# Effects of Self-explanation and Game-reward on Sixth Graders' Algebra Variable Learning.

Introduction

Algebra variable learning

Many students struggle with algebra variable concepts (Novotna & Hoch, 2008; Warren, 2003). MacGregor and Stacey (1997) noted that students usually do not understand how to use variables, the basic and essential concepts of algebra, to help them solve problems accurately and efficiently. Thus, Stacey and MacGregor (1997) suggested that students could be better prepared for further algebra learning by frequently using algebra notation and symbolism (i.e., the variable system) in learning contexts. According to the suggestion, Ross and Willson (2012) conducted an instructional experiment and found that engaging students in the given content with practical examples led to the better understanding of algebra variables. However, to date, issues regarding students' algebra variable learning, such as use of certain strategies and design of examples, are still only partially explored (Bush & Karp, 2013).

As to proper understanding of algebra variables (i.e., the symbol system), Lucariello, Tine, and Ganley (2014) pointed out some key concepts. First, a symbol must be interpreted as representing an unknown quantity (Kuchemann, 1978; McNeil et al., 2010), meaning a student must realise that a symbol represents a unit that does not have a certain value. Second, a student must interpret a symbol as representing a varying quantity or range of unspecified values (Kieran, 1992; Philipp, 1992). This concept is named as the multiple values interpretation of literal symbols (Knuth, Alibali, McNeil, Weinberg, & Stephens, 2005; Lucariello et al., 2014). To explore learners' understanding of these two concepts, Knuth et al. (2005) tested seventh and eighth graders with a problem: "The following question is about the expression '2n+3.' What does the symbol (n) stand for?" The correct responses should represent ideas that (1) the symbol could represent an unknown value (e.g., "the symbol is a variable, and it could stand for any value") and (2) the symbol could represent more than one value (e.g., "it could be 3, 74, or even 123.4567"). The results, however, showed that approximately 39% of seventh graders and 25% of eight graders answered incorrectly. The researchers thus suggested that an appropriate training would be necessary before middle-school education (e.g., when in sixth grade) for students' better understanding of variables.

The third concept involves the relationships between symbols, and in such relationship the values change in a certain approach (e.g., as X increases, Y decreases, Lucariello et al., 2014). For example, Kuchemann (1978) invited and tested 3000 high-school students who had learnt variables with the following problem: "Which is the larger, 2n or n+2? Explain." The results revealed that just only 6% of the students correctly realised the concept that the relation between 2n and n+2 is actually changing with n. Knuth et al. (2005) also explored this understanding of algebra variables by presenting middle-school students with similar problems. Likewise, the results showed that only approximately 18% of sixth graders, 50% of seventh graders, and 60% of eighth graders were aware of the relationship between symbols because their values change systematically.

Self-explanation strategy

Self-explanation, a concept-oriented learning activity, is defined as generating explanations to oneself to make sense of new and known information to be correct, and it has been regarded as an effective strategy in students' algebra learning (Chi, 2000; Chi, de Leeuw, Chiu, & Lavancher, 1994; Neuman, Leibowitz, & Schwarz, 2000). For example, Aleven and Koedinger (2002) facilitated students' self-explanation by using intelligent instructional software for algebra, and they found that students who explained their steps of problem-solving outperformed those who did not. According to Roy and Chi (2005), the process of self-explanation can encourage four forms of cognition: (1) recognise what information is missing while generating inferences, (2) integrate information into what learnt from a lesson, (3) relate new information to learners' prior knowledge, and (4) identify and correct information received. This means that self-explanation strategy not only immerses students in cognitive processes but also engage them in the meta-cognitive activities to monitor what learning content they feel confused about.

While positive impacts of self-explanation have been consistently identified on producing robust learning gains in algebra learning (Aleven & Koedinger, 2002; Chi et al., 1994; Neuman et al., 2000), one problem still remains: the meta-cognitive activity could pose a challenge to middle-school students, such as sixth and seventh graders (e.g., Hausmann & Chi, 2002; Kuhn & Katz, 2009; Nokes, Hausmann, VanLehn, & Gershman, 2011). In self-explanation, students are usually required to deal with new and known information via question answering and answer explaining that could be challenging to be accomplished in a period of time (De Koning, Tabbers, Rikers, & Paas, 2011). Griffin, Wiley, and Theide (2008) encouraged students to self-explain through providing them with reading material and similar questions to practise, and they found that self-explanation could facilitate students' better learning performance and deeper thinking. However, they also noted that if students were not required to record their self-explanations in any forms, there was no significant difference between selfexplanation and non-self-explanation groups. This was supported by McEldoon, Durkin, and Rittle-Johnson (2013). In their study, students were encouraged to generate their explanations via rephrasing and writing to deepen their understanding of learning content, and the results indicated positive effects on learning achievement.

Another issue is the method of recording students' self-explanation because it could affect the representation and structure of explanations (Hausmann & Chi, 2002). For example, generating explanations via typing (or writing) facilitate students' monitoring and reflection on learning progress because it enables them to keep records and modify their explanations. Such method could be more appropriate for learners to employ than verbal ones that are generally less filtered (Chou & Liang, 2009). These typed explanations would encourage learners to properly self-monitor their comprehension, boost their reflective activity, and raise their meta-cognitive awareness (Griffin et al., 2008; Nokes et al., 2011). In the current study, therefore, all the participants were guided to type their explanations by the system, and they could review and modify these records during the learning activity. While above-mentioned studies pointed out the design issue of self-explanation activity and suggested the possible solution that facilitate student to self-explain with less challenges and obstacles, to date, few studies have been done on this issue.

Game-reward strategy

Gamification is the application of game elements to motivate target's involvement in specific contexts such as business and education (Deterding, Dixon, Nacke, O'Hara, & Sicart, 2011; Landers & Callan, 2011; McGonigal, 2011; Sun-Lin & Chiou, 2017). By employing game-reward in teaching and learning, students could be encouraged to participate in tedious tasks with raised learning interest (Glover, 2013). However, Hanus and Fox (2015) noted that the concept of gamification encompasses many different game elements and their applications, thus it is difficult to distinguish and explore every possible facet of gamification. They attempted to apply gamereward strategy by providing students with a leader-board, coins, and badges; the results, however, showed that the non-game-reward group outperformed game-reward group. They mainly attributed such results to gamification forms and whether students were interested in the learning materials. They thus suggested that gamification research should investigate specific elements/types first, such as points, coins, or badges, rather than an overarching concept so that learning effects can be taken more effectively.

The game-reward strategy takes only a small part of game design and can be used to benefit students' learning achievement and learning attitude (Dicheva, Dichev, Agre, & Angelova, 2015; Hattie & Timperley, 2007). Nicholson (2015) noted that a game-reward learning system could be designed around the concept of "point" to represent the basic unit of exchange. Such exchange shows that how well and how much players earn for taking on the specific behaviour. The underlying concept of game-reward is simple: offering points to motivate players (Dicheva et al., 2015; Hanus & Fox, 2015). Nicholson (2015) stated that motivating players through points is similar to motivating people through other forms of incentives such as money or grades. From perspective of learning game design, points provide students with feedback and encourage their certain behaviour. Therefore, game-reward can be used to promote several learning activities, for example, motivating students to read given material, answer questions, explain answers, and review concepts.

Although game-reward is often employed in business and marketing, its educational applications are partially explored. Hamari, Koivisto, and Sarsa (2014) conducted a comprehensive review of empirical studies of gamification across different contexts (e.g., education and consumer science), but only 24 studies were identified. The authors noted methodological problems such as a lack of empirical evidence with the studies, and of the 24 reviewed studies, few actually compared the effects of gamified and non-gamified strategies in terms of students' learning experience. Sun-Lin and Chiou (2017) attempted to examine the effects of gamified tasks on students' algebra learning, and positive results were reported. Although some benefits of gamification such as game-reward have been found on students' mathematics learning, empirical studies are still scant.

The combination of self-explanation and game-reward

According to Roy and Chi (2005), self-explanation could help to produce average learning gains via texts (22%), diagrams (44%), and multimedia presentation (20%). However, there are insufficient empirical evidence of game element effects with self-explanation. Johnson and Mayer (2010) ever reported that adult students who received self-explanation prompts in a learning game outperformed those who did not. Likewise, Hsu, Tsai, and Wang (2012, 2014) found that when students engaged in responding to prompts with game-reward, fourth graders' science concept learning was significantly improved. Despite positive findings reported by these studies, a problem remains: learning-within-gaming may burden students with many different sub-activities or subprocesses (e.g., play games and answer questions) in a period of time.

O'Neil et al. (2014) conducted an experiment in which they added self- explanation prompts requiring players to answer one of three questions after completing each level of a math game regarding addition of fractions. They found that the requirement of self-explanation might result in extraneous cognitive process that distracted students' attention and slowed their learning progress. Another issue was also proposed that students might respond quickly in order to return to the gaming section, but at the same time, they probably missed the opportunities of deeper thinking (e.g., reflection via self-explanation). Their findings indicated possible interaction between self-explanation and game elements, and a better combination should not interrupt the coherence of students' learning and gaming process. Game-reward, as a form of gamification, provides students with incentives to focus their attention on learning tasks without interrupting the coherence, might be a possible solution. As to the design of self-explanation, Johnson and Mayer (2010) and Mayer and Johnson (2010) stated that explanation prompts could be useful as responding prompt might be easier for students to generate selfexplanations than explaining without any guidance. They also mentioned that such design of prompts and learning tasks should be carefully considered because they may still require high cognitive effort that usually affect students' engagement with learning content. While some positive impacts of self-explanation and gamification were reported respectively, little research has been done on their interaction effects on beginners' algebra variable learning.

Research questions

In this study, a learning system was designed to support the combination of self-explanation and game-reward. A 2x2 quasi-experiment was conducted on a mathematics course of a primary school to explore the interaction effects on students' algebra variable learning via investigating the following research questions:

* What are the effects of self-explanation and game-reward strategies, and the interaction between them, on students' learning achievement of algebra variable?

* What are the effects of self-explanation and game-reward strategies, and the interaction between them, on students' learning attitude toward algebra variable?

* What are the effects of self-explanation and game-reward strategies, and the interaction between them, on students' meta-cognitive awareness of algebra variable learning?

Method

Participants

The participants of this instructional experiment included four classes of sixth graders, who were in the 11-13 age range (M = 11.93, SD = .06). A total of 97 students (47 females and 50 males) participated in this instructional experiment, each one class was assigned to a group: S-G group (n = 24), S-NG group (n = 24), NSG group (n = 24), and NS-NG group (n = 25) (see Table 1).

The learning system

A system, which combined self-explanation and game-reward strategies to support students' algebra variable learning, was developed via software Construct 2. It provided nine learning tasks for students, and each unit comprised (1) a reading page introducing an algebra concept (Figure 1), (2) a sample question and the solution demonstrating the application of the concept (Figure 2), and (3) a question for practice helping students reflect on what they learnt (Figure 3).

The self-explanation strategy in this study required the participants to rephrase the algebra concepts they learnt, and to explain their answers to practice questions and each step of their solutions. The learning system was to represent the entire learning procedure and materials, including reading material that introduced given algebra concepts, sample questions and the solutions that helped students understand how to apply the concepts, and practice questions that required students to answer and explain their answers with each step of solutions.

The game-reward strategy in this study was to reward students by points. These points were given according to their self-explanations of algebra concepts after answering questions (e.g., rephrasing concepts, which they just learnt, by using their own words to identify all steps of their solutions). Correct answers and logical explanations would earn them points that could be used to enhance their aircrafts in a mini game.

Instruments

In this study, two algebra achievement tests (pre-test and post-test), an algebra learning attitude scale (pre-survey and post-survey), and an algebra meta-cognitive awareness questionnaire (pre- survey and post-survey) were used. The purpose of the pre-test/pre-surveys was to understand whether the four groups had an equivalent prior knowledge of algebra variables, prior learning attitude, and prior meta- cognitive awareness of algebra variable learning before attending the learning activity. Such results would be used as the covariate to employ appropriate statistics techniques if they were unequal between groups.

Algebra variable achievement test

The achievement tests were modified from an algebraic assessment developed by Lucariallo, Tine, and Ganley (2014). The pre-test and the post-test both consisted of nine multiple-choice items that covered three concepts of algebra variable: (1) the variable must be interpreted as representing an unknown quantity (three items), (2) students must interpret the symbols as representing a varying quantity (three items), and (3) the relationship exists between symbols as their values change in a systematic manner (three items).

To ensure the quality of measurement, 133 sixth graders, who had not participated in the instructional experiment, were recruited to answer the test questions. The reliability of the tests had been estimated based on Cronbach's a measure for the pre-test a = .855 and for post-test a = .939; there was a strong positive correlation between the pre-test and post-test (r = .397, p =.000). In addition, the tests were reviewed and revised by two elementary-school mathematics teachers with more than five years of teaching experience, and this suggested the content validity. The tests also showed significant correlations with the regular school mathematics exam, which involved algebra variable concepts, and the significant positive correlations (between the exam and the pre-test, r = .350, p = .000; between the exam and the post-test, r = .214, p = .010) indicated that the tests had good criterion-related validity.

Algebra learning attitude scale

The scale regarding algebra learning attitude, which covered students' enjoyment, motivation, self-confidence, and perceived value, was modified from the inventory developed by Lim and Chapman (2013). It comprises 19 items using a four-point Likert rating schema. Examples: "I enjoy learning algebra concepts", "I am willing to learn more algebra concepts", and "I am confident that I could learn algebra well."

To ensure appropriateness of the questionnaire, the invited teachers helped to review and revise all the items. To examine the reliability, the 133 sixth graders were also invited to respond every item of the scale. The Cronbach's a value for the entire scale was .927, showing acceptable reliability in internal consistency.

Algebra meta-cognitive awareness questionnaire

The meta-cognitive awareness questionnaire was modified from the instrument created by Panaoura, Philipou, and Christou (2003). It consisted of 27 questions that were developed based on a four-point Likert scale. Here are some examples of question items: "I know how well I have understood the algebra concepts I have learnt", "I know what makes me find it difficult to solve algebra problems", and "I examine my own performance while learning new algebra concepts."

Like previous instruments, the two elementary-school teachers reviewed and revised all the questions, and the 133 students were invited to answer the questionnaire. The Cronbach's a value for entire questionnaire was .959, showing acceptable reliability in internal consistency.

Experimental procedure

The experimental instruction took about 320 minutes in four weeks. First, the researchers explained purpose of the instruction (10 minutes), and all participants were required to answer the pre-test of algebra variable concepts (25 minutes) and responded the pre-surveys of learning attitude scale (15 minutes) and the meta-cognitive awareness questionnaire (15 minutes).

During the learning activity, the students learnt algebra variable concepts and accomplished nine learning tasks in 200 minutes. The differences of learning strategies between groups were the tasks that combined selfexplanation (or non-self-explanation) with game-reward (non-game-reward).

After the learning activity, all groups answered the post-test of algebra variable concepts (25 minutes), and responded post-surveys of learning attitude scale (15 minutes) and the meta- cognitive awareness questionnaire (15 minutes).

Results

Learning achievement

The assumption of homogeneity was tested by Levene's test (F = .991, p = .401), and the two-way ANCOVA results (see Table 2) indicated a significant interaction effect found between the game-reward and selfexplanation (F = 8.312, p = .005, [[eta].sup.2] = 083) on students' learning achievement of algebra variable concepts. It was necessary to conduct a simple main effect analysis to investigate specific learning effects. The descriptive data of students' adjusted test scores for all groups are shown in Table 3.

As to the results grouped by self-explanation strategy, firstly, a significant difference (F = 4.197, p = .048, [[eta].sup.2] = . 102) was found between the S- G group the S-NG group, as shown in Table 4. The learning achievement of the S-G group (adjusted M = 5.397) was significantly greater than that of the S-NG group (adjusted M = 4.797). According to the principles of effect size proposed by Cohen (1988), the partial eta squared ([[eta].sup.2]) of the results of the simple main effect analysis represented a moderate effect ([[eta].sup.2] = . 102 > .059).

Secondly, Table 4 presents that a significant difference (F = 8.340, p = .006, [[eta].sup.2] = 130) was found between the NS-G group and the NS-NG group. To be specific, learning achievement of the NS-G group (adjusted M = 5.968) was significantly higher than that of the NS-NG group (adjusted M = 4.497), as shown in Table 3. The partial eta squared represented a moderate effect ([[eta].sup.2] = .130 > .059).

As to the results grouped by game-reward strategy (see Table 4), there was a significant difference between the S-NG group and the NS-NG group (F = 21.010, p = .000, [[eta].sup.2] = .296). The former gained significantly higher score than the latter, and the partial eta squared indicated a strong effect ([[eta].sup.2] = .296 > .138). However, no significant difference of the learning achievement was found between the S-G group and the NS-G group (F = .135, p = .716).

Learning attitude

After verifying the assumption of homogeneity of regression (F = .997, p = .398), the two-way ANCOVA results (see Table 5) indicated an interaction effect of the strategies (F = 9.549, p = .003, [[eta].sup.2] = 094) on students' learning attitude toward algebra variable. It was thus necessary to conduct a simple effect main effect analysis. The descriptive data of the adjusted students' survey scores are shown in Table 6.

As for the grouped results by self-explanation strategy, a significant difference (F = 12.083, p = .001, [[eta].sup.2] = 251), as Table 7 presents, was found between the S-G group and the S-NG group. The learning attitude of the S-G group (adjusted M = 61.904) was significantly more positive than that of the S- NG group (adjusted M = 46.990). The partial eta squared of the results indicated a strong effect ([[eta].sup.2] = .251 > .138). However, no significant difference was found between the NS-G group and the NS-NG group (F = .595, p = .444).

As shown in Table 7, in terms of grouping results by game-reward strategy, although no significant difference (F = 1.431, p = .238) was found between the S-G group and the NS-G group, but the NS-NG group (adjusted M = 49.468) reported significantly more positively (F = 6.107, p = .017, [[eta].sup.2] = 111) than the S-NG group (adjusted M = 46.990). The partial eta squared represented a moderate effect ([[eta].sup.2] = .111 > .059).

Meta-cognitive awareness

With the verification that the assumption of homogeneity of regression was not violated through Levene's test (F = 2.624, p = .055), the two-way ANCOVA result was shown in Table 8. The interaction effect between the gamereward and self-explanation was not significant (F = .349, p = .556). Thus, it was sensible to directly examine the main effects of the independent variables. The results of main effects analysis, as shown in Table 9, indicated that significant effects were confirmed for self-explanation strategy (F = .5.465, p = .022, [[eta].sup.2] = .059) but not confirmed for game-reward strategy (F = .159, p = .691, [[eta].sup.2] = .002). This implies that the post- survey score of students were significantly different due to self-explanation strategy, as described in Table 8. To be specific, students who learnt with self-explanation strategy (the S-G and the S-NG groups; adjusted M = 81.867) reported significantly higher meta-cognitive awareness of algebra variable learning than those who learnt with non-selfexplanation strategy (the NS-G and the NS-NG groups; adjusted M = 73.795). The partial eta squared of the results of self-explanation strategy indicated a moderate effect.

Discussion and conclusions

Overall, this study explored the interaction effect between self-explanation and game-reward on sixth graders' algebra variable learning, and a 2*2 quasi-experiment was conducted. The experimental results showed that (1) a significant interaction effect was found on students' learning achievement: the S-G group performed significantly better than the S-NG group, and the NS-G group and the S-NG group gained significantly higher scores than the NS-NG group; (2) a significant interaction effect was found on students' learning attitude: the SG group reported significantly more positive results than the S-NG group, and the NS-NG group responded significantly more positively than the S-NG group; (3) no significant interaction effect was found on students' meta-cognitive awareness: self-explanation strategy significantly affected students' responses, but game-reward strategy did not.

A significant interaction effect on algebra variable learning achievement

First, the S-G group performed significantly better than the S-NG group, and second, the NS-G group scored significantly higher than the NS-NG group. These results showed that students could be encouraged to learn given concepts by receiving points as rewards. Although students were not required to explain their answers, they were willing to read the instructional materials and try to understand the algebra variable concepts to correctly answer questions. This means that although game-reward (i.e., earn more points for game-play), as an extrinsic motivator, does not directly relate to learning content, it could still motivate students to leam and gain higher scores than those who do not receive game-reward. These results are in accord with the views of Dicheva et al. (2015) and Nicholson (2015). The game-reward strategy can work to promote several learning activities, including reading learning material, answering questions, and reflecting on their performance. The results of this study could also address issues proposed by O'Neil et al. (2014) and Hanus and Fox (2015). O'Neil et al. (2014) noted that self-explanation would interact with game elements and thought the combination should not interrupt the coherence of students' learning and gaming; Hanus and Fox (2015) stated that gamification research should investigate specific elements, such as points or badges, rather than an overarching design so that the learning effects can be effectively taken. In this current study, the benefits of game- reward strategy were found to encourage students to learn without distracting them from learning process, and thus their learning achievement would be improved more effectively.

Third, the S-NG group gained significantly higher test scores than the NS-NG group. The result revealed that supporting students to explain their answers would have a positive effect on their learning achievement. Students can deepen their understanding of given concepts through the process of self- explanation. They read the instructional material and rephrase it via their own expressions, that is, from input to output in the learning process. Such process would promote students to reflect on what and how they have learnt, and it also provides them with the access to re-read the material when they find it difficult to answer questions and generate explanations. Through this process they could improve the understanding of algebra variable concepts. This result supports findings of Griffin et al. (2008) that self-explanation activity, which are implemented via providing reading instructions and related examples for practice, could benefit students' learning achievement. This result also evidences the ways of explanation generation mentioned by McEldoon et al. (2013). They suggested that promoting students to generate their explanations in a certain manner, such as writing or typing, might effectively help them relate self-explanations to instructional material. This study found that typing explanations helped students keep records and provide them with an opportunity to easily revise, and such approach seems feasible to be used in class.

A significant interaction effect on algebra variable learning attitude

Firstly, the S-G group reported significantly more positively than the S-NG group. The experimental results revealed that game-reward strategy could enhance both students' learning attitude and their learning achievement rather than divert their attention. This is consistent with many previous studies which reported positive effects of game-reward on learning attitude (e.g., Glover, 2013; Hattie & Timperley, 2007). This result also provides further exploration for Attali and Arieli-Attali's (2015) findings that point manipulation showed only minor effects on students' learning performance. The points, used to show students' accuracy in their study, probably lack following applications of the reward, that is, the usage or function of these points. In this current study, points that students gained via accomplishing learning tasks connected to game- play. Students could use these points to enhance attributes of their roles (i.e., aircrafts) in a mini game. According to the result, such gamereward successfully encouraged students to participate in and accomplish learning tasks, and this seems to be a feasible application of gamification for teaching.

Secondly, the NS-NG group responded significantly more positive results than the S-NG group. This indicated that self-explanation did not significantly improve students' attitude toward algebra variable learning. The result supports the finding of Kuhn and Katz (2009). It is possible that students could learn better via self-explanation, but they might find it difficult and make more effort to accomplish learning tasks. They may thus be under some pressure that results in negative learning attitude. This is probably because without appropriate incentives (e.g., game-reward or other rewards), students might feel bored and be unwilling to stick to learning especially when they are required to accomplish challenging tasks (e.g., generate explanations), even if such activity could help them understand the learning material better. A possible explanation may lie in students' abilities in selfexplanation. They found it difficult to generate explanations probably because of a lack of mastered selfexplanation skill. This could be a threshold that students need to cross. If they have an opportunity to practise self-explanation until they could generate self-explanations without too many difficulties, they would be able to get a sense of achievement in the learning process, and their learning attitude might be more positive than that in this study.

Self-explanation enhanced meta-cognitive awareness of algebra variable learning

Although no interaction effect between self-explanation and game-reward strategies was found, self-explanation showed a significantly positive impact on students' meta-cognitive awareness of algebra variable learning. It not only provided students with opportunities to deepen what they learnt but help them raise their meta-cognitive awareness. By explaining the answer with students' own words, they could reflect on what and how much content they could understand. The results are consistent with the findings of several previous studies revealing that guiding students to explain answers with their own expressions can boost meta-cognitive awareness (Kuhn & Katz, 2009; Roy & Chi, 2005).

Moreover, the results may also address a proposed issue that the meta-cognitive requirements of self-explanation lead to higher difficulty for middle-school students (e.g., Hausmann & Chi, 2002; Nokes et al., 2011). In this study, there was a similar practice question following each example in the system, and when students found it difficult to answer questions or generate explanations, they could go back to previous pages to re-read the instructional material without any limitation. The practice question was to encourage students to test themselves, and the access to demonstration pages was to deepen their understanding through review if necessary. The design may reduce the difficulty of accomplishing learning tasks and lead students to reflect on their learning performance and learning process.

References

Aleven, V, & Koedinger, K. (2002). An Effective metacognitive strategy: Learning by doing and explaining with a computerbased cognitive tutor. Cognitive Science, 26(2), 147-179. doi:10.1016/S0364-0213(02)00061-7

Attali, Y, & Arieli-Attali, M. (2015). Gamification in assessment: Do points affect test performance? Computers & Education, 83, 57-63. doi:10.1016/j.compedu.2014.12.012

Bush, S. B., & Karp, K. S. (2013). Prerequisite algebra skills and associated misconceptions of middle grade students: A Review. The Journal of Mathematical Behavior, 32(3), 613-632. doi:10.1016/j.jmathb.2013.07.002

Chi, M. T. H. (2000). Self-explanation expository texts: The Dual processes of generating inferences and repairing mental models. In R. Glaser (Ed.), Advances in instructional psychology (pp. 161-238). Mahwah, NJ: Lawrence Erlbaum Associates.

Chi, M. T. H., de Leeuw, N., Chiu, M.-H., & Lavancher, C. (1994). Eliciting self-explanations improves understanding. Cognitive Science, 18(3), 439-477. doi:10.1207/s15516709cog1803_3

Chou, C., & Liang, H. (2009). Content-free computer supports for self- explanation: Modifiable typing interface and prompting. Educational Technology & Society, 12(1), 121-133.

Cohen, J. (Ed.). (1988). Statistical power analysis for the behavioral science. Hillsdale, NJ: Eribaum.

De Koning, B. B., Tabbers, H. K., Rikers, R. M. J. P., & Paas, F. (2011). Improved effectiveness of cueing by selfexplanations when learning from a complex animation. Applied Cognitive Psychology, 25(2), 183-194. doi:10.1002/acp.1661

Deterding, S., Dixon, D., Nacke, L. E., O'Hara, K., & Sicart, M. (2011). Gamification: Using game design elements in nongaming contexts. In Proceedings of the 2011 Annual Conference Extended Abstracts on Human Factors in Computing Systems (CHIEA'11), Vancouver, BC, Canada. (pp. 2425-2428). New York, NY: ACM.

Dicheva, D., Dichev, C., Agre, G, & Angelova, G (2015). Gamification in education: A Systematic mapping study. Educational Technology & Society, 18(3), 75-88.

Glover, I. (2013). Play as you learn: Gamification as a technique for motivating learners. In Proceedings of world Conference on educational Multimedia, Hypermedia and Telecommunications (pp. 1999-2008). Chesapeake, VA: AACE.

Griffin, T. D., Wiley, J., & Thiede, K. W. (2008). Individual differences, rereading, and self-explanation: Concurrent processing and cue validity as constraints on metacomprehension accuracy. Memory & Cognition, 36(1), 93-103. doi:10.3758/MC.36.1.93

Hamari, J., Koivisto, J., & Sarsa, H. (2014). Does gamification work? A Literature review of empirical studies on gamification. In System sciences (HICSS), 2014 47th Hawaii International Conference (pp. 3025-3034). Hawaii: HICSS. doi:10.1109/HICSS.2014.377

Hanus M. D., & Fox J. (2015). Assessing the effects of gamification in the classroom: A longitudinal study on intrinsic motivation, social comparison, satisfaction, effort, and academic performance. Computers & Education, 80, 152-161. doi:10.1016/j.compedu.2014.08.019

Hattie, J., & Timperley, H. (2007). The Power of feedback. Review of Educational Research, 77(1), 81-112. doi: 10.3102/003465430298487

Hausmann, R. G M., & Chi, M. T H. (2002). Can a computer interface support self- explanation? Cognitive Technology, 7(1), 4-14. Retrieved from http://chilab.asu.edu/papers/hausmannchi2002.pdf

Hsu, C.-Y, Tsai, C.-C., & Wang, H.-Y (2012). Facilitating third graders' acquisition of scientific concepts through digital game-based learning: The Effects of self-explanation principles. The Asia-Pacific Education Researcher, 21(1), 71-82. Retrieved from https://ejournals.ph/article.php?id=4301

Hsu, C.-Y, Tsai, C.-C., & Wang, H.-Y (2014). Exploring the effects of integrating self-explanation into a multi-user game on the acquisition of scientific concepts. Interactive Learning Environments, 24(4), 844-858. doi:10.1080/10494820.2014.926276

Johnson, C. I., & Mayer, R. E. (2010). Applying the self-explanation principle to multimedia learning in a computerbased game-like environment. Computers in Human Behavior, 26(6), 1246- 1252. doi:10.1016/j.chb.2010.03.025

Kieran, C. (1992). The Learning and teaching of school algebra. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 390-419). New York, NY: Macmillian Publishing Co.

Knuth, E. J., Alibali, M. W., McNeil, N. M., Weinberg, A., & Stephens, A. C. (2005). Middle school students' understanding of core algebraic concepts: Equivalence and variable. Zentralblatt fur Didaktik der Mathematik (International Reviews on Mathematical Education), 37(1), 68-76. doi:10.1007/BF02655899

Kuhn, D., & Katz, J. (2009). Are self-explanations always beneficial? Journal of Experimental Child Psychology, 103(3), 386-394. doi:10.1016/j.jecp.2009.03.003

Kuchemann, D. (1978). Children's understanding of numerical variables. Mathematics in School, 7(4), 23-26.

Landers, R. N., & Callan, R. C. (2011). Casual social games as serious games: The psychology of gamification in undergraduate education and employee training. In M. Ma, A. Oikonomou, & L. C. Jain (Eds.), Serious Games and Edutainment Applications (pp. 399-424). Surrey, UK: Springer.

Lim, S. Y, & Chapman, E. (2013). Development of a short form of the attitude toward mathematics inventory. Educational Studies in Mathematics, 82(1), 145-164. doi:10.1007/s10649-012-9414-x

Lucariello, J., Tine, M. T., & Ganley, C. M. (2014). A Formative assessment of students' algebraic variable misconceptions. The Journal of Mathematical Behavior, 33, 30-41. doi:10.1016/j.jmathb.2013.09.001

MacGregor, M., & Stacey, K. (1997). Students' understanding of algebraic notation: 11-15. Educational Studies in Mathematics, 33(1), 1-19. doi:10.1023/A:1002970913563

Mayer, R. E., & Johnson, C. I. (2010). Adding instructional features that promote learning in a game-like environment. Journal of Educational Computing Research, 42(3), 241-265. doi:10.2190/EC.42.3.a

McEldoon, K. L., Durkin, K. L., & Rittle-Johnson, B. (2013). Is self-explanation worth the time? A Comparison to additional practice. British Journal of Educational Psychology, 83(4), 615-632. doi:10.1111/j.2044-8279.2012.02083.x

McGonigal, J. (2011). Reality is broken: Why games make us better and how they can change the world. New York, NY: Penguin.

McNeil, N. M., Weinberg, A., Hattikudur, S., Stephens, A. C., Asquith, P., Knuth, E. J., & Alibali, M. W. (2010). A is for apple: Mnemonic symbols hinder the interpretation of algebraic expressions. Journal of Educational Psychology, 102(3), 625-634. doi:10.1037/a0019105

Neuman, Y, Leibowitz, L., & Schwarz, B. (2000). Patterns of verbal mediation during problem solving: A Sequential analysis of self-explanation. Journal of Experimental Education, 68(3), 197-213. doi:10.1080/00220970009600092

Nicholson, S. (2015). Exploring the endgame of gamification. In M. Fuchs, S. Fizek, P. Ruffino, & N. Schrape (Eds.), Rethink gamification (pp. 289-303). Lueneburg, Germany: Meson Press.

Nokes, T. J., Hausmann, R. G M., VanLehn, K., & Gershman, S. (2011). Testing the instructional fit hypothesis: The Case of self-explanation prompts. Instructional Science, 39(5), 645-666. doi: 10.1007/s11251-010-9151-4

Novotna, J., & Hoch, M. (2008). How structure sense for algebraic expressions or equations is related to structure sense for abstract algebra. Mathematics Education Research Journal, 20(2), 93- 104. doi:10.1007/BF03217479

O'Neil, H. F., Chung, G K., Kerr, D., Vendlinski, T. P., Buschang, R. E., & Mayer, R. E. (2014). Adding selfexplanation prompts to an educational computer game. Computers in Human Behavior, 30, 23-28. doi:10.1016/j.chb.2013.07.025

Panaoura, A., Philippou, G, & Christou, C. (2003). Young pupils' metacognitive ability in mathematics. European Research in Mathematics Education, 3, 1-9. Retrieved from http://www.dm.unipi.it/~didattica/CERME3/proceedings/Groups/TG3/TG3_Panaoura_cer me3.pdf

Philipp, R. A. (1992). A Study of algebraic variables: Beyond the student- professor problem. Journal of Mathematical Behavior, 11, 161-176.

Ross, A., & Willson, V (2012). The Effects of representations, constructivist approaches, and engagement on middle school students' algebraic procedure and conceptual understanding. School Science and Mathematics, 112(2), 117-128. doi:10.1111/j.1949-8594.2011.00125.x

Roy, M., & Chi, M. T. H. (2005). Self-explanation in a multimedia context. In R. Mayer (Ed.), Cambridge handbook of multimedia learning (pp. 271-286). New York, NY: Cambridge University Press.

Stacey, K., & MacGregor, M. (1997). Ideas about symbolism that students bring to algebra. Mathematics Teacher, 90(2), 110113.

Sun-Lin, H.-Z., & Chiou, G.-F. (2017). Effects of comparison and game-challenge on sixth graders' algebra variable learning achievement, learning attitude, and meta-cognitive awareness. Eurasia Journal of Mathematics, Science & Technology Education, 2017;13(6), 2627-2644.

Warren, E. (2003). The Role of arithmetic structure in the transition from arithmetic to algebra. Mathematics Education Research Journal, 15(2), 122-137. doi:10.1007/BF03217374

Hong-Zheng Sun-Lin * and Guey-Fa Chiou

Graduate Institute of Information and Computer Education, National Taiwan Normal University, Taipei City, Taiwan//hzsunlin@gmail.com//gueyfa@ntnu.edu.tw

* Corresponding author

(Submitted March 15, 2016; Revised May 17, 2016; Accepted June 15, 2016)

Caption: Figure 1. A screenshot of algebra concept introduction

Caption: Figure 2. A sample question and the solution

Caption: Figure 3. A practice question
```Table 1. The treatments (learning tasks) for the four groups

Game-reward (G)       Non-game-reward (NG)

Self-explanation (S)   The S-G group: self   The S-NG group: self
-explanation with     -explanation with
game-reward           non-game-reward
Non-self-explanation   The NS-G group: non   The NS-NG group
(NS)                   -self-explanation     (control): non-self
with game-reward      -explanation with
non-game-reward

Table 2. The two-way ANCOVA results of learning achievement

Source                 SS      df     MS        F      [[eta].sup.2]

Self-explanation      .280     1     .280     .097         .001
Game-reward           4.265    1    4.265     1.485        .016
Self-explanation *   23.882    1    23.882   8.312 *       .083
game-reward
Error                264.328   92   2.873

Note. * p < .05.

Table 3. The descriptive data of each group's learning achievement

Self-explanation       Game-reward          Post-test        n
strategy               strategy

Self-explanation       Game-reward         5.397      .451   24
Non-game-reward     4.797      .383   24
Non-self-explanation   Game-reward         5.968      .359   24
Non-game-reward     4.497      .309   25

Table 4. The simple main effect analysis results

Pair                  SS     df     MS        F       [[eta].sup.2]

S-G versus S-NG     13.013   1    13.013   4.197 *        .102
NS-G versus NS-NG   32.909   1    32.909   8.340 *        .130
S-G versus NS-G      .514    1     .514      .135         .003
S-NG versus NS-NG   72.027   1    72.027   21.010 *       .296

Note. * p < .05.

Table 5. The two-way ANCOVA results of learning attitude

Source             SS       df      MS         F      [[eta].sup.2]

Self-            190.405    1    190.405     1.033        .011
explanation
Game-reward      816.360    1    816.360    4.427 *       .046
Self-           1760.734    1    1760.734   9.549 *       .094
explanation
* game-
reward
Error           16963.492   92   184.386

Note. * p < .05.

Table 6. The descriptive data of each group's learning attitude

Self-explanation       Game-reward          Post-test         n
strategy               strategy

Self-explanation       Game-reward         61.904     3.617   24
Non-game-reward     46.990     3.066   24
Non-self-explanation   Game-reward         52.336     2.872   24
Non-game-reward     49.468     2.478   25

Table 7. The simple main effect analysis results

Pair             SS      df      MS         F       [[eta].sup.2]

S-G versus    1522.885   1    1522.885   12.083 *       .251
S-NG
NS-G versus   128.075    1    128.075      .595         .011
NS-NG
S-G versus    280.289    1    280.289     1.431         .033
NS-G
S-NG versus   1087.568   1    1087.589   6.107 *        .111
NS-NG

Note. * p < .05.

Table 8. The two-way ANCOVA results of meta-cognitive awareness

Source             SS       df      MS         F      [[eta].sup.2]

Self-           1445.483    1    1445.483   5.465 *       .059
explanation
Game-reward      41.994     1     41.994     .159         .002
Self-            92.256     1     92.256     .349         .004
explanation
* game-
reward
Error           23013.310   92   264.521

Note. * p < .05.

Table 9. The descriptive data of each group's
meta-cognitive awareness

Strategy                   Post-test        n

Self-explanation         81.867     2.681   24
Non-self-explanation     73.795     2.246   24
Game-reward              77.129     2.452   24
Non-game-reward          78.532     2.475   25
```
COPYRIGHT 2017 International Forum of Educational Technology & Society
No portion of this article can be reproduced without the express written permission from the copyright holder.