Discusses the level of knowledge, understanding, and ability relative to mathematics for the college graduate in the 21st century. Focuses on mathematical literacy and abstraction such as the definition of abstraction, difficulties in addressing abstraction, and one approach to helping students learn to address abstraction. (Contains 12…

Introduces a study on primary school children's reasoning about the concepts of probability and choice. Concludes that relatively few children are sufficiently advanced in their thinking about chance situations to be able to see that different chance situations can have the same probabilistic nature. (ASK)

Proposes a model of mathematics conceptual development consisting of two phases: abstraction and generalization. Illustrates model by references to the development of multiplication and angle concepts. (19 references) (MKR)

This submission contains the Proceedings of the 2007 Annual Meeting of the Canadian Mathematics Education Study Group (CMESG), held at the University of New Brunswick in Fredricton, New Brunswick. The CMESG is a group of mathematicians and mathematics educators who meet annually to discuss mathematics education issues at all levels of learning.…

This paper discusses what the theory of abstraction is about, the need for a theory of abstraction in mathematics education, and the requirements that such a theory should meet. All three are reconsidered from a personal point of view. (KHR)

This paper presents a developmental model of students' acquisition of competence in quantitative and algebraic problem solving. A key notion underlying the developmental model is a distinction between grounded and abstract representations. Grounded representations, like story problems, are more concrete and familiar, closer to physical objects and…

This paper presents a cognitive theoretical framework for the learning of mathematics which has generic implications for other disciplines. The framework has been developed using a combination of established theories about learning and the authors' research into the understanding of some specific types of learning. It is based on the integration…

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…

Reacts to the publications "Curriculum Guidance for the Foundation Stage" and "National Numeracy Strategy: Framework for Teaching Mathematics from Reception to Year 6" published by the Department for Education and Employment in Great Britain. Makes the case for the marrying of formal and informal, abstract and concrete…

Meaningful educational activities and cognitive tools might improve students' active involvements in the teaching-learning process and encourage their reflections on the concepts and relations to be investigated. It is claimed that usage of manipulatives not only increase students' conceptual understanding and problem solving skills but also…

The framework for this paper is a recently developed theory of abstraction in context. The paper reports on data collected from one student working on tasks concerned with absolute value functions. It examines the relationship between mathematical constructions and abstractions. It argues that an abstraction is a consolidated construction that can…

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)