{"id":661,"date":"2013-07-29T16:03:46","date_gmt":"2013-07-29T21:03:46","guid":{"rendered":"http:\/\/linesoftangency.wordpress.com\/?p=661"},"modified":"2013-08-03T00:55:34","modified_gmt":"2013-08-03T00:55:34","slug":"inconvenient-truths","status":"publish","type":"post","link":"http:\/\/blog.chrislusto.com\/?p=661","title":{"rendered":"Inconvenient Truths"},"content":{"rendered":"<p style=\"text-align: justify;\">As happens with amazing frequency, <a title=\"Overthinking My Teaching\" href=\"http:\/\/christopherdanielson.wordpress.com\/\" target=\"_blank\">Christopher Danielson<\/a> said something interesting today on Twitter.<\/p>\n<blockquote class=\"twitter-tweet\">\n<p>Dear children&#39;s publishing industry: It is NOT a property of rectangles that they have &quot;two long sides and two short sides&quot;. <a href=\"https:\/\/twitter.com\/search?q=%23ThatIsAll&amp;src=hash\">#ThatIsAll<\/a><\/p>\n<p>&mdash; Christopher (@Trianglemancsd) <a href=\"https:\/\/twitter.com\/Trianglemancsd\/statuses\/361883799790956544\">July 29, 2013<\/a><\/p><\/blockquote>\n<p><script async src=\"\/\/platform.twitter.com\/widgets.js\" charset=\"utf-8\"><\/script><\/p>\n<p>\u2014 Christopher (@Trianglemancsd) <a href=\"https:\/\/twitter.com\/Trianglemancsd\/statuses\/361883799790956544\">July 29, 2013<\/a><\/p><\/blockquote>\n<p style=\"text-align: justify;\">And, as also happens with impressive regularity, <a title=\"the Max Ray Blog\" href=\"http:\/\/mathforum.org\/blogs\/max\/\" target=\"_blank\">Max Ray<\/a> chimed in with something that led to <a title=\"Rectangle Chat\" href=\"http:\/\/storify.com\/Lustomatical\/rectanglechat\" target=\"_blank\">an interesting conversation<\/a> --- which, in the end, culminated in my assertion that not everything that is mathematically true is pedagogically useful. \u00a0I 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.<\/p>\n<p style=\"text-align: justify;\">By way of an example, <a title=\"Cal Armstrong on Twitter\" href=\"https:\/\/twitter.com\/sig225\" target=\"_blank\">Cal Armstrong<\/a> took a shot at me (c.f. the Storify link above) for my <a title=\"Twitter Math Camp\" href=\"http:\/\/www.twittermathcamp.com\/\" target=\"_blank\">#TMC13 <\/a>assertion that it is completely defensible to say that a triangle (plus its interior) is a cone. \u00a0Because he is Canadian, I think he will find the following sentiment particularly agreeable: we're both right. \u00a0A triangle both is, and is not, a cone, depending on the context. \u00a0It might be helpful to think of it as Schr\u00f6dinger's Coneangle: an object that exists as the superposition of two states (cone and triangle), \u00a0collapsing into a particular state only when we make a measurement. \u00a0In this case, the \"measurement\" is actually made by our audience.<\/p>\n<p style=\"text-align: justify;\">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,\u00a0<em>X,\u00a0<\/em>we can build a cone over\u00a0<em>X<\/em> by forming the quotient space<\/p>\n<p style=\"text-align: center;\"><p style='text-align:center;'><span class='MathJax_Preview'><img src='http:\/\/blog.chrislusto.com\/wp-content\/plugins\/latex\/cache\/tex_65fbfe71b00782072f93d40644f38f3f.gif' style='vertical-align: middle; border: none;' class='tex' alt=\"\" \/><\/span><script type='math\/tex;  mode=display'><\/script><\/p><\/p>\n<p style=\"text-align: justify;\">with the equivalence relation ~ defined as follows:<\/p>\n<p style=\"text-align: center;\"><p style='text-align:center;'><span class='MathJax_Preview'><img src='http:\/\/blog.chrislusto.com\/wp-content\/plugins\/latex\/cache\/tex_5fc2f91247cc693b30541f06d59a38dc.gif' style='vertical-align: middle; border: none;' class='tex' alt=\"\" \/><\/span><script type='math\/tex;  mode=display'><\/script><\/p><\/p>\n<p style=\"text-align: justify;\">If we take\u00a0<em>X<\/em> to be the unit interval with the standard topology, we get a perfectly respectable Euclidean triangle (and its interior). \u00a0Intuitively, 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. \u00a0Boom, coneangle. \u00a0Which, like <a title=\"Sharknado\" href=\"http:\/\/www.youtube.com\/watch?v=iwsqFR5bh6Q\" target=\"_blank\">Sharknado<\/a>, poses no logical problems.<\/p>\n<p style=\"text-align: justify;\">Of course, it\u00a0<em>is<\/em> a problem when you're talking to a middle school geometry student. \u00a0In that situation, saying that a triangle is a cone is both supremely unhelpful and ultimately dishonest. \u00a0What 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. \u00a0But to that student, in that phase of mathematical life, the particular domain is the\u00a0<em>only<\/em> domain, and so we rightly omit the details. \u00a0In an eighth-grade geometry class, there is absolutely no good reason to introduce anything else.<\/p>\n<div style=\"width: 270px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/s1363.photobucket.com\/user\/Lustomatical\/media\/cone_zpsf6e0fb1f.gif.html\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" style=\"border: 0;\" title=\"Topological Cone\" alt=\" photo cone_zpsf6e0fb1f.gif\" src=\"http:\/\/i1363.photobucket.com\/albums\/r711\/Lustomatical\/cone_zpsf6e0fb1f.gif\" width=\"260\" height=\"255\" border=\"0\" \/><\/a><p class=\"wp-caption-text\">Constructing a topological cone over the unit interval<\/p><\/div>\n<p style=\"text-align: justify;\">We do this all the time as math teachers. \u00a0\"Here, kid, is something that you can wrap your head around. \u00a0It will serve you quite well for a while. \u00a0Eventually we're going to admit that we weren't telling you the whole story --- maybe we were even <a title=\"Pretty Little Lies\" href=\"http:\/\/linesoftangency.wordpress.com\/2012\/02\/22\/pretty-little-lies\/\" target=\"_blank\">lying a little bit<\/a> --- but we'll refine the picture when you're ready. \u00a0Promise.\"<\/p>\n<p style=\"text-align: justify;\">Which brings me back to Danielson's tweet. \u00a0From 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). \u00a0But how bad is this lie? \u00a0Better yet, how bad is the <em>spirit\u00a0<\/em>of this lie? \u00a0I think it depends on the audience. \u00a0I'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 <em>not<\/em> squares. \u00a0It doesn't seem all that different to me, on a fundamental level, from saying that cones are 3-dimensional solids. \u00a0Or that you can't take the square root of a negative number. \u00a0Or that the sum of the interior angles of a quadrilateral is 360 degrees. \u00a0None of those statements is strictly true, but the truths are so very inconvenient for learners grappling with the concepts that we actually care about <em>at the time<\/em>. \u00a0It's not currently important that they grasp the complete picture. \u00a0And it's probably not feasible for them to do so, anyway.<\/p>\n<p style=\"text-align: justify;\">Teaching mathematics is an iterative process, a feedback loop. \u00a0New information is encountered, reconciled with existing knowledge, and ultimately assimilated into a more complete understanding. \u00a0Today you will \"know\" that squares and rectangles are different. \u00a0Later, when you're ready to think about angle measure and congruence, you will learn that they are sometimes the same. \u00a0Today you will \"know\" that <em>a\u00a0<\/em>times\u00a0<em>b\u00a0<\/em>can only be 0 if either <i>a<\/i> or\u00a0<em>b<\/em> is zero. \u00a0And tomorrow you will learn about the ring of integers modulo 6.<\/p>\n<p style=\"text-align: justify;\">I will tell you the truth, children. \u00a0But maybe not today.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>As happens with amazing frequency, Christopher Danielson said something interesting today on Twitter. Dear children&#39;s publishing industry: It is NOT a property of rectangles that they have &quot;two long sides and two short sides&quot;. #ThatIsAll &mdash; Christopher (@Trianglemancsd) July 29, 2013 \u2014 Christopher (@Trianglemancsd) July 29, 2013 And, as also happens with impressive regularity, Max [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"class_list":["post-661","post","type-post","status-publish","format-standard","hentry","category-math-teaching"],"_links":{"self":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts\/661","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=661"}],"version-history":[{"count":5,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts\/661\/revisions"}],"predecessor-version":[{"id":718,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts\/661\/revisions\/718"}],"wp:attachment":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=661"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=661"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=661"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}