Developing mathematical modelling competency through problem based project work - experiences from Roskilde University Morten Blomhøj IMFUFA, Department of Science, Systems and Models, Roskilde University
[email protected] Abstract: At Roskilde University we have 35 years of experience with problem based student projects. In particular from the two-year introductory study programme in the natural sciences, we have extensive experiences with interdisciplinary student projects setting up, analysing, applying, and criticising mathematical models. The finding, formulation and delimitation of problems within certain themes are important parts of the students’ project work. Two types of modelling projects are described and analysed in order to illustrate how the problem based study programme supports the students’ development of mathematical modelling competency. 1. Developing mathematical modelling competency Mathematical modelling is a complex interdisciplinary activity, which is intricately connected to the learning of mathematics. In fact mathematical modelling competency is seen as one of eight competencies spanning mathematical competency in general (Niss & Jensen, 2007, ch. 4). In this context mathematical modelling competency can be defined as: By mathematical modelling competence we mean being able to autonomously and insightfully carry through all aspects of a mathematical modelling process in a certain context. (Blomhøj & Højgaard Jensen, 2003, p. 126). It follows from this definition that in order to pursue modelling competency as an educational goal one needs teaching approaches where students sufficiently often are challenged to work with the entire modelling process as discussed in Blomhøj & Højgaard Jensen (2007). In other words modelling competency does not follow from knowing the relevant mathematics and having insights into the relevant part of reality. Teaching mathematical modelling competency also involves developing the students’ autonomy in carrying through a modelling process and reflecting critically upon the validity of the model, and its possible functioning through applications. As is extensively discussed in the didactical literature on the teaching of mathematical modelling it is important to establish a balance between working with the more mathematically demanding inner parts for the modelling process (that is the processes of mathematization and mathematical analysis), and a more holistic approach where the students’ are working with the entire modelling process1. The problem based project work at the science programme at Roskilde University can be seen as an excellent example of how to organise a study programme, which challenges the students to work with all phases of a mathematical modelling process. The aim of this paper is to illustrate and discuss how modelling competency is spanned by two different types of mathematical modelling
1
In Blomhøj & Højgaard Jensen (2007) this dilemma is discussed with the focus on upper secondary mathematics teaching.
projects in this institutional context. However, before doing so, it is appropriate to give some brief general background information on project organised studies at Roskilde University. 2. The project work at the science study programme at Roskilde University Roskilde University was founded in 1972 with the particular aim of being an alternative to the traditional universities in Denmark. From the beginning all study programmes were based on four pedagogical principles: interdisciplinarity, problem orientation, exemplarity and group organised project work (Illeris, 1999). The two-year introductory study programme in natural science is still governed by these principles. During the four semesters in this programme each student should complete one project each semester (15 etcs) and two courses (7.5 etcs each) chosen from a list of 24 courses covering broadly the natural sciences. The students are organised in houses with up to 120 students in each. The physical facilities for the project work are very important. Every project group has access to a room equipped with computer and a blackboard and in addition the house also holds a plenary room where all the students can gather for seminars, project presentations and social activities. For supervising the projects and organising the academic activities in the house a team of 10-12 professors are allocated to the house. The team of supervisors represents the main subjects in the natural sciences including mathematics and computer science. In the beginning of the semester the students form project groups of 4-7 students. The groups are crystallized around suggestions for problem-areas put forward by the students or by the supervisors. In the first three semesters the projects are contained by a theme. These themes are introduced and discussed with the students in short introductory courses during the first two weeks of the semester, in which period the project groups are also formed. The supervising team allocates a supervisor to each project according to the supervisors’ expertise and interests in the best possible way. In addition to other teaching obligations each supervisor is assigned one or two projects. The supervisor meets with each group for 1-2 hours a week and reads and comments their writings. Normally the project results in a written report of 60-80 pages. The students’ project is evaluated on the basis of the project report and an individual oral examination2 . The project themes are: 1st semester: Application of natural sciences in technology and society 2nd semester: Models, theories and experiments in the natural sciences 3rd semester: Reflection on the natural sciences and the dissemination of knowledge in the field of natural sciences. th 4 semester: No particular theme. The project should be within the field spanned by the three previous themes. (Study guide 2006, p. 28) In daily life, the logic of the three themes are referred to as project work with, in and about the natural sciences and it is an important educational aim that the students understand these three different perspectives. Through their projects the students are supposed to get exemplary experiences with these three different perspectives.
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See Legge (1997) for more details on the organisation of the project work and the supervising process.
4. Mathematical modelling projects in the two-year study programme in the natural sciences In all four semesters it is possible to work with projects that involve different aspects of mathematical modelling. In the following I shall focus on projects within theme 1 and 2, which as said, also include projects made in the fourth semester3. In average 25% of the projects in semester 1, 2 and 4 contain a substantial element of mathematical modelling and related reflections. This adds up to around ten mathematical modelling projects each year. Many of the students that continue their studies in mathematics, physics, chemistry or biology have made two projects within mathematical modelling in the introductory programme. During the last 20 years I have supervised close to 20 such projects. Based on this experience I shall try to characterise with concrete examples two different types of modelling projects related respectively to the themes for semester 1 and 2. 4. 1 Projects emphasizing the application of a particular model in a societal context In post modern societies where production, administration and societal planning involve high technology, mathematical modelling plays a crucial role. Mathematical models are developed in interplay with science and technology with the purposes of describing, predicting or prescripting societal or technological problems or systems. Questions such as: How will the Danish population develop the next 10 year? How large a percentage of our electricity consumption can be produced by windmills? How to vaccinate against influenza most effectively? Where and how should we invest in the transportation infrastructure? can not be dealt with without mathematical modelling. These problems are characteristic for the first semester theme and are examples of modelling projects carried out in first or fourth semester. However, the use of mathematical models in relation to such societal and technological issues is not at all neutral. Whenever a mathematical model is introduced in a political and technological investigation it tends to cause (1) a reformulation of the problem at hand in order to be adequate for modelling, (2) changes in the discussions about the problem in the direction of pros and contras the model and changes in the model, (3) a limitation of the actions under consideration to those that can be evaluated in the model and (4) a delimitation of the group of citizens that can take part in the discussion and act as the base of critique for decisions based on the model results. As the following projects illustrate, students can get exemplary experiences with this kind of reflections through modelling projects under the theme of the first semester: The use of a traffic model in the city of Roskilde – the case of “Ny Østergade”. This project, which was a fourth semester project fulfilling the first semester theme examined a current political conflict between the local administration in Roskilde and a group of citizens living very close to where a new road was planned. Together with other traffic regulations this new road was projected as a result of a mathematical modelling process. The city was divided into 57 zones and based on interviews and traffic counting a 57x57 tour matrix was set up to describe the total flow of traffic within and in/out of the city per day. The tours were distributed on a road net representing all the main roads including the planned new one. In the model this was done according to three principles: (a) for all roads, the traffic allocated should be less than the estimated capacity of the road in question, (b) all tours should be placed on the 3
For an example of a third semester project involving modelling, see the paper by Tinne Hoff Kjeldsen in these proceedings where it is illustrated and discussed how students can work with the history of mathematics and science in the third semester.
net so that the calculated time of transportation for each tour is minimized and (c) for each road element of every tour the time of transportation is calculated in minutes according to the formula: L Ti = 60 ! i + ci Li , vi where Li is the length of the i-piece of road measured in kilometres, vi is the average speed at that road segment in kilometres per hour, ci signifies that some extra time is used depending on the type of road and measured in minutes per kilometres. The students analysed the model, the empirical basis for the tour matrix and the estimation of the model parameters. They were able to document that the model was used in two different occasions to predict the traffic on the new road to be respectively above and below 10.000 cars per day. The high estimate was used in an argumentation for the effect of the new road in order to reduce the traffic in the centre of the city and hereby for the construction project to get support from national governmental sources. While the low estimate was used to prevent that the new road was made subject to an EU-procedure for evaluation of the effects on the environment. This procedure, which the group of citizens requested, is only compulsory for road projects with a predicted traffic above 10.000 cars per day. The two results were obtained by simply using two different values for the parameter vi for the new road – a change for which there were no valid foundation since the parameter vi for a not yet constructed road can not be made subject to empirical estimation. Based on interviews with one of the modellers (from a consultant bureau), local politicians and the spokesperson for the group of citizens, the students analysed the discourse about the new road and they concluded that the model played a central role in the discourse and that results from the model was used beyond its scoop of validity. The objective in this type of projects is that the students get an insight into an often very complex mathematical model and the actual or possible applications of the model in a societal or technological context. In the best cases the students also gain first hand experiences with reflecting critically on the validity of model. The point is not to criticize some concrete political decision but to understand mathematical modelling as an often indispensable but also not unproblematic tool for investigating complex problems. 4.2 Projects emphasizing the modelling process Mathematical modelling is playing an important role in many scientific disciplines. Not only in connection with applications of theories from other disciplines but, even more importantly, mathematical modelling is often weaved together with the formulation and definition of theoretical key concepts and with standard experimental practices. Quantitative experiments or interpretation of experimental measurements always involves some kind of mathematical model explicitly or implicitly. Of course this is especially evident in the natural sciences. The general idea with mathematical modelling projects under the second theme is that the students get insights into the interplay between theoretical concepts, mathematical modelling and experiments. To achieve this, the students need to set up a mathematical model or to analyse very closely the modelling process behind an existing model. The following project exemplifies the last type.
Conjugative plasmid transfer. This project was a second semester project with a substantial experimental element. Inspired by a scientific paper (Andrup et al., 1998) the students investigated the rate of transfer of plasmids with an antibiotic resistance gene in two bacterial systems by means of conjugation. The hypothesis presented in the paper is that the rate of transfer will follow the well-know Michaelis-Menten model for the kinetic of first order enzyme catalysis: [S ] , V = Vmax ! [S ] + km where the rate of catalysis V, varies with the concentration of substrate [S]. Vmax [time-1mol-1] denotes the maximum rate and km denotes the substrate concentration corresponding to the rate ½ Vmax. The group analysed all the assumptions behind the theoretical deduction of this model for enzymatic catalysis and discussed for each particular assumption to what degree the conjugative system could be expected to comply with these assumptions. They argued that, from what is known about the conjugation process, the model should be valid with approximations also for plasmid transfer. The analogy to enzyme catalysis means that the donor cell with the plasmid is acting as an enzyme transforming the recipient into a transconjugant without undergoing any transformation itself (Andrup, 1998, p. 37). Hence, V denotes the conjugation rate measured in number of plasmid transfers per time per donor cell. The students tested the hypothesis experimentally in a laboratory and they confirmed the model - at least for some of their experimental series and they were able to estimate values for the model parameters by linear regression. In this project, the students worked intensively with theory (in micro-biology), mathematical modelling and experimental investigation. They experienced that the modelling process and the model was necessary for forming the hypothesis and for designing the experiment, and that the experiment on the other hand was necessary for testing the model and for estimating values for the parameters. They even experienced that the model with its theoretical foundation and the experimental results gave raise to new research questions about the characterisation of different conjugative systems in terms of the model parameters. Conclusion As illustrated, the problem based project work in the two year study programme in natural science at Roskilde University provides excellent opportunities for the students to develop important elements of mathematical modelling competency. However, the project work can not do the job alone. The students also need to work more systematically with the inner parts of the modelling process. This is taken care of in an elementary modelling course covering two semesters (15 etcs)4. Of course for the students to develop the technical level of their modelling competency they also need to learn more advanced mathematics in more traditional mathematical courses. The project work and the focus on mathematical modelling competency come with a price – less time for teaching important and fundamental disciplines. However, in the programme mathematical modelling together with experimentation are seen as the two main constituents in a general introductory study programme in natural science - and for good reasons we think. 4
This course is described and discussed in Blomhøj & Højgaard Jensen (2003) and in Ottesen (2001).
References Andrup, L., Smidt, L.,Andersen, K. and Boe, L. (1998). Kinetics of conjugative transfer: A study of the plasmid px016 from Bacillus thuringiesis subsp. Israelensis. Plasmid 40, p. 30-43. Blomhøj, M. & Højgaard Jensen, T. (2007). What’s all the fuss about competences? Experiences with using a competence perspective on mathematics education to develop the teaching of mathematical modelling. In: W. Blum (red.): Modelling and applications in mathematics education, pp. 45-56. The 14th ICMI-study 14. New York: Springer-Verlag. Blomhøj, M. & Højgaard Jensen, T. (2003). Developing mathematical modelling competence: Conceptual clarification and educational planning, Teaching Mathematics and its applications 22 (3), pp. 123-139. Legge, K. (1997). Problem-orientated group project work at Roskilde University. What is it, how is it performed and why? IMFUFA-tekst 336. Department of Science, Systems and Models, Roskilde University. Illeris, K. (1999). Project work in university studies: Background and current issues. In Højgaard Jensen & Olesen (eds.): Project studies – a late modern university reform. Niss, M. & Højgaard Jensen, T (eds.) (2007): Competencies and Mathematical Learning – Ideas and inspiration for the development of mathematics teaching and learning in Denmark. English translation of part I-VI of the report from the Danish KOM-project. Under preparation for publication in the series Tekster fra IMFUFA, Roskilde University, Denmark. To be ordered from
[email protected]. Ottesen, Johnny (2001). Do not ask what mathematics can do for modelling. Ask what modelling can do for mathematics! In Holton, D. (ed.): The teaching and learning of mathematics at university level. An ICMI-study, pp. 335-346. Dordrecht: Kluwer Academic Publishers. Study guide (2006). The basic studies in natural science. Study guide 2006-2007. Roskilde University.
