From Concrete to Abstract in the Measurement of Length

The concatenation of units of length is widely viewed as the paradigmatic expression of fundamental measurement. Survey, assessment, and test scores in educational and psychological measurement are often interpreted in ways that assume a concatenation of units to have been established, even though these assumptions are rarely stated or tested. A concatenation model for measurement is shown to be equivalent to a Rasch model: any two units of measurement placed end to end must together be of the same length as either one of them added to itself. This additive principle and a concatenation model of measurement together serve as a heuristic guide for organizing two experimental approaches to calibrating instruments for measuring length. The capacity to reproduce the unit of measurement from theory with no need for repeated empirical calibration experiments, as in the geometrical bisection of the line and the resultant halving of the length measure, is highlighted as essential to demonstrating a thorough understanding of the construct.