Common terms

The familiar expression "steep learning curve" may refer to either of two aspects of a pattern in which the marginal rate of required resource investment is initially low, perhaps even decreasing at the very first stages, but eventually increases without bound.

Early uses of the metaphor focused on the pattern's positive aspect, namely the potential for quick progress in learning (as measured by, e.g., memory accuracy or the number of trials required to obtain a desired result)[5] at the introductory or elementary stage.[6] Over time, however, the metaphor has become more commonly used to focus on the pattern's negative aspect, namely the difficulty of learning once one gets beyond the basics of a subject.[citation needed]

In the former case, the "steep[ness]" metaphor is inspired by the initially high rate of increase featured by the function characterizing the overall amount learned versus total resources invested (or versus time when resource investment per unit time is held constant)—in mathematical terms, the initially high positive absolute value of the first derivative of that function. In the latter case, the metaphor is inspired by the pattern's eventual behavior, i.e., its behavior at high values of overall resources invested (or of overall time invested when resource investment per unit time is held constant), namely the high rate of increase in the resource investment required if the next item is to be learned—in other words, the eventually always-high, always-positive absolute value and the eventually never-decreasing status of the first derivative of that function. In turn, those properties of the latter function dictate that the function measuring the rate of learning per resource unit invested (or per unit time when resource investment per unit time is held constant) has a horizontal asymptote at zero, and thus that the overall amount learned, while never "plateauing" or decreasing, increases more and more slowly as more and more resources are invested.

This difference in emphasis has led to confusion and disagreements even among learned people.[7]