We report an empirical study of elementary algebra errors, conducted in three separate schools. The errors are diagnosed using mal‐rules, as proposed by Sleeman (1984, 1,985). Our analysis uncovers the following properties of algebra mal‐rules: The frequency of mal‐rules is severely skewed, there are many infrequent mal‐rules and few frequent ones; mal‐rules are very unstable, students typically use mal‐rules very irregularly; different mal‐rules have explanatory power in different schools (many of our most powerful mal‐rules are previously unreported); mal‐rule diagnosis Is (...) more successful with more skilled students; students' confidence ratings do not partition the total set of mal‐rules, every mal‐rule we find is associated with high confidence ratings by at least one student. The Implications of our data for cognitive theories of error generation are discussed. Contrary to commonplace assumptions, we argue that It is impossible to make a clear distinction between slips and mistakes; most errors depend on properties of the knowledge base and the cognitive architecture. Errors In a procedural skill cannot be assumed to be purely syntactic In orgin. (shrink)

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. (shrink)

Since the publication of Fodor's (1983) The Modularity of Mind, there have been quite a few discussions of cognitive modularity among cognitive scientists. Generally, in those discussions, modularity means a property of specialized cognitive processes or a domain-specific body of information. In actuality, scholars understand modularity in many different ways. Different characterizations of modularity and modules were proposed and discussed, but they created misunderstanding and confusion. In this article, I classified and analyzed different approaches to modularity and argued for the (...) unity of modularity. Modularity is a multidimensional property consisting of features from several dimensions specifying different aspects of cognition. Among those, there are core features of modularity, and these core features form a cross-dimensional unity. Despite the diverse and liberal characterizations, modularity contributes to cognitive science because of the unity of the core features. (shrink)