How might a 4-year-old pupil come to conceive a line, created by himself with a continuous gesture, as constituted by a series of points? In this paper, we theoretically introduce and discuss a design study of the educational path in which this possibility emerged. The path aimed at leading pupils to "experience," in the sense of the…

This paper initiates a discussion around the potential of Dynamic Measurement, an alternative approach to geometric measurement that focuses on how space is generated, and thus measured by the lower-dimensional objects that define it. I use data from a series of design experiments with fourth-grade students to illustrate some forms of reasoning…

Critical to constructing and interpreting graphs is an individual's understanding of the underlying coordinate systems, yet coordinate systems are often overlooked or taken-for-granted in both mathematics education research and curricula. In this paper, we foreground coordinate systems and present a distinction between two uses of coordinate…

In this article we compare two frameworks for analysing young children's responses to the task of copying and extending a 6-dot triangle pattern. We used Mulligan & Mitchelmore's Awareness of Mathematical Pattern and Structure (AMPS) and then Biggs & Collis' SOLO taxonomy, both of which provide criteria for assigning levels. In comparison…

This paper approaches the concept of the circle through the framework of mathematics-as-becoming. This paper focuses specifically on how a concept can be thought of as a process, and on the implications that this might have for mathematics learning. Contrary to long-standing assumptions about mathematical concepts as ideal, inert, Platonic forms,…

To encourage students to do geometry, the art of Islamic geometric ornamentation was chosen as the central theme of a lesson strand which was developed using the newly presented didactical tool called "Learning by Acting". The Dutch students who took these lessons in 2010 to 2013 were challenged to act as if they themselves were Persian…

In this paper I argue my opposition to the consensus which has dominated the literature that young children view shapes as a whole and pay no attention to shape structure and that geometrical thinking can be described through a hierarchical model formed by levels. This consensus is linked to van Hiele's weok by van Hiele-based research. In the…

This paper introduces the notion of "crystalline concept" as a focal idea in long-term mathematical thinking, bringing together the geometric development of Van Hiele, process-object encapsulation, and formal axiomatic systems. Each of these is a strand in the framework of "three worlds of mathematics" with its own special characteristics, but all…

The author shares a story of how Luis González, an iron artisan, helped the author build a wooden and iron toy truck. The knowledge required to build the skeleton for the parallelepiped in the construction of the truck is not in the mathematical high school curriculum in Venezuela. Although Luis never received a degree beyond high school,…

In this article, the author comments on the article "Solid geometry in the works of an iron artisan,'" Castro, 23(3). The author finds it interesting to read about the implicit mathematical knowledge of ironworkers, weavers, tailors and other practitioners as they undertake their various crafts and professions. In the article, Fernando…

The roles of skills and meaning in mathematics are discussed, with concern expressed over the little time spent on developing understanding. Approaches to proportion are described, levels of meaning are proposed, and proportion as a unifying idea for geometry are among the topics discussed. (MNS)

The mathematics curriculum should be imbedded into the cultural environment of the student. Discussed is the mathematical educational potential of decorative motif. (PK)

Four methods of filling a square using programing with Logo are presented, with comments on children's solutions. Analysis of the mathematical or programing concepts underlying a few simple algorithms is the focus. (MNS)

How students are convinced that they have the correct solution to a problem, free of contradiction, is discussed. The role of counterexamples and the need for a situational analysis of problem-solving behaviors are each included. (MNS)

Described are four hour-long sessions of children working with the definition and use of a procedure for triangles in Logo programming. The conclusion is that turtle geometry is not a trivial activity. (MNS)