Learning engagement is highly valued by Chang (2012). According to some scholars, only by engaging in learning activities, students can promote their learning and development; as such, learning engagement plays an essential role in the learning process. Learning engagement is a key factor for successful learning and highly related to students’ academic performance.

Despite the high value placed on learning engagement, no empirical research has clarified the factors that influence students’ learning engagement, the effects of learning environment and teachers’ autonomy support on learning engagement, or the mechanism through which an individual’s self-efficacy and task value are influenced by teachers’ autonomy support. Evidently, further investigation is required.

This study investigated whether the effects of teachers’ autonomy support on learning engagement were moderated by students’ self-efficacy and task value while learning mathematics. The study can supplement control value and selfdetermination theories and fill an empirical research gap by elucidating the effects of teachers’ autonomy support on students’ learning engagement.

This study focused on the intrinsic and extrinsic factors involved in promoting junior high school students’ learning engagement in mathematics. The study investigated whether junior high school students’ perceived autonomy support would indirectly affect their learning engagement through self-efficacy and task value.

The following hypotheses were proposed:

Hypothesis 1. Students’ perceived autonomy support from teachers would predict self-efficacy, task value, and learning engagement.

Hypothesis 2. Self-efficacy would predict learning engagement.

Hypothesis 3. Task value would predict learning engagement.

Hypothesis 4. The theoretical model used in this study would fit the observed data related to Taiwanese junior high school students.

The participants were 1697 eighth-grade Taiwanese students from 30 classes in 10 junior high schools. In total, 869 participants were girls (51.21%) and 828 were boys (48.79%).

The instruments used in the study included the teachers’ autonomy support scale, self-efficacy scale, task value scale, and learning engagement scale. Participants answered questions by indicating their agreement on a 6-point Likert-type scale ranging from 1 (strongly disagree) to 6 (strongly agree). The collected data were analyzed through structural equation modeling.

Information on each scale used is presented as follows:

Teachers’ autonomy support scale: Students’ perceptions of teachers’ autonomy support has been assessed using the learning climate questionnaire (LCQ; Shih, 2009). This questionnaire has sustained reliability and validity when used in the Taiwanese context (Shih, 2009). Furthermore, a confirmatory factor analysis of the data pertaining to the 1697 participants produced the following results: χ2(9, N = 1, 697) = 152.45, p < .05, root mean square error of approximation (RMSEA) = .097, goodness of fit index (GFI) = .97, adjusted GFI (AGFI) = .93, normed fit index (NFI) = .98, nonnormed fit index (NNFI) = .97, comparative fit index (CFI) = .98, incremental fit index (IFI) = .98, relative fit index (RFI) = .97. The Cronbach’s α value was .87.

Self-efficacy scale: Students’ self-efficacy has been assessed using the motivated strategies for learning questionnaire (MSLQ; Wu & Cherng, 1992). The measure has sustained reliability and validity when used in the Taiwanese context (Wu & Cherng, 1992). The results suggested that this model had a reasonable fit with the proposed scale structure [χ2(5, N = 1, 697) = 92.12, p < .05, RMSEA = .10, GFI = .98, AGFI = .94, NFI = .99, NNFI = .98, CFI = .99, IFI = .99, RFI = .98; Cronbach’s α = .92].

Task value scale: Students’ task value has been assessed using the MSLQ (Wu & Cherng, 1992). This measure has sustained reliability and validity when used in the Taiwanese context (Wu & Cherng, 1992). The results indicated that this model had a reasonable fit with the proposed scale structure [χ2(6, N = 1, 697) = 44.84, p < .05, RMSEA = .06, GFI = .99, AGFI = .97, NFI = 1.00, NNFI = .99, CFI = 1.00, IFI = 1.00, RFI = .99; Cronbach’s α = .92].

Learning engagement scale: Students’ learning engagement has been assessed using the learning engagement scale (Lai & Wu, 2017), which contained effort, attention, and persistence subscales. This measure has sustained reliability and validity in the Taiwanese context (Lai & Wu, 2017). This model had a reasonable fit with the proposed scale structure [χ2(87, N = 1, 697) = 921.66, p < .05, RMSEA = .075, GFI = .93, AGFI = .91, NFI =.99, NNFI = .99, CFI = .99, IFI = .99, RFI = .99; the Cronbach’s α for the effort, attention, and persistence subscales were .93, .95, and .92, respectively].

A correlation analysis revealed that teachers’ perceived autonomy support was positively related to self-efficacy, task value, and learning engagement. Additionally, self-efficacy and task value were positively related to learning engagement.

The statistical results indicated that all of indices, except for the chi-squared test value [χ2(31, N = 1, 697) = 1186.74, p < 0.05], demonstrated a reasonable fit between the theoretical model and the observed data (GFI = .88, AGFI = .78, and RMSEA = .15). All relative fit indices [NFI = .94, NNFI = .91, CFI = .94, IFI = .94 and RFI = .91] exceeded the standard value of .90. Finally, the parsimony normed fit index (PNFI) was 0.65 and the parsimony goodness fit index (PGFI) was .50. These results indicate that this theoretical model had excellent overall fit with observed data, indicating that the model is suitable for characterizing Taiwanese junior high school students.

Regarding the fit of the model’s internal structure, the statistical results indicated that all of the estimated parameter values of factor loadings achieved statistical significance, with standardized estimated values being between .80 and .95. The individual item reliability valueswere between .64 and .90, conforming to the measurement standard of .50. The composite reliability values of the four latent variables of teachers’ autonomy support, self-efficacy, task value, and learning engagement were .90, .91, .88, and .89, respectively; all these values were higher than the measurement standard of .60. The average variances extracted (AVE) from these variables (teachers’ autonomy support, self-efficacy, task value, and learning engagement) were .81, .84, .71 and .73, respectively. Again, all these values were higher than the measurement standard of .50. Therefore, this theoretical model exhibited good fit with the model’s internal structure.

According to our results, students’ perceived autonomy support from teachers positively affected their self-efficacy (γ11 = .52, t = 19.44, p < .05), task value (γ21 = .55, t = 20.53, p < .05), and learning engagement (γ31 = .26, t = 8.85, p < .05). Students’ self-efficacy positively affected their learning engagement (β31 = .46, t = 5.49, p < .05). Finally, students’ task value positively affected their learning engagement (β32 = .13, t = 16.99, p < .05).

The study results can be summarized as follows: (a) The theoretical model had good fit with the empirical results. (b) Teachers’ autonomy support had direct effects on self-efficacy, task value, and learning engagement and indirect effects on learning engagement through self-efficacy and task value. If teachers can effectively communicate with students, understand their learning status, provide them with the opportunity to make independent decisions, give timely feedback, and reduce teacher’s control teaching styles, such teachers have a high likelihood of enhancing students’ self-efficacy and task value. (c) Self-efficacy had direct effects on learning engagement. (d) Task value had direct effects on learning engagement. Thus, to enhance students’ learning engagement in teaching practice, teachers should cultivate students’ self-efficacy and task value and help them develop their confidence and interest in learning activities. On the basis of these results, this study has several implications for educational practice. Relevant suggestions for future research are provided.