{"id":792,"date":"2014-03-20T14:19:03","date_gmt":"2014-03-20T14:19:03","guid":{"rendered":"http:\/\/blog.chrislusto.com\/?p=792"},"modified":"2014-03-20T14:19:03","modified_gmt":"2014-03-20T14:19:03","slug":"reality-check","status":"publish","type":"post","link":"http:\/\/blog.chrislusto.com\/?p=792","title":{"rendered":"Reality Check"},"content":{"rendered":"

In accordance with the 7th Commandment of Blogging (If thy comment exceedeth two cubits in length, thou shalt write thine own damn post<\/em>.), here is my personal response to Dan's latest question<\/a>.\u00a0 To be clear: even though the question was put to Mathalicious<\/a> generally, and even though I bodily occupy a nontrivial fraction of that particular organization<\/a>, it would be presumptuous of me to write anything approaching an official opinion<\/a>, especially given the humbling brains attached to the people I spend my days with.\u00a0 But I have thoughts.<\/p>\n

We do thump the real-world drum pretty steadily around the office.\u00a0 We have, as it's known in the biz, a niche.\u00a0 But what exactly constitutes 'real-world' is an interesting question.\u00a0 I don't think it's a particularly important<\/em> 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.\u00a0 And insofar as that term appears with some regularity in the CCSS<\/a>.<\/p>\n

A Line in the Sand<\/h1>\n

There's a philosophically defensible sense in which nothing we'd call a mathematical object is real (or, for that matter, an object<\/em>).\u00a0 They're abstract and causally independent and yadda yadda yadda.\u00a0 There's another philosophically defensible sense in which everything<\/em> we'd call a mathematical object is real.\u00a0 Actually, a couple of senses, with varying definitions of reality.\u00a0 And of course there are positions in between.\u00a0 You could spend lifetime trying to untangle all the competing ideas of what (or whether) a number is.\u00a0 And basically who cares.\u00a0 In my mind, there's a simple way to draw a (note: not\u00a0the<\/em>) line on the curricular map between That Which Is Real and That Which Is Not:<\/p>\n

Is this question self-referential?<\/em><\/p>\n

In other words, are we using math to examine itself, or are we using it to inspect something outside its own borders?\u00a0 So my working definition of 'real-world' math is a mathematical task that is not self-referential.\u00a0 Which means, and I suppose you saw this coming, that my answer to Dan's question is none of the above<\/em>.\u00a0 None of those problems is real-world in any appreciable sense.\u00a0 They are all questions about a circle, a square, and their respective areas: math looking at math.\u00a0 A and B are obvious.\u00a0 C and D are just promising hypothetical candy (which is the absolute worst<\/em> kind of candy) for solving A or B.\u00a0 I suppose E and F are both dipping their toes into the real-world, but self-consciously.<\/p>\n

Okay, so those are some counterexamples.\u00a0 Maybe a countercounterexample will help shed light on my litmus test.<\/p>\n

For Instance...<\/h1>\n

Here are two problems:<\/p>\n

1.\u00a0 Is it true in general that P(A|B) = P(B|A)?\u00a0 If not, can you express P(A|B) in terms of P(B|A)?<\/strong><\/p>\n

2.\u00a0 Should innocent people be worried about the NSA's PRISM program?\u00a0 How much should they be worried?<\/strong><\/p>\n

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).\u00a0 Of course they are the same question, and they are both excellent.\u00a0 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.\u00a0 Also, you get to do some really good math qua math.\u00a0 I find that both professionally compelling and pedagogically useful.\u00a0 That's why I do what I do.\u00a0 Without speaking too forcefully for the rest of the team, I think that's part of the reason we do what we do.<\/p>\n

Unpacking Circles<\/h1>\n

As I understand Dan's position (or at\u00a0 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.\u00a0 In fact, for kids, a circle may be real (Platonist objections notwithstanding) in a much clearer and more visceral way.\u00a0 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.\u00a0 I think that's about right.<\/p>\n

But I also think it's valuable to make just this sort of distinction from time to time.\u00a0 Learning mathematics (or maybe just learning<\/em>) has a lot to do with forming connections.\u00a0 You can know something about addition.\u00a0 You can know something about subtraction.\u00a0 But when you --- a much younger you --- begin to wrap your head around the connection between the two operations, important things are happening.\u00a0 And a connection's impact on understanding is inversely related to its obviousness.\u00a0 The guy who understands the connection between multiplication and division has learned an important thing.\u00a0 The guy who understands the connection between the zeros of a complex function and the distribution of prime numbers has revolutionized an entire field.<\/p>\n

I'm personally interested in helping people make non-obvious connections.\u00a0 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<\/em>., ask real-world (as I've defined it) questions.<\/p>\n

Silver Bullet<\/h1>\n

So essentially my job boils down to finding interesting non-mathematical questions that are isomorphic to interesting mathematical questions, but not obviously so.\u00a0 (How's that<\/em> for a resume bullet!)\u00a0 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.\u00a0 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.\u00a0 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.\u00a0 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.<\/p>\n

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.\u00a0 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.\u00a0 We should, as a community, take the position that a poorly executed idea ought to be avoided.\u00a0 We should question the circumstances and mechanisms that lead to poor execution.\u00a0 I also think we shouldn't dismiss the good idea outright.\u00a0 The real world isn't a silver bullet, but it's a perfectly good bullet to have in the magazine.<\/p>\n

P vs. NP<\/h1>\n

I mentioned supra<\/em> that, while I find this question of to be intellectually interesting, I don't think it's especially important.\u00a0 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.\u00a0 There is only the illusion of dichotomy here.\u00a0 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.\u00a0 But I also realize that the work I do at Mathalicious represents a small (though valuable) part of the mathematical experience students should have.<\/p>\n

I think that work like mine and work like Dan's approach the same target from essentially opposite directions.\u00a0 Dan is trying to reify mathematics by treating it as a properly first-class citizen in the world as we know it.\u00a0 I'm trying to expand mathematical thinking to comprise those parts of the world we may not realize are already within its purview.\u00a0 Somewhere in the middle we create a situation in which mathematics and the real world end up occupying essentially the same space.\u00a0 Isn't that what we're all doing?\u00a0 I hope so.\u00a0 I'd really like that world.<\/p>\n","protected":false},"excerpt":{"rendered":"

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.\u00a0 To be clear: even though the question was put to Mathalicious generally, and even though I bodily occupy a nontrivial fraction of that […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-792","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts\/792","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=792"}],"version-history":[{"count":10,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts\/792\/revisions"}],"predecessor-version":[{"id":817,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts\/792\/revisions\/817"}],"wp:attachment":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=792"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=792"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=792"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}