The number sequences describe a hierarchy of students' concepts of number. This research uses two defining cognitive structures of the number sequences--units coordination and the splitting operation--to model middle-grades students' abilities to write linear equations representing the multiplicative relationship between two unknowns. Results…

In "Making Sense of Math," Cathy L. Seeley, former president of the National Council of Teachers of Mathematics, shares her insight into how to turn your students into flexible mathematical thinkers and problem solvers. This practical volume concentrates on the following areas: (1) Making sense of math by fostering habits of mind that…

Problem solving skills and abilities are critical in life and more specifically in the engineering field. Unfortunately, significant numbers of South African students who are accessing higher education lack problem solving skills and this results in poor academic performance jeopardizing their progress especially from first to second year. On the…

This article explores theoretical and methodological issues associated with task-based interviews conducted with pairs of children. We explore different approaches to interviews from sociological, psychological and subject-based perspectives. Our interviews, concerning mathematical questions and carried out with pairs of 10 and 11-year-olds, are…

Seeking more effective mathematics instruction, this author decided to incorporate Cognitively Guided Instruction (CGI) into first-grade classroom lessons. Students in CGI mathematics classrooms are prompted to use their prior knowledge to solve new problems, establish cognitive structures to which new learning can be connected, and be driven by…

This synthesis paper discusses the research exploring analogical reasoning, the role of analogies in the engineering design process, and educational applications for analogical reasoning. Researchers have discovered that analogical reasoning is often a fundamental cognitive tool in design problem solving. Regarding the possible role of analogical…

During the last few decades, the author and his colleagues have developed an instructional framework for infusing thinking into content instruction in every subject and at every grade level. In thinking-based learning, teachers explicitly teach students thinking strategies and important habits of mind and then give students opportunities to apply…

Explores the development of understanding of the concept of average with students from grades 3 to 9 through interviews. Observed six levels of response based on an hierarchical model of cognitive functioning. Documents usage of ideas associated with the three standard measures of central tendency and representation as strategies for problem…

This paper presented various issues about pedagogical and cognitive aspects of problem solving and explored ways to lessen the heavy cognitive load of a problem solving task. It established a problem type schema for students at different levels. It recognizes the role of modern technology as a cognitive tool that promotes learning mathematics with…

Presents evidence indicating that spontaneously generated analogies can play a significant role in expert problem solving. Focuses on an evaluation strategy known as bridging that has been observed in solutions to both science and mathematics problems. Contains 23 references. (DDR)

Reports on a study designed to assess the value of the Tailored Response Test (TRT) as a learning device. Results suggest that the TRT, which involves active cognitive processing, has a positive effect on learning and is valuable as an evaluation and a learning tool. (DDR)

Describes a study that investigates the relationship between secondary students' written responses and the mathematical and cognitive processing used when solving a complex problem. Contains 40 references. (DDR)

This paper proposes a model for teaching and learning complex thinking skills developed from a synthesis of theories and research. The model functions like a concept map or a graphic organizer with five major components, each discussed in this paper: (1) presence of complex authentic life situations within a context; (2) activation and execution…

Children's analogical reasoning has traditionally been measured by classical four-term analogy tasks or problem-solving tasks. Current theories of analogical development and the evidence on which they are based are reviewed. It is concluded that structural views of analogical development are wrong, and knowledge-based accounts of what develops are…

Described is research which sought to prove the hypothesis that mental models tend to preserve their autonomy with regard to the originals they are meant to represent. The results of this investigation involving 200 Israeli students are presented. (CW)