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### Abstracts

(updated 26 October)
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**Main speakers:**

**Knut Mørken, University of Oslo:**
*Some reflections on how computing and mathematics can mutually enhance learning of mathematics.*
Abstract: At the University of Oslo we have for many years worked to integrate computing in the elementary modules in mathematics, based on a first semester programming module that is compulsory for all students in mathematics and physics. One way to argue for this integration is simply that it is inevitable, given what is happening in research and industry. A more constructive motivation is that it opens up new and interesting applications to the students. Yet another motivation can be that the integration of mathematics and computing leads to new and enhanced ways of learning both mathematics and computing. In this presentation we will discuss all these perspectives.

**Morten Brøns, Technical University of Denmark:**
*Do engineers need mathematics?*
Abstract: In 2013, The Technical University of Denmark and Copenhagen University College of Engineering merged. Each institution had several 3½ year BEng programs which required rethinking as a part of the merging process. I will review the discussions we had about the role of mathematics in engineering education and describe some attempts to unite the rather different learning philosophies at the two institutions. Finally, I will summarize what we have learned in the five years since the fusion, and describe how we, right now, teach introductory mathematics for 1000 students/year.

**Carl Winsløw, University of Copenhagen:**
*Mathematics for non-mathematicians: Rooms for improvement*
Abstract: There is no disagreement about the importance of mathematical knowledge within a wide range of educations and professions; this is also closely linked to mathematical components within natural, technical and social sciences. At the same time, student failure (to pass or to learn, or both) in “service courses” is a widespread problem. For these reasons, it is important to develop still more subtle approaches to what and how mathematics should be delivered to these students.
Mathematics education research has recently made a number of advances in this area, where research is increasingly carried out by university mathematics teachers who specialize in such issues. It In my talk I will outline what I consider the most important for university mathematics teachers at large:
$ $ A. the need to change mathematics courses from “visiting monuments” to deliberate design for inquiry - in particular, new approaches to task design
$ $ B. the need to question the curriculum (at several levels), not just the “pedagogy” - in particular, the traps of “applicationism”, often deeply entrenched in curricula.
My talk will be focused on principles, with just a few examples and references to the vast literature (see [1],[2] for current syntheses).
References:
[1] Winsløw, C. and Rasmussen, C. (in press). University Mathematics Education. To appear in S. Lermann “Encyclopedia of Mathematics Education” (Springer, 2018).
[2] Winsløw, C., Gueudet, C., Hochmut, R. and Nardi, E. (2018). Research on University Mathematics Education. In: T. Dreyfus, M. Artigue, D. Potari, S. Prediger & K. Ruthven (Eds.), Developing research in mathematics education - twenty years of communication, cooperation and collaboration in Europe. Oxon, UK: Routledge - New Perspectives on Research in Mathematics Education series, Vol. 1, pp. 60-74.

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**Other speakers:**

**Brigit Geveling, University of Twente**: *Team-based Learning in a mathematics course*
Abstract: In each study program there is a course that students prefer to avoid. In the first year of Applied Mathematics there is such a course: Analysis. In Analysis students are faced with many new definitions, with theorems and proofs. As if that is not bad enough: they are expected to write proofs at the end of the course themselves.
I did some research on the grades of Analysis about the last seven years. Very often, students use a compensation possibility for Analysis, grades are low.
In presentations and workshops, I learned about the didactic concept Team Based Learning (TBL). In TBL-sessions, students need to talk with each other about Mathematics and students learn to get confidence in their own solutions. So, the question came up: Can TBL improve the performance of students in Analysis?
Last year I introduced three Team-based Learning sessions in Analysis. These sessions stimulate students to discuss about their solutions and knowledge about the topic of the day.
In my presentation I will explain more about TBL and I will give you the results of TBL in Analysis and the reactions of students on this model.

**Frode Rønning, Norwegian University of Science and Technology:**
*A mild form of flipped classroom in large courses for engineering students*
Abstract: At NTNU we have over the past five years made significant changes to the way the basic courses in mathematics and statistics for engineering students are being organised. Traditionally, lectures have been given in large classes in a presentation format without much interaction. Every year, about 1700 new students are enrolled in the Master of Technology programmes and they all take a basic course in Calculus in the first semester.
We wanted to make some changes to the traditional structure but partly based on data from how the students regard the role of the lectures and partly because of the large volume, we did not want to make too dramatic changes. We have replaced one of the traditional lectures with a so-called interactive lecture where the students get the chance to go deeper into central topics of the week by working on carefully designed tasks, and where they have the possibility to discuss with each other, and with the teacher.
In this talk I will describe our new lecture structure and give examples of tasks that we use in the interactive lectures, as well as report on experiences based on data collected from the students.

**Henrik Bang, University of Copenhagen/CMU:**
*The D-Matematik project with samples from high school*
Abstract: Following Felix Klein, mathematics at a certain level in education should be taught in light of more advanced mathematics. This of course applies to high school mathematics and ideally as well to the use of digital technology. In practice, the last can be difficult
At CMU we have worked with ways to connect “mental artefacts” (mathematics without computers) and digital artifacts (computer based mathematics) and the suitable mathematics to facilitate this connection.
A possible way is to introduce elements of discrete mathematics –algorithms, mathematical logic, graph theory, etc. and eventually elements of numeric differentiation and integration before the corresponding continuous elements.
We will present examples of small courses for high school students working with discrete mathematics in explorative ways.

**Håkan Lennerstad, Blekinge Institute of Technology:**
*A calculus course in knowledge feedback format*
Abstract: In this calculus course design, students give mathematical feedback every week via quizzes, activating their knowledge and providing essential information to the teacher which sharpens lectures. After having answered, for each question the student obtain immediate correct answer and explanation. The quiz questions together with the explanations contain a lot of the theory in a rather informal way,
inviting formal proofs and complementing the calculation practice.
I will also discuss cooperation with applied subjects to import mathematical problems.

**Jan-Fredrik Olsen, University of Lund:**
*Turning calculus on its head — some ideas of what to do if your students know how to program*
Abstract: Since 2015, a course in scientific computing with Python has been obligatory for first semester students in mathematics and physics at Lund University. My goal in life has been to transform my introductory course in Calculus, which runs in parallel with the Python course, to allow my students take advantage of their newfound skills in numerical computations to more effectively learn more advanced mathematics. The purpose of this talk is to report on my experiences so far, and to contribute to a broader discussion of how (and why) we should teach first semester Calculus.

**Karsten Schmidt, Technical University of Denmark**:
*Authentic problems from engineering in introductory mathematics*
Abstract: Towards the end of the first year bachelor course Mathematics 1 at DTU (20 ECTS) the standard teaching is replaced by project work. Over four weeks, the students work in groups on a project assignment relevant for their study program, write a report and defend their work in an oral exam. In this talk I will discuss 1) how this work relates to the overall competencies we want our students to obtain, 2) how the project assignment are created in a collaboration with other departments and 3) a recent attempt to identify the important “didactical variables” to consider when designing a project assignment (a joint work with Carl Winsløw).

**Steen Markvorsen, Technical University of Denmark**:
*Mathematics and Maple: A two-component adhesive for building up learning, teaching, and research*
Abstract: We consider the following two statements about intuition – one Maple based and one Mathematics based:
D. J. Bailey and J. M. Borwein (2011):
“Never have we had such a cornucopia of ways to generate intuition. The challenge is to learn how to harness them, how to develop and how to transmit the necessary theory and practice.”
J. Hadamard (1928):
“The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there was never any other object for it.”
In the talk we will illustrate the combined relevance of these statements – not just for research, but also for learning and teaching.

**Stig Larsson, Chalmers/University of Gothenburg:**
*A computationally oriented textbook on calculus and linear algebra*
Abstract: Together with Anders Logg and Axel Målqvist, I am writing a textbook on calculus and linear algebra. It is based on our teaching in the first year of the mechanical engineering program at Chalmers University of Technology. Our goal is to integrate numerical analysis and programming in the mathematics course and at the same time raise the level of the mathematical content. Is this possible? Yes, we claim that it is precisely the use of numerical computation that allows us to increase the mathematical level. For example, in the first week, we can construct the real numbers as Cauchy sequences, because the students generate convergent sequences of rational numbers. In the first course, we can also prove Bolzano’s theorem and Banach’s fixed point theorem, because the students program the bisection algorithm and the fixed point iteration, and these algorithms are identical to the proofs of the respective theorems. In the last course, multivariable calculus, we use the divergence theorem to derive the heat equation with general boundary conditions and the students solve the boundary value problem by the finite element method. In this way, numerical computation and mathematical analysis complement each other on a higher level and the teaching is not limited to formula manipulation.