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Social Behavior and Personality , Volume 48, Issue 6, e9094 https://doi.org/10.2224/sbp.9094 www.sbp-journal.com

Differentiated instruction enhances sixth-grade students’ mathematics self-efficacy, learning motives, and problem-solving skills

Chih-Pin Lai1, Wanpeng Zhang2, Yu-Liang Chang3

1Physical Education and Arts School, Chengyi University College, Jimei University, People’s Republic of China 2Department of Educational Management, Faculty of Education, East China Normal University, People’s Republic of China 3Graduate Institute of Educational Administration and Policy Development, Teachers College, National Chiayi University, Taiwan

How to cite: Lai, C.-P., Zhang, W., & Chang, Y.-L. (2020). Differentiated instruction enhances sixth-grade students’ mathematics self- efficacy, learning motives, and problem-solving skills. Social Behavior and Personality: An international journal, 48(6), e9094

We examined the effectiveness of a differentiated instruction intervention in promoting 6th grade students’ mathematics self-efficacy (MSE), mathematics learning motives (MLM), and mathematical problem-solving skills (MPSS). The relationships among MSE, MLM, and MPSS were also assessed. We employed a longitudinal approach with a pretest and posttest design with 25 students, and used 3 instruments for quantitative data collection. Findings showed that the application of the differentiated instruction learning environment was significantly beneficial in advancing the students’ MSE, MLM, and MPSS. In addition, MSE significantly predicted MLM and MPSS, such that the mediating effect of MLM on the effect of MSE on MPSS was partial. This finding shows that the stronger the MSE of 6th grade students is, the better their MLM are, which, in turn, advance their MPSS.

Keywords differentiated instruction; mathematics learning; mathematics learning motives; mathematical problem-solving skills; mathematics self-efficacy; sixth grade

Academic diversity is the worldwide phenomenon that exists in today’s classrooms, namely, there are similar- or same-age students with mixed ability in every classroom (Tomlinson, 2017). Students come from dissimilar cultures, possess different learning styles, and arrive at school at various levels of emotional and social maturity. This is, therefore, a serious concern for all teachers and teacher educators as they face multiple challenges at every grade level. Because students’ needs differ, teachers are required not only to understand their needs but, by responding effectively to academic and neuropsychological differences, also to offer the students greater success rates and reduce the number of failing students (Tomlinson, 2004).

From Diversity to Differentiation

In view of the numerous factors involved, teachers need to think and plan in terms of multiple learning approaches for individual students’ varied needs to maximize their growth and potential. Teachers who consider and understand the learning position of each student can adjust their teaching methods so that the learning matches the learner (Tomlinson, 2017). The implementation of differentiated instruction will achieve the goal of addressing of both individual differences and learning in the whole-class environment. Tomlinson and Imbeau (2010) pointed out that “differentiation is a philosophy—a way of thinking about teaching and learning” (p. 13). Differentiated instruction is a student-centered integrated teaching approach that encompasses a set of principles connecting concepts raised by educators and psychologists, and integrates beliefs of constructivism, viewpoints of learning styles, and developmental evidence from empirical brain-based studies (Stager, 2007; Tomlinson, 2017).

CORRESPONDENCE Yu-Liang Chang, No. 85 Wunlong Village, Minsyong Township, Chiayi County 621, Taiwan. Email: [email protected]

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In the classroom, advanced learners may feel that the learning content is too easy and poses no challenge, whereas slower learners may be continually struggling to comprehend basic concepts, and tend to give up. Differentiated instruction thus provides a balanced solution for teachers to attend to each learner’s needs by furnishing him or her with appropriate learning tasks and essential social interaction (Small, 2017). This approach not only helps struggling learners to see the big picture and key concepts of a topic as well as its governing principles, it also helps establish a scaffolding of meaning, a requisite framework for future success (Tomlinson, 2004).

From Differentiation to Mathematics Learning

Tomlinson (2017) identified four classroom elements that teachers can differentiate based on student readiness, interest, or learning profile: (a) content—what the students need to learn, or how they are going to access this; (b) process—instructional activities that engage the students in meaningful, sense-making learning; (c) products—culminating tasks that require the students to review, apply, and extend what they learn; and (d) learning environment—how the classroom works and feels. Thus, differentiated instruction means teaching to address diverse learning needs, where a blend of whole class, group, and individual instruction is employed to reach the goal of students’ access to equity (Schoenfeld, 2014). To help students to better understand mathematics learning, the core principles of differentiated instruction with the use of multiple teaching strategies and representations must be applied. The use of formative assessment tools to monitor students’ learning behaviors will ensure effective teaching and learning processes.

Empirical evidence regarding mathematics teaching and learning indicates that implementing differentiated instruction is beneficial not only for mathematics teachers’ professional development (Butler & Van Lowe, 2010), but also for increasing the students’ learning outcomes, interest, and confidence (Tomlinson & Moon, 2013). In Taiwan, much attention has recently been given to the preservice and inservice training of high- quality mathematics teachers to promote students’ meaningful learning. However, according to the results of recent international assessment comparisons, there are potential problems with Taiwanese student learning achievement, interest, and confidence in mathematics. Findings of the 2012 Program for International Student Assessment (PISA) indicate that although the mathematical performance of 15-year- old Taiwanese students ranks fourth in the participating countries, there is an achievement gap of 245 points between high and low achievers; that is, the difference may be equal to receiving a 6-year education (OECD, 2014; Taiwan PISA National Center, 2014). It is stated in the Taiwan PISA National Center (2014) report that, of the participating countries, Taiwan has the highest percentile of low achievers, and the lowest learning interest and confidence in the world. These are potential problems for Taiwan, as low achievers tend to give up, and their teachers may also give up on them, in traditional classrooms.

Previous researchers (e.g., Chamberlin & Powers, 2010; Muthomi & Mbugua, 2014; Tambaoan & Gaylo, 2019) have shown that the use of differentiated instruction favors the promotion of students’ learning achievement or performance in different subject areas (e.g., English, mathematics) at both elementary and secondary levels. Forsten, Grant, and Hollas (2002) proposed 101 practical, easy-to-implement classroom- tested strategies, including curriculum compacting, tiered activities, learning centers, flexible grouping, and mentoring, associated with reproducible and supplementary materials and resources for students from kindergarten to grade 8. A variety of teaching strategies that match basic principles of differentiated instruction is employed to influence the students’ achievement or successful performance in the understanding of different mathematical concepts, such as numbers, geometry, and algebra (Bal, 2016; Muthomi & Mbugua, 2014; Stager, 2007). As a result, these teaching principles and multiple strategies are collectively employed in the professional development program to assist the elementary mathematics teacher to design and implement differentiated interventions in the classroom.

From Differentiation to Self-Efficacy and Learning Motives and Performance

Social Behavior and Personality: an international journal

The findings of Trends in International Mathematics and Science Study: 2011 International Results in Mathematics (TIMSS; Mullis, Martin, Foy, & Arora, 2012) show that the phenomenon of learners’ low confidence and interest in mathematics is a serious problem requiring attention (Mullis et al., 2012). As learning interest and confidence are critical to a student’s readiness to learn mathematics, low interest and confidence may lead to a student giving up because of learned helplessness (Fay, Bickerstaff, & Hodara, 2013). Learned helplessness—an individual’s perception of complete lack of control in mastering a task—is closely related to self-efficacy. Bandura, in his exposition of social cognitive theory, defines self-efficacy as the “belief in one’s capabilities to organize and execute the courses of action required to produce given attainments” (1997, p. 3). Self-efficacy, therefore, significantly affects an individual’s task choices, effort, persistence, and achievement. Further, “efficacy beliefs played [sic] a significant role in influencing not only one’s action but also thought processes, motivations, and affective and psychological states” (Chang, 2010, p. 275). Thus, students who are self-efficacious in learning are likely to make more effort, persist longer when facing obstacles, and eventually attain higher levels of achievement than other students (Bandura, 2006; Pajares, 2006). Researchers have found a significant positive correlation between students’ self- efficacy for learning and consequential motivation and capable performance (Pajares, 2006; Schunk & Meece, 2006; Zimmerman, 2000). Researchers have also found empirical evidence of a positive correlation between students’ self-efficacy and academic achievement in various content domains and at different grade levels (Bandura, 1997; Lent, Lopez, & Bieschke, 1991; Pourdana & Rad, 2017; Schunk & Meece, 2006).

Mathematics self-efficacy (MSE) has a powerful effect on the academic achievement and performance level that a student can achieve in learning mathematics (Chang, 2012; Kitsantas, Cheema, & Ware, 2011; Ocak & Yamac, 2013; Pajares & Miller, 1994), that is, MSE predicts mathematics achievement. Bandura (1997) rated self-efficacy as one of eight influential factors for learning performance. Therefore, educators need to explore and implement ways to help students develop MSE to achieve their potential in the classroom (Chang, 2015; Kitsantas et al., 2011).

Bandura (1997) claimed that students’ learning motives are a very important factor affecting their self- efficacy development, such that highly efficacious students possess strong learning motives, which may assist them in pursuing a better understanding of the content taught and superior learning achievement in the classroom. In contrast to self-efficacy, learning motives refer to the reasons why an individual engages in a specific task (Eccles & Wigfield, 2002), that is, the will to perform the behavior. This comprises an intrinsic emotional state or aptitude that arouses learning behaviors and determines the direction, level, and strength of those behaviors (Bandura, 1997; You, Dang, & Lim, 2016). An individual’s learning motives play a decisive role in promoting his or her novelty, participation, persistence, and learning performance. As with self-efficacy, students with stronger (vs. weaker) learning motives take on more challenging tasks and persist longer in solving problems (Bandura, 1986). Because a motive functions as a stimulant to effort, it leads individuals to perform exploration and learning of their capability, which, in turn, generates better learning achievement or performance (You et al., 2016). Regarding mathematics learning, students with stronger mathematics learning motives (MLM) than their peers accomplish their tasks diligently and persistently. This results in better mathematics learning achievement and performance (Fuqoha, Budiyono, & Indriati, 2018; Stevens, Olivarez, Lan, & Tallent-Runnels, 2004; You et al., 2016).

Further, students’ self-efficacy and learning motives are essentially related to their school learning achievement and performance (Kim & Seo, 2018; Mazumder, 2014; Ocak & Yamac, 2013; You et al., 2016). This is evident in strategies used during the learning process, such as positive learning behaviors, or higher level thinking and problem-solving skills. There is also empirical evidence (Pajares, 2006; Schunk & Meece, 2006) of significant positive correlations between students’ self-efficacy for learning, subsequent learning motives, and mathematical problem-solving skills (MPSS).

In a differentiated learning environment, as students’ learning needs are satisfied, this may lead to successful learning experiences and positive emotional reactions. Mastery experience and emotional arousal

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are essential for students’ positive self-efficacy development (Bandura, 1997; Chang, 2015). Moreover, as multiple teaching strategies are used in the differentiated instruction learning environment for designing and implementing diverse learning activities, students’ learning interests will be piqued and promoted. We expected that this environment would promote students’ self-efficacy development and learning motives, as well as their academic performance (e.g., MPSS).

Why Sixth Grade?

Urdan and Midgley (2003) empirically showed with an American sample that student self-efficacy begins to decline in grade 7 or earlier. This is particularly evident in mathematics when students transition from elementary to middle school for ages 10 to 15 years (Jacobs, Lanza, Osgood, Eccles, & Wigfield, 2002). Students in the fifth and sixth grade not only face dramatic psychological, physiological, and social changes (Chang, 2015) but also confront new challenges in this fast-growing stage (Schunk & Meece, 2006). This is especially true for Taiwanese students, because grade 6 mathematical content, which integrates elementary level teaching, is more complicated and demanding than what is taught in grades 1 to 5 as a whole. Also, students face continuous and intense pressure during middle school (i.e., grades 7 to 9). In contrast to being taught basic understanding, they face a stressful learning schedule of a series of tests over three years in preparation for the high school entrance examination. Thus, sixth graders’ self-efficacy in learning mathematics may decrease because of these changes and challenges, and their learning interest and motives may decline as well. As few researchers have conducted empirical studies to determine the solution to the polarized problem of teaching and learning elementary mathematics, the question of how teachers can avoid this possible decline in learners’ self-efficacy, motivation, and interest, has become more essential to effective teaching and learning of mathematics.

Figure 1. The hypothesized model.

Our main purpose was to determine the effectiveness of a differentiated instruction intervention in promoting sixth graders’ MSE, MLM, and MPSS. We examined the relationship among the MSE and MLM (psychological state) of the sixth graders and their learning performance (i.e., MPSS) after one year of differentiated instruction intervention, and assessed the mediating effect of MLM on the effect of MSE on MPSS (see Figure 1). Therefore, we proposed the following hypotheses: Hypothesis 1a: There will be significant growth in sixth graders’ mathematics self-efficacy after one year of differentiated instruction intervention. Hypothesis 1b: There will be significant growth in sixth graders’ mathematical learning motives after one year of differentiated instruction intervention.

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Hypothesis1c: There will be significant growth in sixth graders’ mathematical problem-solving skills after one year of differentiated instruction intervention. Hypothesis 2a: Mathematics self-efficacy will have a significant effect on, and significantly predict, mathematical problem-solving skills. Hypothesis 2b: Mathematics self-efficacy will have a significant effect on, and significantly predict, mathematics learning motives. Hypothesis 2c: Mathematical learning motives will have a mediating effect on the effect of mathematics self-efficacy on mathematical problem-solving skills.

Method Participants and Procedure

In this longitudinal study, an elementary school classroom with 25 students in the sixth grade in southern Taiwan was selected as the research field. Pretests and posttests were employed to verify the effectiveness of the design and implementation of the differentiated instruction intervention. The elementary teacher and her colleagues voluntarily participated in the professional development programs provided by the teacher educators of a public university, where they emphasized the use of differentiated instruction in teaching mathematics to promote students’ MSE, MLM, and MPSS.

During the 2016–2017 academic year, the elementary teacher and her colleagues formed a mathematics teacher learning community, taking part in differentiated instruction professional development for mathematics teaching, during which the teacher designed and implemented two lessons during each semester in her classroom. The content of these lessons was related to algebra fractions and related concepts. There were 25 student participants (14 boys and 11 girls). At the beginning of their sixth grade year, their age range was 10.2 years–11.3 years (M = 10.56, SD = 0.35). They completed a pretest at the beginning of the 2016–2017 academic year and a posttest at the end of the year.

Measures

We employed the Mathematics Self-Efficacy Instrument, which was developed and validated by Chang (2012, see p. 523), with two subscales, to assess the participants’ MSE (see Figure 2 for sample items). We used the General Self-Efficacy–Related Mathematics subscale (24 items) to assess elementary students’ general self-efficacy, as related to mathematical learning. The subscale comprises four dimensions: (a) enlisting social resources and parental support, (b) academic achievement, (c) self-regulated learning, and (d) meeting others’ expectations. The Self-Efficacy for Mathematical Learning subscale (23 items), which was designed to correspond to the practical mathematics learning context inside and outside school, comprises three types of items to contextually assess students’ realistic learning circumstances: (a) mathematics cognitive, (b) strategy, and (c) test preparation items. As both subscales are closely related to the participants’ learning context, we used them (especially Self-Efficacy for Mathematical Learning) with the main scale to assess the effect of the differentiated instruction intervention on participants’ MLM and MPSS, and to assess their efficacy growth after the intervention. Responses are rated on a 100-point scale, ranging in 10-unit intervals from 0 = cannot do at all, through intermediate degrees of confidence, 50 = moderately certain can do, to complete confidence, 100 = highly certain can do (see Figure 3). Participants rated their current degree of confidence in their ability, in response to each MSEI item. This 0–100 scale response format is recommended by Bandura (2006) as a stronger predictor of performance than one with a five-interval scale (Pajares, Hartley, & Valiante, 2001). This instrument has high internal consistency of .96, .93, and .95 for the total scale, General Self-Efficacy–Related Mathematics subscale, and Self-Efficacy for Mathematical Learning subscale, respectively (Chang, 2012). General Self–Related Mathematics and Self- Efficacy for Mathematical Learning subscales accounted for 27.68% and 20.41% of the variance, respectively, and were both significantly correlated (r = .74, p < .001).

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Figure 2. Sample items from the two subscales of Mathematics Self-Efficacy Instrument. MSEI = Mathematics Self-Efficacy Instrument. Taken from Chang (2012, p. 523).

Figure 3. Response scale for Mathematics Self-Efficacy Instrument

To measure the participants’ MLM, we used the previously validated (Chang, 2010) Mathematics Learning Motive Instrument, which comprises 26 items related to elementary school students’ mathematical learning (see Table 1), and the dimensions of goal orientation, self-efficacious attribution, belief in mathematical learning, and anxiety about learning mathematics. The items are also rated on a 100-point scale (see Figure 3). This instrument has an internal consistency of .89 and .88, according to a pretest and posttest experimental study. As a correlation coefficient of .65 was also obtained between the pretest and posttest, results between pretest and posttest were stable.

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Table 1. Sample Items From the Mathematics Learning Motives Instrument

We measured students’ MPSS using the Mathematical Problem-Solving Skills Rubric, which was designed by teacher educators and externally reviewed and validated by three curriculum and instruction experts. This rubric system comprises six parts: (a) reading, (b) analysis, (c) exploration, (d) planning implementation, (e) verification, and (f) transition (see Table 2 for sample indices). Each part is composed of three indices, each of which is graded on a rubric system, with a score from 0 to 3 points, corresponding to the differentiated instruction learning environment in this study, with a highest possible total score of 54 points. The teacher educators completed the rubric at the beginning and end (pretest and posttest) of the 2016–2017 academic year, and confirmed the rubric with the teacher. We then applied corresponding statistical analysis, namely, descriptive analysis, paired-sample t tests (for Hypotheses 1a to 1c, performed separately for each) and regression analysis (for Hypotheses 2a to 2c).

Table 2. Sample Indices of the Mathematical Problem-Solving Skills Rubric

Results Psychological State and Learning Performance

Development of mathematics self-efficacy. The descriptive scores of the participants’ MSE, which are reported in Table 3, include the average score for the total MSE scale and the two subscales. There were significant differences between the pretest and posttest MSE scores, t(24) = -2.85, p < .05, in the total scale. There was no significant difference between the pretest and posttest scores, t(24) = -0.48, p > .05, in the General Self-Efficacy–Related Mathematics subscale. However, there was a significant difference in the pretest and posttest scores for the Self-Efficacy for Mathematical Learning subscale, t(24) = -4.27, p < .001.

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Posttest (vs. pretest) scores for the whole scale and for the Self-Efficacy for Mathematical Learning subscale were significantly higher. Thus, Hypothesis 1a was supported.

Table 3. Descriptive Analysis and Paired Sample t-Test Results for Mathematics Self-Efficacy and Learning Motives and Mathematical Problem-Solving Skills Scores

Note. MSE = mathematics self-efficacy, MLM = mathematics learning motives, MPSS = mathematical problem-solving skills. * p < .05, ** p < .01, *** p < .001.

Development of mathematics learning motives. Descriptive scores of the participants’ MLM are reported in Table 3. The paired sample t-test results showed that there were significant differences between the pretest and posttest scores for MLM, t(24) = -6.57, p < .001), and the MLM posttest scores were significantly higher than the pretest scores. Thus, Hypothesis 1b was supported.

Growth of mathematical problem-solving skills. Descriptive results of the participants’ MPSS are reported in Table 3. The paired sample t-test results showed that there was a significant difference between the pretest and posttest MPSS rating, t(24) = -3.77, p < .01. The posttest score was significantly higher than the pretest score. Therefore, Hypothesis 1c was supported.

Direct Effect of Mathematical Self-Efficacy and Mediating Effect of Mathematics Learning Motives on Mathematical Problem-Solving Skills

Effect of mathematics self-efficacy on mathematical problem-solving skills. MSE significantly predicted MPSS, F(1, 23) = 79.646, p < .001, with 77.6% of the variance in MPSS explained by MSE. The standardized regression coefficient showed that MSE (β = .881, t = 8.924, p < .001) had a significant effect on MPSS. Of the participants who had higher MSE compared with the others, being taught by differentiated instruction had a positive effect on their MPSS. Thus, Hypothesis 2a was supported. A more detailed inspection of the two MSE subscales showed that together they significantly predicted MPSS, F(2, 22) = 21.081, p < .001, with 65.7% of the variance explained. The standardized regression coefficient showed that Self-Efficacy for Mathematical Learning (β = .812, t = 6.493, p < .001) had a significant effect on MPSS, and the effect of General Self-Efficacy–Related Mathematics (β = -.052, t = -0.419, p > .05) on MPSS was not significant.

Effect of mathematics self-efficacy on mathematics learning motives. MSE significantly predicted MLM, F(1, 23) = 40.530, p < .001, with 63.8% of the variance in MLM explained by MSE. The standardized regression coefficient showed that MSE (β = .799, t = 6.366, p < .001) had a significant effect on MLM. Therefore, Hypothesis 2b was supported. A more detailed inspection of the two MSE subscales showed that

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together they significantly predicted MLM, F(2, 22) = 18.925, p < .001, with 64.7% of the variance explained. The standardized regression coefficient showed that Self-Efficacy for Mathematical Learning (β = .797, t = 6.151, p < .001) had a significant effect on MLM, but the effect of General Self-Efficacy–Related Mathematics (β = -.031, t = -0.238, p > .05) on MLM was not significant.

Mediating role of mathematics learning motives. MSE and MLM together significantly predicted MPSS, F(2, 23) = 133.903, p < .001, with 92.5% of the variance explained. The standardized regression coefficient showed that MSE (β = .370, t = 3.789, p < .01) had a significant effect on MPSS, and there was a significant effect of MLM (β = .640, t = 6.553, p < .001) on MPSS. MSE had a significant effect on both MPSS (β = .881, p < .001) and MLM (β = .799, p < .001), and MLM had a significant effect on MPSS (β = .640, p < .001). The mediating effect of MLM on MPSS (see Figure 4) carried a standardized parameter estimate of .511 (p < .001). These findings showed that MSE significantly predicted MLM, which, in turn, significantly predicted MPSS, revealing a partial mediation effect. Thus, Hypothesis 2c was supported.

Figure 4. Standardized regression coefficients of the mediating role of mathematics learning.

Discussion Our results showed that the sixth grade student participants who received differentiated instruction in mathematics scored significantly higher on MSE, MLM, and MPSS at the end of a year-long intervention. Educators hope that every student in the classroom will actively engage in the learning process and achieve the concept of access to equity while learning mathematics (Schoenfeld, 2014). According to our longitudinal findings, the application of the differentiated instruction learning environment was significantly beneficial to the advancement of the participants’ psychological state (i.e., MSE and MLM) and learning behavior and performance (i.e., MPSS). As Tomlinson (2017) contended, a classroom in which instruction is differentiated is dynamic, student-centered, and organic. It is therefore the teachers’ and educators’ responsibility to provide a well-designed learning environment where the needs of all students can be attended to, and where their diverse capabilities can consequently flourish and be maximized. Teachers and educators are responsible for making a commitment to their students by furnishing different learning options in all classrooms. The implementation of differentiated instruction for teaching and learning mathematics, such as multiple teaching strategies and flexible grouping, multiple representations, formative assessment tools, and the provision of various dimensions of learning content (Small, 2017; Tomlinson, 2017) will ensure that classrooms are an environment where all students can obtain equal learning opportunities. They will thus enjoy the learning process in an environment where their MSE and MLM are enhanced and their MPSS are subsequently significantly advanced.

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The Self-Efficacy for Mathematical Learning subscale was designed to contextually assess students’ realistic learning circumstances in the classroom. In Taiwan, there is greater emphasis on memorizing and calculating for students learning mathematics in the higher grades at elementary school, because of the test- oriented educational system. We found that participants’ mathematical learning self-efficacy substantially increased after their year-long differentiated instruction. They thus perceived themselves as more efficacious in future learning tasks in cognitive, strategic, and test preparation aspects of mathematics. That is, they were more confident in facing individual differences inconsistencies in small groups or whole class tasks during collaboration to solve a mathematical problem. Participants who also felt more confident with their role as an effective study partner when asked to study mathematics with a peer, felt more comfortable before taking a mathematics test, and were also able to remember what they had studied. Therefore, as students in the period of transition from elementary to middle school face dramatic psychological, physiological, and social changes and new challenges (Chang, 2015; Schunk & Meece, 2006), differentiated instruction in mathematics teaching and learning is recommended to prevent possible detrimental self- efficacy decline (see Jacobs et al., 2002; Urdan & Midgley, 2003).

Our finding that participants’ MSE significantly predicted their MPSS is consistent with previous results (e.g., Chang, 2012, 2015; Kitsantas et al., 2011; Ocak & Yamac, 2013; Pajares & Miller, 1994), which indicated that participants’ MSE had a significant effect on their mathematical achievement. Our result also corresponded to Bandura’s (1997) viewpoint that self-efficacy is one of eight influential factors in students’ learning behavior and performance. Further, we found it remarkable that MSE had a significant effect on MLM as well, namely, there was a partial mediating effect of MLM on the effect of MSE on MPSS. This finding indicates that the more efficacious that sixth grade students are in learning mathematics, the better their MLM will be, which subsequently advances their MPSS.

In practical terms, differentiated instruction intervention is influential in promoting students’ psychological state (i.e. MSE and MLM), which has a positive direct and mediating effect on behavioral performance (i.e., MPSS). This is particularly important for students who lack readiness, and who have little interest, and/or confidence in learning mathematics. Echoing the phenomenon reflected in the results of PISA 2012 and TIMSS 1999–2011, active advancement of students’ mathematics self-efficacy beliefs through differentiated instruction is critical to overcome learned helplessness (Fay et al., 2013). In addition, this method of instruction offers support for students to be more confident in future learning tasks (Bandura, 1997; Chang, 2015; Kitsantas et al., 2011; Ocak & Yamac, 2013; Schunk & Meece, 2006).

Analysis of participants’ responses to the Self-Efficacy for Mathematical Learning subscale also yielded a significant positive effect on both MLM and MPSS. As the items are designed to contextually examine and reflect participants’ mathematics learning, this finding shows that the more efficacious the mathematics cognitive, strategic, and test preparation aspects are, the better the MPSS are. We also found that the sixth grade student participants with higher MSE than their peers, tended to have stronger MLM. This may lead them (vs. other students) to pay more attention to what is taught during the differentiated instruction teaching and learning process. Therefore, our results showed that differentiated instruction is essential for sixth grade students regarding the substantial mathematics learning context. They also echo Bandura’s (2006) viewpoint and contribute to possible future predictive explanatory results (Pajares et al., 2001). In sum, given its positive promotion of elementary students’ MSE, the use of differentiated instruction for their mathematical learning is persuasive for teachers, educators, and the students.

Although this study has yielded several significant findings, there are some limitations. First, because of the low birth rate in Taiwan, there are currently around 25 to 29 students in each classroom in 63% of all elementary schools in Taiwan (Republic of China. Ministry of Education, 2017). As a result, the sample size (N = 25) drawn from one classroom was small. Thus, although the distribution of the participants’ MSE, MLM, and MPSS scores is approximately symmetric, and the skewness coefficients are between -0.5 and 0.5

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according to the normality tests of the data corps, the generalizability of our findings may still be limited. We recommend that future researchers recruit more students in the experimental group, such as from more than one classroom. Second, in this longitudinal study we used a one-group pretest–posttest design that may threaten its internal validity. Future researchers can compare experimental and control groups with a covariate of pretest scores. Third, examination of the mediating effect needs sound theoretical grounds or sequent data collections to justify the possible causal relationships among MSE, MLM, and MPSS, which may provide further support for the causal relationships among psychological state and learning performance in this study. Finally, previous results (e.g., Bal, 2016; Jacobs et al., 2002; Zimmerman, 2000) have shown that there are gender differences in MSE, MLM, and MPSS. Therefore, future researchers can assess if gender is a significant moderator in the effect of differentiated instruction on MSE, MLM, and MPSS, or in the causal relationships among these three variables.

Acknowledgements

This research was partially supported by the Ministry of Science and Technology, Taiwan. The authors are grateful to the editors, reviewers, and participants.

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