  A transformation called normalized gain (ngain) has been acknowledged as one
of the most common measures of knowledge growth in pretest-posttest contexts in
physics education research. Recent studies in math education have shown that
ngains can also be applied to assess learners' ability to acquire unfamiliar
knowledge, that is, to estimate their "learning rate". This quantity is
estimated from learning data through two well-known methods: computing the
average ngain of the group or computing the ngain of the average learner. These
two methods commonly yield different results, and prior research has concluded
that the difference between them is associated with a pretest-ngains
correlation. Such a correlation would suggest a bias of this learning
measurement because it implies its favoring of certain subgroups of students
according to their performance in pretest measurements. The present study
analyzes these two estimation methods by drawing on statistical models. Our
results show that the two estimation methods are equivalent when no measurement
errors exist. In contrast, when there are measurement errors, the first method
provides a biased estimator, whereas the second one provides an unbiased
estimator. Furthermore, these measurement errors induce a spurious correlation
between the pretest and ngain scores. Our results seem consistent with prior
research, except they show that measurement errors in pretest and posttest
scores are the source of a spurious pretest-ngain correlation. Consequently,
estimating learning rates might effectively provide unbiased estimates of
knowledge change that control for the effect of prior knowledge even in the
presence of pretest-ngain correlations.
