“If You Can Turn a Rectangle into a Square, You Can Turn a Square into a Rectangle ...” Young Students Experience the Dragging Tool

This paper describes a study of the cognitive complexity of young students, in the pre-formal stage, experiencing the dragging tool. Our goal was to study how various conditions of geometric knowledge and various mental models of dragging interact and influence the learning of central concepts of quadrilaterals. We present three situations that reflect this interaction. Each situation is characterized by a specific interaction between the students’ knowledge of quadrilaterals and their understanding of the dragging tool. The analyses of these cases offer a prism for viewing the challenge involved in changing concept images of quadrilaterals while lacking understanding of the geometrical logic that underlies dragging. Understanding dragging as a manipulation that preserves the critical attributes of the shape is necessary for constructing the concept images of the shapes.     

Learning the indefinite integral in a dynamic and interactive technological environment

The present study was designed to identify the objectification processes involved in making sense of the concept of an indefinite integral when studied graphically in a dynamic technological environment. The study focuses on 11 pairs of 17-year-old students familiar with the concept of differentiation but not of integration. The students were asked to explain the possible connection between two linked dynamic graphs: the function and the primitive function graphs. The present study was guided by objectification theory, which considers artifacts to be fundamental to cognition and which views learning as the process of becoming aware of the knowledge that exists within a cultural context. In the course of two rounds of data analysis we identified six elements in the processes of objectification: objectifying the relationships between segments based on the location, inclination, and concavity of the function graph; and on the relationships between the zero, the extreme, and the inflection points in the function graph and the corresponding points in the primitive function graph.     

Embodied Semiotic Activities and Their Role in the Construction of Mathematical Meaning of Motion Graphs

This paper examines the relation between bodily actions, artifact-mediated activities, and semiotic processes that students experience while producing and interpreting graphs of two-dimensional motion in the plane. We designed a technology-based setting that enabled students to engage in embodied semiotic activities and experience two modes of interaction: 2D freehand motion and 2D synthesized motion, designed by the composition of single variable function graphs. Our theoretical framework combines two perspectives: the embodied approach to the nature of mathematical thinking and the Vygotskian notion of semiotic mediation. The article describes in detail the actions, gestures, graph drawings, and verbal discourse of one pair of high school students and analyzes the social semiotic processes they experienced. Our analysis shows how the computerized artifacts and the students’ gestures served as means of semiotic mediation. Specifically, they supported the interpretation and the production of motion graphs; they mediated the transition between an individual’s meaning of mathematical signs and culturally accepted mathematical meaning; and they enable linking bodily actions with formal signs.     

Reaching the Unreachable: Technology and the Semantics of Asymptotes

This work is part of a larger attempt to explore the nature of symbolic understanding involving graphic technology. This study describes learning advanced mathematics that occurs through constructing qualitative reasoning methods using graphic technology. Data was gathered from a precalculus class who, for a few weeks, investigated and explored asymptotic behavior of rational functions. The analysis is based on observations of group discussions and written works. Learning about asymptotes using software which serves as tool box for numerical evaluation and graphic representation amplifies epistemological complexities related to the infinity concept. Using the software to watch examples of rational functions, generated by symbolic and graphic operations between polynomial functions, enabled students to leave their own traces on the formalization of asymptotes – on its definition, its symbolic structure, and its computational procedures. The discrepancy between technology as a support for visual perception but a tool that cannot support theconception of approaching infinities, makes the study of asymptotes an intriguing domain for investigating what is being manipulated with software and how.This revised version was published online in September 2005 with corrections to the Cover Date.     

Online assessment of students’ reasoning when solving example-eliciting tasks: using conjunction and disjunction to increase the power of examples

ZDM – Mathematics Education - We argue that examples can do more than serve the purpose of illustrating the truth of an existential statement or disconfirming the truth of a universal...     

Functions of Interactive Visual Representations in Interactive Mathematical Textbooks

The paper explores changes in technology that have implications for the teaching and learning of school mathematics. To this end, it examines aspects of interactive mathematical textbooks; specifically it analyzes functions authors may intend to be carried out by embedded interactive diagrams. The paper analyzes theoretical as well as practical lessons that I learned while designing such a book. It is the purpose of this article to provide a rough, preliminary collection of categories of diagram function that would allow an orderly discussion of the subject. While this is not an empirical study, the hypotheses about student practices with interactive diagrams are based on a long series of studies related to the learning of algebra in a technologically-rich environment.     

Solving algebra problems with interactive diagrams: Demonstration and construction of examples

We investigated how students use the representation of data in a given example appearing in an interactive diagram (ID) and how they create additional examples with the ID. Students who worked with the ID that offered limited representations and tools (illustrating ID) looked for ways to bypass the designed constraints: they changed the representation of the data or built new representations, but did not create new examples in any form. Working with another type of diagram (narrating ID), students treated the specific given example as a generic example and were able to reach a generalization in a process of systematic change and comparison. The variety of tools and representations offered in the design of the third type of ID (elaborating ID) yielded diverse strategies: students constructed, without guidance, various examples and initiated further inquiry that sometimes resulted in the systematic construction of examples, although the ID provided no tools for systematic change.