If you have to think about 3x = 18 . . .
Thinking about everything is hard, writes Greg Ashman, author of Cognitive Load Theory, on Filling the Pail. It exhausts students' extremely limited working memory. What teachers want is for to activate a web of knowledge -- a "schema" -- in long-term memory to solve problems.
Alfred North Whitehead, a mathematician and philosopher, observed:
“It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.”
Ashman critiques Peter Liljedahl’s book, Building Thinking Classrooms in Mathematics. The author wants students to figure out how to solve problems themselves without "mimicking" the teacher's explanation.
Liljedahl’s ideas are "a recipe for confused and overloaded students," not the basis of a sustainable teaching plan, Ashman concludes. "We would quickly run out of fresh horses."
Reformers assert that they're improving math learning in ways that can't be measured, writes Ashman. He's dubious. He also doesn't believe in "math zombies" who don't understand math but can solve all the problems.
Some students "are not inclined to solve a problem three different ways or witter on in English about how they did it or whatever else we have prioritised over actual mathematics," writes Ashman. Their ability to ace tests is explained away as "rote memorization" of procedures. "When deployed to explain away the success of students in the Far East on international assessments such as PISA and TIMSS, this takes on a frankly racist tone."
It is not up to teachers to prove that a new approach is unlikely to work, writes Ashman. The math ed guru bears the burden of proof.
Natalie Wexler explains how cognitive-load theory applies to teaching reading on Forbes. It's an excellent explanation of why people need to move information from short-term to long-term memory, organized in a way that enables them to retrieve it.