Mathematics Education

This paper reports on three commognitive conflicts that were identified in a larger research study conducted on university first year students’ discourse on continuous functions. The study is the first to look at continuity related aspects under a participationist, discursive lens in contrast to previous studies on continuity that have used cognitive theories that hold an acquisitionist view on learning. The study adopts Sfard’s commognitive framework to analyze data. Among findings are different ways in which students use the word ‘domain’ and how they struggle with inconsistent ‘realizations’ arising from different definitions presented in text books.

In this paper I present a case study to illustrate conflicts between different ‘realizations’ of the concept of ‘continuous function’ held by a university first year student. Sfard’s commognitve framework is used in the analysis of a student’s work on continuity. I point out how these conflicting realizations have arisen from the inconsistent definitions presented in text books and other mathematical resources. The study also points to the need of extending the notion of “commognitive conflict” in the framework.

This study looks at the interplay between the concept image and concept definition when students are given a task that requires direct application of the definition of continuity of a function at a point. Data was collected from 37 first year university students. It was found that different students apply the definition to different levels, which varied from formal deductions (based on the application of the definition) to intuitive responses (based on rather loose and incomplete notions in their concept image).

This paper provides a detailed analysis of the mathematical communication involving a pair of high school calculus  students who are English language learners. The paper  focuses on the word-use, gestures and dragging actions in the student-pair communication about calculus concepts when paper-based static and then touchscreen dynamic diagrams. Findings suggest that the students relied on gestures and dragging as multimodal resources to communicate about dynamic aspects of calculus. Moreover, examining the interplay between language, gestures, dragging, and diagrams made it possible to uncover English language learners’ competencies in mathematical  communications. This paper points to an expanded view of bilingual learners’ communication that includes gestures, dragging and diagrams.

This paper discusses the importance of considering bilingual learners’ non-linguistic forms of communication for understanding their mathematical thinking. In particular, I provide a detailed analysis of communication involving a pair of high school bilingual learners during an exploratory activity where a touchscreen-based dynamic geometry environment (DGE) was used. The paper
focuses on the word-use, gestures, and dragging actions in student-pair communication about calculus concepts as they interacted with a touchscreen-based DGE. Findings suggest that the students relied on gestures and dragging as non-linguistic features of the mathematical discourse to communicate dynamic aspects of calculus. Moreover, by examining the interplay between language, gestures, dragging and diagrams, it was possible to identify bilingual learners’ competence in mathematical communications. This paper raises questions about new forms of communication mobilised in dynamic,  touchscreen environments, particularly for bilingual learners.

How "fundamental" is the Fundamental Theorem of Calculus, and what does it really mean? In this session, I discuss a teaching approach that has helped students learn the FTC visually, conceptually and enthusiastically with the use of Dynamic Geometry. This session will challenge you to reflect upon the teaching and learning of functions, pre-calculus and calculus in engaging and "dynamic" ways.

This paper discusses the use of dynamic geometry environment (DGE) for facilitating three Sfardian processes for creating new discursive objects: saming, reification, and encapsulation. In particular, I provide an analysis of communication about a DGE sketch involving three high school students who had been enrolled in a calculus course where DGEs were consistently incorporated into the lessons. Findings suggest that the students used a combination of saming, encapsulation, and reification for exploring calculus ideas. Moreover, encapsulation was needed in order to develop an object-level discourse around the area-accumulating function. This paper raises implications about teaching mathematics with technology, particularly by attending to communications and changes in communications about functions as an ordered pair and a graph.

In this article, a thinking-as-communicating approach is used to analyse calculus students’ thinking in two environments. The first is a ‘static’ environment in the sense of static visual representations, such as those found in textbook diagrams, while the second is a dynamic environment as exploited by the use of dynamic geometry environments (DGEs). The purpose of the article is to compare calculus students’ communication as it is facilitated by each of these two environments, and to explore the role of paper- and digital-mediated representations for positioning certain ways of thinking about calculus. The analysis provides evidence that the participants employed different modes of communication – utterances, gestures and touchscreen-dragging – and they communicated about fundamental calculus ideas differently when prompted by different types of representations. The study presents implications for teaching dynamic aspects of functions and calculus, and argues for a multimodal view of communication to capture the use of gestures and dragging for communicating dynamic and temporal mathematical relationships.

This paper discusses the importance of considering bilingual learners’ non-linguistic forms of
communication for understanding their mathematical thinking. In particular, I provide an analysis of
communication involving a pair of high school calculus bilingual learners, who interacted with a
touchscreen-based DGE (dynamic geometry environment). The paper focuses on the word-use,
gestures, and touchscreen dragging actions in student-pair communication. Findings suggest that
the students relied on gestures and dragging as multimodal features of the mathematical discourse to
communicate dynamic aspects of calculus. This paper raises questions about new forms of
communication mobilized in dynamic, touchscreen environments, particularly for bilingual learners.