Much problem solving and learning research in math and science has focused on formal representations. Recently researchers have documented the use of unschooled strategies for solving daily problems -- informal strategies which can be as effective, and sometimes as sophisticated, as school-taught formalisms. Our research focuses on how formal and informal strategies interact in the process of doing and learning mathematics. We found that combining informal and formal strategies is more effective than single strategies. We provide a theoretical account of (...) this multiple strategy effect and have begun to formulate this theory in an ACT-R computer model. We show why students may reach common impasses in the use of written algebra, and how subsequent or concurrent use of informal strategies leads to better problem-solving performance. Formal strategies facilitate computation because of their abstract and syntactic nature; however, abstraction can lead to nonsensical interpretations and conceptual errors. Reapplying the formal strategy will not repair such errors; switching to an informal one may. We explain the multiple strategy effect as a complementary relationship between the computational efficiency of formal strategies and the sense-making function of informal strategies. (shrink)

The Travail of Nature shows that the theological tradition in the West is neither ecologically bankrupt, as some of its popular and scholarly critics have maintained, nor replete with immediately accessible, albeit long-forgotten, ecological riches hidden everywhere in its deeper vaults, as some contemporary Christians, who are profoundly troubled by the environmental crisis and other related concerns, might wistfully hope to find. This is why it is appropriate to speak of the ambiguous ecological promises of Christian theology.

Islamic philosophy; Hanbalites; doctrines; creation (Islam); early works to 1800.

This paper defends a 'punctivist' conception of consciousness from recent attacks by Ian Phillips and Matthew Soteriou. As we intend it, 'punctivism' is the view that a subject's experience over some interval is determined by their experiential states at each instant during it. Phillips and Soteriou both offer ingenious arguments purporting to show that the punctivist is unable to make sense of motion perception; and that only by adopting an 'holistic' conception -- whereby a subject's instantaneous experiences are determined by (...) their experience over the interval -- can we make sense of the puzzles such phenomena pose. We contend that both arguments invoke dubious claims, their proffered solutions come with highly controversial commitments, and, if we take punctivism seriously, it is difficult to see why the so-called puzzles should be puzzling in the first place. A punctivist model of motion perception is proposed in response, and objections anticipated. (shrink)