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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". 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 can't work any problems, it is difficult to work many of them. Meeting the complaint "I can't 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.

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.(2)(3) Elementary school teachers can generally understand enough about place-value to teach most children enough to eventually be able to work with it; but they don't often understand place-value conceptually and logically sufficiently to help children understand it conceptually and logically very well. And they may even impede learning by confusing children in ways they need not have; e.g., trying to make arbitrary conventions seem matters of logic, so children squander much intellectual capital seeking to understand what has nothing to be understood.

And a further problem in teaching is that because teachers, such as the algebra teachers referred to above, tend not to ferret out of children what the children specifically don't understand, teachers, even when they do understand what they are teaching, don't always.............

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