I am - I confess - a professional purveyor of tricks. A mnemonic here, a shortcut there - whatever it takes to get my students the skills they need to get the grade they want.
As such, I wasn't expecting to enjoy Nix The Tricks, a free e-book offering alternatives to some common rules of thumb; I was expecting everything I do to be ruthlessly shot down. I was one step away from getting the Mathematical Ninja to hunt the authors down, just in case.
To everyone's relief, though, the book is good. It even picks up on some of my major bugbears: the deadly minus-and-a-minus-makes-a-plus, the problematic PEMDAS (or BODMAS, as we call it here), the 'moving' model for linear algebra, the distance formula... all of these are taken down in roughly the same way as I would.
I'm completely in agreement with the idea that students should have a deep understanding of how maths works if they want to go on to do work that relies on it. I even agree with the case for teaching understanding from the start.
However, there are two problems the book brushes under the carpet: firstly, it implicitly assumes there's enough slack in the curriculum that teachers have time to teach the deep stuff; and secondly, it ignores the problem that students - like all good mathematicians - are short-sighted and lazy. Almost every GCSE student I've ever met would prefer - and get better results from - a trick method of expanding brackets (FOIL, crab claws, smiley face) over learning the distributive law. Certainly at GCSE, where two-terms-in-a-bracket is the worst it ever gets, FOIL is unobjectionable (in that it's memorable, and tends not to get messed up very often). If I have a student who knows how to use FOIL competently, I'm not going to spend half a class fixing something that isn't broken.
Indeed, if I think back to how I learned (and learn) maths, I've always learned tricks first and got the deep understanding afterwards - if I was interested enough.
Still: as a blueprint for how maths skills ought to be taught, Nix the Tricks is a great, thought-provoking effort. I imagine it'll get some slightly defensive reactions, like mine, and some very positive ones; in any event, it's definitely worth a read. (It comes in at about 64 pages on my Kindle, so it's hardly War and Peace.)
One thing I did take from the book was the idea that 'cancel' is a very dangerous word. Errors in simplifying fractions are a terrible scourge, and the idea of saying "No - you need to multiply or divide the top and bottom by the same thing" should make life a lot easier.