The present era is a digital one where youth and learners have been exposed to digital technologies since birth, growing up in the generation of the internet. These technologies offer students opportunities to make researches and gather information through the internet. The swift transformation of internet technologies allows for greater educational opportunities, a chance for advancement. According to Prensky (2001), our students have changed radically; today's students are no longer what the educational system was designed to teach. In this digital age, youth prefer graphical contents to texts. They like being rewarded for tasks, have little or no patience for lectures, step-by-step logic and instructions. Hence, there is the need for teaching modes to be modified to meet the needs of the students.

Digital game based learning approach is a panacea for solving many problems in schools and training such as enhancing students' learning motivation and enabling hands-on learning experience (Westera, 2015). The growth of entertainment and increase in the production of smart phones and tablet computers has increased the expectations of learners. As a result, learning environments that are highly interactive and visually stimulating are now increasingly desired. The increasing pervasiveness of computers and mobile digital devices within homes has opened a new market for game-based learning. The growth of haptic interfaces is also exposing learners to more interactive interactions with computers, this has created a sort of parallel school. Squire et al. (2005) introduced a new technological trend where students get to author and design their own game-based learning environments. Del Moral and Fernández (2015) said that digital game-based learning assists the elementary classroom students to improve in both mental and calculation learning and also allows them to apply that knowledge to daily life.

Filsecker and Hickey (2014) emphasized the extent to which digital video games increase students' degree of involvement in their tasks. The adoption of a methodology that revolves exclusively around using digital games has the motivating effect of capturing the learners' attention.

Digital game-based learning is a type of game played with defined learning outcomes. Most learning approaches that are in accordance with the traditional way and textbooks often seem esoteric with their many steps, theories and case studies. The game-based on the other hand has abundant characteristics such as representation, fun, play, goals, outcomes, feedback, win states, competition/challenges, problem solving, task, story and so on to increase learning motivation of students. Gamebased learning allows students learn by bringing the entertainment of game, fun and interactiveness into education, achieving the purpose of edutainment. This gaming technology has brought a lot of educational packages that focused on steering hands-on-practices, critical thinking and constructive knowledge.

Several subject areas especially mathematics have been developed with the use of gaming technology such as learning fractional ordering using Lego game (Agbonifo & Ogunmoroti, 2015); learning simultaneous equation (elimination by substitution) using chess game (Agbonifo & Onu, 2017) and learning algebraic factorization using game theory of tic-tac-toe (Agbonifo & Ofueu, 2018). Hence, the paper presents the development of digital game based learning of the determinant of a matrix using snake and ladder game.

There are various ways by which computers can be used to support education. The two common use of technology in education is e-learning and edutainment which make use of the traditional “watch and practice” approach. Even though it is widely accepted that e-learning and edutainment tools can aid the learning of facts, it has been shown that these tools can only do so much when it comes to helping students to develop perception of facts, boost cognition and skills. This is mainly due to the fact that these tools have been poorly designed, boring, simplistic and repetitive and does not allow the students to try out new things or explore new means (Prensky, 2008) Technology such as digital video games is widely used to aid education.

Trybus (2012) stated that game-based education has many advantages: It is cost-effective, has low physical risk or liability to the learner, has standardized assessments for student-to-student comparisons, is highly engaging, has a learning pace tailored to the individual needs of the students immediate feedback responses for students' mistakes, can easily transfer learning to a real-world environment, and is engaging for the learner.

Kickmeier-Rust and Albert (2010) said that there are many advantages to teaching with the technology, such as the rich potential of visualizations and animations that provide new insights and perspectives, information access via ubiquitous means, possible self-directed and selfregulated learning or exchange and collaboration. Huizenga et al. (2009) ascertained that game-based learning is highly motivational and engaging to learners, causing learners to become completely immersed in their learning. Game-based learning actively engages the participant (Bloom, 2009). According to Twigg (2011) game-based learning is fun, motivating for learners, and encourages learners to learn from their mistakes along the way. Game-based learning provides learners with opportunities for competition and engagement and provides immediate rewards to the learner.

Understanding mathematics is crucial for students yet mathematics is learned mostly by routine with very little understanding or possibility for transfer. There is a need for teachers to make math fun and meaningful for their students to motivate them to learn and study mathematics (Ahmad et al., 2010). It was also noted in their study that many students have problems learning mathematics for the following reasons: lack of motivation, boredom, lack of personal meaning to them, little or no encouragement for selflearning and lack of continuity and focus.

Arbaugh et al. (2010) ascertained a need to improve mathematics education in schools today, and improving student learning as the main aim. They also stated that maximizing the technology use in schools and classrooms could help to improve the student learning. Mathematical and communicative or collaborative technologies are two technology types described as useful tools for learning and teaching of mathematics in today's classrooms.

Twigg (2011) said that there was a necessity to deploy technology into math curricula because technology is useful for student learning in society. Furthermore, there are key components such as interactive software and computers that enable students to learn math by doing. Even more importantly what mathematics offer is a way of thinking and doing things. In order to solve mathematical problems, there must be a right attitude for problem solving in a systematic manner. This calls for an approach that is ambitious and coherent that teaches mathematics effectively. The approach should be ambitious in the sense that it seeks to achieve a higher aim rather than a narrow one. Students should feel the need to solve problems, and the approach itself must be activity oriented.

Van-Eck (2006) expressed the need to match digital game-based learning with the needs and experiences of the learner to maintain the effectiveness desired in the learning experience. The researcher also suggested the need to align Game-Based Learning (GBL) experiences with the necessary curriculum and the content being learned. The researcher emphasized that games should “exemplify well-known principles and models” of learning and such digital games should provide opportunities for situated cognition for learners and apply play theory as a means of learning.

Ke (2008) conducted a mixed-methods case study using qualitative and quantitative data to determine the effectiveness of using game in learning math at the upper elementary school level. The study determined that students' attitudes were positively enhanced from using games to learn mathematics. The researcher also determined that the game design plays a significant role in students' interaction with the game.

Gillispie et al. (2009) carried out a study and used participants of five hundred (500) middle school students in rural North Carolina to determine their achievement and attitudes of using the games - Dimension-M and Xeno Island. These games are focused on problem-based concepts using digital games and utilizing concepts in prealgebra and algebra. The researchers used a pre- and post- test design as well as an attitude survey to formulate results. Students increased in math achievement with an average of seventeen percent. The researchers also noted that students were willing to repeat the game-based learning mission to make improvements in their scores on the computer.

Kim and Chang (2010) performed a quantitative study to determine the effectiveness of digital math games for fourth-grade students. The researchers used data from the 2005 National Assessment of Educational Progress data and item response theory scale scores to measure math achievement. It was observed that there is a greater overall math achievement level for male students compared with female counterparts. Furthermore, English-language learners had higher math performance than non-Englishlanguage learner students who did not play games prior to this experience.

Brom et al. (2010) developed a digital game called Europe 2045 which is effective for learners because it has intelligibility, promotes social role-playing, makes use of real data, incorporates storytelling, and provides teacher support. Likewise, Huizenga et al. (2009) discovered that learners were highly motivated, had fun and gained knowledge in playing the project-based history game.

Sardone and Devlin-Scherer (2010) performed a mixedmethods study with twenty-five (25) undergraduate students in teacher education to determine the 21^st century skills utilized in educational games. The participants reviewed fifty (50) games for specific criteria of motivation, critical thinking, problem solving, collaboration and communication. The researchers discovered that digital games possessed many of these 21^st century features.

Yien et al. (2011) conducted a quantitative study to determine the influence of game-based learning on knowledge of nutrition, attitude and habits of students via computer game usage. The participants were sixty-six (66) third-grade students from two elementary schools in Taiwan. It was discovered that there is a significant difference in achievement for the experimental group on nutrition knowledge. However, there is no significant difference in student attitudes or gender performance.

Erhel and Jamet (2013) investigated the effects of digital game-based learning (DGBL) on learning and motivation whose main objective was to identify the conditions under which digital game based learning is most important by carrying out analysis of the effects of learning instruction versus entertainment instruction. The outcome showed that the learning instruction produced deeper learning than the entertainment without impacting negatively on motivation. It was shown that if learners are given regular feedback about their performance, the entertainment instruction would result in deep learning. Although it was demonstrated that a serious game environment can promote learning and motivation, providing it includes features that prompt learners to actively process the educational content.

Holz et al. (2018) designed a mobile game for dyslexic primary school children in German which incorporates gaming elements such as narrative, pedagogical agents, tutorials, feedback and reward mechanisms. The gaming elements are evaluated using sixty-three (63) children who interacted with the game at home for a period of 9-10 weeks. The results showed overall positive perception of the game's graphics and elements to children. The results encouraged the use of the game elements in digital game-based learning environments.

Chen and Chang (2020) conducted two empirical studies to investigate the effectiveness of virtual and real competition in digital game based learning. A case of academic English competition close test is used for this purpose. The first study emphasized on the impacts of the real competition game while the second study anchored on difference between the virtual competition and the real competition game. A total of fifty eight (58) students participated in the game with twenty eight (28) participants assigned for the first study and thirty (30) participants are assigned for the second study. The outcome from the studies was that the first study showed that students made improvement between each task in the real competition game. The outcome from the second study demonstrated that students with virtual competition significantly had better learning performance than those with real competition.

Yeh et al. (2019) carried out empirical evidence of the benefits of mindfulness by incorporating mindful learning experience in digital game-based learning of creativity (IMLED-DG). The elements such as achievement goal, selfdetermination, mindful learning and mastery experience are investigated through experimental instruction in digital game-based learning of creativity. One hundred and eighty-one (181) of 3^rd to 6^th graders are used in the development of IMLEDG, and ninety-five (95) of 3^rd and 4^th graders are included in the six-week experimental instruction through the DGLC. The results showed that the IMLE-DG has good reliability and validity with Cronbach's coefficient value of 0.974. Furthermore, analytical results showed that achievement goal and self-determination have positive influence of mastery experience through mindful learning experience, which suggest that mindful learning is a critical mediator of mastery experience in digital game-based learning of creativity.

Several other applications on gaming technology have been developed that focused on these specific subject matters: learning fractional ordering using Lego game (Agbonifo & Ogunmoroti, 2015); learning simultaneous equation (elimination by substitution) using Chess game (Agbonifo & Onu, 2017); and learning algebraic factorization using game theory of Tic-Tac-Toe (Agbonifo & Ofueu, 2018).

The system design presents description of the game, system architecture, conceptual diagram and system modules.

3.1 Game Description

The story of the digital game describes the case of a powerful kingdom on the verge of collapse in an alien invasion and the possibility of the kingdom emerging as the conqueror resting solely on a powerful weapon which is in a very strange location. Hence, with the knowledge of this weapon known to the king, he has to make a decision to select the most powerful and brave warrior in his kingdom to embark on a journey to retrieve this weapon. The route to this powerful weapon is filled with snakes that delays the journey of anyone passing through and also ladders that make the journey faster and easier. Nevertheless, there exists three black tunnels in which the warrior has to pass through inevitably and the possibility of the warrior passing through the three black tunnels successfully rests on the warriors ability to resolve the causal factor of a two dimensional matrix (3x3), also known as the determinant of the matrix which is dynamically generated when the warrior starts his journey.

The method the warrior is expected to follow is the co-factor expansion method which consists of three main steps. The first step which has to be computed on getting to the first black tunnel is for the warrior to compute the three minors of the first row which are represented by these variables such as m11, m12 and m13 respectively. After supplying correct values for the minors, the warrior continues his quest for the powerful weapon and the status evaluator for the first step is updated as completed. The second black tunnel is where the warrior is expected to compute the co-factors denoted with variables such as c11, c12 and c13 respectively using the values of the minors computed at the first black tunnel. If the co-factor values are correct, the warrior is allowed to continue his quest and the status evaluator for the second step is updated as completed. The last black tunnel represents the last step for determining the causal factor for the 3 x 3 matrix and the warrior is expected to compute the determinant of the matrix using the values of the co-factors that were generated at the second black tunnel and the first row of the 3x3 matrix. If the determinant value is correct, the status evaluator for the third step is updated as completed, and the warrior can once again continue his quest for the powerful weapon. The warrior is expected to get to the location of the weapon after completing all these tasks before his time is up. If all the above conditions are met, the warrior is presented with the ancient weapon and the aliens can then be destroyed with the weapon.

3.2 System Architecture

The system architecture is depicted in Figure 1 which consists of game interface, input, game logic, javafx platform, database and output. Each of the components is described as follows:

Figure 1. System Architecture

3.2.1 Input

Input in the context of this game architecture ranges from the dice values to the step answers that are computed at predefined states. The input is the entity that the game logic evaluates to check if they satisfy their requirements or are in compliance to the rules of the game.

3.2.2 Game Interface

The game interface is simply an intermediary between the game and the player. The game interface consists of both the visual components and non-visual feedback. The visual components include components on the screen that can be read such as graphic icons, text and others, and the visual area that can be interacted with are characters, buttons and such, while the non-visual feedback includes sound, haptic feedback from controllers and so on. The interface also includes input devices that are not necessarily shipped with the game, but are used for interfacing with the game, like keyboards, mice, controllers, touch screens etc.

3.2.3 Game Logic

The game logic is the set of instructions used to instruct the game on how to perform all the underlying tasks properly. It consists of all the rules of the game and the steps that are to be followed to arrive at the intended point or predetermined end.

3.2.4 Input Evaluation

Input evaluation aims to check the validity and correctness of the input using rules defined in the game logic. Evaluation is important to access the quality of the input value before the programming platform carries out the final parsing and before values are stored in the database or processed as outputs.

3.2.5 JavaFX Platform

The JavaFX platform is the underlying system serving as the game engine on which the game is built. It is tasked with interpreting all the program code to a language the computer can understand and execute.

3.2.6 Database

The database is to store the scores of players and also the states of the game.

3.2.7 Output

The output refers to where the learning outcome of the game is displayed.

3.3 System Flowchart

The system flowchart is shown in Figure 2 which begins with an oval component signifying the start of the game environment with a process menu from which the user can select a process. After the user has made a decision on what process to select, the flowchart design goes to the appropriate states. If a user selects a process other than the start game process, then the user is either shown the score board or he exists the game. If the decision from the user is to start the game, a prompt that mandates a user to input his/her name is displayed to the user and the game starts after which the game countdown begins. After all these steps have been followed in the right order, the main game environment loads up and the user rolls the dice to start the game play. A number between one and six is generated after the dice has being rolled which allows the user to make a move and the game state is updated. At specific intervals, there is a test to ensure that the game tasks have been completed and if this test fails, the player rolls the dice again and keeps playing until all the tasks are completed within the time frame given to all players. If the player can complete all the tasks before the time runs out and get to the finish position, the player wins the game.

Figure 2. System Flowchart

3.4 System Modules

The game modules describe all the different game stages in the development of the game. Detailed description of how all the modules interconnect and the importance of each module are shown with the aid of necessary screen shots. The entire learning platform, the integration which the game used and how it should be played for the student to get to the ultimate in the learning concept are also discussed in this section.

3.4.1 Game Entry Point

The game entry point is a stage as shown in Figure 3 with three buttons which serves as the starting point where the student begins to interact with the learning platform. The first button as the name denotes allows the students to start the game by opening up another game stage where each student inputs his name before the main game stage loads up and the second button shows a stage that contains all the scores of every player who has played and has won the game. This allows the new student to be aware of the leading scores and instils a sense of competition into him or her before the game starts. The third button which is the exit button allows the player to leave the game environment without having to start the game at all.

Figure 3. Game Entry Point

3.4.2 Main Game Page

When the student has successfully entered his name, the main game learning environment loads up which describes the starting state where the player is in position of one of the game boards and the quest to retrieve the ancient weapon by solving for the causal factor of a 3 x 3 matrix using the co-factor expansion method as shown in Figure 4. The co-factor expansion of solving for the causal factor is a mathematical method which has three steps. The first step of this method is to find the minor values of the matrix which is used in the second step to compute the co-factor values and the last step of the co-factor expansion method is to find the causal factor, also known as the determinant of the matrix by using the co-factors calculated in the second step. This method follows a stepby- step procedural method which is shown distinctly in the game play. Stepwise manner of solving mathematical questions is one of the main goals of the digital gamebased learning in the area of mathematics because it teaches the students the steps it takes to solve the problem as opposed to skipping to the final answer. This greatly develops the problem solving skills of the student and helps them to approach other related problems in a stepwise solvable block manner. The steps are intuitively integrated into the logic of the game so as to bring the fun of playing games together with the main aim of playing the game which is to help the student learn the concept in an easy and simplified manner where each step represents a tunnel that must be crossed before the game play can continue. The time allotted for the game play is 600 seconds (10 minutes). The reason for incorporating time constraint is to take away the lethargic mindset most students have where they basically think time management is not something to be taken seriously. It also helps them learn how to solve the problem faster in case they have to do it in a time constrained environment, for example, in a typical examination setting or class test.

Figure 4. Main Game Page

When the player rolls the dice, the question is randomly generated and the countdown begins. Random questions are generated so that the student would not have to solve a particular question multiple times which would defeat the purpose of the whole system. Having to solve different sets of questions each time helps them to understand the concept as much as possible with different use cases. The story of the game earlier described starts where the player aims to get to the finish line to uphold the glory of his kingdom. The first step which is to compute the minors is shown in Figure 5 and a hint to guide the player to arrive at the answer is shown in Figure 6.

Figure 5. Game Stage showing Minor Values Input Dialog

Figure 6. Game Stage showing Minor Hint

When the minor values have been supplied, there are two possible cases, which are, either the values are correct or the values are wrong. If the values are correct, the player is allowed to continue playing and the correct value popup is shown to the user which informs the player that he or she can continue playing the game and the status for the first step is updated. If the values are wrong, the player descends back to check point one which is board position one.

The next step when the minor values have been validated to be correct is for the player to continue game play and the second mathematical step is to find the co-factors using the values of the minors earlier computed at tunnel one. It is mandatory to get the right step to the finishing line before moving ahead to the next one as the output of the preceding step serves as the input to the succeeding one. This helps the student to focus on solving problems in a stepwise manner. Figure 7 shows the stage where the player is expected to input the correct values of the co-factors computed, and an optional guide shown in Figure 8 shows how to solve for the co-factors. The guide is available in case the student gets stuck or forgets how to solve a particular step. The primary aim of the game is to teach and learn, so it is important to provide guides where and when necessary.

Figure 7. Game Stage showing Cofactor Values Input Dialog

When the step has been followed and the right co-factor values have been computed, the player is allowed to continue his/her journey and the correct value stage is displayed to the user, but if some or all the values are wrong, the player goes back to the second check point and an error message is displayed.

Figure 8. Game Stage showing Cofactor Hint

The third and last mathematical step of the game is for the player to compute the causal factor using the value of the co-factor computed in step two. The third step can only be calculated using the output of the previous steps which addresses the need to follow the steps as it should be followed and not try to skip steps. Figure 9 shows the stage where the player is expected to input the correct value of the causal factor, and a hint button that provides a guide to the solution in case the student has forgotten how to solve for this step is shown in Figure 10. Hints and guides introduce simplicity and also give the players the impression that they are not just interfacing with computers; it helps to fill the void of physical instructors in the whole process.

Figure 9. Game Stage showing Determinant Input Dialog

Figure 10. Game Stage showing Determinant Hint

To compute the causal factor, the player finds the sum of the product of the co-factors. If the computed causal factor is correct, it implies that the co-factor expansion process has been completed, but if the causal factor is wrong, the player goes back to the third checkpoint and the error popup is also displayed.

When the player completes all these mathematical steps, the learning phase can be said to have been completed at this point. The player can then focus on the fun the game offers. The last task is for the player to make effort to get to the finish line before his/her time runs out and the ancient weapon is presented to the player with his/her score as shown in Figure 11. At this point, the player can decide to download the text file containing all the steps that was followed to solve the question which opens up a dialog that allows the user to select a directory where the file will be saved with the “txt” file extension. The text file serves as a revision note that can be referenced at any time to relearn the entire process or to teach someone else. It contains the question and the stepwise solution.

Figure 11. Game Stage showing a Completed Quest

3.4.3 Game Result

Twenty students of Senior Secondary School (SSS) III, Federal University of Technology, Akure were gainfully engaged and participated in the learning process. The scores of students' learning outcome are shown in Figure 12.

Figure 12. Leader Board Stage

The score is determined by the current time a player takes to win the game. Figure 13 shows the work sheet of one of the students who is gainfully engaged in the playing of the game which demonstrates the stepwise solving of the problem. The player would select his/her preferred directory to save the text file and the text file can be opened with a text editor. The usefulness of this feature is that the students can later refer to this file to get more understanding of the concept they are trying to learn; it can also serve as a class note.

Figure 13. Game Stage showing the Worksheet of the First Player

Performance evaluation was carried out using a survey to determine the following variables/criteria such as: objective of the game; critical thinking and motivation; usefulness of game features in solving the problem; ease of use and learning outcome. The survey consists of twenty questions which was administered to twenty participants from senior secondary school (SSS) III students, Federal University of Technology, Akure. There were eleven (11) males and nine (9) females and their age was between 14- 16 years. The survey used five-point likert scale such as Excellent (5), Good (4), Average (3), Poor (2) and Very Poor (1). The responses from the participants were captured and analyzed using frequency distribution and weighted mean. The points shown in Table 1 were obtained from the analysis of the survey with twenty respondents.

Table 1. Responses of the Participants of the survey

Based on the responses of the respondents with respect to variables' weight assigned to the questions, Table 2 shows the analyzed values which were computed by finding the product of the points and the number of respondents for a particular question.

Table 2. Response of Participants Based on Variables Weight Assigned to Questions

The combination of survey questions which explicitly form each of the evaluating criteria for the performance of the system is depicted in Table 3. This shows the number of question items in the survey that uniquely describe the features of each criterion. Each of the evaluating criteria has significant contributory factor to the overall learning performance on the part of the students.

Table 3. Survey Variables with respect to Questions

The summation of the points is based on the classification of the questionnaire items into variables which underline the objective that determine the significance of the system on students' learning performance using method of percentage representation of values. The method used aligns with the mathematical representation of values as in Equation 1.

(1)

Percentage points values is further defined relatively as in Equation 2:

(2)

The percentage points for the variables were computed as shown in Table 4. A high, moderate and low percentage point values indicate that the evaluating criteria have great, moderate and low contributing effect on the overall students' learning performance respectively.

Table 4. Percentage Point Values for the Survey Variables

The result of the percentage point values as in Table 4 shows that all the variables or evaluating criteria conformed to the underlined foundation of learning theories that aligned with gaming technology based on the fact that all have high values. Hence, it is inferred that the developed system effectively integrates the game objective and supports critical thinking and motivation. The game features assists in achieving the desired result and most importantly, learning outcome variable supports that overall learning achievement is enhanced.

Conclusion

The world is constantly changing, and technology has found a way to improve the way things are done in almost every sphere of life. Education must keep up with the trend if the needs of the students are to be met. From the literature, there are many learning systems found on learning theories that support the use of technology as a means to improve education. Research has also shown that digital game based learning is not only a needed tool or paradigm to teach mathematics but also improves overall mathematics achievements.

Matrix as the subject matter is a very important topic in mathematics due to its wide application in various fields such as flight modeling, military combat, gaming and others. Matrix always comes as a challenge to some students mainly because of the way it is being taught. Hence, this research paper developed a digital gamebased learning environment for learning matrix that focused on the game theory of Snake and Ladder that depicts a stepwise problem solving process. A survey was carried out using five evaluating criteria to determine the performance of the system. The result of the survey demonstrated that the system is significantly suitable to realize the underlined purpose of the design.