Konrad Zusearchitect of the Z3 computer, which uses a bit binary floating-point representation.
This work is available here free, so that those who cannot afford it can still have access to it, and so that no one has to pay before they read something that might not be what they really are seeking. But if you find it meaningful and helpful and would like to contribute whatever easily affordable amount you feel it is worth, please do do.
I will appreciate it. The button to the right will take you to PayPal where you can make any size donation of 25 cents or more you wish, using either your PayPal account or a credit card without a PayPal account. The Concept and Teaching of Place-Value Richard Garlikov 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". 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. 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.
What is necessary to help a student learn various conceptual aspects of algebra is to find out exactly what he does not understand conceptually or logically about what he has been presented. There are any number of reasons a student may not be able to work a problem, and repeating to him things he does understand, or merely repeating 1 things he heard the first time but does not understand, is generally not going to help him.
Until you find out the specific stumbling block, you are not likely to tailor an answer that addresses his needs, particularly if your general explanation did not work with him the first time or two or three anyway and nothing has occurred to make that explanation any more intelligible or meaningful to him in the meantime.
There are a number of places in mathematics instruction where students encounter conceptual or logical difficulties that require more than just practice. Algebra includes some of them, but I would like to address one of the earliest occurring ones -- place-value.
From reading the research, and from talking with elementary school arithmetic teachers, I suspect and will try to point out why I suspect it that children have a difficult time learning place-value because most elementary school teachers as most adults in general, including those who research the effectiveness of student understanding of place-value do not understand it conceptually and do not present it in a way that children can understand it.
And they may even impede learning by confusing children in ways they need not have; e. There are at least two aspects to good teaching: The passages quoted below seem to indicate either a failure by researchers to know what teachers know about students or a failure by teachers to know what students know about place-value.
If it is the latter, then it would seem there is teaching occurring without learning happening, an oxymoron that, I believe, means there is not "teaching" occurring, but merely presentations being made to students without sufficient successful effort to find out how students are receiving or interpreting or understanding that presentation, and often without sufficient successful effort to discover what actually needs to be presented to particular students.
That is not always easy to do, but at least the attempt needs to be made as one goes along. Jones and Thornton, p.
His [sic; Her] investigation showed that despite several years of place-value learning, children were unable to interpret rudimentary place-value concepts. It should not be surprising that something which is not taught very well in general is not learned very well in general.
The research literature on place-value also shows a lack of understanding of the principle conceptual and practical aspects of learning place-value, and of testing for the understanding of it. Researchers seem to be evaluating the results of conceptually faulty teaching and testing methods concerning place-value.
And when they find cultural or community differences in the learning of place-value, they seem to focus on factors that seem, from a conceptual viewpoint, less likely causally relevant than other factors. I believe that there is a better way to teach place-value than it is usually taught, and that children would then have better understanding of it earlier.
Further, I believe that this better way stems from an understanding of the logic of place-value itself, along with an understanding of what is easier for human beings whether children or adults to learn. A teacher must at least lead or guide in some form or other.
How math, or anything, is taught is normally crucial to how well and how efficiently it is learned. It has taken civilization thousands of years, much ingenious creativity, and not a little fortuitous insight to develop many of the concepts and much of the knowledge it has; and children can not be expected to discover or invent for themselves many of those concepts or much of that knowledge without adults teaching them correctly, in person or in books or other media.
Though many people can discover many things for themselves, it is virtually impossible for anyone to re-invent by himself enough of the significant ideas from the past to be competent in a given field, math being no exception.
Potential learning is generally severely impeded without teaching.The power to understand and predict the quantities of the world should not be restricted to those with a freakish knack for manipulating abstract symbols.
When most people speak of Math, what they have in mind is more its mechanism than its essence. This "Math" consists of assigning meaning to a set. Explore a wide range of recent research in mathematics. From mathematical modeling to why some people have difficulty learning math, read all the math-related news here.
The Concept and Teaching of Place-Value Richard Garlikov.
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 .
Computer and Mathematics News. From quantum computers to the value of statistics, read the latest math and computer news. Updated daily. The power to understand and predict the quantities of the world should not be restricted to those with a freakish knack for manipulating abstract symbols.
When most people speak of Math, what they have in mind is more its mechanism than its essence. This "Math" consists of assigning meaning to a set.
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