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Mathematical modeling across cultural and national borders
Thomas Lingefjrd
University of Gothenburg, Sweden
Introduction
To share different educational materials, courses or even full programs across national or cultural boundaries is becoming more and more common around the world. One may even say that it is a clear trend at present time to share educational experiences in a global world. A number of factors are related to the growth in institutions offering courses to distant students in other countries, whether to one other institution half way around the world, or to dispersed students in many distant countries, or to their own citizens that are resident in other countries. Some factors are financial, other factors could be technological, and some might be mainly practical. Sometimes researchers and/or teachers enjoy the idea to share educational resources across borders for a variety of reasons. In the European project Developing Quality in Mathematics Educations II (DQME II) we decided to do it because the project has several common goals across national and cultural borders.
The research group that is part of the project and which supervises and support a specific strand of the DQME II project decided in their last meeting in Bratislava in November 2007 to use a mutual mathematical modeling problem in the local partner schools in the four project countries Denmark, Hungary, Sweden, and Romania during spring 2008. Even if these four countries all are European countries and not that far from each other in terms of geographical distance, they definitively differ from each others in values, traditions, norms, life style and culture. In school it is possible that the cultural differences in terms of teaching methods, views on learning theories, class room atmosphere and independency of teachers may very well be huge. So how should we convince our teachers in our respectively country to use this very specific modeling activity? What about the content of the modeling activity?
In Sweden there are arguments for teaching mathematical modeling in the curriculum, just as there most likely are in other curriculums:
The importance of mathematical models has increased in the society of today. Everything that takes place in a computer for instance, is a result of some sort of model. It is very important that this area is part of the mathematics we teach. (Skolverket, 1997, p. 19, my translation).
Connections between mathematics and the sciences often become apparent when students engage in the modeling of physical phenomena, such as finding the speed of light in water, determining proper doses of medicine, or optimizing locations of fire stations in the forests. (National Council of Teachers of Mathematics, 1998, pp. 327-328)
But that one country has national guidelines that support mathematical modeling is of course no guarantee that other countries have the same views of the value of mathematical modeling. When we started to discuss modeling activities in the DQME II project, it was quite obvious from start that Denmark and Sweden are rather close to each other in terms of views on modeling activities in the gymnasium. Not so surprisingly, since we are close in culture, language, and geographical location. Would that also be true for Hungary and Romania? A large part of the Romanian populations speak Hungarian, I knew that. But what about other significant issues with regard to teaching mathematics? When we actually started to work with this project, it showed that even if one might have a theoretical view of the benefit and procedure of international comparisons and collaboration, there are always practical issues that are hard to foresee and predict.
One of the outcomes for the project meeting in Bratislava in November 2007 was a set of descriptors for how a good mathematical modeling problem should look like. It should be motivating, thereby maybe responding to a challenge, and being authentic of some sort, and so on. It should also ask for good mathematical thinking in the sense that the students should need to make abstractions, assign variables, make assumptions, formalizing and analyzing the math problem, using data and make approximations and estimations. Finally we considered that a good problem should also call for validation of the solution both mathematically and in the real world.
So far, there is of course no agreement in the community of mathematics and mathematics education on what mathematical thinking really is. In The Nature of Mathematical Thinking (Sternberg & Ben-Zeev, 1996), Sternberg wrote in the culminating chapter, In reading through the chapters of this volume, it becomes clear that there is no consensus on what mathematical thinking is, nor even on the abilities or predispositions that underlie it (p. 303).
Schoenfeld (1992) used the assembled word mathematization in his endeavor to describe mathematical thinking:
Learning to think mathematically means (a) developing a mathematical point of view -- valuing the processes of mathematization and abstraction and having the predilection to apply them, and (b) developing competence with the tools of the trade, and using those tools in the service of the goal of understanding structure -- mathematical sense-making. (p. 335)
The process of mathematization might be seen as stoutly connected to the situation in which we are applying mathematics in a real-world problem. When working on such a problem, we are dealing with a collection of objects, relations between these objects, and structures belonging to the area that we are studying. Next we have to translate these into mathematical objects, relations, and structures as representations for the original ones. This entire process, mathematical modeling, leading from the real problem situation to the mathematical model, might be seen as mathematization. But de Lange (1996) gives the term mathematization a more restricted meaning by defining it as the translation part of the modeling process. When transferring a problem, especially an investigational problem, to a mathematically stated problem, one engages in some important activities. These might include formulating and visualizing the problem in different ways, discovering relations, and transferring the investigational problem to a known mathematical model. The next step is to attack and treat the mathematical problem with mathematical tools in activities like representing a relation in a formula, refining and adjusting models, combining and integrating models, and generalizing. The students conceptions of the mathematics involved in the modeling process are essential and are revised through their activities and their reflections on their actions.
Mathematization always goes together with reflection. This reflection must take place in all phases of mathematization. The students must reflect on their personal processes of mathematization, discuss their activities with other students, must evaluate the products of their mathematization, and interpret the result. (de Lange, 1996, p. 69)
After some negotiation the Danish and Swedish members of the DQME II research group decided to use a mathematical modeling problem from medicine, a problem concerning the process of Asthma medication coming directly from a medical doctor associated to the project (see appendix). It is a fact that many students seem to appreciate modeling problems from the area of medicine as more real and more interesting than others (Lingefjrd 2006, p. 111).
But what would our Hungarian and Romanian colleagues think? What would the gymnasium (upper secondary) teachers in respectively country think about the problem? Would they approve to the level of mathematical thinking? Would they approve to the directions of inquiry? Mutual teaching experiments across cultural and national borders require a lot of work and concern some very delicate issues. It is a fact that no one in the research group knows much about the actual teaching that takes place in Danish, Hungarian, Romanian or Swedish classrooms a part from their own countries. The research group has an intention to film the experiments in all the different classrooms. Would that become an obstacle or an opportunity to run the project even better? What about the actual teaching and learning that will take place? How will that be assessed and evaluated in the different schools. Would the modeling project be part of the regular mathematical training for the students in the different countries or would it be an extra and not so serious experiment?
Research questions:
What can pass as a good mathematical modeling project in four different countries?
How will mathematics teachers in upper secondary schools in four different countries actually teach a mathematical modeling project?
What will the students in these four countries learn and how will it be assessed?
Procedures:
Since the Asthma project problem in a way was invented or at least had its origin in Denmark, it came as no surprise that the Danish teachers were the first to teach, document and assess how the Danish students worked with the Asthma project. The teachers in all the other countries had harder to make the project fit it with the more expected teaching assignments in mathematics. When teachers start to plan for the teaching activities that need to be associated with a project like this, they suddenly realize that students may very well be of the same age, but are they similar in mathematical background and training?
The Asthma project requires quite much of the students in order to accomplish what is expected. Evidently this was also acknowledged by the teachers in different countries. So teachers started to think about how students in other countries are trained in different area of mathematics. For some teachers in the project it could very well be seen as natural to use technology, for others it might not. One teacher wrote to me:
Dear Thomas. I have some questions concerning our participation in the Asthma project.
1. Can we use computers during the activities (usually we don't use any scientific calculators or computers, so this can be an extra issue)?
2. If we use computers (and I don't see how we could avoid this) can we use special software like Matlab, Mathematica, Maple or at least Excel for data fitting and parameter estimation?
3. How old are the children you work with at this activity in your country? Do they have any mathematical background regarding derivatives, differential equations, regression analysis or is the aim of this project also to develop some skills in handling these areas as well?
The possibility to document students activities and working conditions in the schools in the participating countries is crucial for the DQME II project. One of my main concerns was therefore to make sure that the modeling activities were filmed by video camera. But how can one do that from Sweden when three out of four of the schools in the project are in other countries and the only communication channel is through e-mail?
I hope to be able to report on a successful collaboration regrinding modeling over national borders in Mexico; right now I do not yet have the results that support that expectation.
References
De Lange, J. (1996). Using and applying mathematics in education. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education (pp. 49-97). Dordrecht: Kluwer.
Lingefjrd, T (2006). Faces of mathematical modeling. Zentralblatt fr Didaktik der Mathematik,. Volume 38, Number 2, pp. 96-112.
Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334-370). New York: Macmillan.
Skolverket. (2000) Kursplaner och betygskriterier fr kurser i mnet matematik i gymnasieskolan. [The Swedish secondary school curriculum and syllabus for mathematics]. Electronically published document. Stockholm: Skolverket. Can be downloaded from HYPERLINK "http://www.skolverket.se/kursplaner/gymnasieskola/index.html" http://www.skolverket.se/kursplaner/gymnasieskola/index.html
Sternberg, R. J. (1996). What is mathematical thinking? In R. J. Sternberg & T. Ben-Zeev (Eds.), The nature of mathematical thinking (pp. 303-318). Mahwah, NJ: Lawrence Erlbaum Associates.
Appendix
Humans that suffer from Asthma are often treated with the medicine theophylline. Theophylline, also known as dimethylxanthine, is a methylxanthine drug used in therapy for respiratory diseases such as asthma under a variety of brand names. Patients are often treated with an equally large dose, D mg, over equally large time intervals, T hours. A doctor has measured how the concentration of theophylline in the blood of one patient varies after the patient has being injected with a dose of 60 mg. The results from this measurement are the basic data for the project.
Time
hoursConcentration
mg/liter (mg/L)0
2
4
6
8
10
12
14
16
1810,0
7,0
5,0
3,5
2,5
1,9
1,3
0,9
0,6
0,5
The students will be asked to construct a mathematical model for the whole situation and to write a report to the doctor that addresses the following questions:
How will the concentration of theophylline in the blood decrease over time?
How can we plan a continuously medication schema with a fixed dose D over a fixed time interval T, so that the concentration after a couple of injections is in the interval 5-15 mg/L?
How can we plan a continuously medication schema with a start dose and thereafter a fixed dose D over fixed time interval T, so that the concentration directly will be within the interval 5-15 mg/L?
What considerations must be taken into account before one use this medication plan for a patient?
Lingefjrd Modeling across national borders PAGE 1
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