The mathematics curriculum changes slowly, regardless if we speak about the ... The matrix of schooling in the society from which the students are drawn puts limits ... teach mathematical modeling and use modern technology when doing so.
Mathematical modeling in teacher education – necessity or unnecessarily. Thomas Lingefjärd Göteborg University, Sweden Background The mathematics curriculum changes slowly, regardless if we speak about the intended curriculum, the implemented curriculum, or the attained or realized curriculum. The curriculum in any mathematics classroom is powerfully constrained by the school culture. The matrix of schooling in the society from which the students are drawn puts limits on what is taught, how, and when. Sometimes it takes a driving force from outside school to change the mathematics curriculum, like for instance technology. The vivacious advances in technology that has taken place during the last decades has called both for appropriate changes in the mathematics curriculum as well as in the teaching of mathematics. Mathematical modeling is an example of how a subject can become possible − or at least easier − to teach early in school because of the technical evolution around us. Mathematical modeling is furthermore an interdisciplinary subject bringing together mathematics and many other fields, where it is possible to illustrate how mathematics is used in products and processes all around us. Nevertheless, it was not until the middle of 1990 when the term mathematical modeling started to appear explicitly in the Swedish curriculum. As a matter of fact, it became visible both in the curriculum for compulsory school and for the gymnasium about the same time. The importance of mathematical models has increased in the age of information society. Everything that happens inside a computer is the result of a mathematical model, as one example. It is important that this area is acknowledged in mathematics education. (Skolverket, 1997, p. 19) The school in its teaching of mathematics should aim to ensure that pupils: - develop their ability to design, fine-tune and use mathematical models, as well as critically assess the conditions, opportunities and limitations of different models, (English version of the Swedish curriculum for the gymnasium, 2000, p. 61.) - develop their knowledge of how mathematics is used in information technology, as well as how information technology can be used for solving problems in order to observe mathematical relationships, and to investigate mathematical models. (English version of the Swedish curriculum for the gymnasium, 2000, p. 61.) It is somewhat ironical that the topic of mathematical modeling has been in the focus of research in mathematics education for much longer than it has been mentioned explicitly in the curriculum. Many researchers and mathematics educators have addressed the benefits and obstacles of teaching, learning, and assessing mathematical modeling (Engel, 1968; Pollak, 1970; Mason, 1988; Blum & Niss, 1989; Zbiek,1993; De Lange, 1996; Noss & Hoyles, 1996; Lingefjärd, 2000; Ottesen, 2001; Lingefjärd, 2002a; Lingefjärd, 2002b, Doerr & English, 2003; Holmquist & Lingefjärd, 2003;).
Teacher Education During the nineties all education systems in Sweden have changed to a decentralized system with goal- and achievement oriented structures, and with national authorities as evaluating systems on different levels. The parliament decides which universities and university colleges that may exist. There are 34 at the moment. The government decides which degrees that are going to be established. The National Agency for Higher Education decides which universities that have the right to issue a certain degree. The universities have freedom to arrange e.g. the teacher education within the frame given through the regulation for the degree. At present time, teacher education programs in mathematics are given at 26 geographically different places. Some of these are within large universities, others within small community college universities or situated at local branches of large universities If exposure and training is a significant factor in teachers strategic decisions about what topic to emphasize and teach in school, then it might be expected that, compared to experience teachers for whom technology and mathematical modeling may represent an intrusion into established practice, newly graduating teachers would be better prepared and more likely to teach mathematical modeling and use modern technology when doing so. This conclusion is built on the hypothesis that today’s teacher education provides experiences, skills, and educational attitudes that are needed for the teachers in order to teach mathematical modeling to their future classes. But is this hypothesis right or wrong? In order to find out if courses in mathematical modeling actually was given and managed at the 26 different departments, I decided to undertake a small survey in the spring of 2003. Faculty members at departments of mathematics, departments of mathematics education, and at departments of education, as well as general administrators, were given the possibility to answer to the following questions: Do you, and your department, organize and arrange a course or courses in mathematical modeling for prospective teachers? • If Yes. What training in mathematics and mathematics educations is needed for a future mathematics teacher, so that she or he will be able to teach mathematical modeling with the help of modern technology? • If No. Since you, and your department, at present time do not teach the prospective teachers mathematical modeling, what are the major reasons not to do so? The survey was sent by e-mail to all the faculty members at the 26 different campuses who, according to their web pages, were involved in mathematics education. In addition I also mailed the same questions to administrators that I considered would have some insights over the content in their programs. Within two weeks I received about 200 answers, some short and concise like “No, we don’t”, others more elaborating on the subject, and some even sending me course materials, syllabus, and web page addresses where I could see how the course was organized and examined, and so forth. All universities, or their equivalents, responded and many of the responding faculty members were interested in participating in a discussion about the survey, its purpose and result. Nevertheless that the response rate was good or almost overwhelming, the results in it self was quite disappointing. Only 4 universities could answer directly yes: we do give a course in mathematical modeling, although it turned out that two of them offered the course in
mathematical modeling as an eligible or voluntary course that only some few students chose to follow. Two more universities were planning courses in mathematical modeling that should start in the fall of 2003. But the remaining 20 departments did not give any course in mathematical modeling for their prospective teachers. The main argument from the faculty members at these sites was that the curriculum was to crowded, and that the students first should study algebra; abstract algebra; calculus in one and several variables; discrete mathematics; geometry; linear algebra; logic reasoning, statistics, and so forth. The underlying argument often showed to be the lack of insight in mathematical modeling among the faculty staff, the feeling that mathematical modeling by nature is an interdisciplinary subject, and therefore not “real mathematics”. A third argument was that mathematical modeling often involves technology, which is considered to be “unfair” and “fuzzy” mathematics by many “hard mathematicians.” Even if the result of my survey was disappointing, it was not surprising. It is important to acknowledge that the opinion about mathematical modeling not being pure mathematical of course is true. Mathematical modeling is really not a precise body of mathematical knowledge in the same way that calculus or linear algebra are. Instead mathematical modeling is a process, and as most processes it have a variety of definitions. In general mathematical modeling can be seen as using a complex web of knowledge related to different branches of mathematics, in order to solve an applied problem by mathematical methods. Since mathematical modeling in general needs translating the abstracted solution provided by the mathematics model to concrete reality, it can also be seen as going outside the domain of mathematics. This is probably one major reason why many mathematics departments believe that mathematical modeling is less useful than other branches of mathematics in the preparation of teachers of mathematics. Ottesen (2001, pp. 337 – 338) gives a nice list of arguments why mathematical modeling could be seen as a way to learn more mathematics and to learn the mathematics one thinks one know even better. Blum and Niss (1989, p. 5) define five arguments termed: formative, critical, practical, cultural, and instrumental. The instrumental argument is similar to what Ottesen advocates for: assist students’ acquisition and understanding of mathematical concepts, notions, methods, results and topics, either to give a fuller body to them, or to provide motivation for the study of certain mathematical disciplines. (p. 5) In my communication with many different mathematicians and mathematics educators throughout the survey, I used another argument that might often be left out. Mathematical modeling can be used as a way to summarize and assess the mathematical competencies the students possess. Let me give a short example. The Catwalk problem Bob Speiser presented the so called catwalk problem to us during a doctoral seminar (spring 2002) at Göteborg University and we, my colleague Mikael Holmquist and myself, decided to use it in a modeling course we should teach in the fall of 2002. One aspect is that the problem actually tests how much calculus students actually know or understands. Speiser and his colleagues have tried the problem with college students as well as high school students. We decided that it could also be used with students who have taken several courses
in calculus (both one and several variables), and set up a study inspired by, and partly similar to the one Speiser and his colleagues reported on. Basic calculus is a way to study change and motion. In the catwalk problem, one challenge is to build connections between local rates of change and total changes, based on real-world data. The problem was originally designed to expose some of the complexity inherent in the use of mathematics to examine motion. Work on this problem by college calculus students, and by a study group of university faculty, has been reported in three papers by Speiser and Walter (1994a, 1994b, 1996). The problem is illustrated by a series of photographs that I do not have space enough to reproduce here. The photographs consist of 24 frames of a single cat, entitled Cat in Walk Changing to a Gallop. Eadweard Muybridge made the photos in 1880, by using 24 cameras that were activated successively at intervals of 0.031 second. They show the cat against a background grid, composed of lines spaced 5 centimeters apart. Every tenth line is darker. The 24 photographs show the cat over a total time of action of 0.71 second. We gave the students in our modeling class (fall 2002) copies of the photos, all the information described above, and asked them to construct one or two mathematical models describing how the cat moved over that time period. They were specifically asked to answer the following two questions: How fast is the cat moving in Frame 10? How fast is the cat moving in Frame 20? In the modeling class at Göteborg University, only 2 students out of fifteen in the class managed to reason analogical to the calculus they had studied. Most of the others created mathematical models (with the help of technology) that were unable to do a good description of the transforming from walk to gallop in frame 10. The two successful students plotted the movement versus time in a xy-diagram first and measured the change in slope in frame 10 and frame 24 by a ruler. By that basic approach, they knew a good approximation to the answer long before they started to construct a mathematical model for the catwalk. The calculus needed for this procedure is taught already at the gymnasium level. The catwalk problem proved to be an excellent tool to illuminate how different concepts of calculus connect and how important it is not to forget the origins. After the course, the lecturers had conversations with each of the students and the ones who were to technology oriented in their problem solving approach and sort of got lost in their mathematical models, had to admit that they totally had forgotten the most basic way of measuring the speed of change. With excerpts from this study which I do not have enough space to fully report on in this paper, and by using these arguments – that student’s conceptions and misconceptions in calculus, as well as their beliefs about calculus, can be lifted up to the surface by mathematical modeling exercises – I managed to convince a handful of the mathematicians I communicated with in the survey that it could be both useful and interesting to give a course on mathematical modeling in their teacher training program. The problem was of course that they in general did not have faculty members whom could both master the technology needed as well as create or find enough challenging and complex problems at the right level.
Conclusion To create, maintain, and sustain a course in mathematical modeling is indeed a difficult task. Even when the university teachers are interested in the subject, there are many different hurdles to pass. There is competition from other branches of mathematics, branches that are considered to be more “natural” in a teaching training program. There is a need of university teachers who masters appropriate use of modern technology. There is a substantial demand of skills needed for both mastering a variety of problems and subjects, as well as procedures for handling the teaching and assessment of the mathematical modeling process. Nevertheless, if one is able to overcome all these hurdles and difficulties along the way, there are rewards waiting along the way. One major benefit will come from a better understanding of what the students actually understands of the mathematics they have studied. That many students in the mathematical modeling classes I have studied seemed to have forgotten many topics, even those studied at the gymnasium level, is not so difficult to deal with as the fact that many of the students introduced and defended contradictory ideas despite their records of satisfactory mathematical achievement. These results challenge many of the foundations of how prospective mathematics teachers are taught and assessed. The findings I have done can be seen as supporting the position that prospective mathematics teachers for the secondary grades may not need more mathematics courses as much as they need different learning experiences. Such experiences should engage them in reasoning and in constructing mathematical models, in assessing the extent to which a mathematical argument is valid, and in developing, comparing, and evaluating alternative solution processes. It is clear that the progress of computing technology is far from ended. We can expect the calculator of tomorrow to do at least as much as and maybe more than what the computer software of today does. And courses in mathematical modeling are important for prospective mathematics teachers as well as for other students who study mathematics. It is obvious that teachers of courses on mathematical modeling must pay great attention to they way they set up, conduct, and grade their assessments. With technology, it is sometimes very easy, much too easy, for students to provide answers, and sometimes even the correct answer, without really understanding what the problem is about. Without assessment situations that make use of the technology and involve the students in critical thinking about what the technology offers in terms of possibilities and solutions, we may very well create students who are dependent on technology and not critical and insightful users of it. References Blum, W., & Niss, M. (1989). Mathematical problem solving, modelling, applications, and links to other subjects – state, trends and issues in mathematics instruction. In W. Blum, M. Niss, & I. Huntley (Eds.), Modelling, applications and applied problem solving: Teaching mathematics in a real context (pp. 1-21). London: Ellis Horwood. 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. Engel, A. (1968). Systematic use of applications in mathematics training. Educational Studies in Mathematics, 1, 202-221.
Holmquist, M., & Lingefjärd, T. (2003). Mathematical modeling in teacher education. In Q. Ye, W. Blum, S. K. Houston, & Q. Jiang (Eds.), Mathematical Modeling in Education and Culture ICTMA 10: Applications in Science and Technology (pp. 197-208). Horwood: Chichester. Lingefjärd, T. (2000). Mathematical modeling by prospective teachers. Electronically published doctoral dissertation, University of Georgia. Can be downloaded from http://ma-serv.did.gu.se/matematik/thomas.htm Lingefjärd, T., & Holmquist, M. (2001). Mathematical modeling and technology in teacher education - Visions and reality. In J. Matos, W. Blum, K. Houston, & S. Carreira (Eds.), Modelling and Mathematics Education ICTMA 9: Applications in Science and Technology (pp. 205-215). Horwood: Chichester. Lingefjärd, T. (2002). Mathematical modeling for preservice teachers: A problem from anesthesiology. The International Journal of Computers for Mathematical Learning 7(2), pp. 117–143. Lingefjärd, T. (2002). Teaching and assessing mathematical modeling. Teaching mathematics and its Applications 21(2), pp. 75-83. Mason, J., (1988). Modelling: What do we really want pupils to learn? In D. Pimm (Ed.), Mathematics, teachers and children (pp. 201-215). London: Hodder & Stoughton. Noss, R., & Hoyles, C. (1996). The visibility of meaning: Modeling the mathematics of banking. International Journal of Computers for Mathematical Learning, 1(1), 3-31. Ottesen, J. (2001). Do not ask what mathematics can do for modeling. In D. Holton (Ed.), The teaching and learning of mathematics at the university level. An ICMI study (pp. 335 – 346). Dordrecht, The Netherlands: Kluwer. Pollak, H. O. (1970). Applications of mathematics. In E. Begle (Ed.) The sixty-ninth yearbook of the National Society for the Study of Education (pp. 311-334). Chicago: University of Chicago Press. Skolverket. (1997). Kommentar till grundskolans kursplan och betygskriterier i matematik. [Commentary to the mathematics curriculum for the compulsory school] Stockholm: Liber. Speiser, R. & Walter C. (1994a). Constructing the Derivative in First-Semester Calculus, Proceedings, International Group for the Psychology of Mathematics Education, North American Chapter (PME-NA XVI) (pp. 116-122). Baton Rouge, LA. Speiser, R. & Walter C. (1994b). Catwalk: First-Semester Calculus. Journal of Mathematical Behavior, 13,135-152. Speiser, R. & Walter C. (1996). Second Catwalk: Narrative, Context, and Embodiment. Journal of Mathematical Behavior, 15, 351-371. Zbiek, R. M. (1998). Prospective teachers’ use of computing tools to develop and validate functions as mathematical models. Journal for Research in Mathematics Education, 29, 184-201.
