A whole lot of "I think", "I believe," "I feel". No reference to existing educational research on what works and what doesn't.

>This isn’t a call to give up, though: instead, we should teach maths differently, in a way that doesn’t churn out mathophobes. When five-year-olds first encounter the subject, it’s as a creative, open-ended activity, involving play and exploration. They learn about numbers using colourful blocks that join up in different ways. They fit these shapes together and tell different stories with them. Just a year or two later, though, maths becomes a discipline with strict rules and a forbidding regime of right or wrong answers. Instead we should try to maintain that sense of exploration and open-endedness, of trying out different approaches to a problem and seeing what works. What’s important about times tables, for example, is not the answers, but the different possible relationships between numbers. What’s important about equations is not the solution, but the techniques we use to untangle a problem using logic. Some of these exercises could be presented more like a Sherlock Holmes mystery, where the focus is on how the clues are pieced together and not just on what the answer must be.

Where's the evidence that this would actually be a superior teaching method? The point of drills is to counteract the natural tendency to forget things. Fluency is gained through practice and repetition. People don't want to hear it, they think it's somehow regressive, but it's a fundamental fact about the brain. You can't get around it any more than you can build muscle without consistent training.

Think of an actual Sherlock Holmes mystery you've read before, months or years ago. Do remember the precise sequence of deductions made, leading from evidence to the killer? Probably not. Could you do what Holmes did, having read it? No, you couldn't. You remember enjoying the performance, you remember some quips and some characters, but not the critical details. It was fun, but you didn't learn anything! That's okay for passively enjoying fiction, but mathematics is supposed to be a thing you eventually do yourself, unsupervised. Getting the right answer will actually matter.

>One powerful way of maintaining interest is to link maths to whatever the children already care about, whether that’s food, singing, dancing, drawing, creative writing, sport, Lego, Minecraft or something else. Mathematical concepts can be related to the real world, rather than something abstract: commutativity, for example, might be presented as a dry numerical exercise along the lines of 5+2 = 2+5. Or we can bring it to life by talking about whether or not it makes any difference if tea is made by pouring hot water over tea leaves or by dropping tea leaves into hot water.

>These kinds of approaches will mean covering material more slowly. Some will argue that this is “dumbing down”. On the contrary, we will be teaching maths at a much deeper level, with the likelihood of these lessons being fully absorbed – rather than drilling students on a bunch of algorithms they’re able to grasp only superficially and will forget once they stop using them.

Hahah. She really thinks she's a revolutionary, doesn't she? She actually thinks nobody has thought of this before. They tried this exact thing in America in the 50s-60s, the "New Math". It was a resounding failure and roundly mocked:

https://www.youtube.com/watch?v=UIKGV2cTgqA

But in the new approach, as you know, the important thing is to understand what you're doing, rather than to get the right answer.
....
(And you know why four plus minus one
Plus ten is fourteen minus one?
'Cause addition is commutative, right!)

Most kids aren't at the level of cognitive development where abstract ideas like "commutative property" are helpful. They need a concrete foundation, that is concretions like the times tables, on which to build abstractions later.

For my part, I understood perfectly well what I was doing when I learned algebra and calculus. "Find the root", "find the slope of the curve"; these are actually not difficult ideas. What is difficult is doing them consistently, quickly, without error. You can "know what you're doing" and still be totally unable to actually do it, in practical terms. Recognition ("I get it!") is fundamentally easier than recall ("I did it!"). The only way I got to fluency was by drilling hundreds of exercises. And don't just take my word for it; Alfred North Whitehead was fighting this "anti-actually-knowing-things-ism" crap a century ago.

>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.

Her commutativity example doesn't make any sense. First, it's not clear that mixing water with tea leaves corresponds to addition. They are two different kinds of thing, and usually we can only add things of the same kind. Why can I "add" water and tea leaves, but I can't "add" apples and oranges? And what about subtraction? If water + leaves = tea, then surely tea - leaves = water. But if you take leaves out of tea, you still have tea, not the water you began with. Second, it does make a difference! There are slight chemical changes, and some people can tell from blind taste tests. Third, why tea? I could equally say that I "add" socks to my feet, and "add" shoes on top of those, but that's clearly non-commutative (but at least "subtraction" works, because I can take off the shoes and the socks and get my feet back!). Pedagogically the tea thing is a disaster -- it leaves you more confused than you started. I hated this kind of fake-explanation as a kid. It's sheer laziness on the part of the teacher.

I think it's telling that she describes herself as a mathematician, and relates no personal experience of teaching mathematics. People far out on the right side of the ability distribution tend to have trouble imagining how average people think. And grownups in general tend to forget how hard it was as a kid to learn things the first time, because they can't shed the mental framework they built up and see things with untrained eyes again.

EDIT lol of course, she's a category theorist. that explains everything.