I don't argue that the fundamentals of +|-|/|x are essential, and that multiplication and division tables up to 12 are critical to core math skills.
As someone who has actually taught high school math, including trying to teach algebra II students who could not do basic arithmetic (I'm talking things like 12 minus 5), I dare you to try to think of ways to get students to actually understand how algebra works, what its applications are, etc. when they can't manipulate even the most basic equation without a calculator to tell them that 12 - 5 is 7.
If it wasn't clear by my statement that fundamentals are essential, that is what I meant. I think that 12 -5 = 7 certainly falls below the high water mark of multiplication and division tables. In other words, you are beginning to argue a point that needs not be argued
Is it possible? Yeah, I managed to do it with some kids. But these kids mostly had a fundamental barrier preventing them from doing just about anything in higher level math, because they simply couldn't manipulate even small numbers on a basic level.
And these kids should probably have extra effort put towards then understanding the critical math that is required in day-to-day life and not be pressed into more advanced mathematics studies. Subjects that are hard for students to learn (hard in the sense that they fall behind the other students) is likely to be nearly impossible to retain or recall. These kids skill set may be in another subject. Don't waste any time on the more advanced stuff that they will not learn or retain.
Knowing basic arithmetic is not just memorization -- with it should come some more intuitive understanding about how numbers relate to each other, and elementary schools should try to convey that information along with any memorization task. The issue isn't so much that a particular student doesn't happen to know the exact fact of 12 - 7 as much as that the student has no intuitive grasp of what the relationships between "12," "-," and "7" are, which should at least give him an intuitive sense of what the answer to that relationship might be.
I agree. My point was that memorizing formulas and functions rarely benefits the average student/adult. Understanding the methods is what should be taught. The basics still apply though and swiftly solving any addition or subtraction well into the hundreds is very manageable with very little memorization. Beyond that, the average person will not have the memory when they reach 21 years old to do in their head anyway (some say as little as 7 digits are easily remembered and available for doing mental math). Techniques to overcome these limits would be a good part of the curriculum.
Massive amounts of memorization interferes with cognitive learning.
Not true, unless it becomes too dominant in the curriculum that it doesn't allow time for anything else. I'm not a huge fan of lots of memorization in schools, but actually having some knowledge in your brain is not only incredibly useful for solving problems that require that information, but it also makes it much, much, much more likely that you'll ever meditate upon that information and make higher-level connections within it.
I believe it has been stated, with various degrees of truth and accuracy, that most people do not memorize well in school settings. I believe that interest in a subject matter is the only way students or adults pursue higher level mathematics (or any subject). Those students with aptitude and interest should definitely be moved into classes that challenge them, while disinterested students should probably be guided to subjects they are more apt too (again, not ignoring the basics)
Memorization does not contribute to higher skill sets beyond the basics, cognitive learning allows people to do advanced work with minimal research.
I have no clue how one can do "cognitive learning" without knowing any facts about anything. I've heard hundreds of hours of educational theory BS yammered at me in numerous pedagogy classes and conferences, but the reality is that critical thinking requires something to think about. If your brain is empty, you can never make connections beyond whatever is on the page or the website in front of you, which improverishes your ability to think with any breadth.
Maybe you misunderstand. facts and equations are not the same thing. I consider myself to be exceptionally able to learn and apply that learning to problems. My skills and capabilities were earned through my being interested in the subject matter, being challenged with problems that others were not able to solve, and through a lot of my own learning outside of school. The school system provided my with very little beyond the basics, and rationed that knowledge out over a excruciatingly long time frame. I easily outpaced the average in school, so I sat in the classroom with people that I considered intelligent and capable, that were not able to do the same work as I was. I know that this statement has more than a slight stench of arrogance, but I don't mean it too.
My point is that I excelled in subjects and the schools that I went to, in 5 states, all failed to challenge me in those subjects. Kids that excelled in other subjects sat bored in those classes or caused trouble.
I would add that many hours spend memorizing things in school went unused. I can still remember the periodic table, no use in real life. I can still remember how a frog is put together and comes apart, no use in the real world. The labs were good, learning about elements, electrons, etc etc was all good. But memorizing the data was of such minimal value that it could have been skipped altogether.
I'm sorry, but you assertions are a load of crap.
In your opinion, which you can enjoy without my criticism.
My point of view is based on my experience and America's schools poor testing vs other 1st world nations. If all is well, then everyone else must be doing the same thing and their people are just smarter. I don't believe that for a second.
