{"id":701,"date":"2013-07-30T08:17:59","date_gmt":"2013-07-30T13:17:59","guid":{"rendered":"http:\/\/linesoftangency.wordpress.com\/?p=701"},"modified":"2013-08-03T00:46:24","modified_gmt":"2013-08-03T00:46:24","slug":"consider-the-strawberry","status":"publish","type":"post","link":"http:\/\/blog.chrislusto.com\/?p=701","title":{"rendered":"Consider the Strawberry"},"content":{"rendered":"

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<\/a>. \u00a0In particular, you have to read Max's comment<\/a>. \u00a0I will put on my teacher face and wait for a few minutes.<\/p>\n

***<\/p>\n

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. \u00a0For serious.<\/p>\n

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. \u00a0On 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. \u00a0The situations are structurally identical (in each case we have some set, X,<\/em> which is a proper subset of another set,\u00a0Y<\/em>), but the second one is much more problematic. \u00a0Why might that be?<\/p>\n

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.\" \u00a0But I'm not at all convinced that's the problem. \u00a0They are both ultimately pattern-recognition tasks. \u00a0Without language getting in the way, you (and small children) can probably recognize strawberries and squares with comparable facility.<\/p>\n

Which brings us to the language. \u00a0Even though the strawberry:fruit::square:rectangle situations are\u00a0structurally<\/em> identical, there is an important (and subtle) linguistic<\/em> distinction in the latter case. \u00a0Consider the following story.<\/p>\n

\n

You find your favorite small child\/guinea pig and present a challenge. \u00a0In your left hand you hold a strawberry, and in your right an apple. \u00a0You say to this child, \"Which hand has the fruit in it?\" \u00a0The child blinks at you for several moments, trying to study your face for clues about the answer to what has just got<\/i> to be a trick question, before finally, tentatively, reaching out to point at one of your hands, more or less at random. \u00a0You reward the child with a piece of delicious fruit.<\/p>\n<\/blockquote>\n

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. \u00a0You say to this child, \"Which hand has the rectangle in it?\" \u00a0The child immediately points to your right hand. \u00a0You reward the child with, I guess, a delicious piece of rectangle.<\/p>\n

Why are these stories so different? \u00a0I submit that it's\u00a0not<\/em> a mathematical issue. \u00a0The real problem stems from the fact that, linguistically, there is no unprivileged fruit: every class of fruit gets its own name. \u00a0But \"square\" is privileged relative to \"rectangle.\" \u00a0When presented with a generic rectangle, we have no word for saying that it is \"a rectangle that is not a square.\" \u00a0In fact, I made up the phrase \"generic rectangle\" precisely to try and convey that information.<\/p>\n

So it turns out I lied a little bit before (how fitting) when I said the fruit\/rectangle situations were structurally identical. \u00a0It'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.<\/p>\n

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