Figure 1.

Formation of the mathematical modeling competence using digital technologies

S. Lukáč and J. Sekerák

University of Pavol Jozef Šafárik in Košice / Faculty of Science, Slovakia stanislav.lukac@upjs.sk, jozef.sekerak@upjs.sk

Abstract — In the OECD PISA documents, mathematical modeling competence together with the problem solving competence are listed as two of the eight basic mathematical competences. The mathematical modeling competence and the problem solving competence are closely related, because the use of different types of models that represent real objects and situations is the premise for development of the competence for solving real life problems. The transformation of existing problem into the mathematical language is the basis of mathematical modeling. In this article we describe the possibilities of using different types of models in solving of problems that enable the formation of the modeling competence. We use the MS Excel spreadsheet and dynamic geometric system Geogebra to model real situations.

Keywords — Modeling, Digital literacy, Education, Mathematics, Problem solving.

I. INTRODUCTION The development of the mathematical modeling

competence and problem solving competence is one of the main goals of the mathematics teaching. That is one of the reasons why these competences are listed as two of the eight basic mathematical competences in the OECD PISA documents. The development of the competence for solving real life problems is connected to the use of different types of models that represent real objects and situations. The understanding of the real situation is the basis for identification of objects and their mutual relations. While creating a mathematical model, we transfer objects and characteristics from the real world into the abstract world of mathematical terms and objects, which create the model. In mathematical modeling, the real objects are replaced by mathematical ones and their relations are described in the form of equations, inequations and functional dependencies.

The mathematization of the real problem is the base of the mathematical modeling. The process of the modeling provides the right opportunities for using the different types of representations of real objects and relations. Especially the graphic representations make the understanding of mathematical content in real problems easier. The results received from the solving of a mathematical model have to be interpreted from the perspective of solving of the original real problem. The recognition of their accuracy includes the reviewing of the suitability of the used model. As the model captures only certain aspects of the reality, it doesn't necessarily have to be suitable for using in different context or various real circumstances. The process of modeling can be comprehensibly characterized with the following scheme:

On the base of knowing the process of the mathematical modeling we suggest the following specification of the mathematical modeling competence: • focus on the bases of the modeled situation, • structuralize domains or situations to be modeled, • "mathematization" (the transfer of the "reality" into

the mathematical structures) - find quantitative or space relations and patterns of real situations,

• create mathematical models, • verify model from the perspective of the real

situation, • think, analyze and present the model (including its

limitations or specifications), • "demathematization" (the interpretation of the

mathematical models in the sense of "reality"), • observe and control the process of the modeling.

In the process of creating and solving of the mathematical model it is possible to use the potential of the digital technologies (DT) in many different ways. Effective mathematical tools for modeling are provided mostly by mathematic programs, like CAS. However in this article, we are focusing on the use of the spreadsheet and dynamic geometric system to create and explore various types of models. The spreadsheet disposes of numeric and graphic tools that simplify the modeling process and make it easier. Tables can present a simple form of mathematical model and their creation and modification in the spreadsheet environment develops the image of modeling as a dynamic process. By using the basic calculation operations and built-in tools we can create formulae that express relations between the data in the table cells. The advantage of the spreadsheet is the fact that the data in the table cells change simultaneously with the manipulation with the model and automatically show the results, the rightness of which, according to the solved problem, we can consequently verify.

Dynamic geometric systems offer a suitable environment for exploration of the graphic representations

265

ICETA 2013 • 11th IEEE International Conference on Emerging eLearning Technologies and Applications • October 24-25, 2013, Stary Smokovec, The High Tatras, Slovakia

978-1-4799-2162-1/13/$31.00 ©2013 IEEE

of data and real objects. Manipulation with free elements of the dynamic construction produces matching changes of dependent figures, which simplifies the examination of various specific situations. A simple change of input parameters and the dynamic character of the constructions make the examination of invariant features of geometric objects and relations between them easier.

II. USING OF VARIOUS TYPES OF MODELS TO FORM THE MODELING COMPETENCE

The usage of various mathematical tools and methods for modeling adds to the development of the ability to apply mathematical knowledge in a daily life. Proficiencies needed for successful application of mathematics in the form of mathematical modeling are markedly different than the ones we need to understand the terms, solve equations or proving mathematical sentences. The difficultness of the modeling activities is not in learning and understanding of corresponding mathematical findings, but in knowing where and how to use them. In the teaching process we generally don't try to develop more complicated models describing complex real situations. The mastery ad the strength of modeling can be in the beginning taught by using completely ordinary situations, the solving of which requires the mathematical knowledge of the basic school-taught mathematics.

Mathematical modeling can have various forms in the teaching process. If we add it to the teaching in the right way, we can support active learning based on experimenting and exploratory activity. Without the demand of completeness, we present three forms of modeling activities: • Empirical modeling that includes collecting,

organization and analysis of data collected through experimenting. For example with dynamic constructions, the students can detect the measure of geometric shapes while exploring and discovering of geometric relations.

• Changes of the input conditions when working with finished models, focused on the exploration of the characteristics and behaving of the models. Modeling activities in this case consist of the sequence of the steps focused on the identification of the rules describing the characteristics of the explored model.

• Gradual improvement and generalization of the model. After understanding the structure of the model and the relations between its basic elements, the students can continue to develop and improve the created models to reflect the chosen aspects of the explored reality even more precisely and closely.

According to the mathematical device and types of presentations of the real objects that were used to create the model, we can identify five types of models: • Arithmetic - presented for example in a form of

table of operation. • Geometric - constructed using the geometric shapes. • Graphic - illustrating graphs that represent

functional dependencies. • Algebraic-analytic - based on stating relations

between variables which represent.

• Combined. We can also classify models according to different

criteria. For example, if the quantities included in the model depend on the time, we can divide models to static and dynamic. With continuous dynamic models, the mathematical modeling is based on creating and solving of differential equations.

According to character of the relations between the quantities in the model, we can identify two basic types of models: • Deterministic - we can clearly mark out the state of

output quantities at any given moment, using their previous states and time continuance of input quantities. The dependencies between quantities in deterministic models are presented in a form of functional relations.

• Stochastic - output quantities and the state of the system can be estimated only as a probability. The values of defining quantities and their changes are the result of random processes. We can use the stochastic models in the DT environment in a form of computer simulations that make exploring of random events easier.

In the next part of the article we will provide a few proposals for using DT in creating and exploring various types of models in problem solving. The choice of the model type for problem solving is dependent on the character of the input information and characteristics of the modeled situation. For work with numeric representation of real object you can use the arithmetic model, which, in its simplified form, presents quantitative relations between quantities. The exploration of results of arithmetic operations can be provided by simple tables consisting of two lines or columns or by more complex tables too.

III. MODELING AND SOLVING PROBLEMS IN SCHOOL MATHEMATICS

Simple arithmetic models can be created while exploring the relations between quantities describing the uniform rectilinear motion. If the table shows that the length of the passed trajectory of a body in equal time intervals increases by equal measure, then the length of the passed trajectory in dependence on the time can be described as linear dependence. The slope of the lines describing this linear dependence, represent the speed of the body.

Students can compare the measure of the passed trajectory in dependence on the time for two cars moving with the same speed (see figure 2).

S. Lukáč and J. Sekerák • Formation of the Mathematical Modeling Competence using Digital Technologies

266

Figure 2.

Figure 3.

Figure 5.

Figure 4.

Given that the cars were at the time t = 0 in different distance from the origin of a coordinate system, the graph of the dependence consists of two different parallel lines. If the cars had different speed the lines would be intersecting. The given examples can be further expanded in the teaching of mathematics. As an example we will present the graphic model of a real situation, in which two cars move towards each other from different positions. In the simpler case we can state that the cars start moving from positions A and B at the same time. This case can stand as a specific case of more general model that will describe a situation in which the cars didn't start moving at the same time. The graph of dependence of the distance of cars from the position A on time is shown in the figure 3.

The delay of the car moving from the position B is represented as the time t’. If we wanted to model the situation, where the car from the position B starts sooner than the car from the position A we would enter a negative value of t’ in the table.

An interesting problem for investigation of dependences was used during the PISA testing in 2003. A pizza shop offers two round pizzas with the same thickness, but different size. The smaller one has a diameter of 30 cm and costs 30 zed. The bigger one has a diameter of 40 cm and costs 40 zed. Which pizza is more beneficial? From the assignment we know that if the diameter increased by 10 cm, the price would also

increase by the same value. The system of charging could point to the linear dependence between the price and the diameter of the pizza. To explore the benefits of the purchase in more detail, we will think about pizzas with more different diameters and we will create a graphic model to visualize the dependence of the price, area of pizza and the area of one piece of pizza for 1 zed on the diameter. To create this model we will use the dynamic geometric system Geogebra. The diameter of the pizza is given in dm and it's shown on the x-axe (see figure 4).

The curve for the diameter d bigger than 1 which increases the fastest shows the quadratic dependence of the area of the pizza on its diameter. The ray with the lesser slope shows the dependence of the area of one piece of the pizza for 1 zed on the diameter of the pizza. The bigger the pizza, the bigger piece one can get for 1 zed.

The basis of the geometric models is geometric shapes that represent the relations between the objects. As an example of the use of the geometric model we chose the following problem: In a plane there are given mutually different points S1, S2, X a Y. Find the rotation R1 with the midpoint S1 and rotation R2 with the midpoint S2 so that R2(R1(X)) = Y is valid.

All images of the point X in the rotation R1 around the point S1 lie on the circle k1 with the midpoint S1 and the radius |S1X|. Similarly all points that transform in the rotation R2 around the point S2 to the point Y lie on the circle k2 with the midpoint S2 and the radius |S2Y|. Intersections of these circles (see figure 5) are marked by points P1 a P2, which are images for X in the rotation R1 and the preimage for Y in the rotation R2.

267

ICETA 2013 • 11th IEEE International Conference on Emerging eLearning Technologies and Applications • October 24-25, 2013, Stary Smokovec, The High Tatras, Slovakia

Figure 6.

Figure 7.

One of the possibilities of the composition of the rotations is R1(S1, a1) ○ R2(S2, a2). In the case shown in the picture 5 there are 8 possibilities. Depending on the number of intersections of the circles k1 a k2 we can get four or zero solutions. In solving the problem we should ask the students not only to construct the model based on the awareness of the meaning of the definition of the rotation but also a complete discussion of the number of the solutions.

In the next part, we will give proposals for using the graphic model for solving statistical problems. In teaching of the statistics the students often don't understand the meaning of the basic statistic characteristics of central tendency. In some statistics sets there can be for example markedly different values of the arithmetic mean and median. This fact is caused by the existence of the extreme values of the statistics variable.

As an example, we could use the statistical processing of the information about the bonuses in a company with a small number of employees. Even one extremely high bonus for an employee will misrepresent the average bonus in the company. To model the described situation we can suitably use the graphic model in which we can change the value of the statistic variable and examine the influence on the value of the arithmetic mean and median.

For example we could model a situation where even more than 75% values of the statistics set will be under the value of the arithmetic mean. In such case the median has more validity about the character of the values of the statistics variable. This fact is the reason why, for example, when a sport performance is judged by multiple judges the lowest and the highest value are not considered in the final value.

The next proposal is focused on the exploration of the characteristics of the arithmetic mean if we change all values of the statistics variable. Students can learn about the linearity of average while processing the data about measured temperatures in various scales. The influence of adding the constant to all values of the statistics variable on the arithmetic mean can be explored while converting of temperatures from the Celsius scale to the Kelvin scale. While manipulating with the model the students should find out that the change of the arithmetic mean is given by the constant of 273,15.

More complex case is made by converting the temperatures into the Fahrenheit scale. The students could be given the following assignment: The statistics set consists of the temperatures in °C measured at 12:00 o'clock each day of the week. Convert the measured temperatures into the Fahrenheit scale and explore the relation between arithmetic means f the temperatures in separate scales. The conversion from the Celsius scale into the Fahrenheit scale is more difficult than in the previous case. It is determined by the relation: °F = 1,8.°C + 32.

To explore the arithmetic means we can use the graphic model created based on the values in the table using the Geogebra program (see figure 6).

After creating the tables with the temperature values in both scales the lists of points along with calculated values of arithmetic means were transferred into the graphic window. After the manipulation with the values the students should find the linearity of average. This

hypothesis can be tested by marking a value calculated on the basis of the transfer relation in the model. This finding can be justified by using the definition of the arithmetic mean.

For the end we picked an example of using the graphic model for solving a real life problem. The visualization and exploration of the modeled functional dependency allow us to experimentally guess the cheapest solution to the building of a road. The cart-road between places A and B consists of two perpendicular straight sections. The longer section starting at point A is 12,1 km long, the shorter is 2,8 km long. In the place where the road changes directions there is a detached house. The mayor of the town has decided to build an asphalt road between the points A and B. The experts estimated that the building of 1 km of the road in the course of the old cart-road would cost about 6000 EUR and building of 1 km of the road not along the course of the old one would cost about 7500 EUR. Find the cheapest possible option, given that the road is being built starting from the point A and can run either along the old cart-road or in any other way.

After calculating the boundary cases when the asphalt road would either run only along the old cart-road or just connect the points A and B, the students could come to modeling a more complex situation, when the asphalt road would follow the old road for a short section and then change directions to the point B. The graphic model of the described situation s presented in the figure 7.

The distance of the point of diversion on the cart-road going from the point A from the origin of the coordinate

S. Lukáč and J. Sekerák • Formation of the Mathematical Modeling Competence using Digital Technologies

268

system is presented by the variable c. By using the lengths of separate sections of the asphalt road and the prices we can calculate the full costs of the building of the road. By changing the value of the variable c the costs of the road are recalculated and the calculated costs are represented by the corresponding point in the coordinate system, in which the costs are applied to the y-axe. By transferring the X to A or to a point where the roads meet in the right angle we can model the above-mentioned special cases.

Using the Locus command we can display the graph of the dependence of the complete costs on the distance of the point X from the origin of the coordinate system. In the figure 7 there is adjusted position of the X that is used to estimate the minimal costs of the building of the road. After the creating of the graphic model the students could continue by constructing the algebraic model. The objective function representing the dependence of the total costs on the distance of the point X from the origin of the coordinate system can be presented in a form:

7500.8,26000).1,12()( 22 cccf ++−= By using the tools of the differential calculus the

students can calculate the minimum of the function f and compare the calculated value with the estimate of the minimal costs that they experimentally laid down using the graphic model. By solving the algebraic model we get the values of c = 56/15 and f(c) = 85200 EUR. The created geometric model can be developed further. Establishing new parameters representing the costs of the building of the asphalt road along the old cart-road and off it allows us to explore the position of X depending on the difference between the two costs values.

IV. CONCLUSION Modeling can be learned only by active practical

activity. To create stimuli for forming of the modeling competence it is advisable for students after gaining an experience with modeling specific situations to continually develop and generalize the model into the form usable for solving the explored problem. The students can suitably use the digital technologies to create such models. We gave a few proposals for using the programs MS Excel and Geogebra in this article.

In the teaching of mathematics there should be enough space saved for various types of modeling activities, which can be a part of working on a projects. In planning of modeling activities it is important to prepare ideas for working with various types of models, from arithmetic, geometric and graphic to algebraic-analytic. Sometimes when teaching mathematics we quickly pass to using the algebraic model, even if the students still lack the skills from using the easier models. However, we should realize that simple arithmetic and geometric models give the basis for the more complex and more sophisticated models used in geometry, algebra and analysis.

AUTHORS Stanislav Lukáč: is a graduate of Faculty of Science Pavol Jozef Šafárik University in Košice as the teacher of general subjects: mathematics, physics and informatics. He completed postgradual study at the UPJŠ in Košice in the field: Theory of mathematics teaching. Dissertation topic was: IT

from the viewpoint of the mathematics teacher needs. He was habilitated as associated professor at the UPJŠ in Košice in the field 9.1.8. Didactics of Mathematics in 2012. Currently works at the UPJŠ in Košice as an assistant lecturer.

Jozef Sekerák: is a graduate of Faculty of science Pavol Jozef Šafárik University in Košice, department of teaching of general subjects: mathematics and chemistry. In 2008, completed postgradual study at the University of

Pavol Jozef Šafárik in Košice Faculty of science in the field: Theory of mathematics education. Dissertation topic was: Diagnosing and developing of competences of students in mathematics education. Currently works at the Pavol Jozef Šafárik University in Košice, Center for Innovative Education (CIV) as Expert IT.

ACKNOWLEDGMENT This work was supported by the Slovak Research and

Development Agency under the contract No. APVV-0715-12 and by the VEGA grant 1/1331/12.

REFERENCES [1] W. Blum, P. L. Galbraith, H. W. Henn, M. Niss, „Modelling and

applications in mathematics education“, the 14th ICMI study, New ICMI Study Series Volume 10, Springer, 2007.

[2] D. Edwards, M. Hamson, „Guide to Mathematical Modelling“, Palgrave Mathematical Guides, 2001.

[3] J. Hanč, M. Kireš, and D. Šveda, „The digital literacy and key competencies as the cornerstones of primary and secondary education modernization”, 8th International Conference on Emerging eLearning Technologies and Applications, 2010, pp. 429–432.

[4] M. Homola, Z. Kubincová, J. Guniš, M. Cápay, M. Magdin, Ľ. Šnajder, Ďalšie vzdelávanie učiteľov základných škôl a stredných škôl v predmete informatika - Web, Multimédiá. 1. vydanie. 2010.

[5] Z. Kubáček, P. Černek, J. Žabka, et. al., Matematika a svet okolo nás, Zbierka úloh, FMFI UK Bratislava, 2008.

[6] S. Lukáč, „Programy typu CAS vo vyučovaní matematiky“, Matematika Fyzika Informatika, vol. 12, no.1, 2002, pp. 47-54.

[7] PISA - Matematika Úlohy 2003, Štátny pedagogický ústav, Bratislava 2004.

[8] J. Sekerák, „Kľúčové kompetencie v matematickom vzdelávaní“, Matematika Informatika Fyzika, Metodicko – pedagogické centrum Prešov a PF UPJŠ v Košiciach, no. 29, 2007, pp. 132 – 137.

[9] A. Ventress, „Digital Images + Interactive Software = Enjoyable, Real Mathematics Modeling“, Mathematics Teacher, vol. 101, No. 8, 2008.

[10] K. Žilková, „Školská matematika v prostredí IKT“, Univerzita Komenského Bratislava, 2009.

269

ICETA 2013 • 11th IEEE International Conference on Emerging eLearning Technologies and Applications • October 24-25, 2013, Stary Smokovec, The High Tatras, Slovakia