I love that phrase: rote understanding.
I was interested to see that Barry was taught "making ten" when he was in grade school:
The “making ten” method is included in the math program used in Singapore—a nation whose fourth and eighth graders have consistently obtained the highest scores in international math tests. Specifically, in Singapore’s Primary Math textbook for first grade, the procedure for adding by “making tens” is explained. Of particular importance, however, is that the procedure is not the only one used, nor are first graders forced to use it. This may be because many first graders likely come to learn that 8 + 6 equals 14 through memorization, without having to repeatedly compose and decompose numbers in order to achieve the “deep understanding” of addition and subtraction that standards-writers—and the interpreters of same—feel is necessary for six-year-olds.I have a question about the teacher's explanation of the number 6:
“Making tens” is not limited to Singapore’s math textbooks, nor is it by any means a new strategy. It has been used for years, as it was in my third-grade arithmetic textbook, written in 1955...
“So if we can partner 9 to a number and anchor 10, we can help our students see what 9 plus 6 is. So we’re going to decompose our 6, and we know 6 is made up of parts. One of its parts is a 1 and the other part is a 5.How do mathematicians think about whole numbers?
Do they see them as "made up of parts"?
Or as decomposable into parts?
(Or both --- ?)
To me, "made up of" and "decomposable into" seem like two different things.
Another question: if 6 is "made up of parts," is 6 one of the parts?
I bet right this minute there are kids all over America who are royally confused by the ramifications of making ten.