When we "think like a computer scientist," we are able to systematically solve problems in different fields, create software applications that support various needs, and design artefacts that model complex systems. Abstraction is a soft skill embedded in all those endeavours, being a main cornerstone of computational thinking. Our…

Isomorphism and homomorphism appear throughout abstract algebra, yet how algebraists characterize these concepts, especially homomorphism, remains understudied. Based on interviews with nine research-active mathematicians, we highlight new sameness-based conceptual metaphors and three new clusters of metaphors: sameness/formal definition, changing…

Due to their abstract nature, representation of mathematical concepts through different registers favors their understanding. In the case of ''sequences and regularities'', it becomes propitious the exploration of different registers of representation in the institution of topics, such as term, order, formation law, and generating expression.…

Learning trajectories are built upon progressions of mathematical understandings that are typical of the general population of students. As such, they are useful frameworks for exploring how understandings of diverse learners may be similar or different from their peers, which has implications for tailoring instruction. The purpose of this…

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)

Among the 12 documents selected for this annotated bibliography of documents and journal articles in the ERIC database is an expert's argument that the brain's multipath and multimodal capacities are ignored by educators. Another writer fears that the "back-to-basics" movement may have eclipsed the prominence earlier accorded to thinking…

This paper describes the methods and techniques called Conceptual Analysis (CA), a rigorous procedure to generate (without involuntary omissions and repetitions) knowledge bases for the development of knowledge-based systems. An introduction is given of CA and how it can be used to produce knowledge bases. A discussion is presented on what is…

Discusses Vygotsky's theories about concept formation, his distinctions between everyday and theoretical concepts, and how empirical generalizations can lead to misconceptions. Examines the implications of these theories for mathematics instruction and its relationship to the current mathematics reform. (34 references) (MDH)

This study reports the development of a sentence preference test designed to contrast tendencies to abstract, generalise and particularise in thinking. (Editor)

Multilevel visual models can make teaching and learning theories more easily understood. The idea of graphic visualizationas developed in this model utilizes three stages of abstract concept formation dealing with concrete visualization, abstract visualization, and abstract generalization, which would provide the learner with an intuitive grasp of…

Three task context factors that form obstacles to generalizing logical reasoning strategies for adolescents are discussed. The relationship between these obstacle factors and Piagetian theory is explored. Findings suggest that it is possible to make some predictions about how and when logical mechanisms will be generalized to new situations.…

Contrasts students' presymbolic and symbolic procedures in generalizing activities. Uses the semiotic-cultural theoretical approach and focuses on the role of body, discourse, and signs when students refer to mathematical objects. Identifies types of generalizations and discusses a specific kind of rupture in the ostensive gestures and…

Working on the hypothesis that if readers had several alternative correct meanings of a given text set before them they would select the one in accordance with their abstract abilities and intellectual maturity, a study investigated the measurement of levels of meaning as illustrated in taxonomies and focused on the applicability of the taxonomies…

The ADAPT program for freshmen emphasizes building an experiential base in introductory courses before moving to abstract formulations. The goal is to develop thinking processes that are integrated with the student's experiences. Students in the ADAPT program outperformed two comparison groups on a variety of thinking tasks. (Author)

Although mathematics deals with generalizations relating abstract ideas, very little attention has been given in the mathematics education literature to the role of abstraction and generalization in the development of mathematical knowledge. In this paper, the meanings of "abstraction" and "generalization" are first explored by…