Strand 6:

Mathematical modelling

with technology

ICTMT 5, Klagenfurt, 6-9 August 2001

(Schedule, tentative as of 8. 6. 2001)

Chair: Jenny Sharp

Monday 15:15 - 16:00 Chair: Jenny Sharp

A Discrete Introduction to Modelling

Duncan Lawson (UK, d.lawson@coventry.ac.uk)

Monday 16:15 - 17:00 Chair: Jenny Sharp

Laplace transform and electric ciruits - an interdisciplinary learning tool

G.Albano*, C. D'Apice, M. Desiderio (Italy, albano@diima.unisa.it)

Monday 17:00 - 17:45 Chair: Jenny Sharp

Modelling Human Growth

Andre Heck (Netherlands, heck@science.uva.nl)

Tuesday 8:30 - 9:15 Chair: Jenny Sharp

Differential Equations in maths and physics instead of analytical methods

George Adie (Sweden, george.adie@te.hik.se)

Tuesday 9:30 - 10:15 Chair: Jenny Sharp

A Practical Investigation Task with the Computer at Secondary School: Bridges and Hanging Chains

Andre Heck (Netherlands, heck@science.uva.nl)

Tuesday 10:30 - 11:15 Chair: Jenny Sharp

Cross curriculum teaching and experimenting in math & science courses using new technologies

Brigitta & Klaus Aspetsberger (Austria, aspetsberger@aon.at)

Tuesday 15:15 - 16:00 Chair: Jenny Sharp

Model of deformations of fluid particles due to electric field

Iavor Hristov (Bulgaria, iavhri@fmi.uni-sofia.bg)

Tuesday 16:15 - 17:00 Chair: Jenny Sharp

An Internet-Based Course Structure for Teaching Mathematics in an Engineering School

Mithat Uysal (Turkey, muysal@msu.edu.tr)

Tuesday 17:00 - 17:45 Chair: Jenny Sharp

Modelling with Difference equations using DERIVE

Mazen Shahin (USA, mshahin@dsc.edu)

Discrete Delayed Population Models with DERIVE

Mazen Shahin (USA, mshahin@dsc.edu)

Wednesday 8:30 - 9:15 Chair: Jenny Sharp

Mathematica and symbolic-numerical methods for solving first order ODEs

Giuliano Gargiulo (Italy, gargiulo@diima.unisa.it)

Wednesday 9:30 - 10:15 Chair: Jenny Sharp

Mathematical modelling with CABRI

Per Broman (Sweden, pbr@planetarium.euromail.se)

Thursday 8:30 - 9:15 Chair: Jenny Sharp

Importance of Mathematics in Engineering Education

Aynur Uysal (Turkey, uysal@itu.edu.tr)

Thursday 9:30 - 10:15 Chair: Jenny Sharp

Mathematical Application Projects for Mechanical Engineers - Concept, Guidelines and Examples

Burkhard Alpers (Germany, balper@fh-aalen.de)

Thursday 11:30 - 12:15 Chair: Jenny Sharp

Plenary:

The use of technology in developing mathematical modelling skills

John Berry (UK, jberry@ctm1.freeserve.co.uk)

Thursday 15:15 - 16:00 Chair: Jenny Sharp

Software Maple and Matlab in teaching of ordinary differential equations

Pavel Prazak*, Antonin Slaby (pavel.prazak@uhk.cz)

Abstracts:

Plenary: John Berry, UK:

The use of technology in developing mathematical modelling skills

An important part of teaching and learning mathematics at all levels of education is the development of the skills needed to solve "real problems". The process of solving real-world problems in mathematics is called mathematical modelling. It can be

summarised by the following diagram (see page of the strand, hotkey for the plenary). Technology has an important role to play in this process. The use of software and calculators are natural in the solution phase. It is now well established that the formulation phase of mathematical modelling represents the ?bottleneck? stage of the modelling process. Helping students to develop good problem solving skills often involves much time and effort in this phase. Data logging equipment is a powerful means of collecting and analysing data as part of the Interpretation phase of the process. The aim of this plenary lecture is to reflect on ways that we can bring technology to the teaching, learning and assessing mathematical skills.

George Adie, Sweden:

Differential Equations in maths and physics instead of analytical methods

Handheld technology with CAS makes it easier to study common scientific phenomena at undergraduate level directly using differential equations and numerical techniques instead of the conventional analytical methods. We will highlight areas of study where requirements are changing in physics using linear and non linear differential equations in one or more dimensions. This leads to changing requirements in mathematics. These changes will be discussed.

G. Albano*, C. D'Apice, M. Desiderio, Italy:

Laplace transform and electric circuits: an interdisciplinary learning tool

The present work is addressed to high school students with scientific trend and it aims at supporting the pupils in learning two subjects: the solution of second order linear differential equations and the study of electric circuits. The two subjects are correlated because one of the presented methods to solve the differential equations uses the Laplace transform, and this is the best way to solve the integral-differential equations that are met in the study of the electric circuits. A package is created using a CAS as Mathematica. The package provides a theoretical framework and many exercises where the students are leaded step by step to solve the differential equations. Using this package equations describing electric circuits can be solved, and consequently physical quantities evolution (current intensity and voltage) can be obtained.

Burkhard Alpers, Germany:

Mathematical Application Projects for Mechanical Engineers - Concept, Guidelines and Examples

In the article, we present the concept of mathematical application projects as a means to enhance the capabilities of engineering students to use mathematics for solving problems in larger projects as well as to communicate and present mathematical content. As opposed to many case studies, we concentrate on stating criteria and project classes from which instructors can build instances (i.e. specific projects). The main goal of this paper is to facilitate the definition of new 'good' projects in a certain curricular setting.

Brigitta & Klaus Aspetsberger, Austria:

Cross curriculum teaching and experimenting in math & science courses using new technologies

Cross curriculum teaching and learning by experimenting are important objectives for future math & science courses. Various practical as well as mathematical skills of the students are trained by carrying out experiments, analysing the results and finally using functions for fitting data points obtained by the experiments. The students have to combine knowledge about different types of functions with knowledge about chemical and physical theorems. As an additional aspect, the students also have to take care of accuracy in experimenting for obtaining good results. The collection of large lists of experimental data is supported by the TI-CBL system. Mathematical experimenting, complicated computations and visualisation are supported by the graphical pocket calculator TI-92. We report about the experiences made with several groups of students at the age of 17 to 18 and about one group of students of high ability at the age of 14. Skills and abilities of the students for carrying out the experiments

Per Broman, Sweden:

Mathematical modelling with CABRI

I will show some examples how Cabri can be used in order to form functions out of geometrical constructions. For example: Properties and use of directrix lines and circles of the different conics. What if we inscribe a rectangle in an acute angled triangle? How can we use Cabri and Derive in combination? I also want to say a few words about TiM, a Nordic network and conference series on Technology in Mathematics education.

Giuliano Gargiulo*, C. D'Apice, R. Manzo Italy:

Mathematica and symbolic-numerical methods for solving first order ODEs

The use of information technology in addition to traditional lectures affords a means to develop student intuition and curiosity, reaching in the same time a deep knowledge of the subject of study. The aim of this work is to show the didactic use of a Computer Algebra System, as Mathematica 4.0/4.1, to illustrate and compare different symbolic-numerical methods for solving first order ordinary differential equations (ODEs). In particular, we apply, relate and compare the built-in functions of Mathematica, the method of integration by series, the Picard process and the linearization method in solving some first order ODEs. Moreover, numerical solutions are compared with symbolical ones at the various stages of computation. This includes use of numerical methods (internally adaptive) to look for and analyse singular points for maximal solutions.

Andre Heck, Netherlands:

Modelling Human Growth

Many a pupil at secondary school asks oneself questions like 'Am I too thick or too thin?', 'Am I short or tall in comparison with persons of my age?', and 'What adult length may I expect to reach?'. To answer such questions one needs real data. We have used the recent Dutch growth study to create learning material for pupils in upper general secondary education (age 15-16 yrs.) to carry out practical investigation tasks. A mathematical highlight is the ICP-model that models length for age within millimetres. It is used in the medical literature and yet consists of growth models that are studied at school, viz., exponential growth, quadratic growth, and logistic growth. We shall present the learning material and discuss the classroom experiences.

Andre Heck, Netherlands:

A Practical Investigation Task with the Computer at Secondary School: Bridges and Hanging Chains

Almost everywhere you can come across hanging chains and cables. Examples are necklaces, high-voltage cables, and cables that support a bridge surface. Do these cables all hang in the same mathematical shape? The first thought of many a pupil will be: this is a parabola, isn't it? In the computer learning environment Coach you can easily measure this on digital images. It will turn out that the parabolic shape quite often occurs with bridges, but that an ordinary chain does not hang as a parabola. Can you understand this? We shall show that a key idea for solving the problem can be discovered by measuring digital images and that it can be theoretically explained with basic physics afterwards. It also leads to a simple computer model of hanging chains. We shall discuss our learning material and classroom experiences, and in this way present an example of how ICT and context situations can contribute to the realisation of challenging mathematical investigation tasks.

Iavor Hristov, Bulgaria:

Model of deformations of fluid particles due to electric field

A mathematical model of finite deformations of compound drop containing another drop due to electric field are obtained. The fluids are homogenous, incompressible and Newtonian. The cases of concentric and eccentric particles are investigated together.

Duncan Lawson, UK:

A Discrete Introduction to Modelling

In applications focused mathematics degree courses there is an understandable desire to introduce students to the ideas and practice of mathematical modelling at an early stage. However, many mathematical models depend on a level of mathematical sophistication, such as differential equations, which most undergraduates do not have on entry to university. Furthermore, it is often the case with such models that specialist mathematical software is required for the solution of the model equations. This combination of sophisticated mathematics and unknown software can be a considerable deterrent to new undergraduates. This paper describes a way of introducing a range of key ideas within modelling, initially without using any new mathematical concepts, and relying on software which is both familiar and not specifically mathematical, namely the spreadsheet. A short description is given of a number of models which are easily explored with spreadsheets.

Pavel Prazak, Czech Rep.:

Software Maple and Matlab in teaching of ordinary differential equations

Matlab and Maple are the powerful interactive numerical computation programmes. They have efficient built in routines enabling wide variety of computations. They have also easy to use graphical commands to make visualisation available. In our contribution we will focus on selected possibility of using symbolic calculations, numerical and graphical methods for support and illustration of the subject of ordinary differential equations and outline various possibilities of visualisation of the solutions of these equations and show the samples of application of above mentioned problems

Mazen Shahin, USA:

Modelling with Difference equations using DERIVE

In this discussion we share the pedagogy and the methodology of modelling real life situations with difference equations using the computer algebra system Derive. This is a part of a reform finite mathematics course in which students explore and discover mathematical ideas on their own as they complete specially designed tasks whose emphasis on applications helps them see the relevance of the abstract concepts. We will emphasise the use of graphical and numerical techniques, rather than theoretical techniques, to investigate and analyse the behaviour of solutions of the difference equations. We will investigate interesting linear and non-linear models as well as systems of difference equations from such diverse disciplines as business, economics, life sciences and social sciences.

Mazen Shahin, USA:

Discrete Delayed Population Models with DERIVE

In this paper we show how Derive can be used efficiently in modelling and investigating discrete delayed population models. In particular we are interested in some population models represented by non-linear second order difference equations. We will explore the stability of the equilibrium values of the systems. We will apply an interesting method to control the chaos in a dynamical system represented by a first order non-linear difference equation. Some of the pedagogical issues related to the use of a CAS in modelling will be discussed.

Aynur Uysal, Turkey:

Importance of Mathematics in Engineering Education

Two different approaches have traditionally influenced mathematics teaching in engineering education. First one considers mathematics only as a tool for professional practice ,while the second one relates mathematics education with the development of the logical and critical thinking without which no tool will be efficient to the learners for their understanding and interpretation of the world. As well known , the second approach has been receiving a growing importance in the last years. In this paper , the second approach are described with detailed examples. A rich set of experience are also presented from the mathematics teaching in the Technical University of Istanbul.

Mithat Uysal, Turkey:

An Internet-Based Course Structure for Teaching Mathematics in an Engineering School

This study sets out to present a detailed and integrated approach for teaching mathematics using world wide web. Previous works and existing www-based teaching structures are first discussed. Then the concept of a course portal following the comprehensive and integrated approach are presented. Main modules of the portal, namely, the main page, multimedia page, courseware page, contact page and the search page are described. The ways to improve the portal are discussed. Some observations from the ITU model (Istanbul Technical University) are also presented.