The Emergence of the “FlexTech” Orchestration of Inferential Reasoning on Pattern Generalization

The purpose of this study is to further our understanding of orchestrating math-talk with digital technology. The technology used is common in Swedish mathematics classrooms and involves personal computers, a projector directed towards a whiteboard at the front of the class and software programs for facilitating communication and collective exploration. We use the construct of instrumental orchestration to conceptualize a teacher’s intentional and systematic organization and use of digital technology to guide math-talk in terms of a collective instrumental genesis. We consider math-talk as a matter of inferential reasoning, taking place in the Game of Giving and Asking for Reasons (GoGAR).The combination of instrumental orchestration and inferential reasoning laid the foundation of a design experiment that addressed the research question: How can collective inferential reasoning be orchestrated in a technology-enhanced learning environment? The design experiment was conducted in lower-secondary school (students 14–16 years old) and consisted of three lessons on pattern generalization. Each lesson was tested and refined twice by the research team. The design experiment resulted in the emergence of the FlexTech orchestration, which provided teachers and students with opportunities to utilize the flexibility to construct, switch and mark in the orchestration of an instrumental math-GoGAR.

Edge Models with the CAD Software: Creating a New Context for Mathematics in Elementary School

Digital media have become increasingly important in recent years and can offer new possibilities for mathematics education in elementary schools. From our point of view, geometry and geometric objects seem to be suitable for the use of computer-aided design software in mathematics classes. Based on the example of Tinkercad, the use of CAD software — a new and challenging context in elementary schools — is discussed within the approach of domains of subjective experience and the Toulmin model. An empirical study examined the influence of Tinkercad on fourth-graders’ development of a model of a geometric solid and related reasoning processes in mathematics classes.

Scheming and Re-scheming: Secondary Mathematics Teachers’ Use and Re-use of Resources

In this article, we examine secondary mathematics teachers’ work with resources using the Documentational Approach to Didactics lens. Specifically, we look at the resources and a teacher’s scheme of use (aims, rules of actions, operational invariants, and inferences) of these resources across a set of lessons (macro-level analysis) that aim towards students’ preparation for the examinations and how this use emerges in a set of three lessons on the same topic (micro-level analysis) as a response to contingent moments. We propose the terms scheming—a teacher’s emerging scheme of use related to the same set of resources used for the same aim—and re-scheming, namely, shifts in such scheming. Our analysis of lesson observations and the teacher’s reflections on his actions from a post-observation interview demonstrate the interplay between the stable characteristics of the scheme of use and the scheming and re-scheming in individual lessons. We conclude this article with a discussion on the methodological potential of using both macro- and micro-level analyses in the investigation of teachers’ use of resources.

From Argumentation to Proof in Geometry Within a Collaborative Computer-Based Environment

This article is part of a wider research project that has the educational goal of supporting students in the production of conjectures, arguments and proofs, as well as promoting a move from the production of arguments expressed in colloquial registers to arguments expressed in literate registers. In this regard, we Giovannina Albano, Umberto Dello Iacono and Maria Alessandra Mariotti designed and implemented a digital educational environment that allows students to formulate and prove conjectures; three different working areas are available where students can work on a geometrical open problem sometimes individually, sometimes in collaboration. In this article, I report on an empirical study aimed at investigating the functioning of one of these areas, the ‘Working with others’ area, where small groups of students are expected to discuss and formulate a shared solution to a problem. The research question concerns if and to what extent the communication tools, specifically designed to foster students’ collaboration, can promote the production of mathematically acceptable arguments. The qualitative data analysis shows that the ‘Working with others’ area seems to foster discussion within the group and can make students aware of their mistakes. Moreover, it can bring out some students’ misconceptions and can provide useful information upon which the teacher can trigger fruitful discussions. However, this working area does not appear to foster a significant improvement of the production of mathematically acceptable arguments, produced by students in a collaborative and sharing mode. The integration of specific components within this working area seems to be necessary to support the student in moving from argumentation to proof.

Fostering Spatial Ability Through Computer-Aided Design: a Case Study

Spatial ability is considered a major factor of intelligence and is increasingly important in times of digitization. This article explores the fostering of spatial ability through computer-aided design software. Different notions of spatial ability will be discussed, and, finally, a concept consisting of five aspects will be described. In addition, literature reviews on the connection between the use of computers and the fostering of spatial ability, as well as on the use of 3D printing technology in mathematics education, are given. Building on this, a case study is presented which examines the work of two middle-school students using computer-aided design software within a workshop at the University of Siegen. From the data material, basic possible actions within such software are derived. These are, based on theory, connected with the five aspects of the specific concept of spatial ability used. The results show various perspectives for the fostering of spatial ability with computer-aided design software.