THE NATURE AND ORIGIN OF RATIONAL ERRORS IN ARITHMETIC THINKING - INDUCTION FROM EXAMPLES AND PRIOR KNOWLEDGE

Students systematically and deliberately apply rule-based but erroneou s algorithms to solving unfamiliar arithmetic problems. These algorith ms result in erroneous solutions termed rational errors. Computational ly, students' erroneous algorithms coin be represented by perturbation s or bugs in otherwise correct arithmetic algorithms (Brown & VanLehn, 1980; Langley & Ohllson, 1984; VanLehn, 1983, 1986, 1990; Young & O'S hea, 1981). Bugs ore useful for describing how rational errors occur b ut bugs are not sufficient for explaining their origin. A possible exp lanation for this is that rational errors are the result of incorrect induction from examples. This prediction is termed the ''induction hyp othesis'' (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 tha n has been done previously. The first step involved teaching participa nts a new number system called NewAbacus, a written modification of th e abacus system. The second step consisted of dividing them into diffe rent groups, where each individual received an example of only one par t of the NewAbacus addition algorithm. During the third and final step , participants were instructed to solve both familiar and unfamiliar t ypes of addition problems in NewAbacus. The induction hypothesis was s upported by using both empirical and computational investigations.