Solving Problems Reductively

Abstract

Solving problems by reduction is an important issue in mathematics and science education in general (both in high school and in college or university) and particularly in computer science education. Developing reductive thinking patterns is an important goal in any scientific discipline, yet reduction is not an easy subject to cope with. Still, the use of reduction usually is insufficiently reflected in high school mathematics and science programs. Even in academic computer science programs the concept of reduction is mentioned explicitly only in advanced academic courses such as computability and complexity theory. However, reduction can be applied in other courses as well, even on the high school level. Specifically, in the field of computational models, reduction is an important method for solving design and proof problems. This study focuses on high school students studying the unit "computational models"-a unique unit, which is part of the new Israeli computer science high school curriculum. We examined whether high school students tend to solve problems dealing with computational models reductively, and if they do, what is the nature of their reductive solutions. To the best of our knowledge, the tendency to reductive thinking in theoretical computer science has not been studied before. Our findings show that even though many students use reduction, many others prefer non-reductive solutions, even when reduction can significantly decrease the technical complexity of the solution. We discuss these findings and suggest possible ways to improve reductive thinking.

Reductive thinking in computer science

“…Reductive thinking, as demonstrated by high school students' solutions to questions dealing with computational models, was discussed in Armoni, Gal-Ezer, and Tirosh (2005). The findings showed that many students preferred direct, nonreductive solutions, even in cases in which reductive solutions could have significantly decreased the complexity of the solution.…”

Reduction in CS

“…The first study examined reductive thinking of high school students in the context of computational model problems [Armoni et al 2005]. The findings showed that many students preferred direct, nonreductive solutions, even in cases in which reductive solutions could have significantly decreased the complexity of the solution.…”

Reductive thinking in a quantitative perspective

“…The first study examined reductive thinking of high school students in the context of computational model problems [3]. The findings showed that many students preferred direct, non-reductive solutions, even in cases in which reductive solutions could have significantly decreased the complexity of the solution.…”