Barry Garelick says no, not always, and I agree:
A Wall Street Journal opinion article drew considerable attention to Common Core (CC) math standards—particularly the sixth-grade standard for fractional division— in early August. In it, math professor (emerita) Marina Ratner criticized approaches she saw in her grandchild’s classroom, where students were expected to represent fractional problems with pictures. She says:
The teacher required that students draw pictures of everything: of 6÷ 8, of 4 ÷ 2/7, of 0.8 × 0.4, and so forth. In doing so, the teacher followed the instructions: ‘Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for 2/3 ÷ 3/4 and use a visual fraction model to show the quotient . . .’
Ratner then asks: “Who would draw a picture to divide 2/3 by 3/4?”
I tend to agree. Requiring students to draw a picture to prove they “understand” the meaning of 2/3 ÷ 3/4 is likely to confuse more than it enlightens. Many others agree, including Barbara Oakley, a professor of engineering at Oakland University in Rochester, Michigan. She expressed her ideas in another Wall Street Journal opinion article called “How We Should be Teaching Math” published just this week. (Here is a link to a non-paywalled version.) In it, she states, “True mastery doesn’t mean you use crutches like laying out 25 beans in 5-by-5 rows to demonstrate that 5 × 5 = 25. It means that when you see 5 × 5, in a flash, you know it’s 25—it’s a single neural chunk that’s as easy to pull up as a ribbon. Having students stop to continually check and prove their understanding can actually impede their understanding, in the same way that continually focusing on every aspect of a golf swing can impede the development of the swing”…
The standard I have discussed here illustrates the theme of “understanding” and “explanation” that pervades many of the CC math standards. Opinion is divided in the education community about how to teach understanding. It is certainly worthwhile to explain to students why the fractional division algorithm works. Even more important, however, is recognizing that a student who knows what problems fractional division can solve and can perform the procedure possesses some understanding. Marking students down who have enough understanding to solve problems but cannot do the things judged to indicate understanding is placing the cart before the horse. It is tantamount to saying, “If we can just get them to do things that look like what we imagine a mathematician does, then they will be real mathematicians.” (emphasis mine–Darren)
Yes, what he said. Let’s just teach what works, shall we?