My previous post about solving problems in multiple ways has been interpreted in multiple ways. In previous eras, students were taught the standard algorithm first. The standard algorithms for multi-digit addition and subtraction were explained via diagrams and other means to show what was really happening when we “carry” and “borrow” (now called “regrouping”). I.e., it was not taught without understanding.
After mastery of the standard algorithm, students were shown alternative methods such as “making tens” and other short-cuts, which spotlighted the conceptual underpinning behind the standard algorithm. Often, students discovered these methods themselves. Anchoring mastery with the standard algorithm first created a distinction and students could see what was the main dish versus the side dish.
Current procedure under textbook interpretations of Common Core, is to delay teaching the standard algorithm, and to teach the alternative methods first under the belief that the standard algorithms eclipse understanding. In so doing, the distinction between main and side dishes are obscured and students are often confused, sometimes being forced to solve problems by inefficient methods such as drawing pictures when they are clearly ready to move on.
Two quotations from Steve Wilson, math professor at Johns Hopkins come to mind in this regard from an article that appeared in Education Next.
There will always be the standard algorithm deniers, the first line of defense for those who are anti-standard algorithms being just deny they exist. Some seem to believe it is easier to teach “high-level critical thinking” than it is to teach the standard algorithms with understanding. The standard algorithms for adding, subtracting, multiplying, and dividing whole numbers are the only rich, powerful, beautiful theorems you can teach elementary school kids, and to deny kids these theorems is to leave kids unprepared. Avoiding hard mathematics with young students does not prepare them for hard mathematics when they are older.
There will always be people who think that you must be able to solve problems in multiple ways. This is probably similar to thinking that it is important to teach creativity in mathematics in elementary school, as if such a thing were possible. Forget creativity; the truly rare student is the one who can solve straightforward problems in a straightforward way.
Then there are those who say that “if only I had been taught why these algorithms work”, now seeing the math through an adult lens. They fail to see that their ability to understand may have come from being taught in a traditional method that did, in fact, teach the conceptual understanding, but without the same degree of obsession about it that now pervades math education.