Estimation in Measurement
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Uncertainty and Degrees of Precision
One of the key ideas that is essential to an understanding of measurement is that measurement is always an estimate. What does this mean? Can we not obtain an exact measurement of something? This seems counterintuitive. Surely things have exact dimensions; is it impossible to obtain them? In a theoretical sense, the answer to this question hinges on the issue of decomposing. Between any two given partitions, we could always, theoretically, create a new partition. This theory carries us yet again into the realm of the infinite. There are an infinite number of partitions that we can imagine in any given space. Since infinity is a theoretical construct, but measurement is a practical concern, we always find that there is a limit to our practical ability to measure. Our tools are only capable of a certain, limited level of precision. Beyond the limits of our tools is the theory that there are further partitions than we can actually measure with those tools. This is what we mean when we say that measurement is always an estimate. That is why one of our most important measurement questions is always the question of how precise we need to be. How close is "close enough?" The answer almost always depends on the context and the purpose for which we are conducting our measurements.
This idea of measurement being an estimate is very important to keep in mind when we are measuring attributes that are somewhat abstract or constructed. When we create measurements such as the consumer price index or the gross domestic product, these measurements are very imprecise estimates of such things as economic activity. When we use test scores as measurements of children's achievement, we are constructing imprecise estimates of their progress. A very real problem is that no one knows just how accurate any of these constructed units really are. We are satisfied that there are real attributes to be measured, and we are satisfied that we have developed some sort of tool to measure them, but we have no way of knowing how accurately we are measuring. When public policy or strategic decisions are made based on measurements of this sort, we are taking risks. We need to know and remember that all measurement is an estimate.
Elementary Mathematics: Pedagogical Content Knowledge, by James E. Schwartz, is designed to sharpen pre-service and in-service teachers' mathematics pedagogical content knowledge. The five "powerful ideas" (composition, decomposition, relationships, representation, and context) provide an organizing framework and highlight the interconnections between mathematics topics. In addition, the text thoroughly integrates discussion of the five NCTM process strands.
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