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(Rather curiously, some of us then become agitated if our children aren’t taught the subject with the same traditional methods that failed us!
Learning isn’t just a matter of absorbing new information or acquiring automatic responses to stimuli.
Rather, we human beings spend our entire lives constructing theories about how the world works, and then reconstructing them in light of new evidence.
The more they’re given algorithms and told exactly what to do, the farther behind they fall in terms of grasping these concepts.
“Mindless mimicry mathematics,” as the National Research Council calls it, is the norm in our schools, from single-digit addition in first grade to trigonometry in high school.
As a result, they often can’t take these methods and transfer them to problems even slightly different from those they’re used to.
Or perhaps I should say this is what can’t do, in light of how many of us adults cheerfully describe ourselves as hating math or lacking any aptitude for it.
But because such a large proportion of homework practice-oriented, we should evaluate this claim carefully.
There’s obviously some truth to the idea that practice is connected to proficiency.
“If one is to be successful in quantitative thinking, one needs a fund of meanings, not a myriad of ‘automatic responses,’” he wrote. Repetition does not lead to understandings.” In fact, if “arithmetic becomes meaningful, it becomes so drill.” An emphasis on making meaning is directly opposed to the view that learning consists of the acquisition of a collection of behaviors.
Brownell’s insights about math instruction have been expanded and enriched by a long line of experts who have come to realize that the behaviorist model is, if you’ll excuse the expression, deeply superficial.