Interesting My experience with my (now 11 year old) child was that the focus was heavily on trying to get to an intuitive understanding of how things work with zero memorization.
On the flip side, if you don't know e.g. your times tables by heart, you are in for a shitton of trouble when it comes to doing the higher level stuff. The best way to achieve the arithmetical competence it takes to successfully do math at secondary levels and above is drilling. Educational history is littered with the wreckage of alternative attempts: New Math, some of the approaches derided as "Common Core", etc.
> 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...
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.
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.
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.
For the teaching methods, I'll defer to A Mathematician's Lament by Lockhart. Kids are taught to memorize and often by teachers (esp in grade school) who don't understand the math. When I got my teaching credentials in secondary mathematics, one of the points driven home was that a large factor that led many grade school teachers to avoid secondary (high school) was fear of the math.
Case: my daughter's 7th grade teacher taught the rules of exponents. Part of it, he said anything to the zeroth power is 1 and anything to the oneth power is itself and nobody knows why, it is just one of those things. Fast forward a year or two, and kids are lucky to remember a factiod right. A simple pattern matching exercise would have helped kids rediscover the rule if needed.