Strategic Thinking: Educational Use & Cognitive Foundation *
Wilfried Sieg · 2006
Strategic thinking is at the intellectual core of the calculus for the 21 st century, which is not the mathematical calculus that emerged in the 17 th century, but rather the logical calculus that was conceived in the same period by Leibniz, one of the two inventors of the mathematical calculus. Leibniz put great emphasis on a universal language to organize concepts, on rules to guide thinking and on mechanical algorithms to solve problems. The idea of the logical calculus came to theoretical fruition in the first half of the 20 th century; it became absolutely vital during the second half of that century in the context of the computing revolution. Proofs, functions and computations are the fundamental components of the logical calculus, but are scattered in logic, mathematics and computer science. Thorough familiarity with these concepts is no longer a privilege of a first rate education cutting across the boundaries of the three disciplines. On the contrary, it is a practical necessity for computer scientists and for students whose subject involves computational modeling, be they biologists, psychologists or economists. For students who reflect on the social impact of computers or conceive of mental processes as computations, it is equally central. Our work aims to contribute to educational practice and research. It contributes to the former by expanding an innovative introduction to logic with a focus on proofs through two deeply integrated parts on functions and computations. It contributes to the latter by using this expanded, fully web-based course, Computational Logic, as a Learning Laboratory to evaluate the efficacy of pedagogical approaches, in particular, our central one of teaching and tutoring strategic thinking. Our approach has its cognitive foundation partially in the computational model of goal-directed logical reasoning AProS; the broader reflective use of logic in mathematical problem solving can, we conjecture, be modeled in extensions of AProS. Thus, the Learning Laboratory also allows us to test, refine and extend the cognitive foundation for strategic thinking. 0. Computational Logic. We are pursuing two central educational objectives: To give students logical tools for rigorous argumentation and conceptual organization, and to convey substantive knowledge of basic mathematical and computational notions, together with historical and philosophical perspectives. These objectives are to be reached in Computational Logic by carefully integrated and scaffolded material on the one hand, and by its effective and individualized presentation on the other. The presentation is computer-based, highly interactive and supported by sophisticated cognitive tutors.