Even more depressing the same mistake shows up every decade under a new babe, constructivism, inquiry based learning, problem based learning. It gets proposed, propounded in ed schools, tested and proven ineffective and then shows up again under a different name, again.

> Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching

> Evidence for the superiority of guided instruction is explained in the context of our knowledge of human cognitive architecture, expert–novice differences, and cognitive load. Although un- guided or minimally guided instructional approaches are very popular and intuitively appeal- ing, the point is made that these approaches ignore both the structures that constitute human cognitive architecture and evidence from empirical studies over the past half-century that con- sistently indicate that minimally guided instruction is less effective and less efficient than in- structional approaches that place a strong emphasis on guidance of the student learning pro- cess. The advantage of guidance begins to recede only when learners have sufficiently high prior knowledge to provide “internal” guidance. Recent developments in instructional research and instructional design models that support guidance during instruction are briefly described.

http://www.cogtech.usc.edu/publications/kirschner_Sweller_Cl...

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.

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.

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.

Looking from the outside in, serious mathematics seems to require a ton of memorization. I really don't know how they keep all the names of everything straight, and what they mean, and how they relate to one another, and so on. That's all memory. There also seems to be an awful lot of drilled pattern recognition (as in you've gotta have the pattern and what it means when you see it memorized) in both the reading and writing of proofs.

From where I sit the mountains between me and deep mathematical understanding sure look like they're mostly memorization and drilling until various identities and mechanical operations are second-nature. Some of those peaks are surely "ah ha" moments or creative thinking or whatever but... not most of it. Maybe on the far side, but I can't even see that far. And damn is that far side ever removed from what 99% of people need math for. Not the way a literature class might be, but the way a Latin or Attic Greek language class might be. It's that far off and round-about and well into diminishing-returns territory.