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Calculus in upper secondary and beginning university mathematics Conference held at the University of Agder, Kristiansand, Norway August 6-9, 2019 This conference was supported by the Centre for Research, Innovation and Coordination of Mathematics Teaching (MatRIC) Edited by John Monaghan, Elena Nardi and Tommy Dreyfus Suggested reference: J. Monaghan, E. Nardi and T. Dreyfus (Eds.) (2019), Calculus in upper secondary and beginning university mathematics – Conference proceedings. Kristiansand, Norway: MatRIC. Retrieved on [date] from https://matric-calculus.sciencesconf.org/ Table of contents Introduction Calculus in upper secondary and beginning university mathematics 1 Plenary lectures Biehler, Rolf The transition from Calculus and to Analysis – Conceptual analyses and supporting steps for students 4 Tall, David The Evolution of Calculus: A Personal Experience 1956–2019 18 Thompson, Patrick W. Making the Fundamental Theorem of Calculus fundamental to students’ calculus 38 Plenary panel González-Martín, Alejandro S.; Mesa, Vilma; Monaghan, John; Nardi, Elena (Moderator) From Newton’s first to second law: How can curriculum, pedagogy and assessment celebrate a more dynamic experience of calculus? 66 Contributed papers Bang, Henrik; Grønbæk, Niels; Larsen Claus Teachers’ choices of digital approaches to upper secondary calculus 67 Biza, Irene Calculus as a discursive bridge for Algebra, Geometry and Analysis: The case of tangent line 71 Bos, Rogier; Doorman, Michiel; Drijvers, Paul Supporting the reinvention of slope 75 Clark, Kathleen M. Examining students’ secondary-tertiary transition: Influences of history, disposition, and disciplinary engagement 79 Derouet, Charlotte Introducing the integral concept with probability tasks 83 Durand-Guerrier, Viviane; Montoya Delgadillo, Elizabeth; Vivier, Laurent Real exponential in discreteness-density-completeness contexts 87 Ely, Robert Teaching calculus with (informal) infinitesimals 91 Feudel, Frank Required knowledge of the derivative in economics – Results from a textbook analysis 95 ii González-Martín, Alejandro S.; Hernandes-Gomes, Gisela What happens after Calculus? Examples of the use of integrals in engineering: the case of Electromagnetism 99 Hagman, Jessica Ellis A vision for the future of college calculus 103 Herbst, Patricio; Shultz, Mollee How professional obligations can help understand decisions in the teaching of calculus across institutional contexts 107 Jennings, Michael; Goos, Merrilyn; Adams, Peter Teacher and lecturer perspectives on secondary school students’ understanding of the limit definition of the derivative 111 Kidron, Ivy The dual nature of reasoning in Calculus 115 Kontorovich, Igor’; Herbert, Rowan; Yoon, Caroline Students resolve a commognitive conflict between colloquial and calculus discourses on steepness 119 Kouropatov, Anatoli; Ovodenko, Regina Construction (and re-construction) and consolidation of knowledge about the inflection point: Students confront errors using digital tools 123 Mesa, Vilma; Liakos, Yannis; Boelkins, Matt Designing textbooks with enhanced features to increase student interaction and promote instructional change 127 Monaghan, John The place of limits in elementary calculus courses 131 Nilsen, Hans Kristian First-year engineering students’ reflections on the Fundamental Theorem of Calculus 135 Pinto, Alon Towards transition-oriented pedagogies in university calculus courses 139 Sangwin, Christopher J The mathematical apprentice: an organising principle for teaching calculus in the 21st century 143 Simmons, Courtney; Oehrtman, Michael Quantitatively Based Summation: A framework for student conception of definite integrals 147 Stewart, Sepideh; Reeder, Stacy Examining unresolved difficulties with school algebra in calculus 151 Swidan, Osama Construction of the mathematical meaning of the function-derivative relationship using dynamic digital artifacts 155 iii Thoma, Athina; Nardi, Elena Linear stability or graphical analysis? Routines and visual mediation in students’ responses to a stability of dynamical systems exam task 159 Törner, Günter The ‘Signature’ of a Teaching Unit – ‘Calculus’ as the ‘Heart’ within Standard Introductory Mathematical Courses 163 Viirman, Olov; Pettersson, Irina What to do when there is no formula? Navigating between less and more familiar routines for determining velocity in a calculus task for engineering students 167 Wangberg, Aaron; Gire, Elizabeth Raising Calculus to the Surface: Extending derivatives and concepts with multiple representations 171 iv Introduction Calculus in upper secondary and beginning university mathematics The genesis of the conference were discussions between us on what was being taught – and what could be taught – under the name ‘calculus’ in schools, colleges and universities in our countries. Whilst we have nothing against the differentiation and integration techniques of calculus, we saw and see calculus as much more than a set of techniques; at its heart is covariational change, rates of change and accumulations, which can be overlooked by foci on techniques. The idea for the conference came when we realised that MatRIC (www.matric.no), the Norwegian centre for excellence in university mathematics teaching, might support a conference on calculus and allow a bigger audience for our discussions; and MatRIC agreed to support us. The conference had four themes: 1. The school-university transition with a focus on calculus 2. The fundamental theorem of calculus 3. The use of digital technology in calculus 4. Calculus, and its teaching and learning, for various disciplines We comment on these themes but first note that calculus at school and calculus at university are not, at a global level, clearly differentiated areas or approaches to calculus: what is taught at school in one country may be taught at university in another country; university calculus courses, especially for non-mathematics majors, often revisit school calculus content. The latter, revisiting school calculus content at university, is one reason why the school-university transition, our first theme, is an important focus. Another reason why this focus is important is the jump from what might be called informal (or elementary) calculus at school to formal calculus (or analysis) at university, the latter involving e-d definitions and proofs. Not all students will experience this jump but mathematics majors will – and this jump to rigour is a noted area of difficulty for many students. The fourth theme, on teaching and learning calculus in various disciplines, is related to the first theme in as much as this teaching and learning occurs, almost exclusively, at the university level, to students who have come from school. It is an important theme in itself: what are appropriate calculus courses for economists, for engineers, etc. and how might these differ (in content and approach) to calculus courses for mathematics majors? This is also a developing area in terms of practical approaches and mathematics education research. The third theme, on the use of digital technology in calculus, exists because (i) mathematical software has the potential to restructure what and how calculus is taught and learnt and (ii) there are many initiatives that essentially incorporate digital technology in the teaching and learning of calculus. The second theme is the only theme to focus on a specific part of calculus, the fundamental theorem. It is there for many reasons. It was the main focus in our initial discussions on calculus. It is a beautiful and important part of calculus. It is quite often covered scantily and/or badly – including the maximal bad approach, the statement “integration is the reverse of differentiation”. There is at least one 1 approach to teaching it (as part of a full calculus course) that we all respect – Pat Thompson’s Project DIRACC. The themes were built into the plenaries: Rolf Biehler was briefed to address the first and fourth theme and Pat Thompson was briefed to address the middle two themes in their plenary addresses and papers. The themes also partially structured the contributed papers in as much as invitations to present a paper went out with the proviso that the paper should address at least one of the four themes. We now provide an overview of the plenary lectures, plenary panel and contributed papers presented but first repeat that the conference was conceived, from outset to completion, as a discussion conference: the conference was more than the sum of the plenary contributions and the presented papers! The opening plenary was by David Tall. David was not well enough to attend the conference and provided a video which you can watch on https://matric-calculus.sciencesconf.org/. David has, almost certainly, written more papers on the teaching and learning of calculus than anyone else in the world, from 1975 to today. Amongst many other things, he is noted for his software (SuperGraph), the concept of local straightness and his Three Worlds approach. His plenary paper, The evolution of Calculus: A personal experience 1956-2019, addresses issues that touch on the first three conference themes. Rolf Biehler has invested much of his academic thought and effort in the past decade to issues concerned with beginning university studies, in which calculus or analysis play a central role. The main focus of his plenary, The transition from calculus to analysis – conceptual analyses and supporting steps for students, is on the transition problem and concerns the move from school calculus to university
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