We examine eight cases of argumentation in relation to a proportional reasoning task--the "Paint" task--in which the "constant sum" strategy was a significant factor. Our analysis of argument follows Toulmin's (1958) approach and in the discourse we trace factors which seem to facilitate changes in argument. We find that the arguments of "constant…

In this paper we deal with the problem of the genetic relationship between the sensual and the conceptual in knowledge formation. Transcript and videotape analyses of two small groups of a regular Grade 11 mathematics class shed some light on the dialectics between semiotic activity and artefact-mediated kinesthetic actions. The analyses point to…

This report focuses on teachers' individual curricula. An individual curriculum includes contents and reasoning and can be structured in a quasi-logical system of goals and methods, which is the result of teachers' planning of mathematics instruction. There is consent that the planning of individual curricula or the instructional practice is a…

Based on a survey of 3067 Finnish 5th and 7th graders and a task-based interview of 20 7th graders we examine student's understanding of fraction. Two tasks frame a specific fraction (3/4) in different contexts: as part of an eight-piece bar (area context) and as a location on a number line. The results suggest that students' understanding of…

In this paper, the theory of variation in the tradition of phenomenographic research approach is placed in the context of Dynamic Geometry Environment (DGE). Central concepts of discernment, variation, simultaneity and space of learning in the theory of variation are discussed for simple dragging episodes in DGE to illustrate the potential…

The metaphorical nature of first-year calculus students' reasoning about limit concepts is explored using an instrumentalist approach. Analysis of written and verbal language reveals that, while these students used motion terminology profusely when discussing limits, it was typically not intended to signify actual motion and did not play a…

Previous research has shown that many secondary school students improperly apply linear models when solving non-linear problems involving lengths, area and volume of similar plane figures and solids. This phenomenon is called the "illusion of linearity". This paper presents a teaching experiment in which we developed and tested a learning…

The present work has the purpose of showing the evolution of topics or mathematical concepts that are both relevant and with marked grades of abstraction. In this report is specifically described the utilization of metacognitive tools. These include concept maps, the Gowin heuristic vee, and the clinical interview. They are efficient in showing…

The National Council of Teachers of Mathematics recently published "Principles and Standards for School Mathematics." For the first time, these new standards include pre-kindergarten standards, while outlining the mathematics that children should learn as they progress through school. The standards present a broad view of what…

The roles of both positive and negative instances in concept formation are discussed. Based on research results, the article stresses ideas that the classroom teacher could try in order to aid concept learning. Several specific examples, based on the concept of commutivity, are given. (LS)

Performance differences between students given preorganizers and those given postorganizers in a university course (Mathematics in Early Childhood Education) were compared. Organizers were stated in performance terms (i.e. "Given any number of stimuli the students will be able to build sets of objects having like and unlike properties").…

Describes how the use of model rockets can be used to stimulate student interest and facilitate the learning of many science and mathematics concepts. (JR)

Thirty concepts from the areas of sets, division, and expressing relationships were studied with twelve tasks dealing with naming or selecting attributes or concepts involved. A factor analysis indicated that all concepts were measures of a single functional relationship and that all tasks measure a single underlying trait. (LS)