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Developing students’ modelling competence in problem oriented projects

0. Why is this relevant?

1. The context: The problem oriented project work at the natural science bachelor program at RU

2. The research question: How to characterise and support students’ progress in modelling?

3. Examples: Projects modelling epidemic diseases - Influenza and Measles in Denmark

4. Summing up – and time for questions and discussion

Morten BlomhøjIMFUFA, INM, Roskilde University

0. Why is this relevant?

• Modelling competency as an educational aim

• Mathematical modelling as a means for learning mathematics

• Modelling competency for STE(M) teachers

• Project organized and problem oriented studies - a (radical) holistic approach to modelling

• Epidemiology is an exemplary domain for mathematical modelling with respect to

- interdisciplinary interplay with mathematics

- the use of modelling and models in the risk society (Beck et al., 1992)

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1. The natural science bachelor program at RU

• A three year interdisciplinary program in natural science (Nat Bach) leading to a bachelor degree with specialisation in two subjects.

• Half of the students’ study time is devoted to problem oriented projects – one project of 15 ECTS each semester.

• The other half is course work – typically three subject organised courses (5 ECTS) each semester.

• The projects are done in groups of 2-6 students under supervision of a researcher, and assessed individually at an oral group exam based on a project report of typically 50-80 pages.

• The yearly enrolment is around 100 students at the Danish program and around 40 at the international program.

Chemistry

Environmentalbiology

MathematicsPhysics

Molecularbiology

Technological and socio-economic planning

Computerscience

The Nat Bach program gives the students an interdisciplinary introduction to natural science and the opportunity to specialize in two subjects in which they can continue in two-year master programs:

Bridge maker

New integrated master programs in mathematical modelling in biology, chemistry, physics, and computer science are offered from 2019. 1/3

Medical biology

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The International Bachelor Study Program in the Natural Science

Project work(15 ECTS)

Course (5 ECTS)

Course(5 ECTS)

Course(5 ECTS)

6. sem. Bachelor project (Natural Science)

Link

Subject 2 Course

Free choice CourseLink

Free choice Course

5. sem. Subject 2Project

Subject 2 Course

Subject 2 Course

Subject 1 Course

4. sem. Subject 1 ProjectLink

Subject 1 Course

Subject 1 Course

Subject 2 Course

3. sem. Basic project 3About Science

CommonCourse BC3

Subject 1 Course

Course(BC 4-8)

2. sem. Basic project 2In Science

CommonCourse BC2

Course(BC 4-8)

Course(BC 4-8)

1. sem. Basic project 1With Science

Link

CommonCourse BC1

Link

Course(BC 4-8)

Link

Course(BC 4-8)

Project

The projects are conducted in groups of 2-6 students under guidance of project supervisor with research experience.

The projects in semester 1-3 (15 ECTS each) should fulfill the following thematic constrains in order to provide exemplary experiences and insights into problem oriented project work with, in and about natural science:

Project 1: Applications of science in technologyand society.

Project 2: Interaction between model, theory, experiment, and simulation in natural sciences.

Project 3: Natural sciences and theory of science.Schema

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The common courses in the beginning of semester 1-3

The common courses (BC 1-3, each of 5 ECTS) deal with themes related to the project work in the particular semester:

1 Semester: Empirical data

2 Semester: Experimental methods

3 Semester: Theory of natural science

Schema

Courses BC 4-8: Courses in natural sciences.

The objective of these courses is to give a broad introduction to and basic knowledge and competences in natural sciences. The aim is to enabling the students to make a qualified choice of subject modules, and to complete these.

Each semester Nat Bach offers a number of courses from which the student should choose two courses in the two first semesters, and one additionally course later in the program.

Courses that are not taken during the basic part of the Bachelor program may be chosen as (free choice) optional courses.

A subject module description may include recommendationsfor up to two particular courses to be taken before starting at the subject module. Schema

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Mathematics courses in the basic part of Nat Bach:

BC4-8:Calculus;Linear Algebra; Statistical ModelsLogic and discrete mathematics

Free choice courses: Understanding the continuous world

Schema

Subject modules:

Each subject module consists of one project (15 ECTS) and four courses (5 ECTS each) – in total 35 ECTS.

Subject course (5 ECTS) means a course, which is a mandatory part of a particular subject module.

Subject project (15 ECTS) is a mandatory part of the subject module.

Subject module courses in mathematics:

Mathematical Analysis I and II; Mathematical modelling and dynamic systems; and Algebra.

The project should be within mathematical modelling.

Recommended study plans for all combinations of natural science subject modules are available for the students.

Schema

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The bachelor project (15 ECTS).

The Bachelor program in Natural Sciences is completed by the Bachelor project.

The Bachelor project is intended to demonstrate students’ ability to formulate, analyse and work with issues within a defined academic topic in natural science.

The choice of a specific problem, method or a relation to an institution or company, may direct the Bachelor project towards a specific job function, nationally as well as internationally.

The area within which the problem of the Bachelor project is formulated must be approved by The Head of Study

Schema

Two courses of free choice (5 ECTS each) means:

• ECTS covered courses at bachelor level offered by any university

• These course can be placed in year 2 and 3

• Course offered by Nat Bach as BK4-8 courses or subject module courses are also available as free choice courses.

• Nat Bach offers in addition courses, which only can be taken as free choice courses.

Schema

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PPL: ProblemorientedProject Learning

1. Project work

2. Problem-orientation 3. Inter-

disciplinarity

4. Participant directed

7. Inter-nationalisation

5. Exemplarity6. Group

work

The project work is problem and research oriented

In the projects the students formulate researchquestions, which can be answered by means of:

• choosing and applying scientifictheories and methods

• searching, choosing and analyzingrelevant scientific literature

• designing and conducting experimentsand/or analyzing empirical data

• communicating according toscientific norms in reports, posters,papers and oral presentations

• utilizing guidance from a researcher

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Problem orientated project work

What is a good problem? (the problem-formulation)A problem-formulation is a research question – or few connected research questions. It should be as explicit as possible.

What is it good for?A problem-formulation gives direction to the project and helps the project group to steer the process.

A problem “is” an arrow A topic “is” a blob

What can be achieved through project work?

• Motivation for learning theories, methods and concepts within the sciences

• Opportunities for in-depth and exemplary learning and reflections

• Personal and social competencies (self-esteem and self-guidance, autonomy)

• Reflective perspectives on the science disciplines and their role and function in sciences, technology and societies.

• High level subject matter competencies such as mathematical modelling competency

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Timeline with milestones for the project work

Group formationChoice of project

Problem formulation 1.0

Mid-term evaluation

Internal evaluationof the project

Report

Groupexam

Individualevaluation

Problem formulation seminar

Pilot project

0 8 20weeks

Presentation seminar

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A project group (2-6 students) meet with their supervisor for 1-1½ hours every week during the semester. The supervisors allocated to a particular semester collaborate in a team.

• The project work and the social life of the students is framed physically by (part of) a building

• A house coordinating supervisor organising the project work and the collaboration in the team of supervisors

• A team of researchers to supervise the students projects

The study environment

• Rooms are available for the project work during the semester

• The house including the kitchen and the plenum can be used for study and social activities outside teachings hours.

• The new students are introduced to the studies and the culture of project work in a two weeks introduction period organised by student tutors.

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The research question:

How to characterise and support students’ progress in mathematical modelling competency in interdisciplinary problem oriented project work?

The objective for researching this question is to

(1) develop further the practice of supervising modelling projects at Nat Bach and at the new integrated master programs in mathematical modelling, and

(2) to contribute to a better understanding of the development of students’ modelling competency in natural science university studies.

2. Students’ progress in modelling competency

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The framework of mathematical competences

(Niss & Højgaard, 2011)

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Mathematical modelling competency means:

A person’s insightful readiness to autonomously -as well as in collaboration with others - carrying through all aspects of a mathematical modelling process and to reflect on the modelling process and on the actual or potential use of the model in a particular context.

(Blomhøj & Højgaard Jensen, 2007)

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Object

Domain of inquiry

Mathematical system

Model results

Action/insight

(a) Problem formulation

(b) Systematization

(c) Mathematization (d) Mathematical analysis

(e) Interpretation/evaluation

(f) Validation

Data

Experience

System Theory

A mathematical modelling process

Three dimensions in modelling competency:

1) Degree of coverage, according to the sub-processes of the modelling process involved in the students’ work.

2) Technical level, according to the mathematics used by the students and how flexible and reflective they are in their use of mathematics and modelling techniques.

3) Radius of action, according to the domain of situations in which the students can perform modelling activities and their level of reflection across domains of applications.

(Blomhøj & Højgaard, 2007)

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Two types of reflections and critique:

One can distinguish between two types of reflections and critique in relation to modelling and applications of models:

a) Reflections and critique which are internal with respect to the modelling process.

b) Reflections and critique which are external with respect to the modelling process.

(Blomhøj & Kjeldsen, 2011)

Reflection and critique as part of modelling competency

The objects in focus for internal reflections and critiqueare decisions taken and assumptions made in the modelling process.

They could be connected to the different sub-processesinvolved in the modelling process, e.g. the formulation of the problem, the use of theory and data, choice of mathematisation, estimation of parameters, mathematical analyses, and interpretation of the model and model results.

Depending on the subject area in which the modelling process is situated different forms of reflections and critique are possible and needed. Here an important distinction is between theory based and a hoc models.

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In the context of application, external reflections and critique are connected to the possible and/or actual functions of a model in an investigation or decision process in a particular societal, technological or scientific context.

Four examples of general issues that can be objects for external reflections and critique are:

(1) Transformation of a original problem to a problem, which is approachable by means of modelling

(2) The model forms the discourse about the problem

(3) Delimitation of the solutions or actions considered to those that can be evaluated in the model

(4) Delimitation of the group of people who can take part in the discourse – the critique base.

The structural relations in an application process

Object

“Theory”

Interests(1)

ModelApplica

-tionMath. system

Interests(2)

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Mathematical modelling is playing crucial roles in the epidemiology of infectious diseases (Anderson and May, 1991).

Models and modelling are used for:

• defining concepts and notions in epidemiology

• describing and explaining data and phenomena

• predicting courses of epidemic infections

• designing and testing vaccination programs or other types of regulations

• providing a basis for designing health care policies

3. Examples: Modelling projects in epidemiology

Typical courses of viral infections

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S(t) I(t) R(t)N

IcS

I

)()´(

)()(

)()´(

)()()´(

tItR

tIN

tItcStI

N

tItcStS

Projects expanding the SIR-model

Notions and assumptions: S: # of susceptible I: # of infectious R: # of recovered (immune) c: The rate of effective contactsv: The rate of recovery * All individuals have the same

behavior and medical response

Numerical solution to the SIR model for:

S(0)=998; I(0)=2; R(0)=0 og c=0,85 =0,4.

Num

bers

0

200

400

600

800

1000

1200

0 5 10 15 20 25 30 35 40

S((t)

I(t)

R(t)

Days

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Simple qualitative analysis of the SIR-model

An epidemic occurs if and only if I´(t)>0, which yields

1)(

N

tSc

10

c

If S(0) N , we have

0 is called the reproduction number. This number is specific for the disease in the particular population and situation where the parameters are estimated.

The number denotes the average number of infections that one infected individual can produce in the particular population if the entire population were susceptible for the infection.

0 200 400 600 800 10000

50

100

150

200

250

300

350

400

450

500

S(t)

I(t)

s(t) = vN/c

Numerical solutions in the phase plan (S,I)

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Herd immunity by means of vaccination

Herd immunity for a disease in a population occurs when the number of susceptible individuals are too small for the disease to spread in the population (too small for coursing I’(t) to be positive). By means of introducing vaccination before the epidemic in the model, the critical immunization threshold (pc ) can be expressed by:

with S0 ≈ N we get:1)1( 0

Nv

Spc c

0

1111

c

vp

c

vp cc

3.1 A project on influenza epidemics

A SIRS model with birth and death was setup to model returning influenza epidemics and pandemics by loss of immunity due to antigenic drift and shifts in influenza virus.

Based on molecular biology the group made assumptions enabling them to estimated the number of mutations likely to occur in the two most relevant virus antigens (hemagglutinin and neuraminidase), during one year. From that an interval for the de-immunization rate was estimated.

Shifts were modelled as stochastic events coursing S(t*)=N.

S(t) I(t) R(t)N

IcS

If N

d S(t) d R(t)d I(t)

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3.2 A project on the Danish policy for influenza

A SIR model with 5 age-groups with specific contact rates was developed for evaluating the Danish policy offering free influenza vaccines for retirees and high risk groups.

The model results indicated that under a societal economical rationality it would be more efficient to use the same resources for vaccines to the age group 6-25 years.

The same number of vaccines to these groups as currently used for the oldest age-group can almost keep the prevalence in oldest group at the same level.

Hence, a dilemma between the personal protection – a individual rationality – and a societal rationality was identified and discussed. Such discussion is only possible by means of mathematical modelling.

3.3 Projects on the child diseases MMR

In 1987 Measles, Mumps and Rubella (MMR) was included in the vaccination program in Dk. The situation can be modelled with an age-structured SIR-model with birth and death (rates f = d ); a cohort vaccination degree (immunization degree, v); and transitions according to numbers of years in each age-group (Li):

S(t) I(t) R(t)N

ISc1 I(1-v)f N(t)

d1S(t) d1R(t)d1I(t)

vf N(t)

S(t) I(t) R(t)N

ISc2 I

d2 S(t) d2 R(t)d2 I(t)

1/L1 S(t) 1/L1 R(t)

A1

A2..

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a.) A project setup and analyzed an age-structured SIR model for measles showing that with current immunization rates of around 82 % (< pc ≈ 95 %) small epidemics will occur and the average age of the infected will increase. The project also analyzed the public debate pro and contra the MMR vaccination program. Measles in Dk

b) A project developed and analyzed a SEIR model and a corresponding simulation model.

c) A project analyzed the medical consequences of the shift in the age profile of the infected individuals for measles and rubella. For example the frequency of the Congenital Rubella Syndrome (CRS).

Other projects on: HIV and HPV, and Gonorrhea and Chlamydia infections.

3.3 Projects on the MMR diseases

Summing up

Maesles i Denmark

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Kilde: SSI

http://www.ssi.dk

Degree of coverage for the second MMR vaccination (age 4 year)

Pc ≈ 95 %

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Numbers of reported cases of measles in Dk 1994-2016

http://www.ssi.dkProjects

Ind

ex f

or t

he

freq

uen

cy o

f C

RS

Degree of immunizationProjects

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Progress in modelling competency

1) Degree of coverage with respect to the modelling cycle:

• Analysing the SIR compartment model and understanding its basic assumptions,

• Formulating and delimiting a problem or problems to be investigated through expansion of the SIR model,

• Working through the modelling cycle setting up and analysing a model for addressing specific problems.

4. Summing up

Progress in modelling competency

2) Technical level, the students develop knowledge and skills for:

• using compartment models for modelling dynamical systems,

• how to transform such models into systems of ODE by means of the “In – Out” principle,

• developing computer models (in MatLab) for numerical analysis and/or for simulation,

• analysing numerical and qualitatively the mathematical behaviour of such models.

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Progress in modelling competency

3) Radius of action:

• To begin with the students’ competency is develop in and limited to the problem area of the project

• Gradually the competency is extended to compartment modelling in epidemiology

• Through subsequent projects and courses the students – to different degrees - begins to understand the generality in modelling dynamical systems by means for compartment models.

Progress in the students’ reflections

Internal reflections:

• The students develop gradually their reflections connected to basic assumptions behind the compartment approach to mathematical modelling in epidemiology.

• The homogeneity assumption for each for the compartments were typically discussed - and often criticized - based on medical or biological knowledge about the processes.

• The basis for estimation of the model parameters and the interpretation of the model results is typically addressed in the students’ reflections in the project reports and/or at the oral project exam.

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Progress in the students’ reflections

External reflections: In several cases the students established the need for mathematical modelling for understanding or regulating the problem or phenomena addressed on the one hand, and on the other hand the shortcomings of the models when it comes to predicting the course of epidemics or the more precise consequences of regulations in the system.

In some projects, students formulate dilemmas in health care policies related to infectious diseases, such as how to optimize vaccination strategies; how to ensure that the critical level of immunization is reached in the population; and how to target information campaigns and other initiatives toward specific groups.

Further readings and related research

Research related to mathematical modelling at Nat Bach or the master program in mathematics:

In project work: Blomhøj & Kjeldsen (2016, 2011, 2009); Kjeldsen & Blomhøj (2013)Niss (2001); Ottesen (2001)

In course work:Blomhøj & Kjeldsen (2013, 2010)

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Interplay between research and development of teaching practices in mathematical modelling

Theory Practice

Development

Research

Thanks for your attention

Time for questions

ReferencesAnderson, R., & May, R. (1991). Infectious Diseases of Humans: Dynamics and Control. Oxford:

Oxford University.

Bailey, N. T. (1986). Macro-modelling and prediction of epidemic spread at community level. Mathematical Modelling, 7(5-8), 689-717.

Beck, U., Lash, S., & Wynne, B. (1992). Risk society: Towards a new modernity (Vol. 17). sage.

Blomhøj, M. and 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 et al. (eds.) Modelling and applications in mathematics education. The 14th ICMI-study 14. New York: Springer-Verlag. 45-56.

Blomhøj, M. and Kjeldsen, T.H. (2017). Interdisciplinary Problem Oriented Project Work: A learningEnvironment for Mathematical Modelling. In Evaluierte Lernumgebungen Zum Modellieren. Springer.

Blomhøj, M., & Kjeldsen, T. H. (2013). Students’ mathematical learning in modelling activities. In Teaching mathematical modelling: Connecting to research and practice (pp. 141-151). Springer Netherlands.

Blomhøj, M. and Kjeldsen, T.H. (2011): Students’ Reflections in Mathematical Modelling Projects. In G. Kaiser, W. Blum, R. Borromeo Ferri, G.Stillman (eds.) Trends in Teaching and Learning of Mathematical Modelling. International Perspectives on the Teaching and Learning, Vol. 1. Springer, 385-396.

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ReferencesBlomhøj, M. and Kjeldsen, T.H. (2010): Learning mathematics through modelling – the case of the

integral concept. In B. Sriraman, C. Bergsten, S. Goodchild, G. Pálsdóttir, B. Dahl and L. Haapasalo (eds.) The first Sourcebook on Nordic Research in Mathematics Education. Montana: Information Age Publishing, 569-582.

Blomhøj, M., and Kjeldsen, T.H. (2009): Project organised science studies at university level: exemplarity and interdisciplinarity. ZDM – International Journal on Mathematics Education, 41(1-2), pp. 183-198.

Blomhøj, M. and Kjeldsen, T.H. (2006): Teaching mathematical modelling through project work -Experiences from an in-service course for upper secondary teachers. Zentralblatt für Didaktikder Mathematik, 38, 2, 163-177.

Kjeldsen, T. H., & Blomhøj, M. (2013). Developing Students’ Reflections on the Function and Status of Mathematical Modeling in Different Scientific Practices: History as a Provider of Cases. Science & Education, 22(9), 2157-2171.

Niss, M. (2001). University mathematics based on problem-oriented student projects: 25 years of experience with the Roskilde model. In The teaching and learning of mathematics at university level (pp. 153-165). Springer Netherlands.

Niss, M., & Højgaard, T. (2011). Competencies and Mathematical Learning: Ideas and inspiration for the development of mathematics teaching and learning in Denmark. IMFUFA, Roskilde university.

Ottesen, J. T. (2001). Do not ask what mathematics can do for modelling. In The Teaching and Learning of Mathematics at University Level (pp. 335-346). Springer Netherlands.

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Methodological reflections

• The analyses are limited to the progress of modelling competency in each single project within epidemiology.

• The analyses of projects within epidemiology could and should be supplemented with analyses of modelling projects in other disciplines.

• In research to follow individual students’ learning trajectories with respect to modelling competency will be analysed based on their project reports and interviews.

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New cases per day during the 1965 influenza epidemic in Leningrad 1965 (Bailey, 1986)