Pdf mathematical modelling

A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships between the variables. Variables may be of many types; real or integer numbers, boolean values or strings, for example. The actual model is the set of functions that describe the relations between the different variables. In business and engineering, mathematical models may be used to maximize a certain output.

The system under consideration will require certain inputs. The system relating inputs to outputs depends on other variables too: decision variables, state variables, exogenous variables, and random variables. Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as parameters or constants. The variables are not independent of each other as the state variables are dependent on the decision, input, random, and exogenous variables.

Furthermore, the output variables are dependent on the state of the system represented by the state variables. Objectives and constraints of the system and its users can be represented as functions of the output variables or state variables. The objective functions will depend on the perspective of the model's user.

Depending on the context, an objective function is also known as an index of performance , as it is some measure of interest to the user. Although there is no limit to the number of objective functions and constraints a model can have, using or optimizing the model becomes more involved computationally as the number increases.

For example, economists often apply linear algebra when using input-output models. Complicated mathematical models that have many variables may be consolidated by use of vectors where one symbol represents several variables. Mathematical modeling problems are often classified into black box or white box models, according to how much a priori information on the system is available.

A black-box model is a system of which there is no a priori information available. A white-box model also called glass box or clear box is a system where all necessary information is available.

Practically all systems are somewhere between the black-box and white-box models, so this concept is useful only as an intuitive guide for deciding which approach to take. Usually it is preferable to use as much a priori information as possible to make the model more accurate.

Therefore, the white-box models are usually considered easier, because if you have used the information correctly, then the model will behave correctly. Often the a priori information comes in forms of knowing the type of functions relating different variables. For example, if we make a model of how a medicine works in a human system, we know that usually the amount of medicine in the blood is an exponentially decaying function. But we are still left with several unknown parameters; how rapidly does the medicine amount decay, and what is the initial amount of medicine in blood?

This example is therefore not a completely white-box model. These parameters have to be estimated through some means before one can use the model. In black-box models one tries to estimate both the functional form of relations between variables and the numerical parameters in those functions.

Using a priori information we could end up, for example, with a set of functions that probably could describe the system adequately.

If there is no a priori information we would try to use functions as general as possible to cover all different models. An often used approach for black-box models are neural networks which usually do not make assumptions about incoming data. Alternatively the NARMAX Nonlinear AutoRegressive Moving Average model with eXogenous inputs algorithms which were developed as part of nonlinear system identification [3] can be used to select the model terms, determine the model structure, and estimate the unknown parameters in the presence of correlated and nonlinear noise.

The advantage of NARMAX models compared to neural networks is that NARMAX produces models that can be written down and related to the underlying process, whereas neural networks produce an approximation that is opaque. Sometimes it is useful to incorporate subjective information into a mathematical model.

This can be done based on intuition, experience, or expert opinion, or based on convenience of mathematical form. Bayesian statistics provides a theoretical framework for incorporating such subjectivity into a rigorous analysis: we specify a prior probability distribution which can be subjective , and then update this distribution based on empirical data. An example of when such approach would be necessary is a situation in which an experimenter bends a coin slightly and tosses it once, recording whether it comes up heads, and is then given the task of predicting the probability that the next flip comes up heads.

After bending the coin, the true probability that the coin will come up heads is unknown; so the experimenter would need to make a decision perhaps by looking at the shape of the coin about what prior distribution to use.

Incorporation of such subjective information might be important to get an accurate estimate of the probability. In general, model complexity involves a trade-off between simplicity and accuracy of the model. Occam's razor is a principle particularly relevant to modeling, its essential idea being that among models with roughly equal predictive power, the simplest one is the most desirable. While added complexity usually improves the realism of a model, it can make the model difficult to understand and analyze, and can also pose computational problems, including numerical instability.

Thomas Kuhn argues that as science progresses, explanations tend to become more complex before a paradigm shift offers radical simplification. For example, when modeling the flight of an aircraft, we could embed each mechanical part of the aircraft into our model and would thus acquire an almost white-box model of the system. However, the computational cost of adding such a huge amount of detail would effectively inhibit the usage of such a model. Additionally, the uncertainty would increase due to an overly complex system, because each separate part induces some amount of variance into the model.

It is therefore usually appropriate to make some approximations to reduce the model to a sensible size. Engineers often can accept some approximations in order to get a more robust and simple model. For example, Newton'sclassical mechanics is an approximated model of the real world. Still, Newton's model is quite sufficient for most ordinary-life situations, that is, as long as particle speeds are well below the speed of light, and we study macro-particles only.

Any model which is not pure white-box contains some parameters that can be used to fit the model to the system it is intended to describe. If the modeling is done by an artificial neural network or other machine learning, the optimization of parameters is called training , while the optimization of model hyperparameters is called tuning and often uses cross-validation. A crucial part of the modeling process is the evaluation of whether or not a given mathematical model describes a system accurately.

This question can be difficult to answer as it involves several different types of evaluation. Usually the easiest part of model evaluation is checking whether a model fits experimental measurements or other empirical data. In models with parameters, a common approach to test this fit is to split the data into two disjoint subsets: training data and verification data.

The training data are used to estimate the model parameters. An accurate model will closely match the verification data even though these data were not used to set the model's parameters. This practice is referred to as cross-validation in statistics. Defining a metric to measure distances between observed and predicted data is a useful tool of assessing model fit. In statistics, decision theory, and some economic models, a loss function plays a similar role.

While it is rather straightforward to test the appropriateness of parameters, it can be more difficult to test the validity of the general mathematical form of a model. In general, more mathematical tools have been developed to test the fit of statistical models than models involving differential equations.

Tools from non-parametric statistics can sometimes be used to evaluate how well the data fit a known distribution or to come up with a general model that makes only minimal assumptions about the model's mathematical form. Assessing the scope of a model, that is, determining what situations the model is applicable to, can be less straightforward. If the model was constructed based on a set of data, one must determine for which systems or situations the known data is a 'typical' set of data.

The question of whether the model describes well the properties of the system between data points is called interpolation, and the same question for events or data points outside the observed data is called extrapolation. As an example of the typical limitations of the scope of a model, in evaluating Newtonian classical mechanics, we can note that Newton made his measurements without advanced equipment, so he could not measure properties of particles travelling at speeds close to the speed of light.

Likewise, he did not measure the movements of molecules and other small particles, but macro particles only. It is then not surprising that his model does not extrapolate well into these domains, even though his model is quite sufficient for ordinary life physics.

Many types of modeling implicitly involve claims about causality. This is usually but not always true of models involving differential equations. As the purpose of modeling is to increase our understanding of the world, the validity of a model rests not only on its fit to empirical observations, but also on its ability to extrapolate to situations or data beyond those originally described in the model.

Besides, theoretical structure self-consistent. Value system of the numbers is an example with regard to the current state of educational programs in for that.

Children conduct activities regarding the qualities of Turkey has been discussed. Mathematical modeling and Instructional Approaches Later, they learn how they will manipulate algebraic Using Mathematical Modeling expressions. Thus, they reach reality, which is the next step. According to Boaler [2], mathematical modelling theory Mathematical terms may have many different meanings.

For focuses on individuals and suggests that knowledge is example, a fraction can be interpreted as a part of a whole, a created as a result of a series of interactions between people ratio between two quantities or division of one number to and the world. This situation requires examination of another. This corresponds to a decimal number, fraction or students' situations with different practices. For this reason, it percentage forms [26].

Mathematical expression of a problem and the of any problem situation into a mathematical model. Understanding the situation problem mathematizing, mathematical working and 2. Sometimes the problem situation 3. Mathematizing that is given is nothing else than a pre-structured 4. Mathematical working mathematical problem or a mathematical problem that is full 5.

Interpretation of real life. This is the classic "word problem" situation that 6. Verification generally occurs in schools. Using mathematics to solve 7. Modelling Cycle [11] Firstly, the problem situation should be understood by the modeling conducted by students in the classroom student, that is to say, "the situation model" is formed.

After environment. The students, who work in small groups during that, the situation is structured and turned into "a real model". During the mathematizing process, which situations that are given. These activities are developed corresponds to the third step, the student turns "the real within the framework of the themes the children are model" into a "mathematical model".

The student conducts interested in and they are organized in a way that will "a mathematical work calculation, solving inequalities encourage children to study and clarify the problem situation.

Real results in the daily life are interpreted and verified models they developed to their friends using various during the fifth step. Finally the possible solutions of the illustration systems such as written symbols, verbal reports, problem are presented and suggestions are made regarding diagrams on paper or pictures [9]. As an example of modeling activities, "big foot problem", According to Galbraith , there are 3 different which is a problem situation regarding real life and which is teaching approaches in modelling.

These are as follows: adapted by Lesh and Doerr a to the second level of 1 "General application approach" focuses on a certain primary education. This modeling activity is organized as application. Generally, the teacher introduces the model follows: and the students use the model in a controlled manner.

This approach is mostly used in secondary schools and Model Example it includes 4th calculation, solving inequalities etc. According to Guinness Book of the World Records, it is the 2 "Structure modeling approach" uses the real life largest shoe of the world with the width of 2. What is the height of the giant that could process form the 1st stage the problem situation should actually wear these shoes? The teacher makes an important effort to make mathematical model used in the 3rd stage mathematizing process.

In this approach, students work with the limited help of the teacher about the problem that is given because teacher does not have to control the students. This approach is not used widely [25].

This since it requires a translation between mathematics and the situation considers not only cognitive process and student reality. In both cases, abstraction is the basic role society, daily life and other scientific disciplines [4]. The modelling activity developed by four high school mathematics teachers were organized as follows in the study 3.

His parents go to work early in modeling in mathematics teaching and learning. This the morning and come back late in the evening. For this importance is based on two ideas which are different but reason, Can does not have the habit of proper nutrition; he certainly compatible with each other. The first idea defended turns to convenience food and high calorie foods. As a result the slogan "mathematics for applications, models and of this, he started gaining weight rapidly.

Can, who is 1. According to this, primary aim and task in tall, comes to weigh 82 kg. Recognizing this situation, his mathematics teaching is to stimulate mathematical activities mother took Can to a dietitian.

Dietitian said that Can is of students of various proficiency levels by using practical classified as an obese person. The second idea defends the slogan "applications, Dietitian suggested an exercise program to Can, which he models and modeling for mathematics". That is to say, being will apply without changing the amounts of daily calories he interested in mathematical activation in contexts that are not takes. According to this, Can will start with an exercise of 20 mathematical increases motivation of the students and feeds minutes per day and 3 days a week and he will continue to do the formation of affective qualities, conceptual thinking and the exercises by increasing the time of exercises 5 minutes mathematical thinking power.

The approach called each week. The table showing which Freudenthal Institute in Holland is an example for this exercise corresponds to how many calories is given below. Note: 9 calories should be spent for 1 gram of oil Non-mathematical contexts have had an important place in mathematics teaching in Holland since Modeling Table 1. Sample table topic has been included in all mathematics programs in Working with weights 30 minutes calories secondary school since [23]. Why is modeling so important for students?

Mathematical Skating 15 minutes 15 calories models and modeling exist all around us, we especially Climbing up stairs 15 minutes 15 calories encounter them in technological devices. It is necessary to Dancing 30 minutes 75 calories form modeling qualifications of the students while preparing Riding a bicycle 30 minutes calories them as citizens responsible for society and become part of society. Development of mathematical modeling Mathematical modeling development is a complex skills of students is underscored as one of the main objectives process in mathematical and scientific application in real.

However, there are still very few examples of about both areas of mathematics and science by associating modeling in mathematics teaching practices in many them with real life events and conducting authentic activities. The reason of the gap between the programs and Such an integrative approach reveals the complexity of the educational practices is that the mathematical modeling is mathematical modeling with one question: "What is the also difficult for teachers. Because it requires real life meaning of abstraction, formulation and generalization in knowledge, the teaching becomes more open and less applications and modeling?

In more traditional mathematics applicable [4]. In other words, successfully apply primary education mathematics program. In reality, mathematical and scientific application, 2 Meaningful learning should be aimed. There 3 Students should establish communication by are not standards about how to teach mathematics and using their mathematical knowledge. Therefore, situations in implementations and modeling? Also, it is important to associate important in the connection between mathematics and real mathematical knowledge with both real life and things event.

This case regards the need to examine not only learned in other lectures. For this reason, problems should be cognitive process and student thinking but also social chosen in a way to help students clearly see the use of implementations in the class.

In both cases, abstraction is the mathematics in the daily life. It is stated in the program that basic function and focal point of inquiry [5]. A student develops mathematical skills such as "problem solving", mathematical knowledge by examining a model appropriate "communication", "association" and "reasoning". In relation for this knowledge, that is to say, he rebuilds it. Models can to problem solving skill, which can be considered as the most be used for three different purposes in mathematics teaching.

The models. Also, modeling emphasizes between concepts and rules and to solve problems with mathematical relations and ensures for students to develop which they can make associations between concrete and their learning styles and understanding of mathematics. Modeling helps teachers communicate with their students The basic vision of both primary school and the updates and motivate them.

Teaching and learning the topic of secondary school mathematical program 5th, 6th, 7th and 8th modeling is difficult. However, a teacher can facilitate and grades [27] is the principle that "Any child can learn develop students' modeling qualifications by developing mathematics".

Principles like "students should be helped in education activities. The basic Secondary School and Secondary Education High principles expected to be developed are listed as follows. Modeling process is defined in the program as Find the 6th item of the number string that is formed by a process "that is completed by mathematizing the starting from 7 and adding 3 each time.

It is underlined that importance is given in the program 3 butterflies to her collection every week 5 weeks later? According to Lesh and Doerr; In recent years, studies on the use of modeling activities in mathematics, most of which As a result, it is seen that while "mathematical modeling" have been mostly carried out abroad, have revealed that skill is not included in primary school and secondary school modeling activities are extremely useful tools for mathematics program, which is developed by MNE and mathematics education [7].

In many countries there is a applied throughout our country, as basic skills and tendency to give a larger place to mathematical modeling in qualifications aimed to be developed, it is included in high curriculums. It is emphasized that mathematical modeling school mathematics program. However, when the relation is one of the main objectives mathematics instruction in between "modeling and problem solving" skill aimed to be secondary school curriculum in Germany.

As a result, it can be argued that mathematical inequalities, functions, similarities in triangles and modeling is not used commonly from elementary schools to trigonometric ratios in right triangles in modeling and universities and teachers do not sufficiently know its problem solving". Even though mathematical modeling is However, when the relation between "modeling and given place in the MNE mathematical education program problem solving" skill aimed to be developed in students by which has been implemented since , samples of the program and the mathematics program of the 10th grade, modeling circle and application do not achieve desired level it is seen that modeling is expressed as "using second-degree of profoundness and significance.

The that gains students should have according to their learning concept of modeling in physics is used to indicate area are given. The topic of highs school mathematics program, it is determined that how modeling is superficially studied in mathematics education "the mathematical modeling" process will occur in programs.

Similarly, it is seen that there are not courses learning-teaching process and whether or not there are about modeling in teacher education or especially in application examples or activities that will help teachers in mathematics teacher education. In its broadest sense, relation to this are not mentioned. Although model examples mathematical modeling is the process of a problem solving are encountered in activities included in primary school and by the mathematical expression of real life event or a secondary school mathematics programs, we can say that problem.

This process enables learners to relate mathematics activity examples with which students will be able to fully to real life and to learn it more meaningfully and apply the steps of the modeling process and which we can permanently. For example, Mathematical literacy was defined in PISA as a gains and a model example regarding the 5th grade "natural person's capacity to use, interpret and formulate mathematics numbers" topic are presented below. Because of the importance given to "mathematical "Mathematical Modelling in Finance" lectures are modeling" in the education programs in countries throughout given in Manchester University.

It is seen that the world and the international examinations such as PISA, mathematical modeling lectures are given in we have to attach necessary importance to this topic in order engineering faculties of some universities in our to not fall behind the modern understanding of education.

However, constructive learning theory, which was put into use in "modeling in mathematics education" lectures should and which was updated later, that students should be made to be made compulsory in primary school mathematics study on model formation activities.