Katherine Beals & Barry Garelick:

“In general, there is no more evidence of “understanding” in the explained solution, even with pictures, than there would be in mathematical solutions presented in a clear and organized way. How do we know, for example, that a student isn’t simply repeating an explanation provided by the teacher or the textbook, thus exhibiting mere “rote learning” rather than “true understanding” of a problem-solving procedure?

“Math learning is a progression from concrete to abstract. The advantage to the abstract is that the various mathematical operations can be performed without the cumbersome attachments of concrete entities—entities like dollars, percentages, groupings of pencils. Once a particular word problem has been translated into a mathematical representation, the entirety of its mathematically relevant content is condensed onto abstract symbols, freeing working memory and unleashing the power of pure mathematics. That is, information and procedures that have been become automatic frees up working memory. With working memory less burdened, the student can focus on solving the problem at hand. Thus, requiring explanations beyond the mathematics itself distracts and diverts students away from the convenience and power of abstraction. Mandatory demonstrations of “mathematical understanding,” in other words, can impede the “doing” of actual mathematics.”

Related: Math Forum: audio/video.