Educational Growth, Learning Progressions, and Metrological Simulacra: Putting the House in (Partial) Order

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Notes

¹ This example is intended for illustrative purposes only and not as a remotely plausible cognitive law, particularly given the conceptual vagueness of “cognitive attribute,” “effort,” and “teaching quality.”

² Humphry (2013b) used the term “quasi-unit.”

³ Boyle’s law was specifically articulated with respect to gases, but the same principle underpinned the use of liquid thermometric substances (Sherry, 2011).

⁴ As it turns out, the linear thermometric expansion of mercury breaks down close to 0°C, but the deviation is negligible for the vast majority of practical applications of a thermometer (Sherry, 2011).

⁵ This final step was the basis for Campbell’s (1920) claim that the Celsius thermometer does not provide real measurements, as the partitioning of the scale into equal spatial intervals was not done on the basis of laws of temperature.

⁶ The Lexile theory of reading comprehension is the one exception to the dearth of substantive quantitative theory and measurement units in educational assessment (Stenner et al., 2013). However, in addition to criticisms raised by Briggs (2013), the claim that the Lexile is a unit of reading ability is dimensionally confused. The fixed points of the Lexile are defined in terms of magnitudes of text difficulty. This would be equivalent to deriving a unit of temperature from fixed points of volume—that is, the wrong quantity.

⁷ Briggs (2013) and Briggs and Peck (this issue) only address double cancellation, but axiomatic conjoint measurement theory includes a hierarchy of cancellation conditions as tests of additivity in data, which are mathematically independent (Michell, 1990). Kyngdon and Richards (2006) explicate deterministic tests of triple cancellation, which require a 4 × 4 conjoint matrix at minimum.

⁸ For example, if a simple rank order of end of school examination results is required for university entrance.

⁹ This point reinforces Briggs and Peck’s need to attend to the conceptual distinction between representation and referent or, specifically in this case, the structure of the data and the theorized structure of the cognitive attribute.

¹⁰ Quantitative relationships are thoroughly homogeneous (Michell, 2012).