I’m jumping around a bit here, but a semi-counterintuitive idea that he advocates, which is apparently backed up by serious research, is what he calls pretesting. This means testing people on material that they have not yet been taught. As long as this is done carefully, so that it doesn’t put students off completely, this turns out to be very valuable, because it prepares the brain to be receptive to the idea that will help to solve that pesky problem. And indeed, after a moment of getting used to the idea, I found it not counterintuitive at all.
In English, “pretesting” as such is often not possible, but it’s useful to attempt to gauge students’s knowledge and go back to wherever the student is confused—which may be very simple aspects of language, like parts of speech. I often had debates about this subject in grad school, when other grad students or professors would lament students’s weak grasp of “basics” or “fundamentals” like comma rules. The stern professors had a point, in that university students should know those things, but I would counter that, if students don’t know them, it’s useful to teach them, even in “advanced” classes. Sometimes students seem to have not been taught much of anything in high-school English classes. Many high-school English classes have devolved into discussions of feelings and vague hand-waving about a given book, and students emerge from them with few concrete skills.
To be sure, sometimes the opposite is true. While teaching in grad school, I had a series of students, all good writers, all of whom had been taught by a particular teacher in a particular high school, and she apparently really drilled students in close reading and essay construction, like someone out of “The Writing Revolution.” The results showed. I meant to send her a letter thanking her but never did. I would guess that she did a form of “pretesting,” albeit without multiple-choice questions, to ascertain students’s skill levels and then base each day in class on what students know. I used to do something similar at times, by doing quick yes/no questions based on raised hands, in order to get a sense of where students were. Now, reading “How Craig Barton wishes he’d taught maths,” I think I should have spent more time and energy on assessment.
In most if not all subjects, it’s not possible to teach (or learn) advanced topics without mastering fundamentals, so an instructor should go back to wherever someone lacks mastery and begin building up from there. If that doesn’t happen, students—in the broadest sense, even outside formal school—at most muddle through and at worst waste everyone’s time. It’s nice to see someone as eminent as Timothy Gowers coming to a similar conclusion.