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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.

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.

 Rectangles = \{Rectangles \cap Squares\} \cup \{Rectangles \cap Squares^C\}

Fruits = Apples \cup Apricots \cdots \cup Strawberries \cup \cdots \cup Watermelons

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:

Rectangles = Squares \cup Nares

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.