Never On Sunday

[Note: unsure whether a square root symbol (√) will appear properly in any given browser, I will use an older typographic convention in its place. sqrt (x) here indicates the square root of x, presumed to be the positive square root unless otherwise stated. It’s a bit cumbersome, and definitely ugly, but not as ugly as an error message.]

My students persist in taking sqrt (x+y) to be equal to sqrt (x) + sqrt (y). For those of you a little rusty on your advanced arithmetic/elementary algebra, this is incorrect; the square root of a sum is not the sum of the respective square roots. To give a concrete example, sqrt (100) = 10; sqrt (100) = sqrt (36 + 64), which emphatically does not equal sqrt (36) + sqrt (64), which equals 6 + 8, or 14. To take this to a ridiculous extreme, sqrt (n), where n is some arbitrary large integer, does not equal sqrt (1) + sqrt (1) + sqrt (1) + … + sqrt (1), for n iterations; if that were so, sqrt (n) would always equal n.

Kinda silly, right? But my kids keep doing it.

The confusion arises from the fact that the square root of a product is the product of the individual square roots. Kids who aren’t paying attention to the difference between addition and multiplication get confused. I can’t tell whether the problem is ignorance, apathy, or fear. Maybe they don’t know the difference, which suggests a failure of earlier schooling that I can either address at the expense of the students who have learned the lesson. Maybe they know the difference but get so tangled up in a phobia towards math that they can’t make the connection between what they know and what they do. Intimidated by the very idea of doing math, some students grab the first idea that comes to hand, write it down, and shove the homework aside as quickly as possible, without pausing to check their work or even to consider whether the methods they’re using are reasonable. Again, this problem I can address, but only at the expense of taking time away from what I’m supposed to be teaching in order to get everyone caught up, which carries some ugly ramifications for a school on the verge of getting its funding cut, or even shut down entirely. Or maybe the students who keep getting it wrong just don’t care, and heaven knows what I can do about that.

I’ve told the class plainly a few times that sqrt (x+y) isn’t sqrt (x) + sqrt (y). When that wasn’t enough, I illustrated the concept with concrete examples and actual numbers. I’ve given them more elaborate examples with variables, in order to demonstrate that variables are numbers, and follow the same rules, even if you don’t know which number they are. I’ve given them the silly case of Simple Simon thinking sqrt (n) = n, described above. I’ve forced them to “help me out” at the dry erase board, drilling them in not making that mistake. I’ve given them supplementary problems—literally one- or two-line calculations, no more than five minutes’ work—in their homework. The kids are starting to roll their eyes, especially the ones who have it right. And yet the class nods and agrees that sqrt (x+y) isn’t sqrt (x) + sqrt (y) and that they all understand, and then go and tell me that sqrt (a + 9) equals sqrt (a) + 3. ‘Cause a isn’t x, I guess.

It reminds me of Bill Cosby’s routine about how all children must suffer brain damage, because they are unable to connect simple commands like “don’t touch that” with action, and, asked why they touched it anyway, respond “I don’t know!” (“My parents never smiled. Because I, too, had brain damage!”) It would be funny if it weren’t so frustrating.

Or, more depressing still, possibly true. How many of these kids suffered malnutrition as kids? How many were raised in lead-painted apartments? How many simply didn’t get enough brain-developing attention? And how should a teacher handle that?