Re:Marvin Minsky: What makes mathematics hard to learn? (2008)
by tokenadult on Tuesday January 26, 2016 @03:40PM
(#10975865)
Kids are ready for formal operations when their earlier experience has prepared them for formal operations. In some places around the world, students in fifth grade--including below-average fifth-graders--are already learning algebra. In quite a few whole countries around the world (for example, Taiwan, where my wife grew up when Taiwan was still a developing country) all seventh graders study algebra and geometry as part of their regular school math lessons, yes, including the below-average seventh graders. (In Taiwan, they also learned enough of the International Phonetic Alphabet to transcribe General American English at that age, but now that skill, which most United States reading teachers lack entirely even with a college degree, has moved into the elementary school curriculum.) Good education at the beginning produces better results in the middle grades.[1]
[1] http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic...
Re:Marvin Minsky: What makes mathematics hard to learn? (2008)
by derefr on Tuesday January 26, 2016 @04:08PM
(#10976092)
"Learning algebra" doesn't necessarily imply understanding the abstract concepts of algebra. Without the developmental substrate, children instead lean mostly on rote memorization. This seems to "work" when the class is taught in terms of rote application of memorized formulae, but will fall over the moment that students are presented with a problem in a novel context, or expected to synthesize multiple component skills into a larger solution.
If you've ever tried to tutor someone who "struggles with word-problems", the majority of the time it's because they don't actually understand the things they've been "learning" at all.
Similarly, this tends to be the difference between people who "get" programming and people who don't. If you ever talked to someone who failed a programming course, it's usually clear from their approach that every new language or API they learned became, to them, another set of rote facts to burn into their mind, never coming together to paint a picture of what the formal act of "programming" is generally.
And, on the converse, in previous centuries, you'd often see people only start learning math from the beginning in their late teens or early twenties—the time when they first attended "university" without previously attending a grammar or seminary school. These 20-year-olds would be able to pick math all up quite quickly, proceeding from addition and multiplication to calculus and beyond within a year or two.
This suggests to me that what we think of as "mathematical aptitude" is far more a measure of developmental age than anything else. I think a large part of why kids hate math is that we try to cram concepts into their brains when those brains aren't yet the right "shape" to take in those concepts. I expect that offsetting every part of the mathematical curriculum at least two years upward—if not far more—would do wonders for attrition rates.
My belief is that we could even fit all the same curriculum in by the end of public high school: although you'd be "squishing" more of the learning into the later years, the stronger minds kids would have at a higher age would allow them to acquire each successive concept both more quickly and more thoroughly, serving as a better base for the learning going forward, which would in turn be accelerated by that base.