The purpose of this study is to investigate students' thinking of direct, inverse and nonproportional problems. Thirty two seventh grade students from three different government schools participated in this study. To collect the data, the participants were asked to solve 9 open-ended problems, including 3 direct, 3 inverse and 3 non-proportional…

Structure sense can be interpreted as an intuitive ability towards symbolic expressions, including skills to perceive, to interpret, and to manipulate symbols in different roles. This ability shows student algebraic proficiency in dealing with various symbolic expressions and is considered important to be mastered by secondary school students for…

Intuitive conceptions in mathematics guide the interpretation of mathematical concepts. We investigated if they bias teachers' conceptions of student arithmetic word problem solving strategies, which should be part of their pedagogical content knowledge (PCK). In individual interviews, teachers and non-teaching adults were asked to describe…

The phenomenon of insight is frequently characterized by the experience of a sudden and certain solution. Anecdotal accounts suggest that insight frequently occurs after the problem solver has taken some time away from the problem (i.e., incubation). However, the mechanism by which incubation may facilitate insight problem-solving remains unclear.…

Understanding contingency table analysis is a facet of mathematical competence in the domain of data and probability. Previous studies have shown that even young children are able to solve specific contingency table problems, but apply a variety of strategies that are actually invalid. The purpose of this paper is to describe primary school…

In the present article, we articulate three assumptions underlying theories proposing that restructuring processes play a key role in insightful problem solving: representational difficulty, representational change, and discontinuity in solution processes. We argue that these assumptions need empirical validation to justify the proposition of…

This article looks at how children in preschool through second grade intuitively solve mathematical problems rather than using textbook strategies with a single path to a solution. The authors discuss Cognitively Guided Instruction (CGI), a curriculum approach that helps teachers understand and encourage children's use of intuitive strategies.…

Considers 33 preservice secondary mathematics teachers' solutions to a famous sampling problem with particular interest on the use of intuition and/or formal mathematics in reaching a conclusion. Considers the relationship of solution strategy to students' background in formal mathematics and gender. Discusses implications for teaching statistics…

In this paper, the author aims to offer an elaboration of simple, yet powerful, mathematical patterns through mathematical games. Mathematical games may serve as colorful instructional tools for teachers and textbooks, and may raise students' motivation and intuition. Patterns are fundamental in mathematics and computer science. In the case of…

This article describes the factors to which the classroom teacher needs to attend to enhance the mathematical problem-solving abilities of students. Emphasis is placed upon the means necessary to develop the attribute of being a problem solver, rather than focusing on the goal of becoming a problem solver. (JJK)

This paper further develops an instructional method called here "process-object linking and embedding". The idea is to link the familiar mathematical processes to objects in a familiar situation, then re-embed the new link through mathematical symbols into their mathematical construction. It makes use of children's extra-mathematical,…

Counterintuitive moments in the classroom challenge common sense and practice and can be used to help mathematics students appreciate the need to explore, reflect, and reason. Proposed are four examples involving geometry, systems of equations, and matrices as counterintuitive instances. (MDH)

Creativity in education often takes the form of concentrated periods of arts-based "light relief" from the rigours of the National Curriculum. In psychology, on the other hand, creativity is often associated with a dramatic moment of "illumination" in solving scientific, mathematical or practical problems. This paper explores a third approach…

This paper presents a historical overview of visualization as a human problem-solving tool. Visualization strategies, such as mental imagery, pervade historical accounts of scientific discovery and invention. A selected number of historical examples are presented and discussed on a wide range of topics such as physics, aviation, and the science of…

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)