**2. Real-life problems, mathematical modeling cycles, and STEM**

One of the first schemes presented as an approach to mathematical modeling is Blum [32]. The mathematical modeling cycle here consists of the real situation and the real world, the mathematical model, and the results in two parallel sections. In the loop, problem-solving is often perceived as a guide for the real situation.

According to Lesh and Doerr [3], it is the basic elements that must be included in a mathematical modeling cycle. There are three basic elements in mathematical modeling (**Figure 1**). According to them, a real-world problem must be started in mathematical modeling. The students generally act in the framework of mathematics and logic with ideas that involve mathematical assumptions and approaches. Then, the mathematics used should be accurate and also in a logical way (**Figure 2**).

The mathematical modeling cycle commonly used in literature is developed by Blum and Leiß [33]. Similar to other models, a distinction is made between the real world and mathematics in this model. A prerequisite for this model is that students

#### **Figure 1.**

*Mathematical modeling cycle [32].*

#### **Figure 2.** *Lesh and Doerr's modeling cycle.*

should understand the mathematical problem and ensure that the model is developed in the real context. Although not mentioned here, it is important to keep in mind that the modeling process is in a repetitive natural loop (**Figure 3**).

Another important element is the existence of basic questions arising from the real-world problem in the mathematical cycle. These key questions can help to solve and study a mathematical modeling activity. Key questions are also very important for solving the problem. A key question can be shown as a real-life example of how long a person may be by spreading from the footprint and the length of the step [3]. Another feature of a key issue is that it allows people to focus on the issue. It can also bring people closer to their jobs or problems. The cycle of Perrenet and Zwaneveld is similar to that of Lesh and Doerr, but it has some differences. We observe that they provide more details and they emphasize three basic elements of mathematical modeling. In the modeling they describe, being outspoken and written communication are of paramount importance.

For example, students can conduct a mathematical modeling study and elaborate their solutions. Students also need to think through the modeling process so that they can clearly explain how well they understand the subject after a certain mathematical use. Thus, this mathematical cycle is repeated in a natural way. The revised solution is required during each cycle. This allows students to progress in different ways throughout the modeling cycle before developing an adequate solution. For the realization of such a process, they argue that the mathematical modeling activities of Perrenet and Zwaneveld must be open-ended (**Figure 4**).

**155**

**Figure 4.**

*Perrenet and Zwaneveld's modeling cycle.*

**Figure 3.**

*Blum and Leiß [33] modeling cycle.*

*The Role of Mathematical Modeling in STEM Integration and Education*

Stohlmann and Albarracin [34] stated that there should be seven items in a mathematical modeling. The first one is that the problem should start with a realworld problem. Second, key questions should be addressed. The third one of these basic elements is the logical thinking of the solution of the problem with mathematical assumptions and approximations. Fourth, the mathematics used must be related to the real situation. The fifth of these elements plays an important role in written communication. The sixth, which is the mathematical modeling process, is an iterative process with open-ended problems. The seventh and last item is the reflections

*DOI: http://dx.doi.org/10.5772/intechopen.88615*

*The Role of Mathematical Modeling in STEM Integration and Education DOI: http://dx.doi.org/10.5772/intechopen.88615*

*Theorizing STEM Education in the 21st Century*

should understand the mathematical problem and ensure that the model is developed in the real context. Although not mentioned here, it is important to keep in

Another important element is the existence of basic questions arising from the real-world problem in the mathematical cycle. These key questions can help to solve and study a mathematical modeling activity. Key questions are also very important for solving the problem. A key question can be shown as a real-life example of how long a person may be by spreading from the footprint and the length of the step [3]. Another feature of a key issue is that it allows people to focus on the issue. It can also bring people closer to their jobs or problems. The cycle of Perrenet and Zwaneveld is similar to that of Lesh and Doerr, but it has some differences. We observe that they provide more details and they emphasize three basic elements of mathematical modeling. In the modeling they describe, being outspoken and written communica-

For example, students can conduct a mathematical modeling study and elaborate their solutions. Students also need to think through the modeling process so that they can clearly explain how well they understand the subject after a certain mathematical use. Thus, this mathematical cycle is repeated in a natural way. The revised solution is required during each cycle. This allows students to progress in different ways throughout the modeling cycle before developing an adequate solution. For the realization of such a process, they argue that the mathematical model-

ing activities of Perrenet and Zwaneveld must be open-ended (**Figure 4**).

mind that the modeling process is in a repetitive natural loop (**Figure 3**).

**154**

**Figure 1.**

**Figure 2.**

*Mathematical modeling cycle [32].*

*Lesh and Doerr's modeling cycle.*

tion are of paramount importance.

**Figure 4.**

*Perrenet and Zwaneveld's modeling cycle.*

Stohlmann and Albarracin [34] stated that there should be seven items in a mathematical modeling. The first one is that the problem should start with a realworld problem. Second, key questions should be addressed. The third one of these basic elements is the logical thinking of the solution of the problem with mathematical assumptions and approximations. Fourth, the mathematics used must be related to the real situation. The fifth of these elements plays an important role in written communication. The sixth, which is the mathematical modeling process, is an iterative process with open-ended problems. The seventh and last item is the reflections

in the mathematical modeling used. However, the most widely used curricula are the modeling activities based on the models and modeling perspective.

For example, let us assume that there is an overflow after a heavy rainfall in a water-filled dam. Thus, in the case of an increase in the water level in the dam, the walls that protect the dam may break and the nearby city may cause a flood. This situation can be overcome by keeping the dam covers open at a certain time after each filling. As the precipitation continues, this situation will be constantly renewed. The important factors in this case are the water level in the dam, the amount of water discharged when the dam covers are open and the time. If necessary, a mathematical formula can be developed by considering the precipitation status for this problem. Even then, different variables or parameters can be found. The formulas that need to be considered here must be adjusted. For example, the amount, size, and time of the caps are important.

Some changes can be made in the model recursively. For such examples and similar representations, the example shown in the figure can be used as a mathematical model (**Figure 5**) [35, 36].

Güder and Gürbüz [38] aimed to improve the ability of interdisciplinary relations in the fields of mathematics, science, and technology in the field of "Energy-Saving Problem" for seventh grade students. In this problem, the concepts of power, motor power, power units (watt-kilowatt), and their transformation into each other are taught. In line with the purpose of the study, they tried to reveal the development of participants from a different perspective in a conceptually enriched environment in line with the multilayered teaching experiment [24, 39]. The multitier teaching experiment is designed to help students understand the modeling activities of teachers and teacher trainers in order to develop models for describing and explaining mathematical structures. The models are the teaching experiments consisting of three stages [24, 39, 40].

In this study, in the first 4 weeks of the study, thoughtful and supportive modeling problems are included. As a second step, the "Energy-Saving Problem" together with the Science teacher was developed by researchers. Finally, in the third stage, the researchers made observations and inferences during the application of the

**157**

**Figure 6.**

*Theoretical framework of the study.*

*The Role of Mathematical Modeling in STEM Integration and Education*

problem and reported the participants' progress in this matter. The general frame-

activities represent the activities associated with mathematics and science.

As mentioned above, the mathematical modeling process is a cyclical process consisting of several steps. Similarly, the IMM process is a cyclical and cascading process. However, unlike mathematical modeling, the inclusion of more than one area of IMM activities leads to a differentiation in the modeling process. Doğan et al. also defined a framework of the interdisciplinary mathematical modeling process for the mathematics and science disciplines in their studies. The IMM process begins in the real world, and first of all the individual needs to understand the real-life problem. The first step, which is expressed as an understanding of the

A conceptual framework proposed by Daniels [42] was included in a study conducted on theories and assumptions developed before STEM training. This framework, which is designed as the theoretical framework of STEM integration, has been shown as three Venn diagram using mathematical modeling applications. While the first circle represents the elements of metacognitive theory (metacognitive knowledge, processes, skills, and strategies), the second circle includes social development theory (social mediated interaction—promoting communication). Lastly, the third circle consists of the teaching elements which are considered as

If a good STEM integration is to be made, elements of the metacognitive [43] and Vygotsky's social development [44, 45] should also be included. As shown in

Real-life problems are discussed in mathematical modeling activities. Real-life situations are complex and cover many areas. Therefore, mathematical modeling is suitable for the different disciplines, and it is seen as an effective tool that can be used in STEM education [19, 41]. This type of activity, which is defined as interdisciplinary mathematical modeling (IMM), includes an understanding of different disciplines. In the understanding of IMM and in the solution of the problems of real-life situation, one or several disciplines are used together with mathematics [19]. In their study IMM is dealt with in mathematics and science. Therefore, IMM

*DOI: http://dx.doi.org/10.5772/intechopen.88615*

problem, enters the STEM world.

basic for education.

work used in this study is as follows (**Figure 6**):

**Figure 5.** *A representation of the mathematical modeling process [37].*

#### *The Role of Mathematical Modeling in STEM Integration and Education DOI: http://dx.doi.org/10.5772/intechopen.88615*

*Theorizing STEM Education in the 21st Century*

caps are important.

ematical model (**Figure 5**) [35, 36].

consisting of three stages [24, 39, 40].

in the mathematical modeling used. However, the most widely used curricula are the

For example, let us assume that there is an overflow after a heavy rainfall in a water-filled dam. Thus, in the case of an increase in the water level in the dam, the walls that protect the dam may break and the nearby city may cause a flood. This situation can be overcome by keeping the dam covers open at a certain time after each filling. As the precipitation continues, this situation will be constantly renewed. The important factors in this case are the water level in the dam, the amount of water discharged when the dam covers are open and the time. If necessary, a mathematical formula can be developed by considering the precipitation status for this problem. Even then, different variables or parameters can be found. The formulas that need to be considered here must be adjusted. For example, the amount, size, and time of the

Some changes can be made in the model recursively. For such examples and similar representations, the example shown in the figure can be used as a math-

Güder and Gürbüz [38] aimed to improve the ability of interdisciplinary relations in the fields of mathematics, science, and technology in the field of "Energy-Saving Problem" for seventh grade students. In this problem, the concepts of power, motor power, power units (watt-kilowatt), and their transformation into each other are taught. In line with the purpose of the study, they tried to reveal the development of participants from a different perspective in a conceptually enriched environment in line with the multilayered teaching experiment [24, 39]. The multitier teaching experiment is designed to help students understand the modeling activities of teachers and teacher trainers in order to develop models for describing and explaining mathematical structures. The models are the teaching experiments

In this study, in the first 4 weeks of the study, thoughtful and supportive modeling problems are included. As a second step, the "Energy-Saving Problem" together with the Science teacher was developed by researchers. Finally, in the third stage, the researchers made observations and inferences during the application of the

modeling activities based on the models and modeling perspective.

**156**

**Figure 5.**

*A representation of the mathematical modeling process [37].*

problem and reported the participants' progress in this matter. The general framework used in this study is as follows (**Figure 6**):

Real-life problems are discussed in mathematical modeling activities. Real-life situations are complex and cover many areas. Therefore, mathematical modeling is suitable for the different disciplines, and it is seen as an effective tool that can be used in STEM education [19, 41]. This type of activity, which is defined as interdisciplinary mathematical modeling (IMM), includes an understanding of different disciplines. In the understanding of IMM and in the solution of the problems of real-life situation, one or several disciplines are used together with mathematics [19]. In their study IMM is dealt with in mathematics and science. Therefore, IMM activities represent the activities associated with mathematics and science.

As mentioned above, the mathematical modeling process is a cyclical process consisting of several steps. Similarly, the IMM process is a cyclical and cascading process. However, unlike mathematical modeling, the inclusion of more than one area of IMM activities leads to a differentiation in the modeling process. Doğan et al. also defined a framework of the interdisciplinary mathematical modeling process for the mathematics and science disciplines in their studies. The IMM process begins in the real world, and first of all the individual needs to understand the real-life problem. The first step, which is expressed as an understanding of the problem, enters the STEM world.

A conceptual framework proposed by Daniels [42] was included in a study conducted on theories and assumptions developed before STEM training. This framework, which is designed as the theoretical framework of STEM integration, has been shown as three Venn diagram using mathematical modeling applications. While the first circle represents the elements of metacognitive theory (metacognitive knowledge, processes, skills, and strategies), the second circle includes social development theory (social mediated interaction—promoting communication). Lastly, the third circle consists of the teaching elements which are considered as basic for education.

If a good STEM integration is to be made, elements of the metacognitive [43] and Vygotsky's social development [44, 45] should also be included. As shown in

**Figure 6.** *Theoretical framework of the study.*

#### **Figure 7.**

*The theoretical framework on metacognition of STEM integration from mathematical modeling perspectives.*

**Figure 7**, STEM integration can be facilitated if the instructors implement these selected theories. These two theories have been proposed based on the following principles for implementation.

When Piaget [45], who explained the theory of cognitive development, explained only the characteristics of the cognitive development age stages, he mentioned the best level of learning and the importance of age for thinking development. Thus, Flavell's [47] theory (metacognitive knowledge, metacognitive experience, and metacognitive strategies) can explain the students' thinking, strategies, and actions to solve mathematical modeling problems [46, 48]. The problem-solving model of Polya may not be sufficient for a STEM practitioner [48]. This is because the problem is defined here in only three ways.

In particular, supporting mathematical and quantitative processes in science, mathematics, and engineering, and thus increasing mathematical reasoning, is the main objective. Technology provides tools to perform quantitative calculations more efficiently or to produce alternative visualization tools for experimental outputs. All modeling processes share the standard features shown in **Figure 8**. It has been demonstrated that there is a capability of researching modeling techniques, mathematical reasoning to model engineering design, and the ability to make scientific inquiry and then produce a structure. Mathematical modeling is of particular importance because it is important to produce appropriate tools to predict how quantification methods, new designs, and new situations will behave [49].

**159**

**3. Discussion**

*Phases of STEM projects.*

**Figure 8.**

*The Role of Mathematical Modeling in STEM Integration and Education*

Niss [50] stated that only theoretical mathematics knowledge is not always sufficient to solve real-life problems. In this case, the importance of mathematical modeling for the transfer of mathematical knowledge emerges. However, because the rapidly developing technologies and science are keys to solving real-life problems in different disciplines, STEM activities have been spread to schools. Thus, STEM integration has been added to the training program of many countries. In the educational programs of some countries, the presence of STEM activities combined

Similarly, when questioning which stages of the mathematical modeling process

In the early 2000s, three and four stage cycles were used in mathematical model-

improve students' problem-solving skills [49], the question of which stages of the "mathematical modeling together with STEM activities" should students use

ing to solve real-life problems. However, over time, due to the need seen, these mathematical modeling cycles have been further elaborated by adding some steps. An example of this is Stohlmann and Albarracin [34] mathematical modeling cycle. Using science, technology, education, and mathematics together with mathematical modeling to solve real-life problems will facilitate to solve these problems [48]. Kertil and Gurel [52] and Sokolowski [49] supported this idea and were among the thinkers of STEM and mathematical modeling together. In the researches, teachers stated that this kind of instruction encourages students, focuses their attention on the subject and they learn the lesson better by leaving a positive effect on them [31, 53, 54]. An example of this is the STEM project conducted in conjunction with mathematical modeling to investigate the impact of student competences on

with mathematical modeling also stands out [51].

problem-solving skills raises too.

*DOI: http://dx.doi.org/10.5772/intechopen.88615*

*The Role of Mathematical Modeling in STEM Integration and Education DOI: http://dx.doi.org/10.5772/intechopen.88615*

**Figure 8.** *Phases of STEM projects.*

#### **3. Discussion**

*Theorizing STEM Education in the 21st Century*

principles for implementation.

**Figure 7.**

**Figure 7**, STEM integration can be facilitated if the instructors implement these selected theories. These two theories have been proposed based on the following

*The theoretical framework on metacognition of STEM integration from mathematical modeling perspectives.*

When Piaget [45], who explained the theory of cognitive development, explained only the characteristics of the cognitive development age stages, he mentioned the best level of learning and the importance of age for thinking development. Thus, Flavell's [47] theory (metacognitive knowledge, metacognitive experience, and metacognitive strategies) can explain the students' thinking, strategies, and actions to solve mathematical modeling problems [46, 48]. The problem-solving model of Polya may not be sufficient for a STEM practitioner [48].

In particular, supporting mathematical and quantitative processes in science,

mathematics, and engineering, and thus increasing mathematical reasoning, is the main objective. Technology provides tools to perform quantitative calculations more efficiently or to produce alternative visualization tools for experimental outputs. All modeling processes share the standard features shown in **Figure 8**. It has been demonstrated that there is a capability of researching modeling techniques, mathematical reasoning to model engineering design, and the ability to make scientific inquiry and then produce a structure. Mathematical modeling is of particular importance because it is important to produce appropriate tools to predict how quantification methods, new designs, and new situa-

This is because the problem is defined here in only three ways.

**158**

tions will behave [49].

Niss [50] stated that only theoretical mathematics knowledge is not always sufficient to solve real-life problems. In this case, the importance of mathematical modeling for the transfer of mathematical knowledge emerges. However, because the rapidly developing technologies and science are keys to solving real-life problems in different disciplines, STEM activities have been spread to schools. Thus, STEM integration has been added to the training program of many countries. In the educational programs of some countries, the presence of STEM activities combined with mathematical modeling also stands out [51].

Similarly, when questioning which stages of the mathematical modeling process improve students' problem-solving skills [49], the question of which stages of the "mathematical modeling together with STEM activities" should students use problem-solving skills raises too.

In the early 2000s, three and four stage cycles were used in mathematical modeling to solve real-life problems. However, over time, due to the need seen, these mathematical modeling cycles have been further elaborated by adding some steps. An example of this is Stohlmann and Albarracin [34] mathematical modeling cycle. Using science, technology, education, and mathematics together with mathematical modeling to solve real-life problems will facilitate to solve these problems [48]. Kertil and Gurel [52] and Sokolowski [49] supported this idea and were among the thinkers of STEM and mathematical modeling together. In the researches, teachers stated that this kind of instruction encourages students, focuses their attention on the subject and they learn the lesson better by leaving a positive effect on them [31, 53, 54]. An example of this is the STEM project conducted in conjunction with mathematical modeling to investigate the impact of student competences on

sustainability in a university classroom [55]. In recent years, it has seen that teachers have been given courses for STEM training based on model-eliciting activities within STEM integration [30, 56].
