The Nature and Origin of Rational Errors in Arithmetic Thinking: Induction from Examples and Prior Knowledge

Students systematically and deliberately apply rule-based but erroneous algorithms to solving unfamiliar arithmetic problems. These algorithms result in erroneous solutions termed rational errors. Computationally, students' erroneous algorithms can be represented by perturbations or bugs in otherwise correct arithmetic algorithms (Brown & VanLehn, 1980; Langley & Ohilson, 1984; VanLehn, 1983, 1986, 1990; Young S O'Sheo, 1981). Bugs are useful for describing how rational errors occur but bugs are not sufficient for explaining their origin. A possible explanation for this is that rational errors are the result of incorrect induction from examples. This prediction is termed the “induction hypothesis” (VanLehn, 1986). The purpose of the present study was to: (a) expand on post formulations of the induction hypothesis, and (b) use a new methodology to test the induction hypothesis more carefully than has been done previously. The first step involved teaching participants a new number system called NewAbacus, a written modification of the abacus system. The second step consisted of dividing them into different groups, where each individual received an example of only one port of the NewAbacus addition algorithm. During the third and final step, participants were instructed to solve both familiar and unfamiliar types of addition problems in NewAbacus. The induction hypothesis was supported by using both empirical and computational investigations.     

Feelings,direction of attention,and expressed evaluations of others

Abstract

Noting the occasional inconsistency in the results of studies of the effects of negative affect on thoughts and actions in social situations, it is suggested that: (1) self-regulatory mechanisms sometimes intervene to determine what people experiencing unpleasant feeling will say (and maybe think) about others around them; and (2) at least some of these mechanisms may have to be activated by relatively high-level cognitive processing and are not always operative. More specifically. the theoretical analysis presented here holds that many different kinds of negative affect will prime thoughts having a negative meaning, including unfavourable judgements of others. However, negative affect does not always lead to openly expressed negative opinions and actions. It is proposed that attention to one's moderately unpleasant feelings tends to moderate the effect of negative affect on subsequent negative thoughts and actions by activating “higher-level” cognitive processing so that consideration is given to different kinds of knowledge pertinent to the given situation, including the social rules defining what is desirable and whatever relevant information is provided in the situation. Two experiments are reported, employing greatly different procedures and subjects of both sexes. in support of this analysis. The seeming disparity between the present conception and self-awareness theory is also discussed. and it is noted that these formulations are actually not at odds. Suggestions are made for further research.