Giving change when payment is made with a dime: the difficulty of tens and ones

ION AND REPRESENTATION Piaget made a distinction between empirical (or simple) abstraction and constructive (or reflective or reflecting) abstraction. In empirical abstraction, we focus on one or more properties of objects and ignore the others. For example, when we categorize objects by color, we focus on color (physical knowledge) and ignore all the other properties. In constructive abstraction, by contrast, we create mental relationships (logico-mathematical knowledge) that do not exist in objects. For example, when we are presented with a dime and a penny and we say that they are different, we are creating a mental relationship that is not in the objects in the external world. This relationship is created by our thinking. The numbers 2, 8, 10, or any other numbers are likewise created by each individual through constructive abstraction. (We prefer to use the term "constructive abstraction" rather than "reflective (or reflecting) abstraction" because the former seems easier to understand.) When the Romans wrote "XL VI" instead of "46," their system of representation (social knowledge) was different from ours. However, their logico-mathematical knowledge of tens and ones, created by constructive abstraction, was exactly the same as ours. In our Arabic system of writing, we use the place (position) of the "4" and the "6" differently from the Romans. When we say that "place value concepts" are hard for children, we must differentiate between the difficulty that children have in representation from the difficulty they have in abstraction. The errors they make in representation usually occur because of their low-level abstraction. The verb "to represent" is often used carelessly, as when we hear that base10 blocks represent the base10 system. Base10 blocks do not represent anything by themselves. Representing is what an individual does. When adults are presented with 4 ten-blocks and 6 one-blocks, we can represent 46 to ourselves because we have already abstracted a system of tens out of our system of ones. When young children see the same blocks, however, they cannot represent forty-six to themselves unless they have constructed a system of tens through constructive abstraction. In Piaget's theory (1945/1962), representing refers to (a) internally evoking or externally expressing an idea and (b) interpreting (comprehending) an external representation. An example of an internal evocation is our thinking about 5 cookies when the cookies are not present. An example of an external expression of this idea is our drawing of 5 circles to represent 5 cookies (which may or may not be present). When we understand the 5 circles thus drawn as being 5 cookies, this is an example of interpreting an external representation. Another example is adults' and children's interpretations of 4 ten-blocks and 6 one-blocks. We can give to these objects only the meaning that our level of abstraction permits. A study by Kato, Kamii, Ozaki, and Nagahiro (2002) demonstrated that young children represent numerical quantities at or below their level of abstraction but not above. In Japanese day-care centers, the authors individually showed to 60 children who were from 3 to 7 years old 4 plastic dishes, 3 metal spoons, 6 pencils, and 8 wooden blocks. Their request to each child was to put something on a piece of paper "so that your mother will be able to tell what I showed you" (p. 35). The children This content downloaded from 207.46.13.58 on Sat, 17 Sep 2016 05:41:53 UTC All use subject to http://about.jstor.org/terms