Learning Progressions, Vertical Scales, and Testable Hypotheses: Promising Intuitions and Points for Clarification

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ACKNOWLEDGMENTS

I am grateful to Ronli Diakow for comments on an earlier draft of this paper.

Notes

¹ Such “structural and mechanistic” details need not be physical, as I have argued elsewhere (Maul, 2013).

² I could find only one specific example of a potentially revisable belief about a learning progression in Briggs and Peck’s manuscript—namely, the proposal that “we [could] discover additional levels, or collapse existing ones” (p. 88). It is not clear how the concept of “distinct levels” (p. 88) within a learning progression—i.e., qualitative, ordinal distinctions—reconciles with the quantitative interpretations endorsed elsewhere in the manuscript, nor is it clear in what sense they are held to be “discoverable.”

³ The conflation of scales and attributes is an example of the representational fallacy described previously. In particular, the claim that an attribute is quantitative is not equivalent to the claim that a scale has interval (or even ratio) properties. The former is a claim about reality, whereas the latter is a claim about a representational system. Further, the connection between the properties of scales and the properties of attributes is not a matter of logical necessity: The existence of a quantitative attribute is neither necessary nor sufficient for a scale to have interval properties. Failure to make this distinction leads to incoherencies such as, for example, the claim that temperature is an “attribute” with “interval scale properties” (p. 94). Temperature is (widely recognized as) a quantitative attribute and as such may be represented on ratio scales (e.g., Kelvin), or interval scales (e.g., Celsius), just as with spatial distance. But more to the point, investigations into the nature of temperature (including the establishment of its quantitative structure and, thereby, its measurability) did not depend on the demonstration that any particular scale of temperature had “interval properties” (cf., Chang, 2004; Sherry, 2011).

⁴ More generally, claims about nearly any attribute will have equivocal interpretations at the level of more-specific facts; elsewhere I have discussed this in terms of multiple realizability (Maul, 2013). But neither vagueness nor ambiguity are equivalent to incoherence.