Wrong answers that aren't

Anya has been doing a lot of arithmetic lately, and there have been several times when she's gotten answers that would be counted simply as "wrong" in school, but for which there were reasonable explanations.

Take the simple addition pages in her workbooks. Usually she is supposed to count the number of objects in each of two groups, write in those addends, and then write in their sum. I was looking over one of these pages, where every answer was either 9 or 10, and noticed she had written 4 + 4 = 8 on one line. In fact the equation should have been 2 + 7 = 9, which I know she knows (there were no other errors), but you see, Anya's favorite number is 8. She resents it when there's no sum of 8 on the page. So she just ignored the instructions on that line and wrote in her own equation. I did not consider this to be "wrong," because my purpose is not to train accurate office workers who obey instructions to the letter; it's for my kids to learn arithmetic. And even if my purpose were to raise dutiful accountants, we have years in which to learn the importance of accuracy. Anya's equation would be marked wrong in school because one teacher cannot divine the thought processes behind the wrong answers of over 20 children-- even supposing they shared my perspective (not likely, with all the emphasis on test-taking).

In another example, Anya told me she added up "one of every number" on her calculator and it added up to 66. Sure, if you add all the numbers from 1 to 11 it does add up to 66, though an adult would probably think of the single digits 1 through 9, which only sum to 45. It takes a little effort to figure out exactly what she's calculated sometimes-- and would a teacher have that time?

Telling a kid their answer is simply wrong is incredibly frustrating to them, and it ignores the opportunity to figure out what the misunderstanding is. Of course humans make typos on the calculator and sometimes we goof on rote memorization tasks, mixing up 6 times 9 and 7 times 9. But a great many "mistakes" in fact point to a conceptual error which it would be extremely useful to investigate. I don't personally remember this happening in school unless I sought the teacher's help myself-- and sometimes not even then.

While we are on the subject of math, there is a heck of a lot of value placed on right answers as opposed to conceptual understanding. Here is an observation I had from undergrad and grad school: people with math degrees are no better at arithmetic or figuring out the tip in a restaurant than non-math majors. In fact, I'm substantially worse at off-the-cuff arithmetic than a lot of other people I know. I have a BS in math and an MS in biostatistics (applied math), and on the SAT, ACT, and GRE, math was in every case my lowest test score. Why? Because the conceptual understanding required to write proofs, conceive of infinite series, and integrate functions in 3-D has pretty much no relationship to the "meticulous" trait that makes for an excellent accountant, schoolteacher, or test-taker.

It's true that even in college I lost points on complex math problems because I'd goofed and missed a minus sign or forgotten to square something. Yes, students of all stripes should try to avoid these little errors, but you know, I made a lot of them and I still got a math degree. I think they're stressed far too much in these early grades. I'm glad my kids won't be in an environment where goof-ups count so much against them, yet real misunderstandings are not identifed or delved into.