Cognitive processes are procedures for using existing knowledge to combine it with new knowledge and make decisions based on that knowledge. This study aims to identify the cognitive structure of students during information processing based on the level of algebraic reasoning ability. This type of research is qualitative with exploratory methods.…

Mathematical problems can be divided into two types, namely, process-open and process-constrained problems. Solving these two types of problems may require different cognitive mechanisms. However, there has been only one study that investigated the differences of the cognitive abilities in process-open and process-constrained problem solving, and…

Although there is no consensus in regard to a unique meaning for abstraction, there is a recognition of the existence of several theories of abstraction, and that the ability to abstract is imperative to learning and doing meaningful mathematics. The theory of "reducing abstraction" maps the abstract nature of mathematics to the nature…

This study investigated the role of broad cognitive processes in the development of mathematics skills among children and adolescents. Four hundred and forty-seven students (age mean [M] = 10.23 years, 73% boys and 27% girls) from an elementary school district in the US southwest participated. Structural equation modelling tests indicated that…

It is claimed that, since mathematics is essentially a self-contained system, mathematical objects may best be described as "abstract-apart." On the other hand, fundamental mathematical ideas are closely related to the real world and their learning involves empirical concepts. These concepts may be called "abstract-general" because they embody…

Interviews 120 children in kindergarten and grades 2, 4, and 6 with five Piagetian tasks to determine the grade level at which most have constructed transitive reasoning, unit iteration, and conservation of speed. Indicates that construction of the logic necessary to make sense of the measurement of time is generally not complete before sixth…

Informal interviews and naturalistic observations indicate that the child often invents novel ways of doing arithmetic and that some type of individualized instruction is necessary. (Author/WM)

Formulates a framework for characterizing elementary children's (n=20) statistical thinking based on a review of research and a cognitive development model, and refines it through a validation process. Proposes four thinking levels which represent a continuum from idiosyncratic to analytic reasoning. Results confirm the four levels of children's…

Considers the cognitive mechanisms available to individuals which enable them to operate successfully in different parts of the mathematics curriculum, such as children's arithmetic shows divergence in performance. Explains how students cope with the transition to advanced mathematical thinking in different ways, leading once more to a diverging…

Discussed are the concepts of intuition, the general properties of an intuitive knowledge, and the classification of intuitions as problem solving of affirmative. An example of intuition using multiplication and division is described in some detail. (MNS)

A brief survey of models in dealing with various types of understanding is given. Then a hybrid model, which proved inadequate for describing understanding, is outlined. Finally, four levels of understanding are discussed: intuitive, procedural, abstract, and formal. The concept of number is used to illustrate these levels. (MNS)

Proposes that children progressively recognize deeper and deeper similarities between their physical angle experiences, and classify them firstly into specific situations, then into more general contexts, and finally into abstract domains. Indicates that the standard angle concept first develops in situations where both arms of the angle are…

Children's understanding of Euclidian space was investigated using three Piaget-type tasks to examine the ability of children to quantitatively locate a point in one, two, and three dimensions. Among the findings were that a disagreement exists between Piaget's data and the results of this study. (BT)

Investigates the concepts of ratio and proportion constructed by grade 9 students by investigating their proportional reasoning schemes and procedures on three types of tasks: missing value, numerical comparison, and qualitative reasoning. Indicates that students frequently used additive reasoning--that is, a comparison of two numbers by…