MATHEMATICAL REPRESENTATION OF REAL SYSTEMS: TWO MODELLING ENVIRONMENTS INVOLVING DIFFERENT LEARNING STRATEGIES C. Fazio, R. M. Sperandeo-Mineo, G.

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1 MATHEMATICAL REPRESENTATION OF REAL SYSTEMS: TWO MODELLING ENIRONMENTS INOLING DIFFERENT LEARNING STRATEGIES C. Fazio, R. M. Speraneo-Mineo, G. Tarantino GRIAF (Research Group on Teaching/Learning Physics) Department of Physical an Astronomical Sciences - University of Palermo (Italy) Abstract: Many research stuies, base on the representational nature of human knowlege an escribing how people construct their knowlege about the worl, focus on strategies of moelling, i.e.,: a facilitating process for the construction of aequate mental moels that will help the unerstaning of physical moels. We report examples where learning of physics follows the same steps than the learning of a new language: as it happens in the case of a new language, the learning of semantic precees the learning of syntax, that is, mathematical representations. Teaching strategies are implemente that take into account that the stuents alreay have the basic tools to generate mental moels, which are the same they use to interpret the worl: to make analogies, iealizations an abstractions. The reporte examples use two ifferent kins of moelling environments: the first one belonging to the category of the so-calle aggregate moelling engines (STELLA) an the secon one belonging to the category of object-base moelling languages (NET LOGO). Introuction Moels are a central topic in iscussion about contemporary science eucation with ebates focuse on incluing a moelling perspective in science curricula an on pragmatic strategies for esigning classroom activities that enable stuents to learn about science as a moelling eneavour. At the core of the ebates are questions connecte with the ifferent ways in which the term moel is use, an in particular: 1. the role of moels an moelling proceures in the process of knowlege construction; 2. the relationship between Physics Moels an Mathematical Moels. Cognitive scientists have ientifie mental moels as funamental tools of human thought use to structure our experience an thereby make it meaningful. Researches on cognitive processes (1, 2) an epistemological analysis of evolution of physics (3) have shown that a moelling approach can constitute a goo frame of reference in orer to esign teaching-learning situations both faitful to the iscipline an relevant for the learner. Moreover, some peagogical strategies have been evelope focusing on transfer of knowlege an competences (4, 5) an allowing pupils to better unerstan many content areas, since they enable them to see similarities an ifferences among apparently ifferent phenomena (6). However, many peagogical questions, concerning methos an strategies making the moelling approach teachable an/or learnable, are to be eepene. It is wiely accepte that the peagogical use of Information Technologies (such as Microcomputer Base Laboratory, multimeia, simulation programs etc.), in mathematics an physics may help pupils to overcome some ifficulties ealing with learning of scientific concepts an proceures. In this paper, we present some peagogical strategies an tools giving pupils the possibility to formalise a problem, not necessarily ealing with physics, following well efine steps tailore to ientify relevant elements of a given observation, efine the associate variables, preict relationships among them an check preictions using tools scaffoling the process of personal knowlege construction. Furthermore, we show how this strategy allows to overcome the well-known ifficulty that stuents encounter in the unerstaning of some mathematical formalism, such as ifferential equations, an their real meaning. The reporte examples are aime to stuents in year range an have been experimente in courses for pre-service teacher preparation. Difficulties in mathematical learning Physics teaching usually use the ifferential calculus in an operational an mechanical way: very soon, even in simple situations (7), stuents o not unerstan what is beeing one an why it is one. Most high school pupils as well as many unergrauate stuents have a inaequate comprehension of the mathematical symbology an, what is surely worse, they o not know when an why ifferential calculus becomes necessary an/or the strategy that it offers in orer to solve problems. A typical case is the unerstaning of the time erivative symbol, /, seen as an operator that, applie on a generic function y(t), gives its rate of change. In this sense / coul be seen as the rate of change operator. The application of the operator / to the variable position is very often well unerstoo by the majority of stuents: probably, because the concept of spee as an inicator of how much quickly a boy is moving belongs to the customary experience. On the contrary, the same cannot be sai for other physical quantities. So, if we apply /, for instance, to the temperature of a cooling boy, very selom stuents are able to grasp the meaning of the symbol, that is rate of variation of temperature. This simple example partially explains why the formalism of ifferential equations, although consiere the starting point in the mathematisation of physical situations, results to be baly unerstoo. In this context moelling activities coul help stuents to evelop competencies in translating ifferent escriptions of real worl into each other. In fact, moelling may be consiere as a translation proceure from verbal escription of real worl to other forms of representation, such as the mathematical or iconic ones. The use of simulation environments, whose components allow to easily visualize physical objects an processes, makes possible the construction of operative thinking forms. 121

2 Aggregate moelling engines: STELLA STELLA is a simulation environment evelope to generally represent the process of thinking (8); its main avantage is to eliminate the nee to manipulate symbols an to make complex mathematics unerstanable an easily manageable. STELLA represents the ynamic changing of a variable by a stock that can fill or rain through incoming or outcoming flows. Its main graphic interface makes available the basic elements assigne to the moel builing. These are: Stocks - containing amount of the variable changing in time, Flows -representing the action to fill or rain stocks, Converters -aitional objects use to complete the logic sentence, Connectors -use to link objects together an efine their relationships. In natural language a mental moel is generally associate to a verbal escription; STELLA allows the users to translate the verbal moel of the system uner scrutiny into a symbolic scheme, by representing carefully all the elements of the iea escribing its evolution. The program automatically generates a coe escribing the scheme built by users an simulates the time evolution of the system representing it through graphs, tables etc. Moelling becomes a translation from verbal escriptions to iconic representations; mathematical equations are, consequently translate from the specific iconic language of STELLA. Some examples with STELLA In orer to introuce the concept of rate (the amount of change of some quantity uring a time interval ivie by the length of the time interval) we usually begin by stuying what happens to the volume of water in a container in the situation represente in fig. 1. We can recognise (see fig.2) the stock use to represent the volume of water an the flows representing the action to fill or rain stocks at efine constant rates R in an R out. The system can be verbally escribe by a statement like: The volume of water is epenent both by the outflow an the inflow of water. In orer to quantify the relationships we can say some more, that is : Its rate of change is given by the algebraic sum of the two rates. The iconic moel of the process an the results for ifferent values of the ratio Inflow-rate/Outflow-rate are represente in fig. 2. Figure 1 Figure 2 The translation of the iconic moel to the mathematical formal moel is straight: = R in R out 1. Decay of pollution: the lake purification The next point is the introuction of the concept of variable rate. We approach the problem by consiering a lake, of constant volume, L, contaminate by some kin of substance. If the lake is well mixe, we can assume that the contaminant concentration is uniform. Assuming that at time t=0 we start to econtaminate the lake by making flow in clean water at rate R an allowing that at the same rate R water containing the polluter flows out, in orer to moel the process of the lake purification an calculate the time epenence of the contaminant volume, C (t), we can begin with a simple verbal escription of the phenomenon. A possible one can be the following: The volume of the contaminant must ecrease, ue to both the outflow of pollute water an the inflow of clean water. The clean water volume consequently increases until it reaches the (constant) volume of the lake. Now, how can we try to schematise the relevant variables an all the logic links between them influencing the process of lake purification? A possible STELLA iconic representation of the lake purification is shown in figure 3. The figure shows that the moel is built by using two stocks, three flows, two converters an some connectors. The relevant variable, the contaminant volume,, is 122

3 The relevant variable, the contaminant volume, C, is represente by the first stock an the other stock oes represent the volume of clean water in the lake, W. One of the converters inicates the lake volume, L, equal to C + W an constant by efinition. The other converter represents the rate of inflow of clean water into the lake an, at the same time, the rate of outflow of pollute water from the lake, equal because of the constant lake s volume. Figure 3 The flow marke Contaminant outflow rate represents the rate of ecrease of the contaminant volume. It must obviously epen from the lake s volume, L, the contaminant volume, C, itself an from the Inflow/Outflow rate an is linke to them by connectors. The two flows connecte to the water stock inicate the two ways the clean water volume has to vary; the Water inflow rate flow represents the constant inflow of clean water into the lake an is linke (or, better, equal) to the Inflow/Outflow rate converter; the Water outflow rate flow, on the other han, must epen from the Inflow/Outflow rate, the lake s volume an the clean water volume. Note that the links between The Contaminant volume stock an the Contaminant outflow rate flow an the Water volume stock an the Water outflow rate flow actually give two feeback loops, inicating that these flows must take into account the instantaneous value of the variable represente by the stocks. As a consequence the two rates Contaminant outflow rate an Water outflow rate are not constant, but epening from the instant values of the volumes. The equations lying behin the moel are: [ ( t) ] C [ ( t) ] = R ( t) W = L Their solutions shoul exhibit exponential time epenences. In figure 4 are reporte the results of the STELLA simulation, with L = 10 8 m 3, C (0) = 10 4 m 3, R = 0,5 m 3 /s. It is worth noting that the first equation contains all the necessary information to solve the problem if we make a more aequate choice of the variable escribing the evolution of phenomenon: i.e. the concentration of the contaminant in the lake, c(t), (the mass of contaminant per cubic meter of water). The equation behin the moel is very similar to the one for the contaminant volume: R L R C ( t) W Figure 4 [ c( t) ] = Rc( t) In this case the STELLA moel is simpler, mainly because we nee not to take into account the water volume time epenence (see figure 5). The solution shown in figure 6 is obtaine with L = 10 8 m 3, c(0) = 0.1gr/m 3 an R = 0,5 m 3 /s 2. Cooling a substance in an environment at constant temperature We now analyse an moel a classical physical problem showing behaviours similar to those analyse in the previous paragraph. We consier a container with a given mass of water at temperature T W set in an environment at a lower. constant temperature T E. The process of water-cooling is easily observe as well as the fact that the cooling 123

4 Figure 5 Figure 6 Figure 7 rate is not constant. The appropriate variable to escribe the phenomenon is the temperature ifference between water an the environment, T(t)= T W (t)-t E. It ecrease as a consequence of the heat flow from the system to the environment. If we call K the cooling coefficient, epening from the physical parameters of the system (liqui an container) 1, we can hypothesize a epenence of the cooling rate T/ from the instantaneous value of T. The STELLA iconic representation of water cooling moel is very similar to the lake s purification moel an consequently to its results (see figure 7). The NetLogo moelling environment NetLogo (9) is a programmable moelling environment for simulating natural an social phenomena. It is particularly well suite for moelling complex systems eveloping over time. Moellers can give instructions to hunres or thousans of inepenent "agents" all operating in parallel. This makes possible to explore the connection between the micro-level behaviour of iniviuals an the macro-level patterns that emerge from the interaction of many iniviuals. The agents can represent animals, cells, trees,.., that is iniviual elements interacting an consequently also molecules of a substance cooling in an environment at low temperature (see figure 8). NetLogo lets stuents open simulations an "play" with them, exploring their behaviour uner various conitions. It is also an "authoring tool" which enables stuents, teachers an curriculum evelopers to moify moels an/or create their own moels. NetLogo is simple enough that stuents an teachers can easily run simulations or even buil their own. An, it is avance enough to serve as a powerful tool for researchers in many fiels. Many real systems are usually so complex that a escription an interpretation at macroscopic level requires mathematical competencies usually not mastere by high school pupils. Their analysis at level of their constituents can, some times, simplify unerstaning by making pupils enable to construct causal explanations of a wie range of phenomena, an provie them with a framework that is useful across a wie range of isciplines. 1 From the Newton s cooling law, the cooling coefficient, K, is equal to hs/c, were h is the external conuctivity coefficient, S is the thermal contact surface between water an the environment an C is the thermal capacity of the sample of water. 124

5 The figure shows the NetLogo interface representing a given number of gas molecules, at a given temperature, containe in a box in thermal contact with the environment at low temperature. The cooling of the gas is represente using ifferent colours for molecules with ifferent energies. A graphical isplay of the gas temperature as a function of time is also represente. Data fittings of the cooling curve can be performe an its mathematical characteristics pointe out. Figure 8 Conclusion The iactic approach, here escribe, has been teste in courses for pre-service teacher preparation, in orer to make prospective teachers aware of the power of informatics tools in facilitating the construction of mental moels aequate to help the unerstaning of physical moels as well as the formalism of ifferential equations. Several parts of the moelling approach have also been teste in some high school classrooms. The STELLA approach offers real avantages in helping users to construct moels an to eliminate the nee to manipulate symbols, making complex mathematics more unerstanable. Stuent teachers, grauate in mathematics or physics, at the beginning encountere some ifficulties in switching the symbolic perspective: they have stuie ifferential equations in their egree courses an were familiar with their formalism. They began to appreciate the new representational formalism when they met more complex systems to moel: they unerstoo the power of the iconic formalism an showe to appreciate its peagogical impact. Some teachers, introuce to STELLA uring a in-service teacher training course, recognize the avantage of using it in the teaching of science, but showe the nee to have a well performe iactic guie introucing to the language. An introuctory guie is in preparation. NETLOGO uses a ifferent approach that allows users to moel systems irectly at the level of their iniviual constituent elements. Using it, stuents can learn to think about actions an interactions of iniviual objects an to escribe complex system properties as the result of iniviual actions. REFERENCES (1) JOHNSON-LAIRD P., Mental Moels. Cambrige University Press, 1983 (2) GUTIERREZ R., Mental Moels an the Fine Structure of Conceptual Change. Proceeings of the XIII GIREP International Conference, PHYTEB 2000, Barcelona, Aug. 27 Sept. 1, (3) NERSESSIAN N. J., Shoul Physicists Preach What They Practice? Thinking Physics for Teaching (Bernarini C. an icentini M. es.) Plenum Press, New York, 1995 (4) GILBERT J. K., BOULTER C. an RUTHERFORD M., Moels in explanations: part 1, horses for courses?. International Journal of Science Eucation, 20, 83-97, (5) SPERANDEO-MINEO R. M., I.MO.PHY. (Introuction to MOelling in PHYsics- Eucation): a Netcourse Supporting Teachers in Implementing Tools an Teaching Strategies, Proceeing of the GIREP Conference: Physics Theacher Eucation Beyon 2000 : Selecte papers, pp (Lonon: Elsevier Eitions). (6) FAZIO C, GIANGALANTI A., SPERANDEO R.M. an TARANTINO G., Moelling Phenomena in arious Experiential Fiels: the Framework of Negative an Positive Feeback Systems. Proceeings of the First GIREP Seminar: Developing Formal Thinking in Physics, Uine, Sept. 2 6, (7) LOPEZ-GAY R., MARTINEZ-TORREGROSA J., GRASS-MARTI A. an TORREGROSA G., On How to Best Introuce the Concept of Differential in Physics. Proceeings of the First GIREP Seminar: Developing Formal Thinking in Physics, Uine, Sept. 2 6, (8) (9) 125