In fact, math syllabi in Poland tend to contain less things (and they are undergoing heavy changes these times), but the teachers are supposed to go in more depth – for instance, to solve atypical problems and to teach creative thinking. It is yet to be seen whether this approach will work.

(Isn’t math the coolest? That post at The Number Warrior is fantastic.)

When I was a student, completing the square was used quite a bit in analyzing quadratic equations in two variables (conic sections), but so far as I can tell all of that analytic geometry content has fallen out of our precalculus and calculus curricula.

Completing the square is still present in the integral calculus…. For example, in finding , one typically starts by completing the square under the radical before either doing a trigonometric substitution or applying a table of integrals.

Also there *is* a problem with really everyday, not artificial, problems… A classical one is to maximize/minimize something using the formula for the vertex of a parabola – not really useful in everyday life, but somewhat closer;).

And once we start teaching it, we reapply it to every new type of equation we encounter. It would be better not to bother with applications.

Actually, one of the nicer applications I’ve encountered was an extended problem involving profit… We had cost per item + fixed cost, and as we raised the price the number of units sold fell… it came from an awful curriculum that I helped get thrown out of my district. (See this

I still maintain that factoring per se has value and should be taught, but I don’t know how much value that translation skill has.

Jonathan