According to Maslow’s Hierarchy of Needs (Maslow & Lewis, 1987), cognitive needs rank as one of the three highest levels of needs. This is consistent with the Self-Determination Theory (SDT) (Ryan & Deci, 2017) which identifies autonomy, relatedness and competence as three innate psychological needs of human beings. When these needs are fulfilled, it is more likely that learners in the classrooms become more self-motivated and independent in learning (Fredricks et al., 2004)

Independent learning, or self-directed learning, is the process in which students set their own learning goals and plan the process of achieving those goals ((Metallidou & Vlachou, 2010); (Knowles, 1975)). Students also make independent academic decisions and engage themselves in activities to achieve these goals. Self-directed learning has been defined as the acquisition of knowledge by individuals for themselves through utilising available resources and taking charge of his own learning, without the help of others ((Herlo, 2017); (Din et al., 2016)). In self-directed learning, teachers serve as facilitators of learning, rather than transmitters of knowledge. Students are given the autonomy to choose their own learning resources and strategies, and they actively participate in self-assessment based on the learning objectives they have set for themselves (Knowles, 1975).

Self-directed learners in mathematics usually exhibit dedication, curiosity and independence in the learning of the subject ((Bishara, 2020); (Sumantri & Satriani, 2016)). They also tend to use their social interactions with their peers and teachers to work on their mathematical problem-solving skills and develop higher mathematical abilities (Bishara, 2020). Furthermore, self-directed learners take responsibility for their own learning ((Khiat, 2017); (Tan & Koh, 2014); (Bagheri et al., 2013)) and display the initiative to monitor their own progress in meeting the learning objectives. They also tend to make connections between different disciplines and form relationships between formal and informal education (Khiat, 2017); (Tan & Koh, 2014);(Bagheri et al., 2013)). Additionally, they are highly inclined to engage in problem identification and are constantly searching for new perspectives of thinking and assigning meaning to what they have learnt (Bishara, 2020).

(Merrill, 2002) proposed the application of the First Principles of Instruction in the classrooms. The First Principles of Instruction consists of five principles: the problem-centred principle, activation principle, demonstration principle, application principle and the integration principle.

The problem-centred principle is enacted when teachers engage their students in solving real-world problems. The activation principle comes into play when teachers use review or a quiz to activate existing knowledge as a foundation of new knowledge. Students are introduced to new knowledge via the demonstration principle. The newly acquired knowledge is then applied to solve problems to enhance the learning process, which is known as the application principle. Through the integration principle new knowledge is integrated into the learner’s schema.

According to the SDT, an intrinsically motivated student will likely engage in learning for its inherent satisfaction (Ryan & Deci, 2017). Researchers found that students who are given the autonomy to learn in classrooms are more likely to be intrinsically motivated, more curious and more likely to be challenged by their own learning (Deci et al., 1981). On the other hand, students who learn in a controlled environment tend to lose the desire to learn or learn less effectively (Utman, 1997).

According to the 2x2 Model of Achievement Goals ((Elliot & McGregor, 2001); (Pintrich, 2000)), students potentially have four types of goals: performance-approach, performance-avoidance, mastery-approach and mastery-avoidance goals (Table 1). Mastery-approach goals are likely to lead to increased engagement ((Harackiewicz et al., 1997)(Harackiewicz et al., 2000)) and achievement in academia (Linnenbrink-Garcia et al., 2008), while performance-avoidance goals are not likely to be beneficial to learning outcomes (Maehr & Zusho, 2009).

Gamification is defined as the deliberate and planned incorporation of game-like elements into a non-game environment, with the intention to increase students’ motivation to learn by making the learning process interesting (Karamert & Kuyumcu Vardar, 2021). Gamification does not equate to game-based learning, that is, using a game to learn concepts (Leong & Toh, 2021). It is a pedagogical approach that is “reflective of a thoughtful approach to integrate characteristics of games into learning through an intentional approach” (Smith, 2018). Many researchers have shown that the inclusion of gamification in mathematics classrooms can help to channel students’ focus towards the learning of mathematics ((Karamert & Kuyumcu Vardar, 2021); (Yanuarto & Hastinasyah, 2023)), and allow learners of different abilities to participate effectively at their own pace ((Sezgin et al., 2018); (Huang & Soman, 2013)).

From a psychological perspective, the aims of gamification include internalising extrinsic motivation of students - for students to become intrinsically motivated and self-directed in their own learning - and providing feedback for students of their learning as they progress ((Karamert & Kuyumcu Vardar, 2021);(Hamari, 2013); (Sailer et al., 2017)). Similar to a man who manages to successfully cross a river does not need the raft which he has built earlier to cross the river, the ultimate goal of gamification is for students to not further rely on gamification once they have become self-regulated in their own learning (Nicholson, 2013).

There are three possible types of feedback that can be incorporated into gamification - granular feedback, sustained feedback and cumulative feedback. Granular feedback means providing feedback directly to students’ actions ((Lo & Hew, 2018); (Sailer et al., 2017)). This can be executed by giving reward points to students when they complete tasks which are representative of a students’ learning progress in a gamified environment ((Werbach & Hunter, 2015); (Sailer et al., 2013)).

Sustained feedback involves the tracking of students’ progress over time ((Lo & Hew, 2018); (Sailer et al., 2017)) and can be implemented by having a progress bar or a levelling system where the points that the students gain contribute to the progress bar or level (Sailer et al., 2013). Cumulative feedback involves assessing a cumulation of students’ actions throughout the course of learning ((Rigby & Ryan, 2011); (Lo & Hew, 2018)). This can be achieved by having a leaderboard to track the points in which each individual student has accumulated. The leaderboard is a form of comparative feedback representing students’ ranking in a gamified environment (Codish & Ravid, 2014), with their successes based on a designated set of criteria (Costa et al., 2013-10). Studies on the effects of gamification have also shown that feedback in the form of points and rewards, immediate feedback and leaderboards could result in greater academic achievement and engagement, satisfaction and enjoyment (Chan et al., 2017) as well as motivation (González et al., 2016).

Acknowledgement in the form of badges or skill levels encourages engagement and provides a clear indication to students of their achievements (Toda et al., 2019). Badges are representations of students’ achievement and merit that students acquire in a gamified environment ((Werbach & Hunter, 2015); (Anderson et al., 2013)). This will likely lead to increased motivation, engagement (Ding et al., 2017) and academic achievement (Pechenkina et al., 2017) in students.

Other possible gamification elements that can be implemented for classroom instruction include the use of objectives, avatars and time pressure. Objectives promote engagement and motivation as it guides students' learning (Toda et al., 2019). Avatars are pictorial representations of students in a gamified environment (Werbach & Hunter, 2015), and they offer the players a freedom of choice in the gamified environment. This feature indeed fuels their need for autonomy in learning ((Annetta, 2010); (Peng et al., 2012)). Lastly, time pressure allows for students to experience a stressful yet exhilarating environment, and can be used to promote engagement, academic achievement (Spires & Lester, 2016) and motivation (Toda et al., 2019).

According to(Cook, 2013), the deliberate incorporation of game-like elements must be seamless. Much thought and intentionality are required to be able to reflect the true characteristics of games while achieving the learning objectives, and to reduce any possible drawbacks. (Arnold, 2014) proposed that by incorporating badges and points systems in a classroom environment does not necessarily mean that the learning environment is gamified. Hence, designing a curriculum that encompasses gamification can be time consuming and must be done with much deliberation to attain the educational objectives intended (Hanus & Fox, 2015).

Other researchers claimed that excessive use of badges, levels, etc., can lead to students being overdependent on extrinsic motivation, allowing intrinsic motivation to falter in comparison ((Hamari, 2013); (Deterding, 2011-05)). Students may be more inclined to achieve more points or a higher level, rather than focusing on what they are learning (Hamari, 2013), resulting in the superficial application of gamification.

Gamification could also lead to “zombification” (Conway, 2014), which is the irrational quest for external rewards. This could also contribute to unnecessary academic stress if students are excessively dependent on it, or recognise their worth by their points, progress and rankings ((Juul, 2013); (Stott & Neustadter, 2013)). (Haaranen et al., 2014) cautioned that some students may become distracted by the interactive elements of the game instead of the content being taught.

The van Hiele’s model proposes five levels of understanding geometry: Visualisation, Analysis, Informal Deduction, Deduction and Rigour – with the abstractness of geometrical concepts increasing at each level (Crowley, 1987). The model made three assumptions about geometric thought: Firstly, geometric thought is hierarchical. Students must proceed from one lower level to the next higher level in order to function well at a certain level. Secondly, a student’s progress through the levels is more dependent on the content and method of instruction rather than age (property of advancement). There is no method of instruction that allows students to skip any level of geometric thought and methods of instruction that facilitate a student’s progress is preferred. Thirdly, “the inherent objects at a particular level become the objects of study at the next level of geometric thought” (Crowley, 1987). For example, at the first level of visualisation, students may only perceive the form of a geometric figure and its properties are not discussed. Once students proceed to the second level of analysis, students become conscious of the properties of the figure (property of extrinsic and intrinsic). Each level of geometric thought corresponds to its own set of linguistic symbols and their relationships ((Crowley, 1987). Consequently, a relation that is classified to be true at one level may be modified at another level. The use of appropriate language and relations is extremely important once students hit the level of analysis. It is noteworthy that the match between teachers’ instruction and students’ level of geometric thought is crucial. If the instruction is more advanced than that of the students’ level, the student may not be able to effectively follow the thought process, leading to the students’ stagnation at a certain level. On the other hand, the students’ learning will not progress if the instruction is less advanced than the students’ level.

Based on the literature review conducted, we synthesized key learning points in the implementation of gamification in our classrooms to nurture self-directed learning in geometry. Firstly, the teacher has to correctly ascertain the level of Van Hiele’s geometric thought that students are working at in order to utilise methods of instruction (mismatch) and language (linguistics) that are suitable to the students’ level. The teacher needs to sequence the materials and topic of study, deliberately choosing what to teach and what not to, according to the level of geometric thought (extrinsic and intrinsic). This is based on the assumption of a good knowledge of students’ level of geometric thought.

In addition, gamification elements could be used to encourage students to set mastery-approach goals instead of performance-avoidance goals. For example, instead of giving badges for hitting a certain number of points, the badge could be used for rewarding skills based on achieving various mathematical competencies (e.g., such as distinguishing acute and obtuse angles or identifying similar and congruent figures).

We propose that in order to utilise gamification as a pedagogical approach in the classrooms, the teacher must be able to incorporate game-like elements seamlessly. The teacher must be able to pick and choose suitable elements for different class profiles.

In answering our research question to integrate gamification to nurture self-directed learnerswe propose that the teachers align their practices in the classroom to the First Principles of Instruction (Merrill, 2002) and van Hiele’s phases of l learning (Crowley, 1987), as shown in Table 2.

In addressing the arguments presented by (Cook, 2013), (Arnold, 2014), and (Hanus & Fox, 2015), we propose a framework to guide teachers in picking and choosing the elements of gamification to facilitate teachers to seamlessly integrate gamification into classroom instruction. Our framework also addresses the concerns of (Hamari, 2013), (Conway, 2014) and (Haaranen et al., 2014) as our model involves the integration of learning principles such as van Hiele’s phases of learning (Crowley, 1987) and Merrill’s First Principles of Instruction, instead of relying solely on the principles of gamification, which can potentially result in the superficial application of gamification as argued by (Hamari, 2013). In this way, we can ensure that elements of gamification in mathematics classrooms are implemented deliberately and suitably at the different phases of learning, with the intention of internalising students’ extrinsic motivation.

We propose the facilitation of the setting of mastery-approach goals instead of performance-avoidance goals via the application of objectives, badges and skill levels. Furthermore, students will also be able to monitor their own progress of learning via the elements of granular, sustained and cumulative feedback.

In succinctly presenting our answer to our research question above, we propose a framework for gamification as shown in Table 3. We next elaborate on the six gamification elements shown in Table 3.

Narrative/storyline allows for the other elements of gamification to be introduced and integrated seamlessly into the classrooms. Narratives can help to tie together all the other elements and the different phases of learning into one coherent context. Teachers could choose to adopt different storylines for different classes to suit their inclination. Narratives and storylines may be used throughout all the five stages of learning.

Explicitly explicated objectives provide students milestones to work with a sense of purpose and direction to work towards these objectives. The objectives set should be coherent with the setting of mastery-approach goals, instead of performance-avoidance goals. Objectives may be used throughout all the five stages of learning.

Avatars are crucial for students to be immersed in the narrative/storyline. Teachers can choose to utilise different avatars for different classes. Similarly, avatars may be used throughout all the five stages of learning.

The point system can serve to reward the students with points and contribute to the progress bars. The accumulation of points can also allow students to earn badges and increase their levels of competency. With the points system in place, the leaderboards may also be implemented to increase engagement from students as they compete with their peers. Through the point system, progress bars and leaderboards, students will receive granular, sustained and cumulative feedback. Feedback may be used from the Activation & Directed Orientation stage to the Integration stage.

Badges and skill levels are the physical manifestation of students’ engagement in demonstrating, applying and integrating new knowledge into their scheme. These badges and skill levels should be aligned to the setting of mastery goals. Badges and skill levels may be used from the Demonstration & Explication to the Integration stage.

We exemplify an application of our proposed framework (Table 3) through the use of a lesson plan (Table 4) for the topic on Congruence and Similarity at the secondary two (age 14) level in the Singapore Mathematics Secondary 2 Ordinary Level Syllabus (Education & Singapore, 2019). The lesson plan (Table 4) allows students to experience the five phases of learning (Table 3). Each phase of learning is accompanied by elements of gamification in accordance with the framework.

In enacting the lessons, students are assumed to have prior knowledge on the properties of triangles, including the angle sum of interior angles in a triangle. They are also expected to be able to identify similar and congruent triangles, list properties of similar triangles, and discern and explain the differences between similarity and congruence by the end of the lesson.