IIIf. Another Aspect, Reading Aloud

Truth emerges more readily from error than from confusion. Bacon

There is no such thing as measurement absolute; there is only measurement relative. Jeanette Winterson.

The Case of the Missing Person Parameters

Eliminating nuisance parameters and #SpecificObjectivity

It was a cold and snowy night when, while trying to make a living as a famous statistical consultant, Rasch was summoned to the isolated laboratory of a renowned reading specialist to analyze data related to the effect of extra instruction for poor readers. There may be better ways to make a statistician feel a valued and respected member of the team than to ask for an analysis of data collected years earlier but Rasch took it on (Rasch, 1977, p. 63.)

If we could measure, in the strictest sense, reading proficiency, measurements could be made before the intervention, after the intervention, and perhaps several points along the way. Then the analysis is no different, in principle, than if we were investigating the optimal blend of feed for finishing hogs or concentration of platinum for re-forming petroleum.

Continue reading . . .IIIf. Reading Aloud

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IIIe: A Spectrum of Math Proficiency and the Specter of Word Problems

In mathematics, one does not understand anything. You just get used to them. Johann Von Neumann

Defining mile posts along the way from counting your toes to doing calculus

The world has divided itself in two factions: those who think they don’t understand math and those who think they do. But we’re not talking about proving Fermat’s Last Theorem or correcting Stephan Hawking’s tensor algebra; we’re talking about counting, applying the four basic operators, and solving the dreaded word problems using basic algebra, geometry, and perhaps a little calculus. That just about covers the range from counting your toes to determining the spot in the outfield where a player should stand to catch a fly ball and should be good enough to get you through freshman math.

Continue reading . . . A Spectrum of Math Proficiency

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IIId. On Any Given Sunday

May the better team have better odds

Pair-wise comparisons and arbitrary labels

All of us have probably thought sometime during the football season that there must be a Rasch analysis in here somewhere. Every team in the National Football League plays a different selection of opponents but rankings are based, mostly[1], on a simple count of games won. We certainly know better than that. Here’s my answer.

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[1]There are more rules for resolving ties in the rankings but these are designed more to create excitement and sell tickets than to ensure that the best team wins.

IIIc. Hot and Cold: making and connecting scales

In educational measurement, we don’t yet know if we are measuring heat or temperature.

A man with one watch knows what time it is; a man with two watches is never quite sure. Lee Segal

Building, calibrating, and equating instruments

Meaning comes from experience and experience comes from ignorance. You learn what hot means by touching the stove; you learn what cold is by not wearing your mittens. Nothing here answers the question of where cold ends and hot begins or what’s the line between “medium” and “medium rare;” those points are subjective, arbitrary, and personal; hopefully not capricious.

Temperature is one of the first lessons we learn: what things are too hot to touch? When is the weather too cool to not wear a jacket? When is it warm enough to go barefoot? How much fever warrants staying home from school? These concepts may define meaningful temperature bands, but “because mom says so” is not very objective, and definitely not measurement. Continue reading . . . Hot and Cold

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IIIb. The Aspect of Color

Roy G. Biv: How many bands in your rainbow?

Art is the imposing of a pattern on experience. Alfred North Whitehead

Performance bands are arbitrary but useful

Qualitative meaning and quantitative precision

We all know our basic colors before we start to school. We learn early on that there are three primary colors (red, yellow, and blue), from which all others can be created, although designers of color printers apparently missed that lesson. The ancients saw five colors (red, yellow, green, blue, violet) in the rainbow. Newton saw seven, adding orange and indigo (perhaps to align with the natural harmony of the universe found in the number of musical notes, days of the week, and known planets; or perhaps he was just buying some vowels.)  Continue reading . . . The Aspect of Color

[While my analogy comparing bands of the rainbow to performance levels may be cute, I probably don’t have the physiology right. While light is a continuous spectrum, our perception of discrete bands may be real, depending on the distribution of cones in our eyes. Was it more important to for our ancestors to discern white animals against a white background or to distinguish ripe fruit and poisonous reptiles.]

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III. Abstracting Some Aspects

  • Measure what is measurable, and make measurable what is not so. Galileo Galilei

Measuring rocks and the significance of being sufficient

The process of measurement begins long before any data are collected. The starting point is a notion, or even better, a theory about an aspect of a class of things we want to understand better, maybe even do some science on. Successful measurement depends on clear thinking about the aspect and clever ideas for the agents. This is much more challenging and much more rewarding than any mathematical gymnastics that might be performed to fit model to data.

All analogies are limited but some are useful. Considering aspects of things far removed from cognitive traits may help avoid some of the pitfalls encountered when working too close to home. Hardness is a property of materials that is hard to define but we all know what it is when it hits us. Color is a narrow region of a continuous spectrum that non-physicists tend to think about as discrete categories. Temperature is an intimate part of our daily lives, which we are quite adept at sensing and more recently at measuring, but the closely connected idea, heat, may actually be more real, less bound to conventions and populations. If I could scale the proficiency of professional football teams and reliably predict the outcomes of games, I wouldn’t be writing this.

Continue reading . . . Hard Headedness: the importance of being sufficient


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