"If the difficulty of an item were not invariant over some useful domain, then the term difficulty would have no useful meaning. The inch on my wooden yardstick is invariant as long as I don't use the yardstick inside the furnace and, of course, am careful to (a) originate the zero end of the yardstick at the place from which I want to measure a difference and (b) align the yardstick parallel to the direction in which I want to measure and (c) hold the yardstick still and (d) look carefully, with my glasses on, to see as sharply as I can what inch mark seems to be reached. And, when I am serious, I will replicate this procedure several times first to exclude wild values and finally to extract one value with an allowance for error which I will then use as the measure."
"I think something will have to be done with this concept that takes into account the fact that the brain/mind is not like anything else. If the maximum weight I can lift with my single arm is 70 lbs, it will also be the same the next day, and the day after etc. It will take a long time before lifting that weight becomes easier. But with problem-solving techniques it is instantaneous. If I learn how to solve a particular kind of problem, no matter what its degree of difficulty, it is a done-deal. Next time it is no longer difficult."
"It seems that the person merely has more ability after learning the new problem-solving technique. Having learned the technique does not change the difficulty of that type of problem relative to the other problems. If, however, there is no hierarchy of problems and problem solving techniques are not acquired in any particular order, then measurement is not possible. For example, if you can perform long division, I assume that you can also do single digit addition. This is because I envision a hierarchy of math operations. When you violate this common understanding of math ability, common because when calibrating various problems they maintain the same relative difficulty, I must decide what to do with your improbable response. Were you careless on the single digit addition? Were you lucky on the long division? Does your math ability conform to the common understanding of math ability?"
John Michael Linacre:
"In fact, the loss of invariance is major challenge to pre-post test equating. If I can't read Chinese, at the pre-test, all sentences in Chinese are equally difficult to read. When I can read some Chinese, at the post-test, some sentences are easy and some are hard. When I'm learning to drive a car, some things are easy and some things are hard. When I've learned to drive, all essential skills are equally easy. So where is the yardstick? We have to decide which situation corresponds to our intention to measure, and then use that situation to quantify the invariant measures that we will apply everywhere. The metrologists follow exactly this procedure when they construct physical measuring instruments."
The Problem of Measure Invariance. Wright, B.D., Huber, M., O'Neill, T., Linacre, J.M. Rasch Measurement Transactions, 2000, 14:2 p.745
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