Many students encounter difficulty in their transition to advanced mathematical thinking. Such difficulty may be explained by a lack of understanding of many concepts taught in early school years, especially multiplicative reasoning. Advanced mathematical thinking depends on the development of multiplicative reasoning. The purpose of this study…

This paper focuses on the changes in thinking involved in the transition from school mathematics to formal proof in pure mathematics at university. School mathematics is seen as a combination of visual representations, including geometry and graphs, together with symbolic calculations and manipulations. Pure mathematics in university shifts…

This paper discusses ways to use worked examples in teaching mathematics. It is argued that neither investigative teaching such as discovery learning nor lecturing and starting from the abstract are helpful as they are based on emotive associations with general labels rather than precise details of pedagogic strategies. (KHR)

This report describes a program for developing and improving critical thinking skills in adolescents in order to prepare them for life-long learning. The targeted population consisted of high school math and social studies students from two middle-class communities located in northern Illinois. The students' lack of critical thinking skills was…

A nonequivalent control group design study, with a math unit on ratio and proportion as the treatment, was conducted to examine the validity of some written tasks as a measure of formal reasoning. Subjects in the study were 68 fifth grade students who enrolled in two classes in a small urban school. The experimental group learned a 12-lesson math…

Provides examples of if-then and combinatorial reasoning situations that have proved successful with college students and can be used with secondary school students. Proposes that practice with these problem solving processes can eliminate the need to memorize formulas that students may not understand. (ML)

Defines the mathematical concept of proportionality. Uses graphs of linear relationships to explore and make generalizations about the characteristics of proportional situations to help students critically evaluate problems involving proportions. (MDH)

Investigates gender differences and the role of spatial visualization in problem solving in high school geometry. Reports that, whereas males and females differed in spatial visualization and in their performance, they did not differ in logical reasoning ability or in their use of geometric problem-solving strategies. (Author)

Higher order cognitive development and success in the study of high school mathematics and science require an understanding of rational number concepts and facility with proportional reasoning and computation. Proportional reasoning is an essential schema for developing formal operational thought. This study involving 16 ninth-grade students was…

To test whether gains in cognitive development can be accelerated, an intervention model was used to give pupils special lessons in science, using ideas of cognitive conflict, metacognition, and bridging in the context of formal operations. Found that intervention led to substantial and long-lasting increased achievement in science, mathematics,…

The purpose of this study was to find out whether students perform differently on algebra word problems that have certain key context features and entail proportional reasoning, relative to their level of logical reasoning and their degree of field dependence/independence. Field-independent students tend to restructure and break stimuli into parts…