Mathematical Modeling and Technology in Teacher Education ...

18 Mathematical Modeling and Technology in Teacher Education – Visions and Reality Thomas Lingefjärd Department of Education/Mathematics, Gothenburg University, Sweden [email protected] Mikael Holmquist Department of Education/Mathematics, Gothenburg University, Sweden [email protected]

Teachers need to know both mathematics and computing tools, understanding the first in light of the second, if they are to teach the mathematical modeling being proposed today. At the University of Gothenburg, prospective teachers took part in a course in which the mathematical content was designed to give them insight into how they could solve extended problems using mathematical modeling by drawing on technology and their background in mathematics. Software for dynamic geometry, spreadsheets, and curve fitting was used. Besides helping to connect teaching and assessment in the course, the technology also offered students new possibilities of documenting, visualizing, and reporting their work. The change in instruction from traditional lecturing to a more problem-based open dialogue resulted in a transformation of authority in which students did not critically question results from technology, which suggested that issues of responsibility and authority need to be made explicit in instruction. The prospective teachers’ view of mathematics as an isolated subject with one correct answer appeared to change to an insight into mathematics as a living tool within all sciences. The way in which technology affects the teaching and learning of mathematics is a growing research field with many variables. We discuss several in this paper.

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Mathematical Modelling and Technology in Teacher Education

INTRODUCTION 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, our 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) It is obvious that teachers need to know both mathematics and computing tools if they are to teach the mathematical modeling being proposed. Their understanding of the subject matter and of the instructional environment surely interact to affect their classroom instruction. Laborde (1995) claims that designing a problem situation in a computer-based environment requires a new analysis of mathematical objects, operations, and didactical feedback. Prospective mathematics teachers therefore must understand mathematics in light of technology. It is no longer enough, if it ever was, for a teacher only to know how to define terms and execute algorithms. Teachers of today need an understanding of mathematics that allows them to produce and interpret technology-generated results, to develop and evaluate alternative solution paths, and to recognize and understand the mathematical limitations of particular technological tools. To exploit new technology in their daily practice, teachers must be well informed about its place and role in a didactical process (Balacheff & Kaput, 1996). The mathematics education community must address many questions, including how to prepare prospective secondary school mathematics teachers to function in a technology-enhanced environment.

BACKGROUND In a technology-enhanced course at the University of Gothenburg, prospective teachers were to become active learners, taking responsibility for their own learning. Students often think that if they fail to learn, this failure is the fault of the teacher, the syllabus, the textbook, the other students, or some other factor. In this course, each student was to assume the entire responsibility for successfully learning the course content. The students were in their second or third year of mathematical studies, preparing to become teachers of mathematics and natural science for Grades 4 to 9 or for Grades 10 to 12. All had taken courses in number theory, Euclidean geometry, linear algebra, real analysis, probability, and statistics. None had apparently had any collaborative experience in solving extended problems requiring approaches unlike those usually practiced in class or in using other mathematical literature than the normally required textbook to support their arguments. The mathematical content of the course was designed to give the students insight into how they could solve extended problems using mathematical modeling by drawing on technology and their background in mathematics. The software used was mainly The Geometer’s Sketchpad (Jackiw, 1995), PC Logo (Daumling, 1997), Excel (Microsoft,

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1997), and CurveExpert (Hyams, 1996). Graphing calculators were available. Theoretical views of mathematical didactics and of mathematical modeling in education were discussed in literature seminars, written assignments, and their work in the computer lab. The students worked in a computer lab with all the software described above and with access to the Internet. Their reports consisted partly of paper documents and partly of computer files on a disk. They could communicate with the instructors from home by electronic mail and fax. The computer lab was reserved for the exclusive use of students in the course each day between 8.00 and 17.00 for 10 weeks. The Geometer’s Sketchpad, Excel, and graphing calculators were introduced through an example or by solving a specific problem. As the instructors introduced graphing calculators, they also discussed mathematical modeling in terms of data analysis, regression analysis, and curve fitting. As well as literature-seminar discussions of the purpose and value of computers and graphing calculators in mathematics instruction, the students had homework assignments and a take-home final examination. The assignments served partly as instructional material and partly as preparation for the final examination. The aim of both the assessment and the take-home examination was to assess the students’ ability to model mathematical problems and to validate their models by reasoning grounded in their content knowledge. A fundamental idea in the assessment of the students was that assessment and teaching were integrated. This was visible in the selection of course material, in how the teaching was conducted, and in how more specific situations of assessment were evaluated. The use of technology as computer programs and graphing calculators naturally affect the evaluation situation and also what we mean by an assessment when technology is involved (Webb 1992). Beside the possibility to connect teaching and assessment, the technology also offers new possibilities of documentation, visualization, and reporting. Another, more hidden, objective is to involve the student in the process of evaluation in order to clarify the characteristics of a good mathematical performance. Evaluation may also be seen as a substantial part of the didactical contract that is being established between student and teacher (Brousseau 1997). Through this interplay the students identify the criteria for a good performance. Further, they learn what is regarded as an unsatisfactory, good, or very good performance. Since the students all are prospective mathematics teachers, it has been natural to discuss the developmental work of assessment in Sweden at the national level. Like in many countries, alternative assessment strategies are discussed and tested with the purpose of adopting a more qualitative perspective when assessing students’ mathematical performance. Sweden runs a national test in mathematics for all students in grade 9, and the criteria recommended for the evaluation of this national test are basically the same as the ones we use in our modeling course. •

The mathematical content – is it correct in terms of notation, figures, diagrams, and conclusions?

•

The report – is it written in a language and style that is structured, clear and distinct?

•

The problem – is it solved, generalized, explored, and investigated to the limit of all available resources?

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In a course of this type there are of course questions regarding how and in what situations the student builds her or his mathematical competence. Clarke (1996) formulates five questions connected to the didactical contract. Teachers and students need to understand: -

What assistance the students can reasonably expect from the teacher; What assistance the students can seek from each other; What level of explanation the teacher is obliged to provide; What questions the teacher can reasonably ask; What form of response will be considered satisfactory.

The use of technology in the course was dominated by two main ideas. First, and most important, the students were given the opportunity to see how technology may affect the teaching, learning, and assessment of mathematics in the classroom. Second, we assumed that the students could deepen their own understanding of geometry, mathematical modeling, and proof if they used technology when solving problems. These objectives challenged the students in their attitude to and knowledge of mathematics.

HOW TO SELECT MODELING PROBLEMS The 80 students in the program for mathematics and natural science teachers at the University of Gothenburg were given problems from different areas from which they would create a model during the course by developing the necessary mathematical theory. Areas such as biology, chemistry, economy, medicine, physics, and sport have been covered in this course during the last 4-5 years, but this paper will concentrate on just two problems given in the course in fall 1998. One of our objectives was to select problems both challenging and interesting. In the report Heeding the Call for Change, published by the Mathematical Association of America, a group of collegiate mathematics educators suggested that “the key [in selecting such problems] is to have the contexts relate to students’ interest, daily life, and likely work settings” (Steen, 1992, p. 100). Another objective was that the student should learn more mathematics in their collaborative work, when discussing and arguing for their models, when searching for literature on the Internet and in libraries, and when writing their reports. George Polya once defined a problem as follows: In general, a desire may or may not lead to a problem. If the desire brings to my mind immediately, without any difficulty, some obvious action that is likely to attain the desired object, there is no problem. If, however, no such action occurs to me, there is a problem. Thus, to have a problem means: to search consciously for some action appropriate to attain a clearly conceived, but not immediately attainable, aim. To solve a problem, means to find such action. (Polya, 1992, p. 117) The problems we chose also need to give the students opportunities to express what they have learned in the course and in previous courses. To work with this kind of problem in an examination should, as much as possible, focus more on qualitative reasoning and less on reproduction of facts and basic routines. The fact that the students are allowed to use

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How to Select Modelling Problems

graphing calculators and mathematical software in their examination further stresses the importance of selecting problems that are relevant in the presence of this aid. At the same time as the problem should remain non-trivial in the presence of the tools, the use of the technology should not be the essential and only success creating component in the performance of the student. A relevant problem should encourage the student to make different assumptions and use different strategies where technology can serve as an aid, never as a goal. Based on these assumptions and perspectives, the following two problems were selected as part of the examination during the fall 1998.

The Swim Problem Table 1 illustrates the time for winning gold medal in the women’s 100-meters freestyle at the Olympic games during the last century. Use the data given and create a mathematical model that can predict future results in general and also answer the question: What will be the winning time in the women’s freestyle in the Olympics year 2000? What kind of responses were we expecting from the students when trying to model this problem? How would they elaborate and generalize? What kind of extra resources would the students employ? First, we expected that some students would try to find additional data, like world championship results and the present world record for the women’s 100-meters freestyle. These findings could be used to compare to the values their model would give for, let’s say, 1998 or to give a better idea about the models mathematical qualities, e.g., a future record limit or horizontal asymptote. We also expected students to investigate several different mathematical models and that the majority would finally arrive at a logistic model.

Year 1912 1920 1924 1928 1932 1936 1948 1952 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996

Time(s) 82.2 73.6 72.4 71.0 66.8 65.9 66.3 66.8 62.0 61.2 59.5 60.0 58.59 55.65 54.79 55.92 54.93 54.64 54.5

Table 1 Students’ responses S1: I definitively think that my model needs a limit larger than 0. To choose a model that approaches 0 is for me as impossible as selecting a parabola. My choice finally becomes a mixture of two models, a rational and an exponential model:

y=

a + bx 1 + cx + dx 2

and

y = a⋅e

b

x

S2: In order to determine the likely winning time for year 2000 I used given data and constructed a suitable model. After going through all possible models in Excel, I decided that a power model (y = axb, b < 0) was the most reasonable. Especially when I looked 50 years ahead, the power model was most reasonable.

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The second student is not aware of the fact that Excel has done the selection of models for her and that she is now trying to convince herself that she is the one who is in control of the situation. Nevertheless the results for the winning time in Sydney 2000 are fairly equal for S1 and S2 − the difference in approach is huge. S1: Model 1

y=

a + bx 1 + cx + dx 2

where

a = 7.89⋅108

b = -437814.26 c = 10381.48 d = -5.59 yields the result 53.7755 seconds. Tool: CurveExpert Model 2

y = a⋅e

b

x

where

a = 0.0079728773 b = 17549.16 yields the result 51.5664 seconds. Tool: CurveExpert

S2: Model y = axb, b < 0

where

a = 4⋅1030 b = -8.7468

yields the result 53.5317 seconds. Tool: Excel The problem was given in the fall of 1998, but the final edition of this article is taking place in December, 2000. We therefore know that Inge de Brujin from the Netherlands won the gold medal in the women’s 100-meters freestyle at the Olympic games in Sydney with the new world record 53.77 seconds.

The Gas Problem (Edwards & Hamson, 1996, pp. 155-156) Recently it has become more and more interesting to use natural gas in order to heat homes. a)

Table 2 gives the weekly gas consumption (m3) and average outside temperature (°C) for a particular house before the installation of cavity wall insulation. Table 2

Temperature ( °C) Gas (m3)

-1 206.6

2 173.2

0 195.6

4 149.4

5 115.7

7 116.0

10 82.4

Construct the simplest possible model to describe the correlation between weekly gas consumption and outside temperature.

How to Select Modelling Problems

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b) Table 3 gives similar data for the same house after insulation. . Table 3 Temperature ( °C) Gas (m3)

-1 134.4

0 127.6

1 120.6

3 110.1

6 89.4

8 72.7

10 59.4

Construct the simplest possible model to describe the correlation between weekly gas consumption and outside temperature after insulation. c)

Table 4 gives monthly averages of the outside temperature at the location of this house from October to May. Table 4

Month °C

O 10.3

N 6.7

D 4.4

J 3.4

F 3.8

M 5.7

A 8.7

M 11.5

Find an appropriate model to describe the annual variation of the average temperature over the year. d) Write down an expression for the amount of gas saved in one year by having insulation and calculate a numerical answer for the amount of saved gas. Even though this problem is substantially more complicated, students might have the impression that it is easy to model with the aid of technology. Several students even used model-generated data to model again, a sort of seduction generated by the technology. The technology does not provide any base for consideration of relevance, and many students seem to have made an inapplicable model of the model. Students’ responses S1: Since the integral is easy to miscalculate I use Excel and CurveExpert to help me. I start to calculate the amount of saved gas per month with the compound formula. Then I copy the 12 values to CurveExpert and apply curve fitting. Just as with the swim problem, most of the students in this study expressed the opinion that a model in every part of the problem is exclusively chosen by the technology. The selection principle is dominated by the ranking of the correlation coefficient values. S2: In order to find a model I used the software CurveExpert. I picked a continuous and periodic function since that would also provide me with temperature values for the summer months. I decided that the function y = a + b ⋅ sin(ct + d) was most suitable. After further investigations I found that this function is not quite periodic.

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The student we quote here gives no explanation why this model is the best and takes no action to change the model despite the fact that she discovers that y is not periodic over the year. This methodology might be compared with the following excerpt from another student: S3: When I enter those values into CurveExpert and apply curve fitting of a model like y = a + b cos(cx + d), in which I define c = 2π/12 since this will force a periodicity equal to 12 months, then I receive the following model… The access to powerful technology allows students to generate quickly, and hopefully accurately, a broad variety of models. It is, however, important that students understand the mathematics they are working with, otherwise the technology may very well create a great deal of new “inaccuracies.” A test of this understanding occurs when the student examines the validity of the model. In this very problem, the validity of each model may be judged when its result is compared to the result from a rough estimate based upon the given values in the problem. Most students did not, however, use this for a validity check. S4: By a rough estimate based on the values from my diagram [from the fitted model] one can see that the area is about 1200 m3. This in turn suggests that the calculated value of 1159,38 m3 is a reasonable value of the amount of saved gas under a year. This student emphasizes a will to control the result of calculations first of all. Naturally this is an important part of the validity of the model, but it does not connect the reasoning back to the original situation and cause of the model situation.

CONCLUSIONS We are convinced that assignments of this form function as a tool for the shift away from facts and standard procedures to conceptual reasoning and also reveal qualities in the students’ opinions concerning concepts and mathematical structures. When the students are forced to explain and argue for their models, they disclose inaccuracies in a way that may very well be hidden otherwise. If we, for instance, ask students to calculate an integral expression, how do we know if eventual errors belong to the routine or the conceptual part of the solving process? With the technology of today the calculating part of solving integrals is only a question of pressing the right button or giving the correct command. The model process in the gas problem eventually led us to an integral expression. The analytical reasoning connected to the evaluation of this integral is something that several students consider as a task for the software, not for themselves. As a result they bypass the important question about the models validity for the whole year. Consequently, they might end up with model describing (and calculating) negative gas consumption for a couple of the summer months.

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S5: Also in this problem I employed CurveExpert to do the heavy work and just entered the given values… Then I take the integral of f(x) from 0 to 52 to get the saved amount of gas over one year. Students who use the possibility to graph their model function often give a more modulated reasoning about their selection of interval for the integration. See Figure 1. S6: When I see the graph and that the curve is “empty” during the warmest months of the summer, then I realize that it would be very stupid to use gas for heating when it is warmer outside than inside and I just exclude this interval from my calculation.

Au gu Se st pt em be r O ct ob er N ov em be D r ec em be r

Ju ly

ne Ju

M ay

Ap ril

ar s M

ry ru a

Fe b

Ja

nu a

ry

18 16 14 12 10 8 6 4 2 0

Figure 1: Illustration of the modeling of outside temperature from given data.

Of course students also use the graphs of the software and the data given as a result they trust and accept in blind faith. We return to the Swim Problem for an example. Se Figure 2 and the regression curve delivered by CurveExpert. S7: I found it remarkable that this model actually delivers really good prognoses up to year 2099. The result then will be 45.55 sec, which is high but I have to live with that. One alternative would be to correct the function by changing the constants, but that is not what I think this course is about, since you want us to use the computers to support us when solving the problems.

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As mentioned above, the rational function delivered by CurveExpert as “best fit” was S = 1.79574887 r = 0.97787379

a + bx 1 + cx + dx 2

where a = 7.89⋅108 b = -437814.26 c = 10381.48 d = -5.59

97 84. 43 79.

Y Axis ( unit s)

y=

89 73. 35 68. 81 62. 27 57. 73 51. 1903.6

1920.4

1937.2

1954.0

1970.8

1987.6

2004.4

X Axis (units)

Figure 2: Regression curve generated by CurveExpert These studies of how students handle modeling situations in the presence of technology, suggest that teachers at all levels need to be cautious about what students actually understand and interpret from the modeling process. We have demonstrated how easy it seems to be for the students to “get lost” and trust the technology far too much, thereby avoiding a necessary validity check. We consider it obvious that a clear focus on the validation part of mathematical modeling is more essential in the presence of technology than maybe ever before. Then the visions of the intended curriculum may become a reality in the mathematical classroom. Acknowledgements This work was initiated during the fall of 1998 and we are very grateful to all the students who willingly let us interview and question them about their assignment work. We are also very thankful to Professor Jeremy Kilpatrick, University of Georgia, Athens, Georgia, USA for reading and discussing the manuscript. REFERENCES Balacheff N, Kaput JJ (1996) ‘Computer-based learning environment in mathematics’ in Bishop AJ et al. (Eds) International handbook of mathematics education. Dordrecht, The Netherlands: Kluwer, 469-501. Brousseau G (1997) ‘Theory of didactical situations in mathematics. Didactigue des mathématiques 1970-1990’ in Balacheff N et al (Eds and Trans). Dordrecht, The Netherlands: Kluwer. Clarke D (1996) ‘Assessment’ in Bishop AJ et al (Eds) International handbook of mathematics education. Dordrecht, The Netherlands: Kluwer, 327-370.

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References Daumling M (1997) PC-Logo. Cambridge, MA: Harvard Associates. Hyams D (1996) CurveExpert: A curve fitting system for Windows. Clemson, SC: Clemson University. Edwards D and Hamson M (1996) Mathematical modeling skills. London: Macmillan. Jackiw N (1995) The Geometer’s Sketchpad. Berkeley, CA: Key Curriculum Press. Laborde C (1995) ‘Designing tasks for learning geometry in a computer-based environment’ in Burton L and Jaworski B (Eds) Technology in mathematics teaching: A bridge between teaching and learning. Lund, Sweden: Chartwell-Bratt, 35-67. Microsoft Co (1997) Microsoft Excel. Stockholm: Microsoft Corporation. National Council of Teachers of Mathematics (1998) Principles and standards for school mathematics: Discussion draft. Reston, VA: Author. Polya G (1992) Mathematical discovery. New York: Wiley. (First published in 1962). Skolverket (1997) Kommentar till grundskolans kursplan och betygskriterier i matematik. [Commentary to the Swedish comprehensive school curriculum]. Stockholm: Liber distribution. Steen LA (Ed) (1992) Heeding the call for change: Suggestions for curricular action. Washington, DC: Mathematical Association of America. Webb NL (1992) ‘Assessment of students’ knowledge of mathematics: Steps toward a theory’ in Grouws DA (Ed) Handbook of research on mathematics teaching and learning. New York: Macmillan Publishing Company, 661-683.