> Teaching is frustrating too. Watching the kids that only care about the tricks ace the exam, and the ones with curious minds screw up a few mechanical things.

Or worse, the majority of kids that don't even realise the subject is more than just the tricks themselves.

And who's fault is that?

It's the system that's at fault. The kids want to cram for the test because the GPA follows them, like the Akashic Records, and that's why you get textbooks that promote un-understood shortcuts. You will have encountered the problem if you have taught university or highschool, and you will have identified it as a hard problem.

That's because these tricks is how you perform the operation quickly and efficiently. I'm not sure if you can even meaningfully derive the algebraic process and meaning from first principles. You should nix the cute mnemonic stories, since those probably serve to confuse people, but the methods themselves are absolutely solid. And the one shown right on the front page (for me) cross multiply is how you divide by fractions. a/b / c/d = a/b * d/c. The fact that 1/a is the reciprocal of a, whatever a is and that 1/(a/b) = b/a should be the first thing you tell people after you tell them about fractions. Then you show why cross multiply works. That shouldn't take more than one lesson.

P.S. On a tangential note, it would help if the names for mathematical terms were in English, rather than some weird amalgamation of Greek, Latin, French and German. In my language, all n-gons are called n-anguloids. So you have threeanguloid, fouranguloid, fiveanguloid, "a proper N-anguloid", etc... And we use the normal names for numbers, rather than Greek ones. And a rectangle is "right-angle-oid" (granted, that's the name in English too, if you squint a bit and know what phonetic shifts to look for and your German is good). Square is quadrat, though. Then again, it's still an everyday word, even if we borrowed it from Greek millennia ago. And the word for "perimeter" also means "a walk around" and "a lap". You literally tell people "the length of the lap of a rightanguloid is the sum of the four sides", which is kind of obvious if you imagine walking the perimeter. And the word for "angle" is the same as "corner". Sadly, that ship has sailed long ago.

I am 99% sure you're talking about Dutch, and in that case: we owe a lot of those non-borrowed terms, including the very term 'wiskunde' (mathematics) to Simon Stevin[1].

[1] https://en.wikipedia.org/wiki/Simon_Stevin

Just for clarity on all my responses in these threads, I'm a (minor) contributor to Nix the Tricks and a personal friend and research partner of the author.

And the one shown right on the front page (for me) cross multiply is how you divide by fractions. a/b / c/d = a/b x d/c.

Exactly! And that's exactly what the book recommends. Remember, the target audience of the book is teachers. If a teacher tells them to just draw a butterfly and multiply the matching parts, the students are not actually learning the mathematics. You can use whatever method you want once you understand the mechanics.

On a slightly tangential note -

In the name of getting rid of "mechanical" learning, most schoolkids these days don't even get to learn multiplication tables. While learning the tables is a pain, they pay rich dividends.

As an example, an 8th-grade kid who I help with math was struggling to find the square root of 529. He tried dividing it by his usual set of numbers, and pretty much hit a brick wall. I tried to get him to see that 529 was between 400 and 625, so its square root had to lie between 20 and 25. He did see it, but it's going to take him a while to figure out that he can do this with any number, if he can remember (or quickly work out) a few squares.

Now, one /could/ argue that computing the square root of a number is in itself mechanical ... but just a nodding familiarity with numbers can make even that a breeze.

And yes, the root cause of the problem is that math is taught as a bag of complicated tricks which need to be remembered, and not understood :/

And since it ends in a 9... :)