The three target articles presented in this special issue converged on an emerging theme: the importance of spatial proportional reasoning. They suggest that the ability to map between symbolic fractions (like 1/5) and nonsymbolic, spatial representations of their sizes or "magnitudes" may be especially important for building robust…

Everyday Mathematics has contributed in important ways to long-standing debates about mathematical concepts, symbolic representation, and the role of contexts in thinking--the latter topic reaching back at least as far as Kant's notion of scheme. The descriptive work plays a role, of course. But it is only by making sense of the observations that…

All educators should be concerned about curriculum. Curriculum is the path to giving students the best possible education enveloping an in-depth learning experience. Dr. Kritsonis's "Ways of Knowing Through the Realms of Meaning" provides a guide for schools to select curriculum that is appropriate, rigorous, and enriched. Through alignment, Dr.…

Uses Lacanian notions to create stories on the mathematics education topics of numeracy, symbolism, and desire. (MDH)

Employing Searle's views, I begin by arguing that students of Mathematics behave similarly to machines that manage symbols using a set of rules. I then consider two types of Mathematics, which I call "Cognitive Mathematics" and "Technical Mathematics" respectively. The former type relates to concepts and meanings, logic and sense, whilst the…

With both high retirement and high attrition rates among educators and a nationwide increasing student population, more teachers are needed. With the impact of a teacher shortage, educational systems began to examine the alternative certification methods. Alternative routes to teacher education profoundly impact the teaching force. The "Ways…

This paper presents an analysis of the different types of meanings that an individual may assign to a collection of algebraic symbols depending on the mathematical context in which the symbols are presented and the mathematical knowledge possessed by that individual. Four contexts for the Quadratic Theorem are used to illustrate the ways in which…

In this commentary on Matthew Inglis' "Three Worlds and the Imaginary Sphere" (see EJ1106688), David Tall develops the theme that the building of theories is not an easy process. A theory in progress is a particularly delicate creation. Theories do not appear fully formed. There is a period of exploration and incubation that precedes the…

I argue that Vygotsky's theory of concept formation (1986) is a powerful framework within which to explore how an individual at university level constructs a new mathematical concept. In particular, this theory is able to bridge the divide between an individual's mathematical knowledge and the body of socially sanctioned mathematical knowledge. It…

It is recommended that schools incorporate literacy in all aspects of the curriculum. In this way, students begin to understand that reading and writing are not just for English class--they are skills that are important for many parts of daily life. This does not mean that teachers need to take time away from their other subjects to achieve this…

Generally, reading is perceived as consisting of identifying words and increasing comprehension skills, but there are a plethora of additional symbols which students need to read and comprehend meaningfully. This paper first discusses reading across the curriculum, emphasizing abstract words, and then considers symbols peculiar to diverse academic…

Discusses algebra teaching by looking back into the history of algebra and the work of George Peacock, who considered algebra from two points of view: symbolic and instrumental. Claims that, to be meaningful, algebra must be linked to real-world problems. (18 references) (MKR)

Concludes that mathematics uses symbols both as processes and concepts. The mathematically oriented student develops flexible ways of using them, but less successful students cling to the security of known procedures to get answers that are less suitable for thinking than flexible symbols which can also be considered as mathematical objects to be…

Attempts to describe a notion parallel to number sense, called symbol sense, incorporating the following components: making friends with symbols, reading through symbols, engineering symbolic expressions, equivalent expressions for non-equivalent meanings, choice of symbols, flexible manipulation skills, symbols in retrospect, and symbols in…

Limitations in children's understanding of the symbols of arithmetic may inhibit choice of appropriate solution procedures. The teacher's role involves negotiation of new meanings for words and symbols to match extensions to solution procedures. (MKR)