Creativity and Challenge: Task Complexity as a Function of Insight and Multiplicity of Solutions

Mathematical problem solving is the heart of mathematical activities at all levels. Problem-solving is both the means and the ends of the development of mathematical knowledge and skills as well as of the advancement of mathematics as a science. Researchers distinguish between problem-solving algorithms, problem-solving strategies and heuristics and problem-solving insight. Insightful and divergent thinking are at the base of mathematical creativity. This chapter analyzes the mathematical challenge embedded in problem-solving tasks from the point of view of evoked mathematical insight and the use of multiple solution strategies. While a variety of variables (such as conceptual density, level of concepts, length of solution or use of different presentations) determine the complexity of mathematical problems, the insight component and the requirement to solve problems in multiple ways increase the mathematical challenge of the task. Researchers distinguish between different types of mathematical insight as they relate to the distinction between mathematical expertise and mathematical creativity. In this chapter, we introduce a distinction between mathematical tasks that allow insight-based solutions and tasks that require mathematical insight. We provide empirical evidence for our argument that tasks that require mathematical insight are of a higher level of complexity.