How can a student experience what happens in the mind of a mathematician while solving a problem? In this paper we discuss a theoretical design of an educational script, based on digital interactive storytelling. Parallel to Docter's 'Inside Out', the cognitive functions occurring in problem solving become characters of a story-problem. Students…

Current conceptualizations of knowing and learning tend to separate the knower from others, the world they know, and themselves. In this article, we offer "relationality" as an alternative to such conceptualizations of mathematical knowing. We begin with the perspective of Maturana and Varela to articulate some of its problems and our alternative.…

An attempt is made to link the domain of mathematics and the natures and abilities of mathematicians to that which is perceptual, presentational, and implicit. (MP)

Makes the analogy between Wilfred Bion's theory of mental processes involving preconception and mathematical problem solving. (MDH)

How the author moved from concern about research to development of prescriptive models of heuristic problem solving and the exploration of metacognition and belief systems is discussed. Student beliefs about problem solving, and their corollaries, are included. (MNS)

How students are convinced that they have the correct solution to a problem, free of contradiction, is discussed. The role of counterexamples and the need for a situational analysis of problem-solving behaviors are each included. (MNS)

How children perceive doubling and halving numbers is discussed, with many examples. The use of calculators is integrated. The tendency to avoid division if other ways of solving a problem can be found was noted. (MNS)

The roles and uses of symbols in mathematical thinking are discussed. The thinking process is further subdivided into specialization, generalization, and reasoning. (MP)

Compared and contrasted are the concepts intuition and logic. The ideas of conceptual thought and algorithmic thought are discussed in terms of the world as a labyrinth, intuition and time, and the structure of knowledge. (KR)

Presents the reactions of a mathematics educator and a mathematician to a seven-year-old student's response for finding square numbers. Reflects on the mathematician's focus on the mathematics of the problem and the mathematics educator's focus on the student's systematic way of working, and discusses the common focus of student creativity. (MDH)

Presents 13 examples in which the intuitive approach to solve the problem is often misleading. Presents analysis of these problems for five different sources of misleading intuitive generators: lack of analysis, unbalanced perception, improper analogy, improper generalization, and misuse of symmetry. (MDH)