We didn’t become good at English by studying a chart: we developed an ear for the language and know how it should sound.

Unless we grew up speaking a different language, or were born to people who didn't have English as their first language. My wife has been in the US since the age of 2 and speaks perfectly but she still has a few minor linguistic tics that a second-generation native speaker wouldn't, and is she shy if she comes across a word she's not sure how to pronounce or whose meaning is unclear.

Having taught English as a second language to people from a wide variety of backgrounds, I don't expect anyone to memorize the royal order of adjectives, but it's incredibly useful for them to know that there is an underlying system and to match patterns of adjectives from their reading material against it, or play games with it (like thinking deliberately wrong phrases such as 'old little lady' for comic value), or be able to refer to it if they're nervous about writing something. The same is true for people who miss out on a proper education because of domestic or socioeconomic problems and maybe come to literacy as adults, but never develop the total confidence of someone surrounded by language since birth. Incidentally, those of you who live in California may have noticed the public information campaign encouraging parents to talk and sing to children, especially babies and toddlers. Research indicates that this has a massive influence on brain development and subsequent success or failure in life. See http://www.economist.com/news/science-and-technology/2159692... for an overview.

So yeah, of course the best way to acquire language is to be in a linguistically rich environment, and just soak it up to develop an intuitive understanding that promotes linguistic creativity and wordplay, as opposed to studying it through formal methods and turning it into a philological exercise. But that does not mean that formalism is bad, or that we should conceal the existence of systematic structures from kids in case it will wreck their creativity or something - if for no other reason, than to spare them the waste of reinventing the wheel should they be inclined to adopt a formalist approach on their own initiative. It's very easy to handwave away such rigid-seeming pedagogical tools if you already enjoy the benefits of total fluency, but for those who do not enjoy the same advantages this is the equivalent of pulling up the ladder behind oneself and then critique the confused for poor listening skills.

I don't have a comment on the math part - I agree with the author that we ought to be open to using multiple learning techniques so that each student can find the best one, but I think he seriously underestimates the utility of formality. When I was 5 I thought chanting multiplication tables every day at school was a bit silly, but 40 years later I greatly appreciate the fact that I can handle everyday trivial math problems reflexively rather than needing to reach for a calculator, pencil and paper, or a mental script of how to perform the calculation. Repetitive drills and formal methods are not the best way to explain new concepts, but they are incredibly useful exercises to retain them and make the basic knowledge feel instinctive in later years.