Teresa Stanbury used to be a Common Core skeptic—until she stepped into a Common Core math classroom where deep learning was taking place. What she saw, struck her into Core dumbfoundedness.
The teacher, Gideon Pelous, buzzed about the room like a shimmering dragonfly while the children—second-graders from the deep inner city—discussed the essence of numerals in small groups.
Before the Core, students would be taught that two plus two equals four, but they would never know why. They would go through their lives not knowing how to explain this basic mystery. Now things were entirely different. The moribund learning of the ossified past had been exhumed and cast away.
“I just had a realization,” said Shelly Thomas, arranging four rectangular blocks in front of her. “I used to think that numerals were quantities. I was trying to figure out what the curve on the 2 meant, and what the double curve on the 3 meant. I even tried measuring them with my ruler. Then I had the insight that numerals aren’t quantities, but rather symbols that represent quantities.”
“You mean to say—“ sputtered Enrique Alarcón as he seized a crayon.
“Yep,” she continued. “This 1 here represents a unit of something. It can be a unit of anything. Now, when we say ‘unit,’ we have to be careful. That’s another thought that came to me, but I haven’t figured it–.”
“I have,” interrupted Stephanie Zill, banging on her Curious George lunchbox. “We use the word ‘unit’ in both a contextual and an absolute sense. That is, a unit is unchanging within the context of a problem, but it may change from problem to problem. Also, certain defined units, such as minutes and yards, have a predefined size that doesn’t change from one context to the next—until you consider relativity, that is.”
“Oh, I get it,” said Enrique. “So, this numeral 1 represents one unit, which could be a unit of anything, but within a given problem, the word “unit” does not change referent unless we are dealing with more than one kind of unit at once. Hey, what color crayon should we use: magenta or seaweed?”
“Magenta,” said Shelly. “So, moving on with this problem, let’s say the numeral 1 represents one of these blocks. The numeral two represents two blocks.” She set two blocks aside to emphasize her point.
“Fair enough,” answered Stephanie. “But how do you get from there to 2 + 2 = 4?”
“OK,” Shelly resumed, swinging her braids. “So, you have these two blocks, and you want to add another two blocks to them. But two blocks, you see, is actually two of one block. So when you add two blocks, you’re actually adding one block twice. Now if you put twenty single blocks together, you get twenty blocks, which isn’t the same as two blocks, but it can be, if you divide those twenty blocks into ten groups of two each. Just try it and you’ll see what I mean. But here we want four blocks, not twenty, so that means that instead of dividing the pile into ten groups of two each, we should divide it into five groups of four each. So we do that. Then we take one of those groups of four and line up the blocks, like this. Then we take our original two blocks and match them up to these four blocks. It turns out that we can do so twice. This means that we are taking two blocks and then two blocks again, which is the same as adding two plus two, and this turns out to be four, which once again, or maybe for the first time, because this is all super-new, is represented by the numeral 4.”
Stephanie and Enrique nodded, rapt. “That was deep,” said Enrique.
“Deep understanding,” Stephanie agreed.
“That’s just the beginning,” said Shelly. “In the old days, we would have left it like that and gone back to dealing with abstract representations of quantities. But thanks to the Common Core, we get to apply this equation to numerous real-life situations. So, say you have a pair of socks and another pair of socks. How many socks do you have?”
“Two pairs,” said Orlando, who had just wiggled his way over from another group that was taking too long to arrive at insights.
“Two pairs, but how many individual socks?”
“They aren’t individual. They’re pairs.”
“But let’s pretend that they’re still individual, even as pairs.”
“Does it matter if they don’t match?”
“Wait,” interjected Stephanie. “I thought you said the units were supposed to be identical.”
“This leads us to question what identity really is,” rejoined Shelly. “Any object in the physical world has a set of attributes. If you consider only certain attributes, such as general shape and purpose, this object may be identical to other objects that otherwise don’t resemble it. However, if you focus on the attributes that differ, they you find yourself confronted with unalike and incomparable objects.”
“I see,” Orlando sighed. “So, if we’re just considering the sockiness of the sock—that is, the property that makes something a sock and not some other object—then we have four such socks.”
“That’s more or less on the right track,” said Shelly. “There are some subtleties that need to be taken into account, but since group time is up, we’ll have to leave that until tomorrow.”
Mr. Pelous called the class back to attention. “Mathematicians, what did we learn in our groups today about two plus two?”
“It equals four!” the students cried.
“Yes, and why?”
The room erupted in voices—all saying different things. Suddenly Orlando began waving his hand frantically.
“Orlando, do you have something to tell us about why 2 + 2 = 4?”
“Yes—if it didn’t equal four, then life would be absurd, or at least very, very strange!”
“And who’s to say it isn’t?” shouted a student from the corner.
“It can’t be that strange, or we wouldn’t be trying to explain it through math,” Orlando said. The bell rang.
Teresa Stanbury thanked Mr. Pelous and wandered dreamily out of the school, marveling at the Common Core and the wonders it had wrought.