Many mathematical concepts may have prototypical images associated with them. While prototypes can be beneficial for efficient thinking or reasoning, they may also have self-attributes that may impact reasoning about the concept. It is essential that mathematics educators understand these prototype images in order to fully recognize their benefits…

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…

The basis of this intervention study is a distinction between numerical calculus and relational calculus. The former refers to numerical calculations and the latter to the analysis of the quantitative relations in mathematical problems. The inverse relation between addition and subtraction is relevant to both kinds of calculus, but so far research…

This paper examines how a person's gesture space can become endowed with mathematical meaning associated with mathematical spaces and how the resulting mathematical gesture space can be used to communicate and interpret mathematical features of gestures. We use the theory of grounded blends to analyse a case study of two teachers who used gestures…

There are many studies on the role of images in understanding the concept of limit. However, relatively few studies have been conducted on how students' understanding of the rigorous definition of limit is influenced by the images of limit that the students have constructed through their previous learning. This study explored how calculus…

The integration of history into educational practice can lead to the development of activities through the use of genetic "moments" in the history of mathematics. In the present paper, we utilize Oresme's genetic ideas--developed during the fourteenth century, including ideas on the velocity-time graphical representation as well as geometric…

Discusses the role of context problems as they are used in the Dutch approach known as realistic mathematics education (RME). Uses an RME design for a calculus course to illustrate that theory based on the design heuristic using context problems and modeling, which was developed for primary school mathematics, also fits advanced topics such as…

Presents a semiotic analysis of the potential of computer algebra systems (CAS) for enabling mathematical activity on a conceptual level higher than usual. Illustrates examples of theoretical points from a development project in the context of a first year university course in calculus. Discusses how CAS may be used in a didactical analysis and in…

Presents an exploration of conversation in a single episode of a 12th grade girls' calculus class. Analyzes Habermasian communicative competence which requires students to show critical thinking and the ability to express their views. Explains learning outcomes by teachers' mode of questioning, cooperative endeavor by students, instructional…

Developed is a working definition of the concept of preformal proof through the analyses of an action proof, a geometric-intuitive proof, and a reality-oriented proof of a differential equation. Instructional problems for teachers and learners in connection with preformal proofs are pointed out. (MDH)