[natestemen]

As someone who majored in math/physics I'm going to heavily disagree. I was quite poor when it came to multiplication tables, and monsters like long division, but was a pretty good student through university. I never felt like I needed any of those things through any higher level maths, and making those a requirement to learn (or at least drill as much as they do) seems counter productive.

[pflats]

The basic times tables are a big sticking point in secondary mathematics. Because polynomial work depends heavily on factorization, students who cannot readily see that 56 = 7 * 8, and then apply their elementary operations skills to say 7 * 8 = (27) (8/2) = (47) (8/4) will struggle more than those that can.

You can develop skills to compensate (e.g., starting with small primes and working up instead) but it's still an extra step in the process.

This does not necessarily affect a student's overall understanding of polynomial functions, but can hang over their head in the actual work with them.

[asark]

I've observed factoring generally to be a big problem. I also recall the official introduction of factoring in our curriculum (9th grade?[0]) to be when they lost an awful lot of people. Perhaps not coincidentally, it's also where things took a big swerve from straight application of memorized facts and the correct algorithms into lateral thinking and intuitive guesswork, and no-one bothered to make that clear at the time. Lots of very angry, discouraged students resulted, all wanting to know exactly WTF they were being asked to do and getting, from their perspective, no straight answer.

Mathematics from there on continued to mostly be the memorizy-algorithmy kind, but with enough of the intuitiony-thinky kind sprinkled in and no clear guide to which was which or even that the teachers were asking us to do something fundamentally different that anyone who hit a big speed-bump at factoring didn't have a hope of keeping up.

[0, EDIT] Maybe 7th or 8th? It's been too long, IDK.