Notes for a history of the teaching of algebra

Abundant literature is available on the history of algebra. However, the history of the teaching of algebra is largely unwritten, and as such, this chapter essentially constitutes some notes that are intended to be useful for future research on this subject. As well as the scarcity of the works published on the topic, there is the added difficulty of drawing the line between the teaching of algebra and the teaching of arithmetic—two branches of knowledge whose borders have varied over time (today one can consider the arithmetic with the four operations and their algorithms and properties taught in schools as nothing more than a small chapter of algebra). As such, we will be very brief in talking about the more distant epochs, from which we have some mathematics documents but little information on how they were used in teaching. We aim to be more explicit as we travel forwards into the different epochs until modern times. We finish, naturally, with some reflections on the present-day and future situation regarding the teaching of algebra.info:eu-repo/semantics/publishedVersio

Teaching and Learning of Calculus

This survey focuses on the main trends in the field of calculus education. Despite their variety, the findings reveal a cornerstone issue that is strongly linked to the formalism of calculus concepts and to the difficulties it generates in the learning and teaching process. As a complement to the main text, an extended bibliography with some of the most important references on this topic is included. Since the diversity of the research in the field makes it difficult to produce an exhaustive state-of-the-art summary, the authors discuss recent developments that go beyond this survey and put forward new research questions

Different Levels of Sophistication in Solving and Expressing Mathematical Problems with Digital Tools

All over the world, several organizations have nurtured the development of students’ problem-solving abilities by organizing competitions and tournaments of different kinds. This is the case of the Mathematical Competitions SUB12 and SUB14, promoted by the University of Algarve, addressing students from grades 5 to 8 (10–14 year olds) in the south of Portugal. To each proposed problem, participants are required to explain their problem-solving process and find ways to express their thinking. They may use any of the digital tools they have available and they find useful for solving a given problem. Our research has uncovered the aptitudes of young competitors in taking advantage of everyday digital tools and its representational expressiveness to give form and substance to their reasoning and strategies. Another emerging aspect is the apparent existence of different degrees of robustness of the solutions submitted, mainly in terms of the strategies that competitors develop, with a particular technological tool, to solve the problems. In this chapter, we are taking a selection of solutions submitted to two problems, in which competitors resort to GeoGebra, in one case, and to Excel, in the other. We offer a proposal for identifying levels of sophistication and robustness of technology-based solutions to the problems, according to the characteristics of the tool use and its connection to the conceptual models underlying students’ thinking on the problems.info:eu-repo/semantics/publishedVersio