two operations, important things are happening.  And a connection's impact on understanding is inversely related to its obviousness.  The guy who understands the connection between multiplication and division has learned an important thing.  The guy who understands the connection between the zeros of a complex function and the distribution of prime numbers has revolutionized an entire field.

I'm personally interested in helping people make non-obvious connections.  There are lots of good ways to do that, and we as educators should pursue all of them, but one way is to connect clearly mathematical ideas to questions that are not clearly mathematical in scope, viz., ask real-world (as I've defined it) questions.

Silver Bullet

So essentially my job boils down to finding interesting non-mathematical questions that are isomorphic to interesting mathematical questions, but not obviously so.  (How's that for a resume bullet!)  I've already touched on why I think non-obviousness is important, but there's another reason: when the contextual link is trivial, the question generally becomes terrible.  And it's really, really easy to create trivial links. And then the real-world problem is no longer isomorphic; rather it becomes both substantially identical to and superficially uglier than the original problem, which is unproductive.  It's easy to pour a thin candy shell of context that does nothing to conceal the shape of the underlying problem, or to improve its flavor.  The real world can definitely ruin a task if your only goal is to incorporate something --- anything --- non-mathematical because for some reason you're afraid to ask a math question about math.

And in that sense I understand the impetus behind the 'fake-world math' backlash, because there's a certain amount of extant conviction that slapping the 'real-world' label on something magically confers awesomeness...which it certainly does not.  Such wanton slapping can also make it seem as though it's somehow desirable to avoid mentioning mathematics while teaching it, which is a lousy way to treat of such a rich subject, and rather unsubtly suggests that math is unpalatable on its own.  We should, as a community, take the position that a poorly executed idea ought to be avoided.  We should question the circumstances and mechanisms that lead to poor execution.  I also think we shouldn't dismiss the good idea outright.  The real world isn't a silver bullet, but it's a perfectly good bullet to have in the magazine.

P vs. NP

I mentioned supra that, while I find this question of to be intellectually interesting, I don't think it's especially important.  Mostly because what I'm interested in, globally, are great math tasks, and the greatness of a task is independent of whether it's situated in- or outside of the real world, however we choose to limn it.  There is only the illusion of dichotomy here.  I drew my own personal line, and I chose to work on one side of it because I'm partial to the view from over here.  But I also realize that the work I do at Mathalicious represents a small (though valuable) part of the mathematical experience students should have.

I think that work like mine and work like Dan's approach the same target from essentially opposite directions.  Dan is trying to reify mathematics by treating it as a properly first-class citizen in the world as we know it.  I'm trying to expand mathematical thinking to comprise those parts of the world we may not realize are already within its purview.  Somewhere in the middle we create a situation in which mathematics and the real world end up occupying essentially the same space.  Isn't that what we're all doing?  I hope so.  I'd really like that world.

How to Think

Teach me how to think.  Better, teach me how to teach someone to think.  It's my job, after all.  And once you've done that, imagine all the sparkling inspective instruments we can set upon the world, keen at all the right edges.  A whole new generation of thinkers.  Is there anything more beautiful?  Did you shiver a a hopeful little shiver just now?  Because this is the kind of bullshit I have no patience for.  As if we weren't already, the both of us.  Thinking.

This story turns up at least once a week in my Twitter feed --- you know the one --- wherein the true value of some or other thing isn't so much about the thing itself, but about how the thing helps students learn how to think.  Or worse, become thinkers.  A story that always smells faintly of parable.

But people like research, so let me lay some on you.  The absolute best predictors of student thinking are respiration, metabolism, and excretion.  Everything else is house money.

That should be an incredible relief.  I mean, wouldn't you feel just a little daunted at the prospect of having to jump-start an inert lump of organic matter every Monday?  Of having that as your moral imperative and professional obligation?  Some days I couldn't even find my purple dry erase marker.

What we don't and can't do is teach our students to think.  Let's not insult them.  What we do is help them learn to pay attention to the myriad little tics and habits that attend thoughtfulness.  To be aware of the shape and sensation of their own cognition.  To be mindful of their rich internal voices.  We don't teach thinking.  Ever.  On our best days, we encourage introspection.

If that's not persuasive, I humbly suggest an experiment.  Want to see someone squeal his emotional tires?  I mean really spin?  Imply that he's failing to control his own brain.  Suggest that something is broken at his locus of fundamental humanity.  Get your face right up next to the spot that provides maybe the only reassurance of his own corporal existence and declare it unsound.

Then stand back.

Keys to a Rubbled Kingdom


Let's acknowledge, at the outset, that it's basically impossible to talk about one's wartime experience without sounding like a prick.  If your stories are too exciting, then you're bragging/embellishing/outright lying, which is prickish on its face.  If your stories aren't exciting enough, then you're being modest — probably falsely so — which is even worse, because not only are you bragging in some implicit, backhanded way, but you're also denying the listener his conventional opportunity for the minor act of hero worship that is fast becoming the only way for a population almost entirely divorced from two decade-long wars to connect with the alien minority that has shouldered their weight.  And should this lose-lose proposition be too exhausting to navigate, or should you have a headache, or should you have recently scraped the roof of your mouth on some weapons grade Cap'n Crunch, or for any reason at all, really, should you have the balls to actually utter the phrase, "I don't want to talk about it," well then you had better have at least a Silver Star and some visible scarring to back that up, otherwise you are the biggest prick on record.  Who do you think you are?

But, in spite of all that, I'm going to talk about my wartime experience anyway — such as it is — because the news about Fallujah falling back into chaos has affected me in a way I didn't expect.  And, because this is the only outlet I have at my disposal, I will dispose of it.  But this isn't really about me.  Just indulge me for a moment.

My War in Six Paragraphs

First, allow me to lower your expectations.  My personal participation in the war was approximately as ordinary as war-type participation can be.  I was an artillery officer in the Marine Corps, a young second lieutenant with Battery G, 2nd Battalion, 11th Marines, whose main job was running the Fire Direction Center, a gig that mostly involved figuring out ways to get hundred-pound bullets fired from great big cannons to land in tactically advantageous places.  For seven months in 2006 I sat inside a bunkered-in trailer just outside of Fallujah and waited for people to shoot rockets and mortars at us.  When they did, a whole slew of very expensive radar devices would calculate the point of origin, and after a little bit of math we would tell the guns which way to point and how much powder to use, &c., and soon they would be booming like mad trying to thunk the guys who wanted us dead.  It was all very loud and exciting for a few minutes out of every day.  One of the ways you could recognize new people around Camp Fallujah was to see who ducked when the artillery started up; if you couldn't tell the good booming from the bad, boy did you look foolish.  It was a source of constant entertainment.

I said thunk back there instead of murder, which is what I meant.  We're all adults here.

How often were we successful?  Honestly, I don't know.  Radar devices, no matter how expensive, are lousy at picking up corpses.  But my Marines were so fast — they could pump four rounds through a gun before the first one hit the ground — and there are only so many ways to avoid supersonic steel in the middle of the open desert.  Plus, I've always been pretty good at math.  It's a small, mean way to feel, hoping you have murdered someone, to have been an aspiring thunker of men.  But the mathematician in me will say, definitively, we killed more than one person.  I can't give you any more significant figures than that.

Besides sitting around and waiting for opportunities to do very intense math problems, I took precisely three convoys between Fallujah and the air base in Al Taqaddum.  The first one was simply to get to our new home in Fallujah after flying in from Kuwait.  I sat in the back of an up-armored 7-ton in the middle of the night and scrunched myself up mentally into a tiny corner of the universe as a precaution against being exploded, which must have worked.  When we pulled into the city an hour before dawn, thousands of rays of light spilled from the thousands of bullet holes in every structure we passed.  I unscrunched myself long enough to wonder at the spillage of so much light.  All I could think was, Man, we fucked this place up.  Of course that's the majestic plural.  I wasn't there for that part — for all of the placing of bullet holes in structures.  At the time, this upset me greatly.

The other two convoys were part of a round-trip to pick up some new electronics, a job for which I volunteered.  The new equipment we were to acquire was for jamming radio signals so that the insurgents couldn't to use them to blow up any of the shit piled alongside seemingly every inch of road in Anbar Province.  I thought it would be ridiculous for someone else to die on the way to or from picking up gizmos intended to keep us from getting killed, and I didn't want that on my conscience.  Also, I was starting to get tired of doing arithmetic while there were all these perfectly good roadside bombs left unexploded by my absence.  That's another strange feeling, wanting to get thunked — but not too severely.

I didn't even have to make the final drive out to Taqaddum on the way back home.  The Army loaned us some helicopters for the trip, which was awfully swell of them.  It would have been embarrassing for the insurgents to blow us up while we were on our way out the door — which is what they wanted anyway.  The Blackhawks helped us all to avoid that little misunderstanding.

Here's the most traumatic bit.  When my part of the war was over, I had to fill out a Post-Deployment Health Assessment Questionnaire.  One of the questions that ostensibly aided in the assessment of my post-deployment health was,   How often did you feel that your life was in danger?  Because the bad guys weren't so good at math, I had to fill in the NEVER circle.  I thought about filling in the OCCASIONALLY circle, but it would have been a stretch.  I got to put DAILY next the the question about exposure to loud noises, but it's not the same thing.

NEVER.  What a shameful thing for a war veteran to have to mark on an official government form.

The Lede, Sufficiently Buried

While the Marines (re)captured Fallujah in 2004 — house by house, and sometimes room by room — from an entrenched enemy with no designs on survival, I was still at The Basic School in Quantico, VA.  I sat in the chow hall every day, staring at the table reserved for pictures of the young officers and enlisted instructors killed in combat since the start of the war.  When I arrived in June, there were a handful of solemn framed faces; by the time I checked out, just before Christmas and the end of Phantom Fury, we were rearranging the furniture to make room for a fifth KIA table.  I listened to the hard-earned lessons of the survivors, sometimes only days removed from the fighting, that I might not have to pay so dearly for my education.  And then, a few months later, I was handed the keys to their rubbled kingdom, mortgaged — in the literal sense of the word — in blood.

I know you don't give a shit about Fallujah.  It's okay, it doesn't make you a bad person.  It's just one of the many miserable places in the world that has nothing to do with anything anymore.  But for a little while it was mine.  I didn't take Fallujah, which would have made it important to me.  I inherited Fallujah, which makes it sacred.

There's no earthly reason I should be upset that the city is again in disarray.  I didn't go because I thought we were going to solve the problems of the Iraqi people.  I sure as hell didn't go to defend my country.  (Rusty mortars only fly so far.)  I went because that's what you do for the dead.  You keep the things they give you.

And for that, I'm so sorry.  To the faces on those tables.  God knows how many tables now.  To their mothers and fathers, to their children and widows, I'm sorry.  That's the only decent thing to be said.  And it is, like all gestures and redresses born of human loss, completely insignificant.

Pretty Big Ideas

So Grant Wiggins threw down the gauntlet.  And Patrick Honner, as Patrick Honner is wont to do, picked it up.  And then Grant Wiggins --  I'm not totally sure what traditionally happens to a gauntlet at this point -- did some other gauntlet-related thing in reply.  It was fast and furious.  Actually, it was incredibly civil and well considered, which spirit I will try to preserve here.

For the tl;dr crowd, Wiggins posted a celebratory 100th-blog-post rant against Algebra (the course, not the content).  In that post he challenged Algebra teachers to name Four Big Ideas contained in the curriculum.  And Honner responded with some pretty solid candidates:

  1. Algebraic Structure
  2. Binary Relations
  3. The Cartesian Plane
  4. Function

Wiggins quasi-stipulates to a couple (binary relation, Cartesian Plane) -- with much qualification -- and more or less dismisses the others (quite politely).  It seems that Wiggins agrees all those ideas are important, but he has a very particular notion of what makes for a Big Idea:

So, I wish to up the ante. To me a big idea is big for both: I am looking for those ideas that are big – powerful and fecund – for both novice and expert.

I always return to this simple example from soccer: Create dangerous space on offense; collapse dangerous space on defense is a big idea at every level of the game, from kid to pro. And it is transferrable to all space-conquest sports like lacrosse, hockey, and basketball. Truly big.  [emphasis in original]

I'm not interested in debating the general lifelessness of high school Algebra, which strikes me as largely uncontroversial at this point in the conversation.  In fact, the same rant could just as easily be applied to mathematical instruction at almost any level (Wiggins even includes the obligatory quotation from A Mathematician's Lament).  I'm also not interested in trying to produce more convincing examples.  Nope, I'm interested in talking about soccer.

I think the soccer analogy is on the verge of making it impossible to have a meaningful discussion about math education.  Not just Algebra, but mathematics.  And here's why: the creating and collapsing of dangerous space might be the only Big Idea in soccer.  I submit that, if that's the standard for bigness, then there just aren't four Big Ideas to be had.

If soccer were taught like math, then you might take a course called Moving Without the Ball I as a freshman.  And in that class there would be a unit about Overlapping Runs.  And you would probably hate it, because it would be an awful lot of running, and you wouldn't ever be sure why you were doing all this goddamn running, because maybe your coach isn't overly concerned about unveiling to you the beautiful truth that a good, long overlapping run pulls a defender way down into the corner and stretches the whole defense and creates some dangerous space for a midfielder to run into.  And you would moan and check the calendar for when you were starting the Passing Into Space unit, because you heard it was totally easy -- mostly just standing around and pushing nice, easy passes toward cones.  But that really says more about your coach's ability/willingness to keep your eye on the Truly Big Idea (because hey, he's pretty much coaching the way he was coached in the first place, back when kids just shut up and did their Backwards Jogging homework without complaint) than it does about the bigness of the Pretty Big Ideas that you're working your way through, because the Truly Big Idea is just too ungodly huge to be useful in making you a better soccer player.  After all, there's just the one.

In an attempt at analogic involution, I'm going to try to come up with a mathematical analogy for the soccer analogy for math:

I always return to this simple example from mathematics: Create structure when you're building; look for structure when you're exploring is a big idea at every level of the subject, from kid to professional mathematician.   And it is transferrable to all structure-having systems like language, chemistry, and logic.  Truly big.

Maybe structure is the only big idea in math.  At least the only one Wiggins might agree to.  And that's just too big.  What we really need in Algebra are some Pretty Big Ideas.  So here's my gauntlet, which is admittedly significantly lighter than the one that's currently being kicked around: what are four Pretty Big Ideas in Algebra?  Honner got you started.

What's in a Circle?

Recently I had the good fortune to both attend #TMC13 and share my very own personal favorite lesson during, appropriately, one of the "My Favorite" sessions.  And now I'll share it with you.

I was getting ready to start a unit on conic sections with my Advanced Algebra kids.  I was planning to start, as I assume many of you do, with circles.  I imagined it going something like this, Lusto's 10 Steps to Circle Mastery:

  1. Get somebody, anybody in the room, to spout the definition of a circle , hopefully including a phrase such as, "the set of all points a fixed distance from a given point..."
  2. Talk about this distance business.  Do we know something about distance?  How do we measure it?
  3. The distance formula.
  4. Why the distance formula is fugly.  Remember the Pythagorean Theorem?  Of course you do.  It's awesome.
  5. In the plane:
  6. In the Cartesian plane, centered at the origin:
  7. This distance, d, has a special name in circles, right?  Right:
  8. Appeal to function families and translations by <hk>.
  9. Boom.

Based on my hopes for Step 1, and based on my need for like five uninterrupted minutes to take attendance, find my coffee mug, &c., I hastily scribbled an extremely lazy and unimaginative warm-up discussion question on the whiteboard.  Four simple words that  led to some surprising and amazing mathematical conversation.

What is a circle?

That's it.  My favorite lesson.  The whole thing.  And here's how it went.

I was walking around looking at/listening to all the different definitions the groups had come up with.  And they were nuts.  There were dubious claims about unquantifiable symmetry, sketchy sketches with line segments of indeterminate provenance, rampant appeals to a mysterious property known as roundness.  Most of the arguments were logically circular but, alas, mathematically not.  The word curvy appeared more than once.  It was a glorious disaster of handwaving and frustration.  I knew, deep in my reptilian brain, that this is what's known in the business as a "teachable moment."

What is not a circle.

Nailed it.

At this point I was basically just walking around being a jerk.  I was drawing all kinds of crazy figures that minimally conformed to what they were telling me a circle was, and getting lots of laughs in the process.  And then I had the thought, even deeper in my reptilian brain, that transformed the whole experience from an interesting activity into a bonafide lessonWhy the hell am I the one doing this?

So here's what the lesson eventually became, Lusto's 6 Instructions to Humans on the Brink of Amazing Mathematical Discussion:

  1. In your groups, answer the question, "What is a circle?"
  2. Absolutely no book-looking or Googling.  If all goes well, you will be frustrated.  Your peers will frustrate you.  I will frustrate you.  Don't rob anybody else of this beautiful struggle.  If your definition includes the word locus, you are automatically disqualified from further participation.
  3. Each group will have one representative present your definition to the class.  No clarification.  No on-the-fly editing.  No examples.  No pantomime.  Your definition will include, and be limited to, English words in some kind of semantically meaningful order.  Introduce variables at your own risk.
  4. If you're going to refer to some other mathematical object (and I suspect you will), make sure it's not an object whose definition requires the concept of circle in the first place.  (Ancillary benefit: you will be one of the approximately .01% of the population who learns what "begging the question" actually means.)
  5. Once a group presents a definition, here is your new job: construct a figure that meets the given definition precisely, but is not a circle.  Pick nits.  You are a counterexample machine.  A bonus of my undying respect for the most ridiculous non-circle of the day.
  6. When you find a counterexample, make a note of the loophole you exploited.  What is non-circley about your figure?

After giving the instructions, I could pretty much just sit back for a while and watch things get awesome.  If there's one thing that's easy to do, it's get teenagers to argue with each other.  Granted, it's a little harder to get them to argue about math, but not much.  (They're basically ready to fight at all times; the MacGuffin is largely unimportant.)  So that's one thing that makes this lesson my favorite.  Another thing is that we ended up with a pretty bullet-proof circle definition by the end of the exercise.  When you spend a whole lot of time crawling through loopholes in the hopes of beating up on your peers, you find an awful lot of loopholes to close.  Talk about a fantastic mathematical habit of mind.  Yet another thing, and maybe the coolest, is that it led to some of the best questions/observations my kids ever came up with.  Here is a brief, paraphrased sampling, with my annotations as to why they're so great:

  • Wait, if the circle is just the points on the edge, then how can a circle have area?  There's nothing that gets kids thinking precisely about mathematical language faster than the realization that every teacher in the world is using it incorrectly.  We should really be saying, the area of the region bounded by a circle, or the area of the circle's interior...  I had never thought twice about that, but now I sure as hell do.
  • What does it mean to be "inside" a circle?  Similar to the above, but even more amazing.  The fact that a circle divides the plane into two disjoint regions is a completely nontrivial result.  It's basically a statement of the Jordan Curve Theorem, which was proved pretty recently in the history of mathematics.
  • If a radius is a distance, then a "circle" depends on how you measure distance.  This one is on my Holy Shit List.  We had spent like a half hour one day talking about the Taxicab Plane, just because I thought it was cool and made the distance formula seem mildly less boring.  But somebody pointed out that circles would look totally different if we measured distances that way.  And yes!  They would!  At that point, I felt like I should probably just retire.

The final reason this lesson is my favorite is probably also the reason that you should care.  There's nothing particularly special about the word circle.  You can take the sentence, What is a [widget]? and pick just about any mathematical widget that kids have some nascent intuition about.  Give it a shot.  Maybe it won't be your favorite, but it'll be pretty great.

Consider the Strawberry

In general, I hate doing this --- because it feels like a self-promotional trick --- but in order for this post to make any kind of sense, you have to go back and read the last one.  In particular, you have to read Max's comment.  I will put on my teacher face and wait for a few minutes.


For two reasons, I'm going to unpack the strawberry analogy a bit more: (1) I am in love with it, and (2) it highlights an important pedagogical point about the relationship between squares and rectangles.  For serious.

As Max pointed out, even very small children have no problem recognizing the rather trivial fact that all strawberries are fruits even though not all fruits are strawberries.  On the flip side, anyone who has ever taught geometry knows, with something like absolute certainty, that much older and more mathematically savvy students have great difficulty recognizing that all squares are rectangles even though not all rectangles are squares.  The situations are structurally identical (in each case we have some set, X, which is a proper subset of another set, Y), but the second one is much more problematic.  Why might that be?

The seemingly obvious answer is that recognizing a strawberry is nearly automatic, and probably evolutionarily encoded, while recognizing a square requires abstract reasoning about the congruence of mathematical objects called "line segments."  But I'm not at all convinced that's the problem.  They are both ultimately pattern-recognition tasks.  Without language getting in the way, you (and small children) can probably recognize strawberries and squares with comparable facility.

Which brings us to the language.  Even though the strawberry:fruit::square:rectangle situations are structurally identical, there is an important (and subtle) linguistic distinction in the latter case.  Consider the following story.

You find your favorite small child/guinea pig and present a challenge.  In your left hand you hold a strawberry, and in your right an apple.  You say to this child, "Which hand has the fruit in it?"  The child blinks at you for several moments, trying to study your face for clues about the answer to what has just got to be a trick question, before finally, tentatively, reaching out to point at one of your hands, more or less at random.  You reward the child with a piece of delicious fruit.

Consider the same story, except now you hold in your left hand a picture of a square, and in your right a picture of a generic rectangle.  You say to this child, "Which hand has the rectangle in it?"  The child immediately points to your right hand.  You reward the child with, I guess, a delicious piece of rectangle.

Why are these stories so different?  I submit that it's not a mathematical issue.  The real problem stems from the fact that, linguistically, there is no unprivileged fruit: every class of fruit gets its own name.  But "square" is privileged relative to "rectangle."  When presented with a generic rectangle, we have no word for saying that it is "a rectangle that is not a square."  In fact, I made up the phrase "generic rectangle" precisely to try and convey that information.

So it turns out I lied a little bit before (how fitting) when I said the fruit/rectangle situations were structurally identical.  It's true that in each case we have a set (square, strawberry) that is a subset of a larger set (rectangle, fruit), but it turns out the larger sets have different linguistic partitions.

So when you ask the child which hand contains the rectangle, she chooses the generic rectangle immediately.  Why?  Because, had you meant the square, then you damn sure would've just said "square" in the first place, even though both hands hold perfectly correct answers to your challenge.  If our language were set up such that strawberries were the only specially named fruits (which seems like something Max would wholeheartedly support), the child in the first story would likewise choose your non-strawberry hand every time, without hesitation.

So what can we do with this?  It seems that strawberries have something to teach us about squares.  Actually, it seems that all the other fruits have something to teach us about rectangles.  It's taken the entire history of humanity to organize fruits into useful equivalence classes, but luckily we find ourselves in a much, much simpler situation with rectangles; after all, there are only two classes we care about!  We already have a name for squares, so let's call non-square rectangles "nares."  Now our partition looks like this:

Which hand has the nare in it?  Easy.  Better yet, unambiguous.  Now, I'm not seriously lobbying for the introduction of nares into the mathematical lexicon (for one thing, nare is already a word for a weird thing), but it might be a fun way to introduce young children to the concept of a non-square rectangle.  After removing the greatest impediment to understanding the square/rectangle relationship (that "square" is the lone special case of this broader class of "rectangles," which word is generally reserved for "rectangles-but-not-squares," since, if someone means "square," we already have a freaking word for it), that scaffolding can eventually be disassembled.

But the cognitive edifice the scaffolding initially supported will have cured a little by then.  In other words, why not make the distinction we actually care about explicit from the beginning, rather than end up in linguistic contortions to get around the fact that the distinction is solely implicit in standard usage?  Make up your own word, I don't care.  Don't want to be cute about it?  Fine.  Just abbreviate non-square rectangles as NSRs or something.  But make them easy to talk about --- as easy as it is to talk about a tangerine or cumquat rather than a "fruit that might be a strawberry, but very often is not."  Because, seriously, if that's the way our fruit classification worked, there would be an awful lot of kids running around with the reasonable and tightly-held belief that strawberries are not fruit.

And that would be a shame.

Inconvenient Truths

As happens with amazing frequency, Christopher Danielson said something interesting today on Twitter.

— Christopher (@Trianglemancsd) July 29, 2013

And, as also happens with impressive regularity, Max Ray chimed in with something that led to an interesting conversation --- which, in the end, culminated in my assertion that not everything that is mathematically true is pedagogically useful.  I would go further and say that a truth's usefulness is a function of the cognitive level at which it becomes both comprehensible and important --- but not before.

By way of an example, Cal Armstrong took a shot at me (c.f. the Storify link above) for my #TMC13 assertion that it is completely defensible to say that a triangle (plus its interior) is a cone.  Because he is Canadian, I think he will find the following sentiment particularly agreeable: we're both right.  A triangle both is, and is not, a cone, depending on the context.  It might be helpful to think of it as Schrödinger's Coneangle: an object that exists as the superposition of two states (cone and triangle),  collapsing into a particular state only when we make a measurement.  In this case, the "measurement" is actually made by our audience.

When I am speaking to an audience of relative mathematical maturity, I can (ahem...correctly) say that cone-ness is a very broadly defined property: given any topological space, X, we can build a cone over X by forming the quotient space

with the equivalence relation ~ defined as follows:

If we take X to be the unit interval with the standard topology, we get a perfectly respectable Euclidean triangle (and its interior).  Intuitively, you can think of taking the Cartesian product of the interval with itself, which gives you a filled-in unit square, and then identifying one of the edges with a single point.  Boom, coneangle.  Which, like Sharknado, poses no logical problems.

Of course, it is a problem when you're talking to a middle school geometry student.  In that situation, saying that a triangle is a cone is both supremely unhelpful and ultimately dishonest.  What we really mean is that, in the particular domain of 3-dimensional Euclidean geometry, when we have a circle (disk) in a plane and one non-coplanar point, we can make this thing called a cone by taking all the line segments between the point and the base.  But to that student, in that phase of mathematical life, the particular domain is the only domain, and so we rightly omit the details.  In an eighth-grade geometry class, there is absolutely no good reason to introduce anything else.

 photo cone_zpsf6e0fb1f.gif

Constructing a topological cone over the unit interval

We do this all the time as math teachers.  "Here, kid, is something that you can wrap your head around.  It will serve you quite well for a while.  Eventually we're going to admit that we weren't telling you the whole story --- maybe we were even lying a little bit --- but we'll refine the picture when you're ready.  Promise."

Which brings me back to Danielson's tweet.  From a mathematical point of view, there are all kinds of problems with saying that a rectangle has "two long sides and two short sides" (so many that I won't even attempt to name them).  But how bad is this lie?  Better yet, how bad is the spirit of this lie?  I think it depends on the audience.  I'm not sure it's so very wrong to draw a sharp (albeit technically imaginary) distinction for young children between squares and rectangles that are not squares.  It doesn't seem all that different to me, on a fundamental level, from saying that cones are 3-dimensional solids.  Or that you can't take the square root of a negative number.  Or that the sum of the interior angles of a quadrilateral is 360 degrees.  None of those statements is strictly true, but the truths are so very inconvenient for learners grappling with the concepts that we actually care about at the time.  It's not currently important that they grasp the complete picture.  And it's probably not feasible for them to do so, anyway.

Teaching mathematics is an iterative process, a feedback loop.  New information is encountered, reconciled with existing knowledge, and ultimately assimilated into a more complete understanding.  Today you will "know" that squares and rectangles are different.  Later, when you're ready to think about angle measure and congruence, you will learn that they are sometimes the same.  Today you will "know" that times can only be 0 if either a or b is zero.  And tomorrow you will learn about the ring of integers modulo 6.

I will tell you the truth, children.  But maybe not today.

Luck(?) of the Draw

What is luck?  Is luck?  And, if you vote yea, is a belief in luck an obstacle to understanding probability?

This question came up on Twitter a couple of nights ago when Christopher Danielson and Michael Pershan were discussing Daniel Kahneman's recent book, Thinking, Fast and Slow.  Specifically, they were talking about the fact that Kahneman doesn't shy away from using the word luck when discussing probabilistic events.  This, of course, is the kind of thing that makes mathematically fastidious people cringe.  And Danielson and Pershan are nothing if not mathematically fastidious.  Spend like five minutes with their blogs.  So Danielson twittered this string of twitterings:

According to Danielson, luck a "perceived bias in a random event."  And, according to his interpretation of Kahneman, luck is composed of "happy outcomes that can be explained by probability."  Let me see if I can define luck for myself, and then examine its consequences.

What is luck?

I think, at its heart, luck is about whether we perceive the universe to be treating us fairly.  When someone is kind to us, we feel happy, but we can attribute our happiness to another's kindness.  When someone is mean, we feel sad, but we can attribute our sadness to another's meanness.  When we are made to feel either happy or sad by random events, however, there is no tangible other for us to thank or blame, and so we've developed this idea of being either lucky or unlucky as a substitute emotion.

But happy/sad and lucky/unlucky are relative feelings, and so there must be some sort of zero mark where we just feel...nothing.  Neutral.  With people, this might be tricky.  Certainly it's subjective.  Really, my zero mark with people is based on what I expect of them.  If a stranger walks through a door in front of me without looking back, that's roughly what I expect.  And, when that happens, I do almost no emoting whatsoever.  If, however, he holds the door for me, this stranger has exceeded my expectations, which makes me feel happy at this minor redemptive act.  If he sees me walking in behind him and slams the door in my face, he has fallen short of my expectations, which makes me sad and angry about him being an asshole.

And, in this regard, I think that feeling lucky is actually a much more rational response than being happy/sad at people, because with random events at least I can concretely define my expectation.  I have mathematical tools to tell me, with comforting accuracy, whether I should be disappointed with my lot in life; there is no need to rely on messy inductive inferences about human behavior.  So I feel lucky when I am exceeding mathematical expectations, unlucky when I'm falling short, and neutral when my experience roughly coincides with the expected value.  Furthermore, the degree of luck I feel is a function of how far I am above or below my expectation.  The more anomalous my current situation, the luckier/unluckier I perceive myself to be.

Let's look at a couple examples of my own personal luck.

  1. I have been struck by lightning zero times.  Since my expected number of lightning strikes is slightly more than zero, I'm doing better than I ought to be, on average.  I am lucky.  Then again, my expected number of strikes is very, very slightly more than zero, so I'm not doing better by a whole lot.  So yeah, I'm lucky in the lightning department, but I don't get particularly excited about it because my experience and expectation are very closely aligned.
  2. I have both my legs.  Since the expected number of legs in America is slightly less than two, I'm crushing it, appendage-wise.  Again, though, I'm extremely close to the expected value, so my luck is modest.  But, I am also a former Marine who spent seven months in Iraq during a period when Iraq was the explosion capital of the world.  My expected number of legs, conditioned on being a war veteran, is farther from two than the average U.S. citizen, so I am mathematically justified in feeling luckier at leg-having than most leg-having people in this country.

Which brings us back to this business of luck being a "perceived bias in a random event."  I'm not convinced.  In fact, I'm absolutely sure I can be lucky in a game I know to be unbiased (within reasonable physical limits).  Let's play a very simple fair game: we'll flip a coin.  I'll be heads, you'll be tails, and the loser of each flip pays the winner a dollar.  Let's say that, ten flips deep, I've won seven of them.  I'm up $4.00.  Of course, my expected profit after ten flips is $0, so I'm lucky.  And you, of course, are down $4.00, so you're unlucky.  Neither of us perceives the game to be biased, and we both understand that seven heads in ten flips is not particularly strange (it happens about 12% of the time), and yet I'm currently on the favorable side of randomness, and you are not.  That's not a perception; that's a fact.  And bias has nothing to do with it, not even an imaginary one.

Now, in the long run, our distribution of heads and tails will converge toward its theoretical shape, and we will come out of an extremely long and boring game with the same amount of money as when we started.  In the long run, whether we're talking about lightning strikes or lost limbs or tosses of a coin, nobody is lucky.  Of course, in the long run---as Keynes famously pointed out---we'll be dead.  And therein, really, is why luck creeps into our lives.  At any point, in any category, we have had only a finite number of trials, which means that our experiences are very likely to differ from expectation, for good or ill.  In fact, in many cases, it would be incredibly unlikely for any of us to be neither lucky nor unlucky.  That would be almost miraculous.  So...

Is luck?

As in, does it really exist, or is it just a perceptual trick?  Do I only perceive myself to be lucky, as I said above, or am I truly?  I submit that it's very real, provided that we define it roughly as I just have.  It's even measurable.  It doesn't have to be willful or anthropomorphic, just a deviation from expectation.  That shouldn't be especially mathematically controversial.  I think the reason mathy people cringe around the idea of luck is because it's so often used as an explanation, which is where everything starts to get a little shaky.  Because that's not a mathematical question.  It's a philosophical or---depending on your personal bent---a religious one.

If you like poker, you'd have a tough time finding a more entertaining read than Andy Bellin's book, Poker Nation.  The third chapter is called "Probability, Statistics, and Religion," which includes some gems like, "...if you engage in games of chance long enough, the experience is bound to affect the way you see God."  It also includes a few stories about the author's friend, Dave Enteles, about whom Bellin says, "Anecdotes and statistics  cannot do justice to the level of awfulness with which he conducts his play."  After (at the time) ten years of playing, the man still kept a cheat sheet next to him at the table with the hand rankings on them.  But all that didn't stop Dave from being the leading money winner at Bellin's weekly game during the entire 1999 calendar year.  "The only word to describe him at a card table during that time is lucky," says Bellin, "and I don't believe in luck."

But there's no choice, right, but to believe?  I mean, it happened.  Dave's expectation at the poker table, especially at a table full of semi-professional and otherwise extremely serious and skillful players, is certainly negative.  Yet he not only found himself in the black, he won more than anybody else!  That's lucky.  Very lucky.  And that's also the absolute limit of our mathematical interest in the matter.  We can describe Dave's luck, but we cannot explain it.  That way lies madness.

There are 2,598,960 distinct poker hands possible.  There are 3,744 ways to make a full house (three-of-a-kind plus a pair).  So, if you play 2,598,960 hands, your expected number of full houses during that period is 3,744.  Of course, after 2.6 million hands, the probability of being dealt precisely 3,744 full houses isn't particularly large.  Most people will have more and be lucky, or less and be unlucky.  That's inescapable.  Now, why you fall on one side and not the other is something you have to reconcile with your favorite Higher Power.

Bellin's final thoughts on luck:

I know in my heart that if Dave Enteles plays 2,598,960 hands of poker in his life, he's going to get way more than his fair share of 3,744 full houses.  Do you want to know why?  Well, so do I.

And, really, that's the question everybody who's ever considered his/her luck struggles to answer.  No one has any earthly reason to believe she will win the lottery next week.  But someone will.  Even with a negative expectation, someone will come out way, way ahead.  And because of that, we can safely conclude that that person has just been astronomically lucky.  But why Peggy McPherson?  Why not Reggie Ford?  Why not me?  Thereon we must remain silent.

Is a belief in luck an obstacle to understanding probability?

I don't see why it should be.  At least not if we're careful.  If you believe that you are lucky in the sense of "immune to the reality of randomness and probabilistic events," then that's certainly not good.  If you believe that you are lucky in the sense of "one of the many people on the favorably anomalous side of a distribution," then I don't think there is any harm in it.  In fact, acknowledging that random variables based on empirical measurements do not often converge toward their theoretical limits particularly rapidly is an important feature of very many random variables.  In other words, many random variables are structured in such a way as to admit luck.  That's worth knowing and thinking about.

Every day in Vegas, somebody walks up to a blackjack table with an anomalous number of face cards left in the shoe and makes a killing.  There is no mystery in it.  If you're willing to work with a bunch of people, spend hours and hours practicing keeping track of dozens of cards at a time, and hide from casino security, you can even do it with great regularity.  There are how-to books.  You could calculate the exact likelihood of any particular permutation of cards in the shoe.  I understand the probabilistic underpinnings of the game pretty well.  I can play flawless Basic Strategy without too much effort.  I know more about probability than most people in the world.  And yet, if I happen to sit at a table with a lot more face cards than there ought to be, I can't help but feel fortunate at this happy accident.  For some reason, or for no reason, I am in a good position rather than a bad one; I am here at a great table instead of the guy two cars behind me on the Atlantic City Expressway.  That's inexplicable.

And that's luck.

Measure Your Blessings

In my last post, we took a look at how our choice of unit has both mathematical and linguistic consequences when we try to talk about one of something, particularly in a few weird cases.  One of the themes (here unit = "theme") that came up in the course of the discussion is the notion that there are certain objects that lend themselves to counting, and others that lend themselves to measuring.  Moreover, the words we use in our reckoning of different objects/substances are informed by our mathematical interpretation of their underlying structure.

Checkout Lines and Carnival Rides

I grew up with two extremely precise parents: a teacher mother who routinely marks public signs with a Sharpie to fix grammatical and spelling errors, and a teacher father who routinely soliloquizes over dubious scientific claims in the media.  Perhaps it's no accident that I both teach (and love) math and write (and love) this blog.  One of the things I apparently learned/inherited from my mom is a visceral, knuckle-whitening cringe induced by express checkout aisles labeled "10 Items or Less" instead of "10 Items or Fewer."  It's a reaction that has lodged itself firmly into the parts of my brain normally reserved for images of poisonous snakes and lion silhouettes.  We experience this discomfort because there is a dissonance between the referent noun items (a countable substance) and the comparative adjective less (a measure word).  It makes no more sense to speak of a number of items less than ten than it does to speak of a paint color taller than red.  Height is not an attribute of paint color; measure it not an attribute of item count.

Of course no one is actually confused by that what the sign means: "If the cardinality of the set of items in your basket exceeds ten, please find another line."  Still, the entire point of a grammar is to avoid ambiguity whenever possible.  Even though there is no appreciable ambiguity in the express aisle, that we're even talking about this highlights an important aspect of our language: we do in fact draw a line in the linguistic sand with the measurable on one side and the countable on the other.  We have different words that signal counting and measuring in the same way that, e.g., German has different words that signal you (singular) and you (plural).  As good citizens, we try to use the right signalling words, and we're perhaps slightly irritated when others don't.

Once we've figured out whether we're measuring or counting, though, the grammatical questions are decided for us, so the important---and sometimes difficult---part lies in gauging which side of line we're on.  In the express aisle, for instance, we have to determine what structure, exactly, underlies the nature of an "item."  Well, we understand that the point of the express aisle bound is to get people through the line as quickly as possible, and the quickness with which one negotiates the line is a function of the number of scans that take place.  In effect, we can measure checkout time in terms of boops.  So, in this case, 1 boop = 1 item.  And, since boops are atomic (it makes no sense to think about what half-a-boop might mean), we model them with cardinal/ordinal numbers.  Your six-pack of Diet Coke?  One boop, not six.  Your ten yogurts, which are conspicuously not linked or priced together? Sorry, 10 boops = 10 separate items.  There is certainly no room for less than here.

At the other end of the spectrum, my mom wouldn't look twice at a Tilt-A-Whirl line with a "No Riders Less Than 48 Inches Tall" sign.  Presumably the point of the height restriction is to prevent too-small human beings from slipping out of the safety restraints mid-tilt or -whirl, and thus any height at all below the 48-inch cutoff is potentially hazardous.  We want to exclude someone who is 45 inches tall, or 45.019 inches, or 42.31 inches.  Since it seems as though we're modeling height with real numbers, we know that we're in measuring country.  Now we don't have room for fewer.

Time Keeps On Slipping/Discretely Clicking Into the Future

Of course it's not always so simple.  Depending on the context---and thus the unit---in question, things might be either countable or measurable.  Consider time.  On the one hand, time seems like the prototypical infinitely-divisible thing, hence calculus gives us exceedingly good predictions about the behaviors of physical bodies moving about the universe.  On the other hand, we are sometimes only concerned with time meted out into discrete chunks, hence survey-of-history courses.

Let's pretend we're about to run a 100m sprint together.  I might say to you, "If you can run this in less than 9.58 seconds, you'll have beat the world record."  We're measuring seconds, so this makes perfect sense.  If, however, we were talking about my teaching career, I might say something like, "Since I've been teaching for fewer than three years, I have to be observed by my principal."  Now we're counting years.  I don't think anyone would freak out if I said "less than three years," but it's extremely considerate for me to use "fewer" in this case, because it signals to my conversational partner that "years" is the relevant unit of account.  In fact, almost any conversation about my teaching experience is likely to be couched in counting terms, because almost all of the important distinctions (e.g., pay) are binned into one-year increments, the fractional parts of which are totally irrelevant.  In other words, we're modeling with cardinal/ordinal numbers, which require counting words.

The amphibious nature of concepts like time can lead to some interesting and confusing consequences.  For instance, the ratio of the human lifespan to the rate at which the Earth orbits the sun makes it often convenient to group time into fairly large chunks.  Large chunks generally demand counting language.  On the other hand, we like to be pretty precise with our reckoning of time, and precision often demands measuring language.  Which interpretation wins out might depend on cultural norms.

Consider the traditional Chinese method of determining age.  You are born into some year.  This is your first year.  When the lunar year rolls over, even if that happens tomorrow, you have suddenly been extant during two different years, which makes you two years old.  We're starting at one and modeling with cardinal/ordinal numbers, i.e., counting.  In the West, we think this is nuts.  The moment you sneak out of the birth canal (or, I suppose, the abdominal cavity), the clock starts running.  Your birth is the zero marker, and everything is measured as a distance relative to that point.  That is, your lifespan is measured from birth to death.  To say that someone is 18 means two entirely different things in the two cultures.  In China it means that he has taken at least one breath on 18 distinct calendar pages; in the West, it means his age measurement lies in the interval [18,19).  Neither one is a priori more correct; it just means that someone who is 18 under the Eastern system might be barely old enough to start driving in most of the U.S.

But even in the West we're not entirely consistent in our choices.  We are by and large measurers of time, but we still count it under certain circumstances.  For instance, the year 72 CE belongs to the First Century of the Common Era, even though 72 has a zero in the hundreds place.  That's a telltale sign of counting.  But then we also might talk about the 1900's instead of the 20th Century, and "1900's" is based on a measurement starting at the zero point of Jesus's possibly apocryphal, possibly entirely literally true birth (well, not really the zero point...we jumped from 1 BCE to 1 CE without any Year Zero...even though we've been measuring for a long time, now, we got off to a lousy counting start).  And a century seems to be about the inflection point: any chunk larger than that and we almost exclusively count (we might talk about the 2nd Millennium, but nobody talks about the 1000's), and anything smaller we almost exclusively measure (ever hear anybody refer to the 90's as "the 200th Decade?").

What We Talk About When We Talk About Math

Like so many things in mathematics, this counting/measuring business really comes down to deciding what domain we're in so that we can choose an appropriate mathematical model.  Notice I've been saying throughout the piece that we "model" counting or measuring with real or cardinal numbers.  And, like all choices of model, it boils down to convenience.  The continuity of real numbers makes them very nice to work with sometimes; it's comforting to think that I can measure out any arbitrary amount of water or time that I might need, even though that's not strictly true.  Nothing in the physical universe really has anything to do with real numbers.  We don't know (almost) any empirical measurements beyond about 7 or 8 decimal places.  And, even if we did, it seems as though the universe itself, being quantized, turns out to be metaphysically countable.  In other words, we could get away with counting language alone, you know, if you didn't mind measuring carnival riders in Planck Lengths.

Of course we don't want to do that, so we've agreed to trade some verisimilitude for the pleasantries of measurement language, even if it slightly increases our grammatical efforts in the process, and even if that increased effort leads to the occasional error in signage.  I'll try to keep that in mind next time I find my heart rate climbing in the grocery store.

Aside: Many of the ideas about the business of measuring and/or counting time, as well as the seeds of my fierce loyalty to a singular "data," can be traced back to John Derbyshire's excellent book, Prime Obsession.

Human Composure

Frequent commenter and child logic expert Christopher Danielson recently contributed a very cool video, entitled One is one...or is it?, to the Ted-Ed project in which he explores  what it is we really mean when we say "one."  The very notion requires an implicit or explicit reference to a unit, and sometimes things aren't quite as simple as they seem.  In particular, we might have units that are either built up of (composed units) or divided into (partitioned units) smaller sub-units.  Or sub-sub-units.  For instance, the loaf of bread in your cupboard is partitioned into individual slices; the pinochle deck you stash in the coffee table is composed of individual playing cards; and Pop Tarts are sold in boxes, which contain packs, which contain individual pastries. One is relative to your choice of unit.

Of course this is all very interesting, but the first thoughts that popped into my head during the dancing apple slices number involved the two conspicuous cases that seem to defy composition and partitioning: (1) units whose sub-units are human, and (2) units whose sub-units are tiny.  It's still possible to smoosh them together and rend them asunder, but not so neatly.

Part the First

For just a moment let's consider the North Carolina State Wolfpack.  Now that is definitely an all-the-way, one-hundred percent composed unit.  It is a singular pack, composed of singular wolves.  Unambiguous.  Until you read some press releases, which are simply pregnant with phrases like, "the Wolfpack are..." Which is very strange indeed.

Consider two fictitious news stories.  In the first, NC State is relocated by the NCAA to South Carolina.  In the second, an actual, literal pack of wolves (Canis lupus) is spotted migrating from Raleigh to Charleston.  The first headline would read, "Wolfpack Head Across the Border," while the second would read, "Wolfpack Heads Across the Border."  These two scenarios are mathematically identical; the only difference is that, in one case, the sub-units are figurative wolves.  So why do we require a different verb conjugation?  It seems that people somehow resist being subsumed by composed units in a way that, e.g., playing cards do not.  Admittedly this is a psycho-linguistic curiosity more than a mathematical one, but still...units can be slippery.

It's maybe more obvious that people resist being partitioned.  After all, if you make it through life without anybody partitioning you, let's call that a nontrivial success.  But it shows up in the language, too.  We are very rarely wont to consider a disembodied finger directly.  When we partition people, we replace the corporeal whole with a possessive placeholder, a pointer (ha!) to the original unit.  We hold the source object in memory in a much more vivid and deliberate way than with other objects.  An apple slice is an apple slice, culinary nuances notwithstanding, but "John's foot" and "your foot" and "the crazy lady in 3B's foot" require modification.  Unless you live under fairly abnormal conditions, indefinite articles no longer suffice: "a foot" or "the foot" rarely come up.  We make grammatical concessions more readily and more often for human "units" (suppressing, here, many jokes) than nonhuman ones.

Part the Second

Consider the strangeness of the following utterance: "I think the rice is done."  Why aren't they done instead?  There are tons of those little buggers!  There certainly exists a plurality of foodstuffs.  But when we deal with tiny sub-units (especially if they're homogeneous), we have a hard time unitizing them naturally.  We can go clumpy: "I think the pot/serving/microwave bag of rice is done."  We can go grainy: "I think the grains of rice are done."  But both of those solutions feel deeply unsatisfactory.  There are contortions involved.  We either have to create a new group name (think pod of dolphins or murder of crows), or a new unit name (think kernel of corn or drupelet of raspberry).  Awkward either way.  It's as if, on a fundamental level, we ache for rice to be singular entity, but of no definite unit membership.

And this weirdness, I think, is no longer merely syntactic, but deeply mathematical.  When we unitize the world in language, we make an important distinction between that which is countable and that which is measurable.  There is an analogy to be made here between discrete and continuous objects, respectively.  It is natural to consider the composed unit of a dozen eggs, because eggs are easily countable, and because it's easy to count to a dozen.  It is much more difficult to create a conventional composed unit out of water molecules, because water molecules are hard to count, and because it would be hard to count to a number of water molecules that would be useful in most situations.  Thus, we measure water and treat it as an un-composed unit.  Somewhere in between those two extremes, we have things like rice, much more countable than water, much less countable than eggs.  In fact, it's closer to water than eggs in its countability, so we treat rice as measurable/continuous, even though it's technically countable/discrete.  Sand.  Salt.  Data.*

*I will fight to the death, via torturously long diatribes on gerunds and loan words, that "data" should be treated as singular in English, even though it's inflected as a plural in Latin, all based on the composed unit argument above.  If you're going to be a total weeny and use it in the plural, at least be consistent.  I had better never hear you talk about "an agenda" (singular), because "agenda" is also inflected as plural; each item is technically an "agendum."  I'm watching you.

So tiny things resist composition, and they resist partitioning even more vehemently.  For one, they're already tiny.  It's inconvenient to let these things get any smaller (and our unit choices, after all, have an awful lot to do with convenience), and there's no natural starting point from which to partition things in the first place.  If I wanted to decompose "sand" into parts, how big is this parent sand?  We've stumbled into a kind of reverse paradox of the heap.  A loaf of bread readily admits slices.  A _________ of sand admits grains.  Tough to fill in the blank.

An Interesting Competition

What happens when these two notions are pitted against each other?  Which one wins out in our brains and on our tongues?  How fortunate for my blog that the Miami Heat are currently playing in the NBA Eastern Conference Finals, and that the Miami Heat are one of the few professional sports teams with a singular name (as an fun exercise, try to list all the others---major sports only, no AA curling or anything).  Did you hear what I just typed?  "The Miami Heat are one of..."  But wait, that's nuts!  I've never in my life heard anybody complain that the humidity are unbearable, so why should the capital-H Heat be any different?  Heat is an abstract and amorphous thing.  In our everyday usage, it's definitely a measurable substance---like water, a singular.  But it's also a unit composed of people.  And when I tell you about the current state of the NBA, I tell you that the Heat are in the Eastern Conference finals.

Our bias against making conglomerations out of people is so strong that it can overcome our natural tendency to treat both composed units and measurable substances as singular.  We hold ourselves in such high regard that we're willing to regularly construct borderline nonsensical phrases to maintain our artificially inflated position.

Go, Pack!