Use of Digital Technology in Integrated Mathematics Education

Abstract

Digital learning environments create a dynamic and engaging learning and teaching context that promotes a deeper understanding of complex concepts, eases the teaching process and fosters a passion for learning. Moreover, integrating interactive materials into pilot courses can assist teachers in better assessing student learning and adjusting their teaching strategies accordingly. The teachers can also receive valuable insights into students’ strengths and weaknesses, allowing them to provide targeted support and intervention when needed. For students from the defence and security fields, digital learning environments can create realistic simulations and virtual training scenarios that allow students to practise their skills in a controlled and safe environment, develop hands-on experience, and enhance their decision-making abilities without the need for real-world training exercises. In this context, the purpose of this paper is to introduce an approach for solving mathematical problems embedded in technical scenarios within the defence and security fields with the aid of digital technology using different software environments such as Python, Matlab, or SolidWorks. In this way, students can visualise abstract concepts, experiment with different scenarios, and receive instant feedback on their understanding. At the same time, the use of didactic and interactive materials can increase the interest among students and teachers for utilising mathematical models and digital technologies in the educational process. This paper also helps to reinforce key concepts and enhance problem-solving skills, sparking curiosity and creativity, and encouraging active participation and collaboration. Throughout the development of this proposal, based on survey analysis, good practices are presented, and advice for improvement is collected while having a wide range of users giving feedback, and participating in discussions and testing (pilot) short-term learning/teaching/training activities.

1. Introduction

Due to advancements in technology and teaching methods, education has become more diverse, accessible, and personalised. The emergence of online learning platforms, interactive tools, and virtual classrooms has revolutionised how students can engage with and absorb information. There is also a greater emphasis on holistic education, including social–emotional learning, critical thinking skills, and cultural competency. Nevertheless, besides memorising facts and figures, education aims to equip students with the necessary skills to succeed in a fast-changing world.

In this sense, teaching Mathematics using digital tools can enhance how mathematics concepts are presented to students, provide opportunities for interactive and engaging learning experiences, and facilitate individualised instruction and assessment. Some examples of technology in mathematics education include online tutorials, interactive simulations, virtual manipulatives, graphing calculators, and educational software. These tools can help students visualise abstract concepts, explore real-world applications of mathematics, and practise problem-solving skills in a dynamic and interactive environment. Additionally, technology can support teachers in designing and delivering effective math lessons and tracking student progress and performance by providing specific instructions to meet the diverse needs of learners. Thus, by incorporating technology into mathematics education, educators can create more engaging and personalised learning experiences that promote critical thinking, problem-solving, and mathematical reasoning skills among students.

In light of the events of recent years, the impact of technology on education has been substantial, as it has revolutionised the way education is delivered: online teaching and learning have become a new reality [1]. The circumstances created by the COVID-19 pandemic have forced schools to adapt and find alternatives to the usual face-to-face teaching. Thus, most classes were conducted online [2], and the Internet and online platforms facilitated new learning experiences across various fields of study [3]. On the one hand, digital readiness has rapidly become a unique opportunity for students with fewer possibilities to participate in the learning process. It expands their chances of success in the labour market. On the other hand, teachers in certain disciplines face challenges when using digital technologies. While the teaching community has demonstrated a remarkable ability to innovate and embrace new approaches, many challenges, such as insufficient connectivity and digital infrastructure, have also become evident [4].

Although technology continues to be an unknown for many, the consensus among teachers is that it has proven to be an invaluable and indispensable tool, providing essential support in the educational process. Through collaborative efforts, students and teachers demonstrated the impact of technology in the educational process, illustrating how passion and a desire for growth can make a significant impact [5,6]. This behaviour supports the continuation of online learning and meets the real needs of students.

As we explore educational technology further, it becomes clear that mathematics education is an area where its impact is truly transformative. Utilising digital tools and resources, educators can involve students in dynamic and interactive learning experiences that accommodate diverse learning styles and abilities. During practical experience, a gap has been identified between teaching and learning mathematics in higher education institutions [7], where various courses based on mathematical models, such as Algebra, Calculus, or Advanced Mathematics, are characterised by a lack of connection between the required knowledge and its application [8].

Mathematics, a critical field in technical sciences, presents particular difficulties in learning. It is often too abstract and theoretical. Moreover, presenting real-world applications or use cases to students can sometimes be an issue if teachers lack practical expertise. Teachers and specialists can better explain how mathematical models, theorems, and other concepts apply to real-world phenomena by using a dedicated digital solution and having the appropriate skills. Consequently, this paper aims to enhance interest in mathematics and improve problem-solving skills by analysing best practices using digital tools and proposing a mixed teaching approach between the traditional method of teaching mathematics and the alternative method based on digital technology.

Teaching mathematics in a contemporary digital environment seems inadequate among young audiences [9]. The traditional method of teaching mathematics involves presenting problems and formulas on a whiteboard using an ink marker, which is familiar and intuitive to teachers [10]. However, this approach is not easily replicable in an online teaching environment. Furthermore, the lack of skills in writing mathematical formulas presents a significant challenge. To provide adequate and engaging examples, careful consideration and selection of the best software for teaching and learning is crucial. This paper is based on the author’s experience in the field of learning education in defence and security education obtained in Erasmus+ KA2 projects, such as [11,12].

Since presenting equations, theorems, and concepts can be challenging, the scope of this paper is to enhance interest in studying mathematics, make mathematical problems and applications more understandable and accessible via the power of digital tools, and develop students’ problem-solving skills by leveraging specialised digital solutions.

The paper begins with an analysis of a series of surveys of best practices among partner institutions and stakeholders [12]. These surveys provide a list of directions in defence and security education. Further, the paper introduces a scenario-based methodology for teaching mathematics on specific scenarios within the defence and security fields, and it explores how various digital environments can be combined. In this context, a mixed approach between the traditional method of teaching mathematics and the alternative method based on digital technology appears to be the optimal solution. The expected outcomes of the mixed approach include improved performance in math-related tasks, better problem-solving skills, and enhanced analytical thinking among students. As a result, there is a need for teaching competencies in digital applied mathematics and for students to be able to work on problems within a digital environment. There are also limited opportunities for students and teachers from different disciplines to communicate using a common language, such as digital (software) mathematics.

The result of the conducted research is expected to be implemented within the European Common Technical Semester for Defence and Security, (EuCTSDS), (2020-1-RO01-KA203-0803752022) [13] and in other similar international programmes of study, in order to improve the quality of the education.

2. Theoretical Framework

The use of digital technology can enhance integrated mathematics education by providing interactive and engaging tools for learning and practice. Digital technology can be integrated into mathematics education with the aid of the following:

-

Online resources: Students can access a variety of online resources, such as video tutorials, interactive simulations, and practice exercises, to supplement their classroom learning [14].

-

Educational apps: There are numerous applications available that allow students to gain mathematical skills in a fun and engaging way [15]. Applications like Khan Academy offer instant feedback and help students monitor their progress platforms [16].

Digital technology can also support constructivist approaches to learning [17]. The constructivist approach emphasises that individuals construct their understanding and knowledge through experiences and interactions with their environment. In this sense, digital tools not only allow but also encourage students to engage in interactive learning experiences to construct their knowledge by providing students access to various resources and information. For example, interactive simulations and online experiments can allow students to actively explore concepts hands-on, enabling them to build their understanding through direct experience. Additionally, digital tools can provide students with access to diverse perspectives and sources of information, empowering them to construct their knowledge based on multiple viewpoints.

Further, from the technology acceptance model (TAM) perspective [18], a series of factors that can influence the students’ acceptance and use of digital technologies can be identified. According to TAM, two key factors influence users’ acceptance of technology:

-

Perceived Ease of Use: This refers to the degree to which users believe using a particular technology will be easy and convenient.

-

Perceived Usefulness: This refers to the extent to which users believe that a particular technology will help them achieve their goals or tasks. Students’ satisfaction with digital technologies is likely to be higher if they perceive them as valuable and beneficial for their academic and personal needs.

Starting from the technology acceptance model (TAM), a series of surveys were conducted to analyse students’ digital competencies in learning mathematics for technical systems within the defence and security fields before and during the COVID-19 pandemic. The research was carried out on the territory of the European Union in several research centres and universities. The survey involved 1047 students in the security and defence fields, subdivided by age and gender, as presented in Figure 1. Most of the students had different countries of residence than the countries of study. Additionally, a few students were from France, Greece, Hungary, Lithuania, Portugal, and Slovakia as countries of residence and study. Furthermore, some students were originally from countries outside the European Union, such as Afghanistan, Saudi Arabia, Burkina Faso, Cameroon, Mali, Moldova, Montenegro, and Senegal.

The majority of the students studied in the institutions were taking part in the project Digital Competences for Improving Security and Defence Education (DIGICODE) [12] but with some exceptions and variability, especially for Romanian students. While most students were pursuing military and/or engineering studies, others were enrolled in diverse fields such as economics, information technology, health, humanities, languages, law, or social sciences. Their distribution concerning the course year shows that many students attended from the third year of their bachelor’s, thus ensuring the capability to confront the pre-pandemic situation with those during and after COVID-19 for most of the sample. The ratio between military students (cadets) and civilian students was about 4:1 (Figure 2).

According to the analysed surveys, the majority of students had a positive perception of their digital skills. Only a small percentage (5.6%) rated their skills as “low” or “very low,” while a moderate portion (22%) expressed caution (Figure 3). Furthermore, the technology and software used for learning resulted in a balanced distribution of satisfaction levels: 27.3% of students reported strong satisfaction, while 28.9% expressed being essentially unsatisfied. Therefore, proper attention must be given to technological aspects, such as the digital tool’s ease of use and usefulness. By understanding and addressing these factors, educators and technology developers can help enhance students’ acceptance and satisfaction with digital technologies.

Most students also recognised the importance of technology in learning during pandemic times, with only 14.2% of them deeming it only “slightly” important or even “not at all” important. They are generally aware of the pivotal role technology played during COVID-19 on remote learning.

Students privileged “independence and self-direction” on the one hand and “collaboration and teamwork” on the other, slightly putting aside “critical thinking and problem-solving skills” and “creativity and innovation,” probably because they perceive the two as less relevant to their learning (Figure 3).

3. Methods

Education in defence and security fields is crucial for preparing individuals to address evolving challenges in the public sector, private sector, or non-governmental organisations. By equipping learners with practical skills and knowledge, this type of education helps strengthen our collective ability to protect against threats and safeguard the well-being of communities and nations [19]. It is important to ensure a balance between different learning methods, with an emphasis on understanding the content, critical selection, focused attention, organising concepts logically, and training and practising skills with broad applicability.

For the surveys, data collection procedure consisted of the following steps: defining the objectives of the survey; clearly outlining the key information necessary for the survey; establishing the data collection method; selection of the sample population; analysing the data; and identifying the patterns or trends in the data that could help to achieve the research objectives.

The methodology employed in defence and security education is specifically tailored to address the unique challenges and requirements of this field. This approach focuses on hands-on, experiential learning, allowing students to apply theoretical concepts in practical scenarios. By incorporating simulations, case studies, and real-world exercises, students are better equipped to handle the complex and dynamic nature of defence and security issues [20]. One key aspect of this methodology is the emphasis on critical thinking and problem-solving skills. Students are encouraged to analyse and evaluate different scenarios, think creatively, and make informed decisions under pressure. This not only enhances their ability to assess risks and develop effective strategies but also prepares them to respond to unexpected challenges in real-world situations. Furthermore, the methodology in defence and security education emphasises the importance of developing strong communication and decision-making skills. Students are given opportunities to practise effective communication with peers, instructors, and stakeholders and collaborate in teams to solve complex problems. This helps them to build relationships, foster trust, and work cohesively towards common goals, which are crucial in the defence and security field.

Consequently, this paper proposes a methodology for teaching specific scenarios or use cases of a technical discipline using digital technology. The proposed methodology for solving a scenario with a mixed approach is organised into three parts. The first part offers a background description of the analysed scenario, the subsequent part provides the mathematical modelling of the scenario, and the final part involves a software implementation of the scenario in a virtual environment (Figure 4).

By using the software implementation part, the proposed methodology can offer graphical capabilities, allowing students to visualise mathematical concepts more intuitively and offering them a deeper understanding of abstract mathematical ideas [21]. Since the computer algebra system can perform calculations much faster than traditional pen-and-paper methods, the students can spend less time on routine calculations and more time understanding and analysing mathematical concepts. Another argument for using a software implementation is that it can help reduce calculation errors by providing step-by-step solutions and allowing students to check their work more efficiently, which can also build confidence in students’ mathematical abilities. Finally, a software implementation can handle mathematical problems that are more complex and time-consuming to solve by hand. This allows students to tackle more challenging problems and develop problem-solving skills. In conclusion, a digital tool is valuable because it can enhance mathematics learning by increasing efficiency, providing visual representations, reducing errors, enabling exploration and experimentation, and facilitating complex problem-solving.

Regarding the engineering side of mathematics used in defence and security education, a series of branches have been identified (Figure 5), in which scenario-based teaching can be a great tool for learning mathematics as it provides real-world context to mathematical concepts. By incorporating scenarios into learning, the trainees can understand how mathematical concepts are applied in practical situations, making mathematics more engaging and relevant, as they can see the value and purpose of their acquired skills, which also fosters a sense of accomplishment, creating a positive learning environment. In addition, scenario-based learning can also be an effective way to develop critical thinking and problem-solving skills. By working through scenarios and making decisions, the trainees must analyse information, evaluate options, and consider the potential consequences of their choices, which can lead to a deeper understanding of the subject matter and enhance retention.

Moreover, scenario-based learning can also be a valuable tool for professors looking to create engaging and effective learning experiences for their students. By presenting students with realistic scenarios that require them to apply their knowledge and skills, this approach can enhance learning outcomes and prepare students for real-world challenges.

Regarding a mathematical scenario, one way to handle and learn from it is to analyse the problem and understand the concepts involved carefully. This may require breaking down the problem into smaller parts, identifying essential information, and applying appropriate mathematical principles to solve it. Practising problem-solving techniques, such as working through similar scenarios, seeking help or guidance from teachers or peers, and persisting even when faced with challenges, is also essential and helpful. Additionally, reflecting on the solution and considering different approaches or strategies can reinforce learning and enhance understanding of the mathematical concepts involved in the scenario.

Furthermore, a series of mathematical-based scenarios specific to defence and security fields are presented and solved through the mixed approach, which consists of 2 methods: the clean mathematical method and the software-based implementation method.

Scenario 1—Automatic analysis of the motion of an aerospace vehicle

The flight of an aerospace vehicle (projectile, bomb, and missile) is a process based on a mathematical model with a high complexity [22]. A series of very important but, at the same time, very complex problems are represented by the determination of the behaviour of the aerospace vehicle during the flight, the determination of the flight performances, and the determination of its general motion, considering the whole system of forces and moments acting on it. In the scenario of general motion in actual flight conditions, a series of difficulties appear, making it difficult to draw up the differential equations of motion in a form that is as complete as possible (Figure 6).

The aerodynamic forces and guidance laws are the most important inputs to the mathematical models. They are represented from the mathematical point of view as non-linear terms, depending on the aerodynamic coefficients. The non-linearity of the differential equations gives the mathematical model its complexity.

Therefore, the analysis of the motion of an aerospace vehicle is conducted by adopting a series of hypotheses designed to simplify the physical model of the phenomenon according to the purpose and reduce the complexity of the equations that describe the phenomenon (Figure 7 and Figure 8). Thus, in the case of small incidents, the hypothesis of the independence between the longitudinal motion (and the motion from the launching plane) and the lateral motion is admitted. For the aircraft firing, as well as for the stable trajectories, the incidence angle is small, and in that case, the trigonometric functions involved in the mathematical model could be linearized. In that case, the motion in the vertical plane could be considered independently from the motion in the lateral plane, and consequently, the number of differential equations from the mathematical model is decreased.

Mathematical solution for Scenario 1

Trajectory calculation can be a difficult task, depending on the level of knowledge of the firing context and the level of detail implemented in the mathematical model of the trajectory. As in any modelling process of a physical phenomenon, in the case of trajectories, there will also be a compromise between the complexity of the model and the ease of obtaining results, as well as in terms of the agreement of numerical results with experimental ones. Reasonable results can be obtained with the help of simple trajectory models, especially to illustrate the application of the principles of the weapons use planning process. Subsequently, the model must be completed and developed to obtain accurate results that are as close as possible to reality.

In the trajectory models presented below, the equations describing the motion of the projectile/bomb are developed based on the hypothesis that the motions in the vertical and horizontal planes are independent. Also, the components of the initial velocity in the two directions are considered to be known.

The coordinate axis systems, as well as the variables that define the position of the aerospace vehicle and its state of motion, are chosen to not limit the generality of the studied problem and lead to equations that, after appropriate transformations and simplifications, can be integrated by known methods [22].

The general laws of mechanics will be used to establish the general dynamic equations that express the motion of an aerospace vehicle’s centre of mass and its motion around the centre of mass. Also, the aerospace vehicle is considered to be a body with six degrees of freedom and absolute rigidity, thus excluding the influence of the structure’s elasticity.

Defining the $\overline{V},\overline{\omega }$ and $\overline{F}$ vectors through the components on the axes of the mobile system connected to the body of the aerospace vehicle, the vector equation of motion of the centre of mass of the aerospace vehicle will be [23]

$$m{\displaystyle \frac{\mathrm{d}\overline{V}}{\mathrm{d}t}}=m\left({\displaystyle \frac{\partial \overline{V}}{\partial t}}+\overline{\omega }\times \overline{V}\right)=\overline{F}$$

(1)

where $m$ represent the mass of the aerospace vehicle, ${\displaystyle \frac{\partial \overline{V}}{\partial t}}$ is the local (relative) derivative of velocity, and $\overline{F}$ is the result of external forces acting on the aerospace vehicle.

Projecting the vector Equation (1) on the axes of the velocity-related system $OXYZ$, the scalar equations of motion of the centre of mass are obtained [24]. These equations are

$$m\left({\displaystyle \frac{\mathrm{d}{V}_X}{\mathrm{d}t}}+{\omega }_Y{V}_Z-{\omega }_Z{V}_Y\right)=m{\displaystyle \frac{\mathrm{d}V}{\mathrm{d}t}}=F_X$$

(2)

$$m\left({\displaystyle \frac{\mathrm{d}{V}_Y}{\mathrm{d}t}}+{\omega }_Z{V}_X-{\omega }_X{V}_Z\right)=mV{\displaystyle \frac{\mathrm{d}{\mathsf{\theta }}_H}{\mathrm{d}t}}=F_Y$$

(3)

$$m\left({\displaystyle \frac{\mathrm{d}{V}_Z}{\mathrm{d}t}}+{\omega }_X{V}_Y-{\omega }_Y{V}_X\right)=mV{\displaystyle \frac{\mathrm{d}{\mathsf{\theta }}_V}{\mathrm{d}t}}=F_Z$$

(4)

Considering the aerospace vehicle to be a symmetrical body of revolution (geometrically and in terms of mass distribution) [25] about the longitudinal axis $O\mathsf{\zeta }$, it turns out that the axes $O\mathsf{\xi }$, $O\eta$ and $O\mathsf{\zeta }$ are the main axes of inertia, and the kinetic moment of the vehicle concerning the centre of mass, $O$, will be written:

$${\overline{K}}_o=J_{\mathsf{\xi }}{\omega }_{\mathsf{\xi }}{\overline{i}}_{\mathsf{\xi }}+J_{\eta }{\omega }_{\eta }{\overline{i}}_{\eta }+J_{\mathsf{\zeta }}{\omega }_{\mathsf{\zeta }}{\overline{i}}_{\mathsf{\zeta }}$$

(5)

where ${\overline{i}}_{\mathsf{\xi }},{\overline{i}}_{\eta }\mathrm{and}{\overline{i}}_{\mathsf{\zeta }}$ are the unit vectors of the axes $O\mathsf{\xi }$, $O\eta$ and $O\mathsf{\zeta }$.

The vector equation of motion of the aerospace vehicle around the centre of mass will be written as below:

$${\displaystyle \frac{\mathrm{d}{\overline{K}}_0}{\mathrm{d}t}}={\displaystyle \frac{\partial {\overline{K}}_0}{\partial t}}+\overline{\omega }\times {\overline{K}}_0={\overline{M}}_0$$

(6)

where ${\displaystyle \frac{\partial {\overline{K}}_0}{\partial t}}$ is the local (relative) derivative of the kinetic moment.

Due to the longitudinal symmetry of the aerospace vehicle [26], the moments of inertia relative to the transverse axes will be equal, $J_{\mathsf{\xi }}=J_{\eta }=J$.

Projecting the vector Equation (6) on the axes of the $O\mathsf{\xi }\eta \mathsf{\zeta }$ system, the scalar equations of motion of the aerospace vehicle around the centre of mass are obtained:

$$J_{\mathsf{\zeta }}{\displaystyle \frac{\mathrm{d}{\omega }_{\mathsf{\zeta }}}{\mathrm{d}t}}=M_{\mathsf{\zeta }}$$

(7)

$$J{\displaystyle \frac{\mathrm{d}{\omega }_{\eta }}{\mathrm{d}t}}+\left(J_{\mathsf{\zeta }}-J\right){\omega }_{\mathsf{\zeta }}{\omega }_{\mathsf{\xi }}=M_{\eta }$$

(8)

$$J{\displaystyle \frac{\mathrm{d}{\omega }_{\mathsf{\xi }}}{\mathrm{d}t}}+\left(J_{\mathsf{\zeta }}-J\right){\omega }_{\mathsf{\zeta }}{\omega }_{\eta }=M_{\mathsf{\xi }}$$

(9)

Based on the above, considering an aerospace vehicle with its own fast rotational motion having the centre of pressure in front of the centre of mass and using the following notations for simplification of writing [27]:

$$K_X={\displaystyle \frac{\mathsf{\rho }S{C}_x}{2m}},K_Z={\displaystyle \frac{\mathsf{\rho }S}{2m}}{\displaystyle \frac{\mathrm{d}{C}_z}{\mathrm{d}\alpha }},K_L={\displaystyle \frac{\mathsf{\rho }S}{2m}}{\displaystyle \frac{\mathrm{d}{C}_l}{\mathrm{d}\alpha }},K_m={\displaystyle \frac{\mathsf{\rho }S\mathcal{l}}{2J}}{\displaystyle \frac{\mathrm{d}{C}_m}{\mathrm{d}\alpha }}$$

(10)

$$K_{Z\omega }={\displaystyle \frac{\mathsf{\rho }S}{2m}}{\displaystyle \frac{\mathrm{d}{C}_z}{\mathrm{d}\omega }},K_{L\omega }={\displaystyle \frac{\mathsf{\rho }S}{2m}}{\displaystyle \frac{\mathrm{d}{C}_l}{\mathrm{d}\omega }}$$

(11)

$$K_{ma}={\displaystyle \frac{\mathsf{\rho }S\mathcal{l}}{2J_{\mathsf{\zeta }}}}{\displaystyle \frac{\mathrm{d}{C}_{m_{ax}}}{\mathrm{d}\alpha }},K_{ma\omega }={\displaystyle \frac{\mathsf{\rho }S\mathcal{l}}{2J_{\mathsf{\zeta }}}}{\displaystyle \frac{\mathrm{d}{C}_{m_{ax}}}{\mathrm{d}\omega }},K_{m\omega }={\displaystyle \frac{\mathsf{\rho }S\mathcal{l}}{2J}}{\displaystyle \frac{\mathrm{d}{C}_m}{\mathrm{d}\omega }}$$

(12)

$$a_R={\displaystyle \frac{T}{m}},K_j={\displaystyle \frac{C_j{Q}_e}{J}},K_j^{\prime }={\displaystyle \frac{C_j^{\prime }{Q}_e}{J_{\mathsf{\zeta }}}}$$

(13)

The following forces and moment indices are taken into account [28]:

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Force of gravity:

$$G=mg$$

(14)

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Thrust:

$$T=m{a}_R$$

(15)

-

Drag force:

$$R=m{K}_X{V}^2$$

(16)

-

Lift force:

$$P=m{K}_Z{V}^2{\alpha }_V$$

(17)

-

Lateral force:

$$F_L=m{K}_L{V}^2{\alpha }_H$$

(18)

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Additional lift due to the pitching motion:

$$P_{\omega }=m{K}_{Z\omega }V{\omega }_{\eta }$$

(19)

-

Additional lateral force due to yawing motion:

$$F_{L\omega }=m{K}_{L\omega }V{\omega }_{\mathsf{\xi }}$$

(20)

-

Main aerodynamic moment (pitching moment):

$$M_{m_{\eta }}=J{K}_m{V}^2{\alpha }_V$$

(21)

-

Yawing aerodynamic moment:

$${M_m}_{\mathsf{\xi }}=J{K}_m{V}^2{\alpha }_H$$

(22)

-

Axial moment:

$$M_{ma}=J_{\mathsf{\zeta }}{K}_{ma}{V}^2{\mathsf{\epsilon }}_a$$

(23)

-

Additional axial moment due to rolling motion:

$$M_{ma\omega }=J_{\mathsf{\zeta }}{K}_{ma\omega }\mathcal{l}V{\omega }_{\varsigma }$$

(24)

-

Damping moment due to gas jet:

$$M_{am\mathsf{\zeta }}=-K_j^{\prime }{J}_{\mathsf{\zeta }}{\omega }_{\mathsf{\zeta }}$$

(25)

$$M_{am\mathsf{\xi }}=-K_{j}J{\omega }_{\mathsf{\xi }}$$

(26)

$$M_{am\eta }=-K_{j}J{\omega }_{\eta }$$

(27)

Using the forces and moments presented above, the following equations are obtained that describe the motion of the aerospace vehicle:

$${\displaystyle \frac{\mathrm{d}V}{\mathrm{d}t}}=a_R-K_X{V}^2-g\sin {\mathsf{\theta }}_V$$

(28)

$${\displaystyle \frac{\mathrm{d}{\mathsf{\theta }}_V}{\mathrm{d}t}}=K_{Z}V{\alpha }_V-K_{Z\omega }{\omega }_{\eta }-{\displaystyle \frac{g}{V}}\cos {\mathsf{\theta }}_V$$

(29)

$${\displaystyle \frac{\mathrm{d}{\mathsf{\theta }}_H}{\mathrm{d}t}}=K_{L}V{\alpha }_H-K_{L\omega }{\omega }_{\mathsf{\xi }}$$

(30)

$${\displaystyle \frac{\mathrm{d}{\omega }_{\mathsf{\zeta }}}{\mathrm{d}t}}=K_{ma}{V}^2{\mathsf{\epsilon }}_a-K_{ma\omega }\mathcal{l}V{\omega }_{\varsigma }-K_j^{\prime }{\omega }_{\mathsf{\zeta }}$$

(31)

$${\displaystyle \frac{\mathrm{d}{\omega }_{\eta }}{\mathrm{d}t}}=\left(1-J_{\mathsf{\zeta }}/J\right){\omega }_{\mathsf{\zeta }}{\omega }_{\mathsf{\xi }}-K_m{V}^2{\alpha }_V-K_{m\omega }\mathcal{l}V{\omega }_{\eta }-K_j{\omega }_{\eta }$$

(32)

$${\displaystyle \frac{\mathrm{d}{\omega }_{\mathsf{\xi }}}{\mathrm{d}t}}=\left(1-J_{\mathsf{\zeta }}/J\right){\omega }_{\mathsf{\zeta }}{\omega }_{\eta }-K_m{V}^2{\alpha }_H-K_{m\omega }\mathcal{l}V{\omega }_{\mathsf{\xi }}-K_j{\omega }_{\mathsf{\xi }}$$

(33)

To complete the system of Equations (28)–(33) a series of kinematic ratios are introduced:

$${\displaystyle \frac{\mathrm{d}{\mathsf{\phi }}_V}{\mathrm{d}t}}={\omega }_{\eta }$$

(34)

$${\displaystyle \frac{\mathrm{d}{\mathsf{\phi }}_H}{\mathrm{d}t}}={\omega }_{\mathsf{\xi }}$$

(35)

$${\displaystyle \frac{\mathrm{d}\mathsf{\gamma }}{\mathrm{d}t}}={\omega }_{\mathsf{\zeta }}$$

(36)

$${\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}=V\cos {\mathsf{\theta }}_V\cos {\mathsf{\theta }}_H$$

(37)

$${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}=V\cos {\mathsf{\theta }}_V\sin {\mathsf{\theta }}_H$$

(38)

$${\displaystyle \frac{\mathrm{d}z}{\mathrm{d}t}}=V\sin {\mathsf{\theta }}_V$$

(39)

$${\alpha }_V={\mathsf{\phi }}_V-{\mathsf{\theta }}_V$$

(40)

$${\alpha }_H={\mathsf{\phi }}_H-{\mathsf{\theta }}_H$$

(41)

The Equations (28)–(39) form a system of 12 non-linear first-order differential equations, representing the complete system of equations of general motion of the aerospace vehicle, which provides by integration time variation of parameters (unknown functions) $V$, ${\mathsf{\theta }}_V$, ${\mathsf{\theta }}_H$, ${\mathsf{\phi }}_V$, ${\mathsf{\phi }}_H$, $\mathsf{\gamma }$, ${\omega }_{\mathsf{\zeta }}$, ${\omega }_{\eta }$, ${\omega }_{\mathsf{\xi }}$, $x$, $y$, and $z$, which determines both the motion of the centre of mass and the motion around the centre of mass of the studied aerospace vehicle [29].

The digital solution for Scenario 1

The spatial motion around the centre of gravity is currently analysed on short trajectory arcs without considering the overall variation of the centre of the gravity velocity vector. The calculation of the trajectory is made considering the corresponding variations of the incidence angle and the drift angle, obtained by the separate study of the longitudinal motion and of the lateral motion; most of the time, however, to determine the trajectory, the well-known hypothesis of ballistic motion is admitted, without incidence, thus disregarding the effect of lift, the variation of which is conditioned, through incidence, by the motion around the centre of gravity. To determine the trajectory, a solution is to use Matlab R2024a [30] for modelling the equations of the trajectory [31] (Figure 9).

The digital solution consists of solving a system of 12 non-linear ordinary differential equations (ODE), Equations (28)–(39). The nonlinearity is given by the aerodynamic forces’ Expressions (16)–(24). The solving process of each individual equation represents the determination of the projectile translations and rotations, meaning the behaviour of the aerospace vehicle on the trajectory.

The interface developed in Matlab [32] offers the possibility of choosing different characteristics of the projectiles and the possibility of defining the parameters of the aircraft. These parameters can be defined by their physical characteristics, performance capabilities, and operating limitations (Figure 10). Establishing these parameters is important for ensuring safe and efficient operations, as well as for regulatory compliance and maintenance planning (Figure 11). One of the main challenges is the identification of the initial conditions. Usually, the initial conditions are obtained by the measurements of the ballistic parameters. These measurements are influenced by different disturbances such as environmental factors, accuracy of the measurement devices, and human factors. All those perturbations could be seen as random numbers governed by a normal distribution, in that way being considered within the modelling process.

Figure 12, Figure 13 and Figure 14 present the evolution of the main parameters that describe the motion during the trajectory.

In 40 s, the aerospace vehicle travels a horizontal distance of about 9 km. During this period, the speed first increases (corresponding to the active period when the rocket engine is running). This period is followed by the active period, where, up to 25 s, the speed decreases due to the effect of the drag force, after which it shows a slight increase as an effect of the gravitational acceleration. The angle of incidence (as the difference between ${\mathsf{\phi }}_{V0}$ and ${\mathsf{\theta }}_{V0}$) has an oscillating damped evolution. The damping of the oscillation takes place in about 1 s (Figure 15). The pitch angular speed has an evolution similar to that of the angle of incidence.

Scenario 2—Determine the velocity of a projectile using a ballistic pendulum

A ballistic pendulum is a device used to measure the speed of a projectile by capturing it in a pendulum and measuring the height to which the pendulum swings. It is often used in physics labs to demonstrate the principles of conservation of momentum and energy. The ballistic pendulum works by allowing a projectile to strike and become embedded in a large block of wood or other material attached to a pendulum. The change in momentum of the projectile can then be used to calculate its initial velocity.

Mathematical solution for Scenario 2

The bullet is fired into the pendulum, which then moves upwards a certain distance and at an angle that will be measured. From the height reached by the pendulum, its potential energy can be calculated. This potential energy is equal to the kinetic energy of the pendulum at the bottom of the swing immediately after the collision with the bullet. The bullet is fired into the pendulum, which then moves upwards a certain distance and at an angle that will be measured.

The approximate method assumes that the pendulum and the ball act together as a point mass. This method does not take into account the rotational inertia. It is slightly faster and easier than the second method but not as precise.

The potential energy of the pendulum at the final position can be calculated as bellow:

$$\Delta PE=Mg\Delta h$$

(42)

where M is the combined mass of the pendulum and the ball, g is the gravitational acceleration, and ∆h represents the change in height. Furthermore, height is computed:

$$\begin{array}{l}\Delta h=R\left(1-\cos \mathsf{\theta }\right)\\ \Delta PE=MgR\left(1-\cos \mathsf{\theta }\right)\\ \Delta PE=0.0206\end{array}$$

(43)

The momentum of the pendulum after the collision is

$$P_p=M{v}_p$$

(44)

which is substituted into the previous equation to obtain

$$KE={\displaystyle \frac{P_p^2}{2M}}$$

(45)

Solving this equation for the momentum of the pendulum, we receive

$$P_p=\sqrt{2M\left(KE\right)}$$

(46)

This momentum is equal to the momentum of the ball before the collision:

$$P_p=m{v}_p$$

(47)

Equating these two equations and replacing KE with the known potential energy, the equation is obtained as follows:

$$m{v}_p=\sqrt{2M^{2}gR\left(1-\cos \mathsf{\theta }\right)}$$

(48)

The digital solution for Scenario 2

To validate the result obtained through the analytical method, Simulink R2024a and SolidWorks Standard 2020 environments were used to simulate the behaviour of the ballistic pendulum.

SolidWorks allows students to design and create 3D models for their projects and assignments. The Student Edition of SolidWorks is a valuable tool for students studying engineering, design, and other related fields, allowing them to gain hands-on experience with a leading CAD software programme. Using SolidWorks, students can design and analyse complex mechanical systems, structures, and components. They can learn how to apply engineering principles to solve real-world problems and optimise designs (Figure 16).

Using SolidWorks for teaching students can enhance their technical skills, creativity, problem-solving abilities, and collaboration skills, preparing them for careers in engineering, design, and other related fields.

The weight of the bullet and the weight of the projectile resulted from the calculation is automatically performed by the Solidworks programme (Figure 17).

As an optional extension of the MATLAB software package, SIMULINK provides a graphical user interface for modelling and simulating dynamic systems. It provides access to various block diagrams, which allows quick and straightforward system modelling, simulating, and deployment without the need to write any simulation code.

Simulink’s interface is easy to use and understand, making it a great tool for students new to the field. It also provides access to a wide range of built-in blocks and libraries for students to create complex models without writing code from scratch. Additionally, Simulink has built-in support for real-time simulation and hardware-in-the-loop testing, allowing students to interface their models with physical systems for hands-on learning experiences (Figure 18). Overall, Simulink is a valuable tool for students looking to gain practical experience in the field of engineering and learn through hands-on experimentation.

The transition from the mathematical model to the graphical model can be made by analysing the various terms of the equation. To start, the effect of the integrative block on the derivative of a function can be analysed. In this regard, if the derivative of a signal is fed into an integrator, the output of the integrator will yield the original signal. If the second derivative of the signal is fed into the integrator, the output will yield the first derivative. Now the simulation scheme of the model can be constructed using Simulink functional blocks. By using the multiplication block, the sine block, and two integration blocks, the equation of motion can be obtained. Having created the simulation scheme, the monitored quantity is connected to an oscilloscope for visualization.

Scenario 3—Automatic people detection within an established perimeter

The proposed scenario for detecting people is integrated into a video surveillance system that performs real-time perimeter monitoring around a firing range and generates an alarm when people are detected inside this monitored area. Thus, the role of this system is to trigger the alarm to notify the human decision-maker responsible for the monitored area. When the alarm is triggered, the system presents the scene where a person was detected.

Mathematical solution for Scenario 3

The people detection will be performed through facial recognition, using Principal Component Analysis (PCA) [33], by reducing the number of variables for representing a face, a technique used in computer vision and pattern recognition. PCA is a method that reduces the dimensionality of data while retaining most of the variation in the dataset.

In facial recognition, PCA is used to extract the most important features from a facial image [34]. These features are then compared to a database of known faces to determine a match. The process involves capturing an image of a person’s face, identifying key facial features such as eyes, nose, and mouth, and then using PCA to extract the most relevant information from these characteristics. Through this method, the system can accurately detect and identify individuals in real-time, making it a valuable tool for security and surveillance applications. Additionally, PCA can also help improve the speed and accuracy of facial recognition systems, making them more reliable and efficient in various environments [35].

We will consider that we have several images of the same size available for several people in a database, and we can have one or more poses for the same person. We will consider that the person’s face is centred in any facial image [36]. Thus, each pixel of an image can be considered a variable, and a face is thus represented as a vector of P variables (where P represents the number of pixels). Given the considerable number of variables involved, we can reduce the complexity of comparing two faces by approximating a face through a smaller number of variables. An image of n pixels can thus be considered a point in image space (n-dimensional space) (Figure 19).

The image matrix turns into a line vector such that the line vector $p_x=(i_1;i_2;\dots i_n)$ where $i_1$,$i_2,$… represents the grey level of pixels. The vector is formed by transforming the image into a line vector; thus, for a 128 × 128 image, we will have the size n = 16,384 (16K):

Given N images, each of n pixels, the entire set of images can be represented as a $\left(N\times n\right)$ matrix D, where each image represents a line:

$$D=\left[\begin{array}{cccccc}150& 152& \cdots & 255& \cdots & 252\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ 144& 171& \cdots & 245& \cdots & 223\end{array}\right],N\times n$$

(49)

The first step of the PCA method is to move the origin of the data to the middle of the n variables, an operation that is equivalent to subtracting the average value of the variables in each image. In our case, the average image (average face) is calculated by averaging over all the columns of the matrix D. This average face is to be extracted from each image of the dataset D. The obtained difference matrix is denoted by U. In the considered example, the average is

$$\left[\begin{array}{ccccccc}120& 140& \cdots & 230& 230& \cdots & 240\end{array}\right]$$

(50)

$$D=\left[\begin{array}{ccc}30& \dots & 12\\ ⋮& \ddots & ⋮\\ 24& \dots & -17\end{array}\right],N\times n$$

(51)

The first step in determining the principal components of the D matrix is to calculate the covariance matrix of the U matrix. It can be easily seen that, due to the extraction of the average value from each column of the D matrix, the $n\times n$ covariance matrix C can be expressed as

$$C=COV\left[U\right]=U^T\times U/n$$

(52)

The rank of the matrix U $\left(N\times n\right)$ is max N, so there are max N nonzero eigenvalues for $COV\left[U\right]$; then, it can be calculated the eigenvectors of the transposed matrix. Thus, if ϕ is an eigenvector of the transposed matrix, then $\mathsf{\psi }=U^T\ast \mathsf{\varphi }$ is an eigenvector of the initial matrix $C=COV\left[U\right].$ The eigenvalues are the same as for the transposed matrix. Following the normalisation of the eigenvectors, their set forms a basis in the $R^N$ space:

$$\forall {\mathsf{\psi }}_i,{\mathsf{\psi }}_j,{\mathsf{\psi }}_i*{\mathsf{\psi }}_j=\left\{\begin{array}{l}0,\mathrm{if}i\ne j\\ 1,\mathrm{if}i=j\end{array}\right.$$

(53)

We will now express all the lines of the U matrix (the former faces from which the average face was subtracted) in the base formed by the eigenvectors. Each eigenvector corresponds to an eigenvalue that represents the variation of the U matrix data along the corresponding eigenvector. Therefore, instead of using all N eigenvectors, we will only use M representatives that correspond to the largest M eigenvalues. Typically, for the facial recognition problem, the value of M is quite small (between 20 and 50). Thus, we will express the lines of the matrix U in the M-dimensional basis formed by the M vectors chosen:

$${\mathsf{\psi }}_{PCA}=\left\{{\mathsf{\psi }}_1,{\mathsf{\psi }}_2,\cdots ,{\mathsf{\psi }}_M\right\}$$

(54)

Each line will have M coefficients that are obtained by projecting that line (scalar product) onto each of the chosen vectors. Thus, an image $p_x=(i_1;i_2;\dots i_n)$ is expressed in the new space of the chosen vectors (the space of the principal components) as follows:

$$p_{\mathsf{\psi }}=\left(p_x-{\mathsf{\mu }}_x\right){\mathsf{\psi }}_{PCA}$$

(55)

We thus managed to reduce the image representation from an n-dimensional space (n = 128 × 128) to an M-dimensional space. The original image can also be roughly reconstructed from the M vectors by applying the inverse transformation:

$$p_x=p_{\mathsf{\psi }}{\mathsf{\psi }}_{PCA}^T+{\mathsf{\mu }}_x$$

(56)

It can be shown that by choosing the M eigenvectors corresponding to the largest M eigenvalues, the squared error of the reconstruction of the matrix D is minimised (any other vectors lead to a larger error). The choice of M, however, depends on the dataset D; for this, an M value can be chosen that allows a reconstruction superior to a given peak signal-to-noise ratio (PSNR), or the following method can be adopted. Given that the sum of the eigenvalues is equal to the sum of the variations in the N directions given by the eigenvectors, one can calculate the percentage of “variation” preserved by choosing K principal vectors as follows:

$$r_k=100\ast {\displaystyle \frac{{\displaystyle {\sum }_{i=1}^k{\mathsf{\lambda }}_i}}{{\displaystyle {\sum }_{i=1}^N{\mathsf{\lambda }}_i}}}$$

(57)

K is chosen so that the percentage is at least equal to a given threshold (e.g., 95%).

Given a test image $p_y$, the average image will be subtracted from it and then it will be expressed (projected) into the M-dimensional basis of the chosen eigenvectors, resulting in a set of coefficients $W=\left\{w_1,w_2\dots w_M\right\}$. The closest image to the test image will be determined by evaluating a distance (e.g., Euclidean [37] or Mahalanobis [38]) between the W vector and the W’ vectors calculated for the other images of the database.

Euclidean distance between $X=\{x_1,x_2\dots x_n\}$ and $Y=\{y_1,y_2\dots y_n\}$ is

$$d\left(X,Y\right)=\sqrt{{\displaystyle \sum _{i=1}^n{\left(x_i-y_i\right)}^2}}$$

(58)

Mahalanobis distance between $X=\{x_1,x_2\dots x_n\}$ and $Y=\{y_1,y_2\dots y_n\}$ is

$$d\left(X,Y\right)=\sqrt{{\displaystyle \sum _{i=1}^n\frac{{\left(x_i-y_i\right)}^2}{{\mathsf{\sigma }}_i^2}}}=\sqrt{{\displaystyle \sum _{i=1}^n\frac{{\left(x_i-y_i\right)}^2}{{\mathsf{\lambda }}_i}}}$$

(59)

where ${\mathsf{\sigma }}_i^2$ represents the variation along the component with index i, and in the present case, it is equal to the eigenvalue ${\mathsf{\lambda }}_i$ corresponding to the eigenvector ${\mathsf{\psi }}_i$.

In a real system, an acceptance threshold T is chosen for the minimum distance between the test and database images. If the calculated distance is lower than T, then the person is recognised; otherwise, they are not. For this purpose, two types of errors are defined as follows:

• False positives (false alarms): a person in the database is not recognised.
• False negatives (missed alarms): an external person is recognised by the system.

The digital solution for Scenario 3

Python is a great programming language for education because it can be used for various applications, including web development, data analysis, artificial intelligence, and more. This makes it a versatile language that can be applied to many different fields. Programming in Python encourages students to think logically and analytically to solve problems. This can help develop critical thinking skills that are valuable in many different fields. Python is also widely used in the tech industry, meaning that students who learn it in an educational setting will have valuable skills to help them in their future careers.

Python is a great language for education because of its simplicity, versatility, and the skills it helps students develop. That is why Python 3.12.2 was the digital environment in which face recognition was implemented. The digital solution was developed and tested on the Olivetti faces dataset.

The algorithm assumed the following steps:

✓

Database initialisation: Each entry in the dataset is composed of a person’s image, a vector of labels (a number is associated with each person) and a vector of strings (the name of the person appearing in the image).

✓

Split into a training and a test set by keeping 25% of the data for testing.

✓

Compute a PCA (eigenfaces) on the face dataset (treated as an unlabelled dataset): unsupervised feature extraction/dimensionality reduction (Figure 20).

✓

Each image is considered a vector in a $R^n$ space dimension.

✓

The PCA algorithm is applied over these vectors, reducing the size of the vector space to a smaller number (Figure 21).

✓

The reduced vectors are introduced into a machine learning algorithm (support vector machines or logistic regression).

✓

Quantitative evaluation of the model quality on the test set.

The first classification algorithm tried is SVM. At the end, the classification report is displayed as a report of some performance parameters that are automatically computed for a predictive analysis (

Table 1. Results obtained with SVM.

	Precision	Recall	F1 Score	Support
Person 1	0.23	0.14	0.18	70
Person 2	0.00	0.00	0.00	30
Person 3	0.48	0.78	0.60	165
Person 4	0.00	0.00	0.00	36
Person 5	0.17	0.10	0.13	41

).

Repeat the previous experiment for the logistic regression algorithm (

Table 2. Results obtained with logistic regression.

	Precision	Recall	F1 Score	Support
Person 1	0.21	0.17	0.19	70
Person 2	0.09	0.13	0.11	30
Person 3	0.47	0.47	0.47	165
Person 4	0.09	0.08	0.09	36
Person 5	0.07	0.07	0.07	41

).

The presented case studies demonstrate the numerous advantages of incorporating digital technology in defence and security education, including increased engagement, realistic training scenarios, enhanced learning experiences, cost effectiveness, accessibility, and performance tracking. These outcomes support using digital technology as a valuable tool for improving the quality and effectiveness of education in defence and security fields. Learning through scenarios, using digital methods, ensures the development of students’ abilities for abstract thinking, generalisation, and synthesis for specific problems in defence and security education.

4. Discussion

Digital technology offers students access to virtual manipulation, such as geometric shapes, number lines, and graphs, allowing them to explore mathematical concepts interactively and practically. When instructing mathematics to students within the defence and security education system, it is essential to utilise real-world applications and practical examples. This article proposes first to ensure that students have a strong fundamental topic of mathematics, such as algebra, calculus, and trigonometry, then connect these topics with engineering subjects relevant to their field. This approach aims to demonstrate the practical relevance of mathematics, fostering critical thinking and analytical skills in students when addressing mathematical challenges.

The future of educational methodology is expected to be more personalised, flexible, and technology-driven, with a greater emphasis on skill development and lifelong learning. In this direction, virtual reality and augmented reality can create immersive learning experiences, enabling students to engage and interact with concepts more effectively [39]. In this sense, Augmented Reality (AR) holds the potential to transform the way mathematics is taught and learned by offering an interactive and engaging platform for students to explore mathematical concepts in a more immersive way. By combining virtual elements with the real world, augmented reality can enrich the learning experience and assist students in comprehending and visualising abstract mathematical equations and theories. One approach to integrate augmented reality into mathematical education is through interactive apps and tools that allow students to manipulate virtual objects and visualise mathematical concepts in 3D [40]. For instance, students can use AR apps to graph equations, explore geometric shapes, or visualise vectors and matrices in a more tangible manner. This hands-on approach can aid students in gaining a better understanding of complex mathematical concepts by providing a more intuitive and interactive learning experience [41]. However, there are also limitations to consider like costs, connectivity, and technical expertise being required by some teachers. Despite these limitations, AR technology can help educators create more engaging and effective learning experiences for students. Additionally, it can boost student motivation and engagement, ultimately enhancing their understanding and appreciation of mathematics.

Artificial Intelligence (AI) learning tools are also a promising direction as they can provide personalised feedback and support to students, enabling them to progress at their own pace and identify areas where they may require additional assistance. Some of the key benefits of using AI learning tools include personalised learning [42], accessibility, immediate feedback, data-driven insights, and engaging [43] and interactive learning experiences [44]. Moreover, AI-driven learning tools have the potential to revolutionise education by providing personalised, accessible, and efficacious learning experiences that can assist students to attain their academic objectives and thrive in the rapidly evolving digital landscape. It is crucial for educators to be aware of the limitations and challenges of AI learning tools. Educators should carefully consider the ethical implications, potential biases, and costs associated with implementing AI tools in mathematics education and work towards creating a balanced approach that leverages the benefits of AI while mitigating its risks.

Further, using game elements in education can also increase student engagement and motivation, making learning more enjoyable and effective [45]. Numerous educational games and applications can help students practise their math skills in a fun and interactive way. These resources serve as a valuable supplement to traditional classroom instruction, reinforcing fundamental concepts and affording students additional practice opportunities [46]. By framing mathematical lessons as quests or adventures, students can perceive themselves as embarking on a mission to conquer problems and attain specific objectives [47]. This approach can make learning more exciting and immersive as students confront and surmount challenges and obstacles in their quest to enhance their mathematical abilities. Students are also encouraged to work collectively to solve problems and engage in friendly competition through mathematical games, which can foster a sense of camaraderie and teamwork while motivating students to strive for improvement [48]. While there are limitations to using game elements in mathematics education, like costs, access, and time constraints, the practical applications and potential implications for learning outcomes and student engagement make it a valuable tool for educators to consider incorporating into their teaching practices.

Blockchain technology also has the potential to establish secure and transparent credential systems, such as digital diplomas and certificates, streamlining the process of verifying academic achievements. Additionally, it can securely store and track students’ progress and achievements in math concepts and exercises [49], enabling educators to assess individual performance and provide targeted support. Furthermore, this technology can facilitate peer-to-peer learning by allowing students to share and verify math solutions with each other, promoting collaboration and the exchange of ideas within the classroom [50]. On the other hand, one limitation of Blockchain technology is the complexity of understanding Blockchain technology itself. It can be challenging for educators and students unfamiliar with Blockchain to grasp and implement the concept in educational settings. This could hinder the adoption of Blockchain technology in mathematics education. Another limitation is the cost associated with implementing Blockchain technology in education. This may prevent some educators from incorporating Blockchain technology into their teaching practices. Regarding potential implications, Blockchain can enhance collaboration among students and educators by providing a secure and transparent platform for storing and sharing mathematical data. It can also enable the creation of decentralised educational platforms that offer personalised learning experiences to students.

Personalized learning platforms are also a valuable option because they use adaptive technology to track students’ progress and provide recommendations for additional resources based on their performance [51]. They can also utilise personalised learning algorithms to propose courses aligned with students’ interests and objectives [52]. In terms of implications for mathematics education, personalised learning platforms can be a valuable tool for differentiating instruction and allowing students to work at their own pace. However, since the main limitation of these platforms is the lack of human interaction and feedback, educators must supplement these platforms with additional resources and support to ensure students develop a deep understanding of mathematical concepts.

Internet of Things devices can be used to create smart classrooms that are equipped with sensors and connected devices, allowing for more efficient and interactive learning environments [53]. While there are limitations and potential risks associated with using IoT in mathematics education, like potential concerns around data privacy and security when using IoT devices in the classroom or costs, there are also significant opportunities for innovation and improvement in teaching and learning. It will be important for educators to carefully consider how to integrate IoT technology responsibly and effectively to maximise the benefits for all students.

Furthermore, 3D printing technology offers the capability to create physical models and prototypes of concepts, allowing students to visualise and understand complex ideas tangibly. Visualisation of geometric shapes such as cubes, spheres, and pyramids enhances students’ understanding of their properties and interrelations [54]. Creating manipulatives for hands-on learning, such as fraction bars, number lines, and base ten blocks, can help students better understand mathematical concepts. Furthermore, exploring complex equations and functions allows students to visually explore how changing variables affect the shape of the graph [55]. Customising learning materials can be used to create custom learning materials tailored to individual students’ needs. For example, teachers can create tactile graphics for visually impaired students or custom puzzles for students who learn best through hands-on activities [56]. In this way, 3D printing technology also holds the potential to revolutionise mathematical education by making abstract concepts more concrete and compelling for students. By addressing challenges such as cost, training, and curriculum integration, educators can harness the power of 3D printing to enhance teaching and learning in mathematics.

Overall, digital technology advancements are anticipated to revolutionise mathematical education by making it more personalised, interactive, collaborative, and data driven [57]. These transformations have the potential to improve student engagement, comprehension, and performance, ultimately leading to enhanced outcomes for students in mathematical education. Accessibility, equity, privacy, teacher training, and digital literacy are the challenges to be considered by schools when implementing digital tools and technologies in teaching practices. There should be a constant interest in professional development to ensure inclusive and effective technology integration in education. Educators should receive ongoing and relevant training on effectively integrating technology into their teaching practices. They should encourage collaborative learning through technology and design their lessons using the principles of universal learning design, which aims to create flexible and accessible learning environments that accommodate the needs of all students. Lastly, schools must ensure that all students have equal access to technology, regardless of socioeconomic background.

5. Conclusions

While analysing the curricula of certain partner institutions [12], we observed that students in the defence and security fields could significantly benefit from an innovative and practical-oriented approach to learning mathematics through various digital tools, some of which are exemplified in this paper. The proposed teaching methodology is a mix of the traditional method of teaching mathematics and the alternative method based on digital technology. It is designed to provide practical knowledge and experience in solving different mathematics-related technical problems, which is particularly beneficial for first- and second-year students. Furthermore, teachers in the defence and security fields can enhance their competencies by researching, proposing, and developing new teaching methods for various scenarios based on mathematics that can be digitally transformed for better presentation and understanding.

Implementing innovative teaching methods with the aid of digital technology and even using different problematisation strategies requires a shift in mindset and a willingness to try new approaches. By fostering creativity, critical thinking, and collaboration in the classroom, students can be prepared to use all the resources from the digital age. Case studies and best practices such as Classroom Model, Virtual Reality in Education, and Digital Assessment Tools demonstrate the effectiveness of using digital tools in the learning process to improve student engagement, personalise the learning experience, and enhance academic performance.

However, innovative teaching methods can be associated with potential challenges or limitations, such as resistance to change, a lack of ability to think outside the box, or the lack of resources and support from the institution while developing new teaching methods. Therefore, institutions need to recognise the value of these innovative methods and provide the necessary resources and support.

New teaching methods can sometimes lead to unforeseen challenges or problems that teachers may have yet to anticipate. Teachers need to be flexible and willing to adapt as they encounter obstacles. Some strategies for overcoming any barrier and ensuring successful implementation could be by providing ongoing training and support for teachers to effectively implement innovative teaching methods, ensuring that the innovative teaching methods align with curriculum standards and learning goals, involving stakeholders in monitoring the implementation of innovative teaching methods, and gathering data on their effectiveness.