Hey YouTube, I kind of started this impromptu travel series and this is the first installment - What's In My Louis Vuitton Toiletry Case 25. I hope you enjoy it. Thanks for watching. If you have any quetions about the bag, feel free to leave them in the comment section below. ~Wil
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An analysis of representative literature concerning the widely recognized ineffective learning of "place-value" by American children arguably also demonstrates a widespread lack of understanding of the concept of place-value among elementary school arithmetic teachers and among researchers themselves. Just being able to use place-value to write numbers and perform calculations, and to describe the process is not sufficient understanding to be able to teach it to children in the most complete and efficient manner.
A conceptual analysis and explication of the concept of "place-value" points to a more effective method of teaching it. However, effectively teaching "place-value" (or any conceptual or logical subject) requires more than the mechanical application of a different method, different content, or the introduction of a different kind of "manipulative". First, it is necessary to distinguish among mathematical 1) conventions, 2) algorithmic manipulations, and 3) logical/conceptual relationships, and then it is necessary to understand each of these requires different methods for effective teaching. And it is necessary to understand those different methods. Place-value involves all three mathematical elements.
Practice versus Understanding.
Almost everyone who has had difficulty with introductory algebra has had an algebra teacher say to them "Just work more problems, and it will become clear to you. You are just not working enough problems." And, of course, when you cant work any problems, it is difficult to work many of them. Meeting the complaint "I cant do any of these" with the response "Then do them all" seems absurd, when it is a matter of conceptual understanding. It is not absurd when it is simply a matter of practicing something one can do correctly, but just not as adroitly, smoothly, quickly, or automatically as more practice would allow. Hence, athletes practice various skills to make them become more automatic and reflexive; students practice reciting a poem until they can do it smoothly; and musicians practice a piece until they can play it with little effort or error. And practicing something one cannot do very well is not absurd where practice will allow for self-correction. Hence, a tennis player may be able to work out a faulty stroke himself by analyzing his own form to find flawed technique or by trying different things until he arrives at something that seems right, which he then practices. But practicing something that one cannot even begin to do or understand, and that trial and error does not improve, is not going to lead to perfection or --as in the case of certain conceptual aspects of algebra-- any understanding at all.