Extrapolation: The sine qua non for abstraction in function learning.

Abstract

ion was investigated by examining extrapolation behavior in a function-learning task. During training, participants associated stimulus and response magnitudes (in the form of horizontal bar lengths) that covaried according to a linear, exponential, or quadratic function. After training, novel stimulus magnitudes were presented as tests of extrapolation and interpolation. Participants extrapolated well beyond the range of learned responses, and their responses captured the general shape of the assigned functions, with some systematic deviations. Notable individual differences were observed, particularly in the quadratic condition. The number of unique stimulus-response pairs given during training (i.e., density) was also manipulated but did not affect training or transfer performance. Two rule-learning models, an associative-learning model, and a new hybrid model with associative learning and rule-based responding (extrapolation-association model [EXAM]) were evaluated with respect to the transfer data. EXAM best approximated the overall pattern of extrapolation performance. Research on conceptual behavior has historically focused on category learning and the application of categorical knowledge. So dominant is this focus that the terms concept and category are often used interchangeably (e.g., see Bourne, 1966; Smith & Medin, 1981). It is useful to distinguish these two terms, however, because there are many types of concepts that can not be adequately characterized as categories (Busemeyer, McDaniel, & Byun, 1997; Estes, 1995). In general, concepts also pertain to causal variables (e.g., intelligence) and relationships between these variables (e.g., income is correlated with intelligence). This article investigates functions, which conceptualize the relationship between causal variables. By definition, a function maps a set of input values (called the domain of the function) into a set of output values (called the range of the function), such that each input value is assigned only one output value. In function-learning situations, the range is composed of a continuous set of response magnitudes (e.g., predict a student's GPA on the basis of IQ scores). In category learning, on the other hand, outputs consist of discrete and nominal response categories (e.g.,

Effects of testing on learning of functions

“…The performance for the test items outside the trained range (transfer) helps to inform us on this issue. Formal exemplar models indicate that if the subject learns only the inputoutput points during training and then applies only the individually learned exemplars to support responses on transfer points (outside the trained range), performance along the transfer region is fairly flat; that is, there is no extrapolation of the slope inscribed by the training points (see DeLosh et al, 1997, for modeling and examples of individuals displaying this pattern of extrapolation). Accordingly, if test/study training promotes only learning of individual training points and these points are used individually to generate transfer responses, then the advantage observed for that training procedure for trained points would be eliminated (or dramatically reduced) on the transfer points (outside the trained range).…”

“…Accordingly, if test/study training promotes only learning of individual training points and these points are used individually to generate transfer responses, then the advantage observed for that training procedure for trained points would be eliminated (or dramatically reduced) on the transfer points (outside the trained range). Furthermore, the slopes of the outputs in the transfer region would be flat (see DeLosh et al, 1997).…”

“…This prediction would be supported if the test/study group showed more accurate performance on trained points but not on transfer points relative to the study-only group, along with near-zero slopes in responses to the transfer points (consistent with an exemplar-learning model; Busemeyer, Byun, DeLosh, & McDaniel, 1997;DeLosh et al, 1997).…”

“…Alternatively, in many cases human learners do display some extrapolation (but not perfectly so) to tested transfer points (Carroll, 1963;DeLosh et al, 1997;Wagenaar & Sagaria, 1975), thereby suggesting that learners have acquired some notion of the relations among the trained input-output pairs. If this were the case in the present study, the observed slopes in the subjects' predictions on the transfer points should generally be nonzero.…”

Rethinking Eliminative Connectionism

“…7 For example, DeLosh et al (1997) tested the ability of an eliminative connectionist model to extrapolate linear, exponential, and quadratic functions, and compared that result with the extrapolation abilities of human subjects. They found that although humans were able to extrapolate these functions beyond the range of trained responses, the eliminative connectionist model that they studied was not able to extrapolate adequately beyond the range of trained responses.…”