Download Deep Thinking: What Mathematics Can Teach Us About the Mind by William Byers PDF

By William Byers

There's a couple of solution to imagine. most folks are accustomed to the systematic, rule-based pondering that one unearths in a mathematical evidence or a working laptop or computer application. yet such pondering doesn't produce breakthroughs in arithmetic and technological know-how neither is it the type of pondering that ends up in major studying. Deep pondering is a special and extra simple means of utilizing the brain. It leads to the discontinuous “aha!” adventure, that is the essence of creativity. it's on the middle of each paradigm shift or reframing of a problematical state of affairs. The identity of deep pondering because the default nation of the brain has the capability to reframe our present method of technological switch, schooling, and the character of arithmetic and technology. for instance, there's an unbridgeable hole among deep pondering and laptop simulations of pondering. many folks suspect that one of these hole exists, yet locate it tricky to make this instinct specific. This ebook identifies the way the real intelligence of deep considering differs from the factitious intelligence of “big information” and “analytics”. Deep considering is the basic element in each major studying event, which results in a brand new option to take into consideration schooling. it's also necessary to the development of conceptual structures which are on the center of arithmetic and technological know-how, and of the applied sciences that form the trendy global. Deep pondering are available at any time when one conceptual process morphs into one other. The resources of this learn comprise the cognitive improvement of numbers in childrens, neuropsychology, the research of creativity, and the historic improvement of arithmetic and technology. The strategy is uncommon and unique. It comes out of the author's long event as a mathematician, instructor, and author of books approximately arithmetic and technological know-how, resembling How Mathematicians imagine: utilizing Ambiguity, Contradiction, and Paradox to Create arithmetic and The Blind Spot: technology and the main issue of Uncertainty.

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Extra resources for Deep Thinking: What Mathematics Can Teach Us About the Mind

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However even at this preliminary stage I note that it would appear as though the properties of “naturalness” and “difficulty” are at odds with one another. If deep thinking were natural, then one would think that it would be simple yet I claim that it is always difficult. Think of a baby learning to walk—walking is natural and yet learning to walk is hard. It takes a great deal of effort and the will to overcome many failures. All significant episodes of learning and creativity will be found to have these two components: something innate that drives one towards making the effort and some obstacle that stands in the way of one’s eventual success.

Of course when you ascend to the world of fractions you learn that it contains what is essentially a copy of the counting numbers, now considered as the fractions n/1. From then on we successfully manage the ambiguity of considering an integer n as both a counting number and (simultaneously) as a fraction. The mathematician would say that we have isomorphically embedded the counting numbers into the fractions, which means that the copying process preserves all of the operations of arithmetic. The mathematician thinks of this as a formal process and so relatively easy, and does not always bear in mind that for the learner it involves the change of conceptual systems and that is always hard.

Management must be willing to relinquish absolute control but first it must acknowledge the difference between systematic and deep thought and make a decision to foster the latter. Finally, there is a paradox here that is basic to what is going on. Technology is always attempting to replicate aspects of intelligence and deep thinking. Of course this could be said of any conceptual system. And this attempt is always doomed to fail in an absolute sense even though some of the attempts are extraordinary.

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