When my DD was a five year old in first grade, she was finishing up a homework assignment, and the last question in the textbook instructed her to explain how she arrived at her answer to the previous exercise. She didn’t know what to write because she just knew the answer. I thought the question was asinine and told her it was o.k. to write that she just knew. She wrote, “because my spirit told me so.” Her teacher wrote that she loved the answer and explained to me that she didn’t expect students to do the “write about it” questions at the end of the exercises. She thought they were silly at this point too. Luckily, DD has had great math teachers who realize the shortcomings of the textbooks they have to use.
DD is now 10 and will be taking seventh grade advanced math this year. She is just starting to really be able explain her thinking. I think this is a result of her brain naturally beginning to develop new cognitive skills. I also find that these explanations don’t emanate from the problems she knows the answer to, but from the problems she has trouble solving. Whenever she faced a problem she wanted help with, I always responded with questions: “what do you need to find out?” “what do you already know?” “what do you think you should do first?” Lately when she’s been stuck, she’ll start explaining her process, explaining what approaches she’s eliminated and why they won’t work, and ultimately explaining why her answer is right without me ever asking a question. What begins with a “mom, come here I’m stuck,” ends with her giving me a mini-lesson on problem solving.
There are two dangers in pushing kids to “understand and explain” before their brains are ready to do so. One is that they can become so frustrated they wind up hating math and believing that they’ll never “get it.” The other is that in trying to explain complex concepts too early many teachers and textbooks try to simplify these complicated concepts for young students and get the math wrong.
This is a great explanation from an old Keith Devlin piece in his “Devlin’s Angle” column sponsored by the Mathematical Association of America. In it, he makes the distinction between “functional” and “conceptual” understanding of mathematics.
“I think it probably is possible to achieve understanding along with skill mastery for any mathematical topic, but it would take far too long, with a likely result that the student would simply lose heart and give up long before achieving sufficient understanding. . . . Thus, whereas conceptual understanding is a goal that educators should definitely strive for, we need to accept that it cannot be guaranteed, and accordingly we should allow for the learner to make progress without fully understand the concepts.”
Here’s the link to the article and the follow up articles that resulted from his suggestion at the end that teachers stop “saying that multiplication is repeated addition,” one of my own pet peeves.
“What Is Conceptual Understanding?”
http://www.maa.org/external_archive/devlin/devlin_09_07.html“It Ain't No Repeated Addition”
http://www.maa.org/external_archive/devlin/devlin_06_08.html“It's Still Not Repeated Addition”
http://www.maa.org/external_archive/devlin/devlin_0708_08.html“Multiplication and Those Pesky British Spellings”
http://www.maa.org/external_archive/devlin/devlin_09_08.html“What Exactly Is Multiplication?”
http://www.maa.org/external_archive/devlin/devlin_01_11.html