7th International Conference on the Teaching of Mathematical Modelling and Applications
Jordanstown 1995
Conference review by:-
Kirsteen Duncan, Paisley University, Scotland (dunc-ms0@paisley.ac.uk)
Judy Goldfinch, Napier University, Edinburgh, Scotland (j.goldfinch@napier.ac.uk)
Susan Jackman, Napier University, Edinburgh, Scotland (s.jackman@napier.ac.uk)
This report has been published in Zentralblatt fur Didaktik der Mathematik (ZDM) (International Reviews on Mathematical Education), 28 (2), pp 67 - 69, 1996.
1. Introduction
ICMTMA 7 was held at the University of Ulster's beautiful Jordanstown campus from 16th to 20th July, 1995. Over 130 university lecturers, education researchers and school teachers attended. The conference was truly international with delegates from the following countries:
England 28 Northern Ireland 17 Germany 13 USA 13 Denmark 10 Scotland 9 Portugal 6 Norway 4 Australia 4 Israel 3 Russia 3 Netherlands 3 Sweden 2 Puerto Rico 2 Japan 2 Finland 2 Eire 2 Spain 2 Poland 1 Austria 1 Switzerland 1 Italy 1 Greece 1 India 1
The Conference Team had put together an excellent series of thought provoking and inspiring plenary speakers, providing something for all interests, and such a wealth of shorter presentations and workshops that the main difficulty was deciding what one could least afford to miss! It was the unanimous feeling of the delegates that the conference team were to be congratulated on a first-class conference from every angle. [One session that no one missed was the modelling of a production process where the product is not sold until 25 years after it is made (a tour of a whiskey distillery!) and of a tessellation problem with giant hexagonal cylinders (a visit to the Giant's Causeway!).]
There were three main themes running through the conference: mathematical models used in teaching and industry, educational motives for using modelling, and methods of assessment. There were also several presentations on the use of computer packages in the teaching of modelling, the most popular ones seeming to be spreadsheets, Mathematica and Derive, with references also made to Matlab and Mathcad.
2. Plenary sessions
2.1 The Assessment Factor.
Professor Leone Burton of the University of Birmingham in England set the tone for the conference with a stimulating plenary talk challenging the conference to think about the wider role of assessment in mathematics teaching. Research has shown that students can perform well in traditional pencil and paper examinations while retaining fundamental misconceptions about key concepts. This is partly because we tend to assess those things that are easily assessed, and students can become quite sophisticated in spotting these areas to learn and ignoring everything else.
Professor Burton suggests that assessment is going on all the time in a classroom, both by the tutors and by the learners, and that all these forms of evaluation should be utilised routinely by the tutors to continually boost both their and the students' knowledge of the students' learning. This assessment should allow recognition of the heterogeneity of students and the way they see things, and provide a platform for improved further teaching and learning. She showed a video of a class who were presented with the sequence:
a row of two x symbols
x x
followed by two rows of three x symbols
x x x
x x x
followed by three rows of four x symbols
x x x x
x x x x
x x x x
and asked to explain, in words, the pattern they saw. Three different, but equally valid, ways of seeing this sequence were described by different students. One saw the pattern as a set of n by n squares with an extra column. One saw it as a set of squares with a row removed. And one as a rectangle where the rows were 1 star longer than the columns. Each visualisation would lead to the same general formula for the number of stars but by different routes: n2 + n, (n+1)2 - (n+1) and n(n+1). If the tutor assumes one particular way of seeing the pattern and develops a general formula following that route, some perfectly able students will be lost or at least disadvantaged. On the other hand, identifying and recognising the different perceptions can enhance the understanding for the whole class.
Professor Burton urged us to be flexible and to encourage variety in our students in order to ensure that there are equal opportunities to succeed; also, to seek multiplicities of information to provide feedback for both us and our students at different levels, in different ways, using different styles. Various ways of introducing this were presented, including question provoking activities before introducing a new topic, as well as reflective activities afterwards.
2.2 Mathematics, Modelling and the Way the Mind Works.
Dr. Susan Lamon from Kansas State University in the USA spoke on the developments in cognitive science in the United States and their implications for mathematics teaching and assessment. She looked at the relationship between mathematical modelling and the perspectives of cognitive science, particularly constructivism. Mathematical models were viewed as external indicators of student cognitive structures that were built and amplified through the tutor's interventions. Delegates were shown how analysis of students' models could inform tutors about their students' understanding, and how this could be used to guide the instructional process: a nice link here with Professor Burton's talk. She suggested that mathematical modelling as a teaching tool is compatible with cognitive model theory and is not just a fad.
2.3 Mathematics as Orientation in a Complex World.
Dr Hans-Wolfgang Henn, from the Staatliches Seminar fur Schulpadagogik (Gymnasien) in Germany, was particularly welcomed as a plenary speaker from the non-English speaking world. Dr Henn spoke eloquently on the problem of getting students to value mathematics. He feels that the most important goal of education in teaching mathematics is to instil a value of the possibilities of using mathematical methods to handle incoming problems from all different parts of life. He bemoaned the fact that mathematics was often still seen as an experience area separated from standard life, and re-emphasised Professor Burton's point that the form of assessment drives the teaching, informing the students and the world what is 'important' in the course. He urged teachers to be willing to ask more open-ended questions of their pupils, and to allow and expect more variety in their answers. He sees modelling as one way of making the connection between the mathematical world and the real world more transparent.
Dr. Henn went on to speak of the 'Mathematics in Practice' units now compulsory in each of the nine school years in his area of Germany (Baden-Wurrttemburg), and introduced some fascinating modelling examples that he finds motivates his students. These included a study of the German Tax formula (which involves quadratics), and a study of the Head Injury Criterion used to quantify the results of crash tests and assess the risk of head injury. He finds that these work particularly well with his most senior pupils. He gives his pupils the model and asks them to discuss it, make sense of it, and to answer various questions on it.
2.4 Modelling Patient Flow Through Hospitals.
Professor Sally McLean of the University of Ulster demonstrated some practical modelling work on the flow of geriatric patients through hospitals on which she is working with Professor Millard in London. Geriatric patients can be classified as 'short-stay' and 'long-stay' and the model she described showed how difference equations could be set up for the numbers in each group. These describe how patients are admitted, move between short- and long-stay states, and leave. The model is in use by London hospitals and being used to answer questions like, "What happens if we convert ten long-stay beds to short-stay?" or "How long will it take to empty a ward if we stop admissions to it?"
Professor McLean went on to show us various developments of the original model, such as converting the original deterministic model to a stochastic one, changing the admission behaviour (from instant replacement to a Poisson distribution), and including community care as an extra patient 'state'. Future plans include the introduction of other variables such as the age and sex of the patients, and the inclusion of other states such as nursing homes etc.
As she said at the start, what happens to geriatric patients is, or will be, of interest to most of us and her fascinating presentation was certainly of interest to all of us! The early model can be found written up in Harrison & Millard, Methods of Information in Medicine, 30, (1991) pp.221-228.
2.5 Computational Modelling of Industrial Processes: Demands of the Real World
Professor Mark Cross of the University of Greenwich in London continued the theme of real-world modelling but impressed on us how real-life modelling of industrial processes is complex and expensive requiring huge computing capacity.
Industrial modelling in practice has time and software constraints that the student modeller may not have met. Professor Cross discussed how to enable the progression from students with basic modelling skills to computational modellers who can handle problems of industrial significance. Modelling in an industrial context has to be in a team of people working together: the modeller's role is to orchestrate the route to the solution, acting as an interface between the engineer (or other client) and the computer scientist. He (or she) has to be able to control the flow of information: this requires both the ability to interrogate subject experts, and a knowledge of the techniques and limitations of the available computer software (or, at least, the ability to learn these quickly).
He used the example of modelling the casting of turbine blades which involves, simultaneously, the phenomena of fluid flow, solidification, heat transfer, stress and strain, together with the coupling between the fluid flow and the electromagnetic fields used to control the flow. He described the computational modelling required to map the mathematical equations into a flexible algorithmic solution framework appropriate for the available software tools.
Professor Cross is seeing a dramatic increase in the use of mathematical models in industry and emphasised the importance of training people who are going to be effective in this.
2.6 Curriculum Development and Assessment in Northern Ireland.
For the final plenary we were joined by several teachers from schools in Northern Ireland. Mrs Catherine Coxhead from the Northern Ireland Council for Curriculum, Examinations and Assessment described curriculum development in mathematics in Northern Ireland. She compared methods, standards and assessment with those in England., In addition, she highlighted the differences between the achievements of pupils at school and the expectations of lecturers in higher education (a problem not unique to Northern Ireland!)
3. Other Highlights.
Following the plenary sessions there was always a choice of between four and six activities ranging from 2-hour hands-on workshops to short half-hour presentations of a new model or experience. This report picks out the highlights as seen by its three authors, but there were countless others that we could not attend and so cannot report on. There was an excellent overall range of talks, some from those new to modelling, some from educationalists, and some from those very experienced but trying out some interesting new models or teaching methods.
Presentations on innovative models included modelling the flow of tourists arriving and departing from a city, the scheduling of frequent-service mini buses so that don't catch each other up, and the effect of toxic drugs on cancer tumours. In each case a very simple model was given which could be tackled by beginners, and gradually built up to something which could form part of a honours level course.
Several presentations looked at the problem of motivating students and attracting them to mathematics. It was agreed that there has always been a conflict between teaching the mathematics first and then looking for applications, versus looking at problems and through them discovering the underlying mathematics involved. It was also agree to be important to distinguish between learning a model and learning to model. One seminar that tried to tackle this issue was presented by Ted Hodgson of the USA. He asked his students to think of a problem that interested them and to have a brain storming session to decide what they wanted to learn about the topic. The class was then split into groups and each group concentrated on 'one piece of the puzzle'. The topic of interest turned out to be bungee jumping and the questions concerned involved basic mechanics. This was a class of high school students with no mechanics background. They looked up articles and studied sets of data. No mechanics laws were used, instead the students built their own 'model' which was sufficient for their purpose of obtaining sensible answers to the questions they had posed. This gave them a feel for the modelling process through their own research and experimentation, together with a tremendous sense of achievement.
John Stone of England described the use of DERIVE and MATHCAD for investigating chaos based on the logistic equation, the calculating power of these packages allowing students to see the patterns and experience the excitement of the problem without getting bogged down in the tedious solution of the equations. in a similar way, Knut Nissen from Denmark also showed how maths can be fun when he demonstrated the art of code cracking in a most entertaining workshop.
The use of spreadsheets such as EXCEL for allowing students to explore a problem was demonstrated by James Nicholson of Northern Ireland. Well designed accompanying worksheets were essential to allow students to develop their critical skills and mathematical common sense. He gave examples in linear programming problems, sampling and regression problems, and in a mortgage problem.
Assessment methods featured largely on the programme. The Conference Organiser, Ken Houston, found time to present a talk on the assessment of poster presentations. Each group of students were asked to research a particular problem or model. Their task was to write a set of lecture notes on their topic and circulate them round the rest of the class. Each group also gave an oral presentation on their topic and produced a poster to illustrate it. There had been class discussion beforehand about what makes a 'good' poster and students we required to come up with their own criteria. The lecturer assessed each poster, but they were also peer assessed and the results compared. It was felt that by using peer assessment students gained a better understanding of what makes for good poster communication. The design of poster presentations was felt to encourage clear and concise thinking and to introduce professional practice.
The development of 'transferable' or 'common' skills such as teamwork and communication was well demonstrated by David le Masurier and Sylvia Dunthorne (England) in a video of students working week-by-week on a group assignment. Participants in the workshop were asked to identify the skills being developed and to consider how they could be assessed. Other transferable skills such as report writing were discussed in a workshop on the use of critical reviews by Bryan Orman (England). His students were required to correct and criticise (or even rewrite) reports by other students. This also develops their skill at making sense of a model report or journal paper.
The FACETS and the QUEST packages for analysing assessment data were demonstrated in workshops by John Izard (Australia). These packages are very useful for anyone undertaking research into assessment methods, or, following the suggestions from the conference plenaries, experimenting with new assessment procedures and wanting to check their validity. They are also useful in today's very large classes where several assessors are involved in the assessment process and the question of different levels of leniency arises. This is particularly important with modelling assessments which can be more subjective than traditional examination questions.
4. Conclusion
The new era of peace in Northern Ireland has inspired its population with a great enthusiasm for showing visitors its many attractions and beautiful countryside, and the real hospitality of its people, from singing bus driver to university vice-chancellor, made the atmosphere in which this conference took place one of unparalleled welcome and warmth. This, combined with first-class accommodation and truly excellent organisation from the conference committee, made this conference one of the friendliest conferences that this reporting team have ever been to, ideal for the exchange of ideas, the sharing of experiences and the formation of new working relationships which are the main benefits of any good conference. The conference delegates themselves contributed to this of course: as one delegate, newly launched into the world of mathematics education, put it, "Everyone here is so sympathetic and supportive towards other people's work."
The conference ended with the announcement that ICTMA 8 would take place at Cairns in Australia. A promotional video of the sunny Queensland beaches was on show almost immediately! Delegates were asked for ideas for this conference, and the following suggestions were made (though not intended as criticisms of ICTMA 7): to schedule informal meetings of people just starting in the field and looking for collaborators; to allow more time for moving between rooms; fewer half-hour sessions; and there was disagreement over whether contributors should classify their presentations by level (Primary, Secondary, Tertiary or General) or by theme, or whether the cross-fertilisation was in fact beneficial.
It was generally agreed that the Ulster conference was going to be a hard act to follow, but everyone had high hopes that the enthusiasm and inspiration engendered would carry through to 1997.