Calculus from around the world
This thematic collection represents the eight papers finally selected for publication following submissions invited from the sixteen presentations within Topic Study Group 13: The Teaching and Learning of Calculus at ICME-12, held in Seoul, South Korea from the 8th to the 15th July 2012. The six countries from which these studies are drawn illustrates the international breadth of the audience and discussions in Topic Study Group 13, at which 16 papers and a further 10 posters from a total of 17 countries were presented. The papers presented at the conference came from: Australia, Brazil, China, Iran, Israel, Korea (2 papers), Mexico and Puerto Rico (collaborative study), New Zealand (a paper which reported on contemporary research from a further seven countries presented at the 2011 Delta conference in New Zealand on the teaching and learning of undergraduate mathematics and statistics), Spain (2 papers), Uruguay and the USA (4 papers from researchers representing 11 institutions). Original submissions for all 16 papers presented in Topic Study Group 13 are available online in the Pre-Proceedings of ICME-12, http://www.icme12.org/forum/forum.asp. In addition to their work in editing the papers presented here, Martinez-Luaces and Oates would also like to acknowledge the contributions of Sunsook Noh, Ewha Women's University, Seoul, South Korea, as co-chair of Topic Study Group 13.
The eight papers elaborated on in here provide a stimulating insight into developments in the teaching and learning of Calculus, at the upper secondary and tertiary level. Collectively, they describe pedagogical advances, new trends, and research in recent years on the teaching and learning processes of Calculus, and consider ways in which we may meet the challenges described. Presentations at the conference were carried out in four sessions under four broad themes, and the papers are published here in the same order as they appeared within these themes, namely: The Derivative Concept; Modelling, Applications and other topics; Other Concepts in Calculus: Integrals, series, etc; and Pre-Calculus and First Calculus Courses.
The first session at the conference presented four papers that considered a range of developments in the teaching of differentiation. These papers inspired considerable debate, principally about the order of topics in the teaching of Calculus and the level of conceptual complexity with which we should introduce differentiation, a theme that was repeated continually throughout the study group sessions, for example in discussions about limits and integration in sessions three and four respectively. In the paper presented here from the first session, Jungeun Park, from the University of Delaware in Newark, USA, describes a study which explores how calculus students talk about the derivative as a function based on the concepts of function at a point and function on an interval. The study examines students' discourse on the derivative using a communicational approach to cognition, with a particular focus on students’ descriptions about the derivative and the relationships among a function, the derivative function, and the derivative at a point. Park provides some fascinating examples of how students think and talk about derivative concepts, and the study reinforces the difficulty students experience with the notion of a derivative as a function. The study suggests that addressing the derivative as a function based on how students previously developed their thinking about the function itself may make the concept more accessible to students.
The next group of four papers are drawn from the third session of the study group considering several important concepts in Calculus, such as integrals, series, differential equations and multivariable Calculus. In the first of these, Kouropatov and Dreyfus from Tel Aviv University in Israel describe the development of a curriculum based on the idea of constructing integration concepts from the didactical perspective of the integral as accumulation. Their presentation continued the considerable discussion started in the first session at the conference, about the order in which we teach differentiation and integration, with many arguing that the integral-as-accumulation is a more natural and accessible way to introduce students to Calculus. Certainly, Kouropatov and Dreyfus conclude in this paper that based on the tentative results described, their proposed curriculum is suitable and effective in helping students develop abstract integral concepts by reorganising previous constructs including function, derivative, and infinite sum. The next paper, presented at the conference by María Teresa González Astudillo, reports on a study by a team of researchers at the University of Salamanca in Spain. It describes the growth of mathematical understanding shown by university students engaged in mathematics classroom tasks about the concept of numerical series. Using the Image-Making theoretical model developed by Pirie and Kieren, they describe how students moved between the different layers of understanding (Image-Making, Image-Having and Property-Noticing). They conclude that repeating these actions with a range of specific examples enriched students’ understanding of such concepts as the convergence of a series. The next two papers both consider higher levels of Calculus; the first, a collaborative study by Rafael Martínez-Planell at the University of Puerto Rico and María Trigueros Gaisman at the Instituto Tecnológico Autónomo de Mexico, describes new observations on students’ graphical understanding of two-variable functions. The implications of their findings have led them to design new activities to improve students’ graphing of functions of two variables, which they regard as critical for students to develop a better understanding of multivariable Calculus. The final paper in this group of four is by a team of researchers at The Ohio State University, presented at the conference by Jennifer Czocher. They use differential equations as an exemplary post-Calculus course to explore the consequences of an analysis of foundational Calculus topics in terms of how such topics are used. There was considerable agreement at the conference with their assertion that Calculus tends to treated by both mathematicians and educational researchers as though it occurs in a vacuum. In their concluding remarks, they argue that it is important to know how prerequisite content is used in later courses, and not just whether it is used, if we want to come to a better understanding of what it means for students to know mathematics for a purpose.
The last group of three papers are drawn from the session focusing on pre-Calculus and first Calculus courses. The first, presented at the conference by David Bressoud from Macalester College, Saint Paul Minnesota, reports on a large-scale study undertaken by the Mathematical Association of America of postsecondary Calculus instruction in the United States. The analysis of some of the data presented here highlights students’ mathematical backgrounds as well as aspects of instruction that contribute to successful programs. However, while the authors note that it is still early days in the analysis of the wealth of data collected by this study, there are also a number of emerging trends that they describe as disturbing, especially the evidence that many students in the USA continue to be turned off Calculus at research-university level. Next, a paper presented at the conference by José Antonio Fernández Plaza, reports on the preliminary findings of a study by a team at the University of Granada in Spain, that investigated the different uses that students make of terms such as “to approach”, “to tend toward”, “to reach” and “to exceed”, terms that describe some properties of the concept of finite limit. They describe the use of a conceptual analysis framework to examine students’ discourse, for example in the errors induced by colloquial and everyday use of specific terms, and the relatively undeveloped and imprecise language characterised in students references to such concepts as non-exceedable or unreachable limits. The final paper in this collection is by L. Vivier, from the University Paris Diderot, France. In a link back to the teaching of derivatives and limits discussed in earlier papers, Vivier describes a pre-Calculus approach to the consideration of the tangent line that adopts graphical and algebraic points of view, as opposed to the common curriculum consideration as a sub-product of the derivative of a function. He considers two historical methods that may help students with the difficult transition, from the graphical frame where arose our early knowledge, to the algebraic, or even analytic, frame. He notes that the notion of tangent is complex and is not confined to calculus, as curricula seem to suggest, and argues that an algebraic approach as suggested in these examples could help better prepare students for more formal Calculus.
We believe the eight papers presented here effectively demonstrate both the range and commonality of issues confronting lecturers and students in the field of contemporary Calculus teaching and learning. We hope they stimulate readers’ interest, and prompt further investigation and attempts to integrate some of the ideas presented here into their teaching practice. Certainly we enjoyed the lively and robust discussion that the papers generated in Topic Study Group 13 at ICME-12. We look forward to continued debate at future conferences.
Victor Martinez-Luaces
Universidad de la República, Uruguay
Greg Oates
The University of Auckland, New Zealand