### Motion Design Problems: From weekly assignments to real research

##### Prof. Burkhard Alpers

##### Aalen University, Germany, Burkhard.Alpers@htw-aalen.de

Motion design is an important activity for mechanical engineers since in many machines machine parts or goods are moved. This holds in particular for packaging machines where products are bundled, shrink-packed and placed in boxes. A short video will be shown to demonstrate this. Motion design is essentially a mathematical activity since mathematical representations of motion have to be created for downloading onto the controller of a machine drive.

In the presentation several examples will be given for including motion design problems in the (mathematical) education of engineers, ranging from simple assignments suitable for weekly assignment sheets to challenging projects or even bachelor theses. Mechanisms which realize certain motions are also taken into account. We present a reproduction of an old hammer mill for which a 'good' motion is to be designed, and we show a motion design environment which has been programmed within a bachelor thesis. Finally, we give an example of current research work in this area. We will also discuss the question in which ways mathematical modelling is involved in motion design tasks.

### Mathematical modeling of biological processes:Should we focus on mathematicians to study biology,or biologists to learn mathematics needed to model?Challenges and case studies from NTNU

##### Dr. Nadav Bar,

##### NTNU, Norway, nadi.bar@ntnu.no

Mathematical modeling and computer simulations became common tools to analyze the mechanisms of gene and metabolic regulation in biology. Developing and applying these tools require excellent mathematical skills, extensive biological knowledge and intuition in information technology (IT). However, this combination is extremely rare in our traditional education system.

More specifically, we seldom find students that have both the mathematical skills and the biological knowledge required to model biological processes. Several study programs were developed in recent years, attempting to solve this problem. These programs were designed to teach students (both in bachelor and master levels) the art of mathematical modeling, mathematical analysis, computer simulations of various biological (micro and macro) processes. One solution involves recruiting students from biology, and teaching them some of the mathematical disciplines required for modeling biological processes. This solution has shown mixed success rates, dominated by several difficulties, including adaptation to exact quantification methods, applying statistical methods and analysis. On the other hand, programs that teach mathematicians to focus on certain fields in biology, and then apply computer modeling, were shown to produce good results. These students were able to rapidly adapt to learning biology, by focusing on very narrow biological processes or fields. However, they were lacking the overall biological intuition to interpret the results. In conclusion, programs that attempt to educate mathematical modeling of biological processes are needed revision in order to adapt to this rapidly growing field.

### Developing Modelling Based Mathematics Teaching by Means of Theorieson Conceptual Learning

##### Morten Blomhøj and Tinne Hoff Kjeldsen

##### IMFUFA, NSM, Roskilde University, blomhoej@ruc.dk

The context for the presentation is an in-service course for Danish teachers on the use of mathematical modelling and project work in upper secondary mathematics teaching (Blomhøj & Kjeldsen, 2006). One of the objectives of the course is to develop the teachers' abilities to deliberately use didactic theories on conceptual learning to develop their students' concept formation through the design and teaching of mathematical modelling projects. In teaching and evaluating this in-service course, we have experienced a need for supporting more concretely the teachers' use of the presented didactic theories (e.g. Sfard, 1991; Steinbring, 1987; Vinner & Dreyfus, 1989) in their design and analyses of the students modelling activities.

In my presentation I present and discuss a schema for spanning the use of different representations of both process and object aspects of selected mathematical concepts involved in a given modelling context. The schema is considered as method that can bridge educational research and the development of teaching practice in mathematical modelling. The use of the schema is illustrated and discussed in relation to a modelling project on alcohol and THC (the active drug in cannabis) that was developed and tested in the course. The schema can function as a mediating link between theory and teaching practices and hereby support research-based development of practice.

Moreover the modelling project on alcohol and THC is an excellent case for how to introduce upper secondary students to the modelling of dynamical phenomena by means of compartment modelling and difference and/or differential equations.

In relation to the presentation we will have the opportunity to carry out experiments for developing and validating a model of the intake and decay of alcohol.

Full presentation can be found here.

Blomhøj, M. & Kjeldsen, T. H. (2006). Learning mathematical modelling through project work: Experiences from an in-service course for upper secondary teachers. *Zentralblatt für Didaktik der Mathematik*, 38, 163-177

Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections of processes and objects as different sides of the same coin. *Educational Studies in Mathematics*, 22, 1-36

Steinbring, H., (1987). Routine and meaning in the mathematics classroom. *For the Learning of Mathematics*, 9, 24-33

Vinner, S. & T. Dreyfus, (1989). Images and definitions of the concept of function. *Journal for Research in Mathematics Education*, 20, 356-366.

### Reflections on the use of NetLogo in teaching ecology

##### Prof. Øyvind Fiksen

##### University of Bergen, Norway, Oyvind.Fiksen@bio.uib.no

Biologists increasingly use computation, programming and mathematics to express and convey ideas about the natural world. Ecology is a quantitative subject and is often written in a mathematical language, often to the surprise and dislike of our students. So how can we make students learn and appreciate the value of acquiring the skills needed to read or even master this subject? Here I will refer our first impressions of using the open source program NetLogo as a tool to introduce computer modelling in a broad, basic ecology course. NetLogo is a practical environment for coding simple individual-based spatial processes, and includes an interactive graphical interface. We used it to illustrate some theoretical concepts, and to train students in generating figures and write about their simulation experiments. The results were encouraging, and a larger group of users and cases will make the use of it even more useful in the future.

Full presentation can be found here.

### Getting drunk and sober again

##### Dr. André Heck

##### University of Amsterdam, Netherlands, A.J.P.Heck@uva.nl

In this presentation I discuss various models that students could implement and use to investigate blood alcohol concentration after consumption of one or more alcoholic drinks. Results from these computer models are compared with measured data that were obtained with breath analysis equipment. The broad range of models for intake and clearance of alcohol in the human body ensures that students have great opportunity to practice evaluation and revision of their models. They can develop the critical attitude that is necessary for successful modeling of biological, chemical or physical phenomena.

### Collaborative learning in mathematical modelling tasks

##### Dr. Paul Hernandez-Martinez

##### Loughborough University, UK,

##### P.A.Hernandez-Martinez@lboro.ac.uk

There is evidence in the literature that collaborative work can contribute positively to students' mathematical learning and in particular in mathematical modelling situations. However, most of the research done on effective collaborative learning in mathematics comes from a cognitive perspective.

Using Cultural-Historical Activity Theory I analyse the social interactions of small groups of undergraduate engineers while working on mathematical modelling tasks. I focus my attention on how the interactions between members of each group (e.g. rules of the group, division of labour) affect the outcome of the task and particularly how they contribute to the collective mathematical sense making process.

I will present the results of the analysis of two episodes of peer group collaboration, one in which the outcome was successful and one in which the group's solution to the problem was mathematically incorrect. I will conclude that effective collaborative work greatly depends on the competency of group members in using skills such as communicative (e.g. active listening, reflection, effective speaking) and inter-personal (e.g. negotiation, assertiveness) skills. Students that are competent in such skills are in better position to gain more from peer collaboration and learn better.

### Modelling and inquiry – how are they related?

##### Prof. Barbara Jaworski

##### Loughborough University, UK, B.Jaworski@lboro.ac.uk

I will address the *meaning and* *nature of modell*ing as set out in the Danish 'KOM' framework (e.g., Niss. 2004) and look for points of contact with *three layers of inquiry* that I have suggested as a theoretical framework for identifying or creating a community of inquiry to support mathematics learning and teaching development (e.g., Jaworski 2008).

I will use, as an example, analysis from the ESUM project in which we studied an innovation to support conceptual understanding of mathematics by first year engineering students in a UK university (e.g., Jaworski, 2014).

Full presentation can be found here.

Jaworski, B. (2008). Building and sustaining inquiry communities in mathematics teaching development. Teachers and didacticians in collaboration. In K. Krainer (Volume Ed.) & T. Wood (Series Ed.) *International handbook of mathematics teacher education: Vol. 3. Participants in Mathematics Teacher Education: Individuals, teams, communities and networks* (pp.309-330). Rotterdam, The Netherlands: Sense Publishers.

Jaworski, B. (2014). Unifying Complexity in Mathematics Teaching-Learning Development: A Theory-Practice Dialectic. In Y. Li et al. (eds.), *Transforming Mathematics Instruction: Multiple Approaches and Practices*, Advances in Mathematics Education, DOI 10.1007/978-3-319-04993-9_20,

Niss, M. (2004). The Danish "KOM" project and possible consequences for teacher education. In R. Straesser, G. Brandell, B. Grevholm and O Helenius, Educating for the future: Proceedings of an International Symposium on Mathematics Teacher Education. Sweden: Royal Swedish Academy of Sciences.

### Student's interpretations of graphical and symbolic models

##### Dr. Thomas Lingefjärd

##### University of Gothenburg, Sweden, thomas.lingefjard@gu.se

Humans meet, adapt and/or interpret mathematical models every day. There are traffic simulations of transportation systems, traffic jam explanations through graphical representations, information provider models, lighting and city safety projects – where light poles or indoor lighting should be placed, drug and medicine models – how often should you take a pain killer or the meaning of exponential growth of Ebola and there are hidden and open economy models which affect our lives in many ways. All humans perceive and interpret graphical representations of models every day and to teach students the ability to do accurate interpretations is an important goal for the educational system. Research in mathematics education related to mathematical modeling is mainly focused on how students, teachers, mathematicians, or work labour do mathematical modeling. My talk will focus on how students interpret and understand graphical and symbolical representations of models and how we - theoretically - can understand their understanding.

### Different personality types in different profession-cultures:

### Dire consequences for math- & stat didactics

### and for science at large!

##### Harald Martens, professor dr.techn

##### Dept. Engineering Cybernetics, NTNU, Trondheim

##### CEO IDLETechs AS, harald.martens@ntnu.no

Based on my 40 years of experience in multivariate data modelling, in many different universities and companies in many countries, I have a dramatic - and perhaps slightly provocative - picture of the status of mathematics and statistics teaching in today's schools and universities:

With today's explosion in Quantitative Big Data, there is a dramatic lack of willing and able data analysts, who can interpret what the real-world data actually tell us, not just give p-values for someone's prior hypothesis. It is therefore essential than the number of scientists capable of useful mathematical modelling and statistical assessment drastically increases. But this is not happening today – rather the opposite. Math and statistics teaching takes a lot of effort from teachers and from students. With meager results, unfortunately. The number of bright, but math-averse people does not seem to decrease.

It appears that different scientific disciplines tend to attract different personality types. This will be illustrated with a summary of empirical data from psychologist Helge Brovold's PhD (NTNU / Renate-senteret 2014): A two-block PLSR summary relating personality types to work types in 2200 Norwegian adults shows distinct associations between personality traits and vocation type. It shows, for instance, a high correlation between being theoretically oriented and being introvert.

Although not yet proven explicitly, I believe this uneven distribution of personality types exists between the mathematizing cultures and most other science cultures, e.g. the biology and medicine. The problem is that the topic of personality differences is not acknowledged in a constructive manner, and that has dire consequences for the distribution of math and statistics in science and technology.

The lecture will e.g. refer to the dramatic findings of prof. Ioannides at Stanford concerning the lack of reproducibility of claims in high-level journal papers, and likewise to prof. McCloskey's finding concerning misuse of statistics in bio-medical papers.

Am I right in insisting on personality difference between the mathematizing and the biologizing cultures? If so, how can we characterize and acknowledge the personality difference, and compensate for them so that we can reach out to a wider range of personality-types among the students?

Full presentation can be found here.

### How can we reach out to the creative entrepreneur types with our statistics, mathematics and physics education?

##### Harald Martens, professor dr.techn

##### Dept. Engineering Cybernetics, NTNU, Trondheim

##### CEO IDLETechs AS, harald.martens@ntnu.no

In this lecture, I shall outline some activities geared towards the personality types that get the least out of traditional mathematical, statistical and physical education styles – the context-oriented, intuitive, creative entrepreneurial students.

1. In our company IDLETech we develop the ISee project: An inductive, explorative motivation-and teaching software- allowing the students to discover the beautiful patterning of natural processes – and in consequence – some of the laws of nature - by themselves, using their mobile phone cameras. This will show them the usefulness of math and statistics by analyzing the world that surrounds us, and motivate them to study hard for their more conventional mathematizing exams.

2. At the Dept. Engineering Cybernetics, NTNU, Trondheim we are have just started to test out several educational experiments for exploratory real-world data-driven modelling, again based on their own multichannel experiments.

Full presentation can be found here.

### Models of alcohol consumption in mathematics courses

##### Prof. Yuriy Rogovchenko

##### UiA, Norway, yuriy.rogovchenko@uia.no

In this presentation we review some simple models for alcohol consumption and discuss their possible use in undergraduate mathematics courses. As the ambiguity of the title suggests, alcohol consumption might, at least theoretically, accompany mathematics teaching. To justify the deliberately chosen title, we conduct an experiment with a group of volunteers with the goal of validating some models or, perhaps, showing their inconsistency.

### Mathematical modelling activities and students' motivation

##### Yuriy Rogovchenko, Olov Viirman

##### UiA, Norway, yuriy.rogovchenko@uia.no

The purpose of this presentation is to report from ongoing research activities within the MatRIC modelling workgroup. First we discuss a number of important didactical aspects related to teaching of mathematical modelling to university students. Then some preliminary results of our collaborative project with BioCEED, the center of excellence in biology education based at the University of Bergen, are presented. We conclude by introducing another research project on mathematical modelling in undergraduate education we have initiated with our colleagues at Loughborough University.

### Homeostasis: the need for modeling. How to reduce the anxiety and skepticism of biologists towards mathematical modeling

##### Prof. Peter Ruoff

##### Centre for Organelle Research, University of Stavanger, Norway, peter.ruoff@uis.no

Students in "Biological chemistry – biotechnology" at the University of Stavanger get their introduction to mathematical modeling by means of reaction kinetics where they solving first-order differential equations, the so-called rate equations. Many of our biology-orientated students show a reservation towards mathematical methods, mathematical applications/programming. In cooperation with control engineers at UiS we have developed concepts how robust homeostasis can be understood both from a kinetic and cybernetic viewpoint. I will show how these concepts can be applied to biological systems and discuss with you how such concepts together with mathematical modeling can be incorporated into the study plan of biology-orientated students.

### From abstract research to weekly assignments

##### Prof. Arvid Siqveland

##### Høgskolen i Buskerud og Vestefold, Norway, Arvid.Siqveland@hbv.no

I will start with an algebraic structure of operators, and show that this corresponds to a dynamic system. This proves that there are at least two ways of learning mathematics by modelling: Mathematical modelling and modelling mathematics. The last one, teaching the pure mathematics, then giving the actual model, is clearly faster. But which is the best?

Full presentation can be found here.

### Students in academia are different. Who do we talk to?

### An empirical study from NMBU

##### Prof. Solve Sæbø

##### NMBU, Ås, Norway, solve.sabo@nmbu.no

A personality test related to both the Myers-Briggs Type Indicator and the Big Five test was given to 288 students who volunteered to take part in a study of the association between personality traits and exam performance. Exam grades from nine undergraduate subjects, some mathematical and some non-mathematical, were used, and a partial least squares (PLSR) analysis revealed a connection between exam grades and some personality characteristics. We take this as an indication of biased teaching style and pedagogical structure in the university, not as systematic differences in skills. The results show that the traditional teaching structure at universities (lectures in large auditoriums, limited dialog, highly structured curriculum, textbook reading and paper-and-pencil tests) tend to disfavor "out-of-box" thinking students. If this is a typical tendency at universities worldwide, academia and society in general may lose resourceful people in positions were their talents really can make a difference, for instance in research. Clearly some adaptive teaching style should be implemented since "one size don't fit them all".

Full presentation can be found here.

### An overview of research on the uses of mathematical modelling in university biology education

##### Dr. Olov Viirman

##### MatRIC, University of Agder, Norway, olav.viirman@uia.no

Mathematics and mathematical modelling are becoming more and more important in biology, and hence also in the education of future biologists. On the other hand, biology has sometimes been seen as the refuge for students interested in the natural sciences, but less mathematically inclined. Over the last decades, increased effort has been spent on addressing this potential conflict. In these efforts, mathematical modelling often plays a double role, both as a topic of study in its own right, and as a means of increasing the perceived relevance of mathematics to biology. I will present a brief overview of the literature on the various uses of mathematical modelling in the teaching of biology at university level. So far, research on these topics is limited, and consists mostly of case studies, often mainly describing a specific course or teaching unit, and a few more theoretical papers highlighting the needs and challenges when working towards strengthening the role of mathematics in biology education. Studies investigating the impact of initiatives in using mathematical modelling in biology education on student learning or attitudes towards mathematics are rare.

### "Authenticity" in the teaching and learning of mathematical modelling

##### Prof. Pauline Vos

##### University of Agder, pauline.vos@uia.no

In this presentation I will discuss authentic aspects in mathematical modelling education. I will present results from a study, asking: what authentic aspects can be identified within mathematics education? Data were collected from an extra-curricular excursion to the university by secondary school students, which dealt with Graph Theory and how it assists in modelling a railway timetable.

Full presentation can be found here.