{"id":18,"date":"2012-01-10T10:56:00","date_gmt":"2012-01-10T16:56:00","guid":{"rendered":"http:\/\/linesoftangency.wordpress.com\/2012\/01\/10\/noe-2-thynges"},"modified":"2012-01-10T10:56:00","modified_gmt":"2012-01-10T16:56:00","slug":"noe-2-thynges","status":"publish","type":"post","link":"http:\/\/blog.chrislusto.com\/?p=18","title":{"rendered":"noe.2.thynges"},"content":{"rendered":"

Thanks to the book Group Theory in the Bedroom<\/em><\/a>, by Brian Hayes, I finally found someone to blame for my geometry students' daily growing hatred of mathematical language.\u00a0 His name is Robert Recorde.\u00a0 For those of you who, like pre-this-week me, have never heard of Robert Recorde, don't worry: you've seen his handiwork.\u00a0 Recorde was a 16th-century Welsh doctor, mathematician, and author of Whetstone of Witte<\/em> (1557), the book in which the modern equals sign first appears.<\/p>\n

<\/p>\n

Hayes only mentions Recorde's contribution in passing, as part of an extremely interesting essay on the mathematical and philosophical nature of identity, but includes a curious quotation from Whetstone<\/em> that explains why<\/em> Recorde chose the particular symbol he did.\u00a0 Here is a reproduction of the original text:<\/p>\n

\"\"<\/a><\/div>\n

And here is the content of the second sentence, in somewhat more recognizable English:<\/p>\n

\"And to avoid the tedious repetition of these words, 'is equal to,' I will set, as I do often in work use, a pair of parallels...lines of one length, thus: ====, because no two things can be more equal.\"<\/p><\/blockquote>\n

On one hand, what a beautiful example of simple, useful notation arising organically from one man's dissatisfaction with the clunky and cumbersome status quo.\u00a0 Here's a guy who was really tired of writing \"is equal to\" over and over again, and so he invented some shorthand to make things more comprehensible in his daily efforts.\u00a0 (If you want an example of just how awkward mathematical ideas can be when posed in natural language instead of symbols, here is an example, drawn more or less at random from Euclid's Elements<\/em>: \"If from a point without a circle, straight lines are drawn to the circumference; of those falling upon the concave circumference the greatest is that which passes through the center, and the line which is nearer the greatest is greater than that which is more remote.\"<\/em>\u00a0 Sheesh.)<\/p>\n

On the other hand, there is something about Recorde's explanation<\/em> of his notational choice that strikes the modern mathematical reader as odd, particularly if that modern mathematical reader is a teacher of high school geometry.\u00a0 Namely, parallel lines are not even close to equal.<\/em><\/p>\n

To be fair to Mr. Recorde, these matters were not so nicely settled in his day, but there is a non-negligible amount of irony in the fact that the \"modern\" equals sign was chosen for a reason completely at odds with the \"modern\" concept of equality.\u00a0 How so?<\/p>\n

To answer that, let's set aside equality for a moment and consider sameness.\u00a0 Specifically, let's consider that there are (at least) two perfectly sensible interpretations of \"the same\" in everyday speech.\u00a0 If I pull into the parking lot at school tomorrow, and in the spot next to me is someone who also<\/strong> drives a blue 2002 Subaru Impreza WRX, I might say something to the other driver like, \"Hey, we have the same car!\"\u00a0 That is, on some fundamental level, our cars share enough defining characteristics to be considered identical, even though we both agree that the two vehicles are composed of distinct sets of molecules.\u00a0 Compare this situation to one in which I say to my sister, \"Hey, we have the same father!\"\u00a0 This time there is only one referent, one object, one guy.<\/p>\n

In our everyday lives, the type of sameness in question is easily discerned by the context.\u00a0 I am very rarely confused about whose car I'm driving, or about how two people can share a parent.\u00a0 In math though, these two concepts of sameness have distinct technical denotations.\u00a0 And, since most students think of \"equal\" and \"same\" as being, well, the same, there arises a creeping discomfort from having to make a seemingly picayune distinction where it doesn't ordinarily exist.\u00a0 And geometry class is the first place this confusion manifests in earnest.<\/p>\n

Geometric objects are considered \"equal\" if and only if they are the same as sets<\/em>.\u00a0 They contain exactly the same points.\u00a0 To say, for instance, that one quadrilateral is equal to another is like saying that my father is the same as my sister's father.\u00a0 There is really only one object in question.<\/p>\n

If, however, we have geometric objects whose fundamental characteristics<\/em> are identical, we replace the word \"equal<\/strong>\" with \"congruent<\/strong>.\"\u00a0 To say one quadrilateral is congruent to another is to say that their corresponding sides have the same length, and that their corresponding angles have the same measure, because side lengths and angle measures are the defining characteristics of quadrilaterals.\u00a0 This is the Subaru scenario.<\/p>\n

The question of equality v. congruence is already confusing to students, but it gets much, much worse.\u00a0 For one thing, the ideas are not strictly distinct.\u00a0 Equality in geometry is actually a subset of congruence: if two things are equal (selfsame), then they are necessarily congruent (characteristically identical), but the converse need not be true.\u00a0 For another thing, objects that are not equal, but are congruent, have measures that are not congruent, but equal.\u00a0 How can that be, since I just said that equality implies congruence?\u00a0 Because the measures are numbers<\/em>, and numbers aren't geometric objects, so they don't play by the rules of congruence.\u00a0 (I apologize to the mathematically fastidious reader who is already typing nasty things in my comments box about modular arithmetic.\u00a0 Yes, numbers can sometimes be congruent, but not really in the same sense.)<\/p>\n

As an example, imagine I have two unique line segments with endpoints A&B and C&D, respectively, and both segments measure 4 units in length.\u00a0 Then I might say, \"Segment AB is congruent<\/strong> to segment CD because the measure of segment AB is equal<\/strong> to the measure of segment CD;\u00a0 furthermore, segment AB is both congruent<\/strong> to and equal<\/strong> to segment AB, but is only congruent<\/strong> to segment CD, which also happens to be both congruent<\/strong> to and equal<\/strong> to itself.\"\u00a0 That sentence, in a nutshell, is why my students hate me.<\/p>\n

More to the original point, now we see why Recorde's choice of parallel line segments to represent equality is so strange.\u00a0 Segments are equal if and only if they define precisely the same set of points, and parallel segments don't even have a single point in common; in fact, that's roughly the definition of \"parallel.\"\u00a0 In a real sense, no two things can be less<\/strong> equal!<\/p>\n

<\/div>\n","protected":false},"excerpt":{"rendered":"

Thanks to the book Group Theory in the Bedroom, by Brian Hayes, I finally found someone to blame for my geometry students' daily growing hatred of mathematical language.\u00a0 His name is Robert Recorde.\u00a0 For those of you who, like pre-this-week me, have never heard of Robert Recorde, don't worry: you've seen his handiwork.\u00a0 Recorde was […]<\/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":[15,19,24,32,55],"class_list":["post-18","post","type-post","status-publish","format-standard","hentry","category-math-teaching","tag-equality","tag-geometry","tag-language","tag-math","tag-teaching"],"_links":{"self":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts\/18","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=18"}],"version-history":[{"count":0,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts\/18\/revisions"}],"wp:attachment":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=18"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=18"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=18"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}