[marksbrown]

A good example in the UK is teaching students the FOIL technique for algebraic expansion. Students typically can expand (ax+b)(cx+d) because they've learnt a recipe but cannot expand say (ax2+bx+c)(dx+e).

Many schools here focus on such tricks (nix the tricks was a great book focusing on such things) as schools here are judged on pass/fail rates.

In general, exams are an excellent way to assess students en masse at their ability to remember similar problems but not inherent problem solving techniques. The latter I've found is possible to teach 1to1 but far harder with a class of varying abilities.

[LargeWu]

That, to me, is not a problem with the exam though. It's a problem of teaching to a special case and not the general case. If you want to find fault, it's in the incentive system. But I don't see how the exam itself is the problem.

[marksbrown]

Well I won't reiterate all of 'bad education' by Bryan Caplan but to my mind exams are imperfect because:

1. Schools are not equal. It's not fair to compare students when they usually have no choice over their teachers. 2. Exams cover an arbitrary syllabus controlled by undemocratic exam boards. 3. Topics are chosen by exam boards that can be examined not by importance. 4. Students who perform poorly under stress of exam conditions are punished for it. 5. Exams serve no real purpose. Children are not chickens being graded for sale. They're at best a weak signal of aptitude.

I would much prefer exams to serve as a prerequisite of sitting a future course rather than an assessment at the end. That way teachers can actually teach rather than continuously repeat the same content.

[pokerface_86]

the inability the generalize the foil procedure to an expression with more than 2 variables speaks more to the non mathematically oriented population just sucking at generalizing things. i have found this to be a very “you have it or you don’t” type of thing, not really something that can be taught

[snowfarthing]

Then again, it may be because FOIL is stupid.

I've always had a difficult time wrapping my head around this acronym. What counts as "outer"? What counts as "inner"? And yes, when there are more than two items (not necessarily variables!) to be multiplied, you suddenly have to ignore this little trick, because now it's confusing to know what to do about the middle stuff -- and it doesn't take into account non-commutativity either.

And yes, some of the problem may be due to my (very recently diagnosed! at least, formally) autistic mind. But I cannot help but think that if someone with a PhD in math struggles with and largely ignores "FOIL", then the problem may be with the technique, and not with the people who don't understand it.

[pokerface_86]

i never found foil stupid, but i also only have a bachelors in math. maybe this was because i had already been exposed to multiplying polynomial expressions, beyond just a 2 term * 2 term by the time i had learned it in school, but i never found it particularly complicated to grasp. foil was never taught to me as the only way to multiply polynomials, rather, an easy algorithm to apply in a certain case. the goal is for you to make the connection that oh, in a 3x2 case, u have to multiply each term in the 3 with each term in the 2, etc.

i think your problems with foil can be extended to the general way math is taught. at least for me, it was always full of tricks, little rules that can be broken sometimes, and i was constantly learning new things that made me realize my old teachers had taught us tricks to shortcut solutions.