Content area
Problem-solving reflection constitutes a crucial component of mathematical problem-solving, yet it has been underexplored in academic literature and educational practice. To identify the factors influencing students’ behavioral intention toward problem-solving reflection, this study developed a behavioral intention model based on the Theory of Planned Behavior. It administered a questionnaire to examine these influencing factors. The valid study sample comprised 479 secondary school students from central and western China, specifically spanning six consecutive grade levels from junior high school first year to senior high school third year in Hunan Province and Guangxi Zhuang Autonomous Region. Results demonstrated that the modified behavioral intention model for mathematical problem-solving reflection exhibited strong reliability, validity, model fit, robustness, and predictive power. In mathematical learning, problem-solving reflection knowledge, problem-solving habits, and subjective norms showed significant positive direct effects on behavioral intention toward problem-solving reflection, whereas problem-solving reflection attitude and self-efficacy demonstrated no statistically significant effects. These findings offer novel insights for related research, and the proposed model provides actionable guidance for educational administrators, schools, and mathematics teachers to identify, predict, and enhance students’ mathematical problem-solving literacy.
Introduction
Mathematical problem-solving ability is a comprehensive manifestation of students’ mathematical competence, and learning mathematics is, to some extent, equivalent to learning to solve mathematical problems. Mathematics instruction aims to develop students’ problem-solving strategies. Enhancing students’ problem-solving ability has been a central focus of research in mathematics education. Polya (1945), a prominent mathematics educator, proposed four stages of mathematical problem-solving: Understanding the Problem, Devising a Plan, Carrying out the Plan, and Looking Back (Schoenfeld, 1987). According to Polya’s theory, problem-solving reflection (Looking Back) constitutes an essential component of the problem-solving process, enabling solvers to refine their problem-solving skills and strengthen their cognitive structure.
However, empirical studies have shown that students seldom review their problem-solving processes, contrary to Polya’s advocated practices. For example, Midawati (2022), employing a descriptive qualitative methodology, highlighted that students struggled to draw conclusions from questions and verify answers during the review phase. Furthermore, researchers have observed that students tend to replicate solutions from classroom examples but frequently omit the critical review step, resulting in inadequately reflected and validated answers (Rahmadania et al., 2024). Schoenfeld (1987), through a teaching experiment on problem-solving with first- and second-year undergraduates at Berkeley University, identified four primary cognitive factors influencing mathematical problem-solving: basic mathematical knowledge and skills (resources), problem-solving strategies (heuristics), knowledge selection and application (control), and mathematical beliefs (belief systems). Later, he introduced a fifth cognitive dimension—practice, which gained broad scholarly consensus. This established a five-factor cognitive framework consisting of the knowledge base, problem-solving strategies, monitoring and control, beliefs and affects, and practices (Schoenfeld et al., 1992). While these advances in mathematical problem-solving research have enhanced the understanding of its pedagogy, reflection on problem-solving remains understudied. This reflective component is overlooked in academic discourse and is often absent from contemporary mathematics classrooms.
Reflection in mathematical problem-solving is a holistic metacognitive process that encompasses the systematic examination, analysis, and evaluation of the problem-solving process, including stages before, during, and after solution implementation. It requires multidimensional critical thinking regarding strategies, cognitive patterns, emotions, and potential improvements. This may include questioning method selection, exploring alternative approaches, and assessing applicability conditions. Unlike related concepts, such as looking back (Polya’s final stage), generalizing (abstracting solutions into universal patterns), or problem posing (creating new challenges by modifying conditions), reflection synthesizes these elements while prioritizing comprehensive self-assessment. Although reflection may culminate in generalization or novel problem creation, its essence resides in the ongoing, critical scrutiny of thought processes and the pursuit of learning optimization throughout all problem-solving stages.
Reflection is widely recognized as an essential component in learning and developing expertise (Hj et al., 2020). It is a process that bridges prior knowledge with new knowledge, enabling students to construct deeper, more integrated knowledge structures, enhancing the accessibility and usability of their knowledge (Moon, 2013). Additionally, reflection is acknowledged as a key strategy for fostering creative skills (Shin, Kim, and Lee, 2017). On the one hand, reflection encourages examining problems from multiple perspectives and raising questions, thereby advancing divergent thinking. On the other hand, reflection, as a metacognitive practice, heightens metacognitive awareness of one’s thought processes, prompting students to pursue unconventional approaches and thereby cultivate creativity (Cohen and Ferrari, 2010). Effective reflection enhances learning, as it strengthens memory and cognitive capacity by linking new content to prior knowledge or experiences (Schön, 1987). This process is critical for developing students’ problem-solving abilities. Investigating reflective strategies in mathematics problem solving is essential because refining these strategies empowers students to improve their problem-solving and reasoning skills. However, most students disregard the Looking Back step, viewing problem-solving as complete upon obtaining an answer and failing to engage in reflective practices. Given reflection’s pivotal role in advancing problem-solving and broader mathematical competencies, further research warrants greater emphasis on students’ problem-solving reflections.
This study investigates the key factors that influence secondary school students’ behavioral intention to engage in problem-solving reflection. The findings will provide a foundation for analyzing how to enhance students’ problem-solving and creative skills, while informing the refinement of teaching methods to promote future mathematics learning.
Theoretical background and hypothesis development
Theoretical analysis model
The Theory of Planned Behavior (TPB), developed by Ajzen (1985), is one of the most widely applied theories in the social and behavioral sciences. This theory provides a framework for understanding how behavioral patterns change. According to the TPB, human behavior is governed by three fundamental belief categories: (1) behavioral beliefs (anticipated outcomes of the behavior), (2) normative beliefs (perceived social expectations), and (3) control beliefs (factors enabling or constraining behavioral execution). The TPB’s foundational model specifies attitude, subjective norm, and perceived behavioral control as the core determinants of behavioral intention (Ajzen, 1991).
Reflective problem-solving behavior is a goal-oriented cognitive process that demands the deliberate allocation of time and cognitive resources. Its key dimensions include attitudinal beliefs regarding reflection’s value for skill enhancement, social influences from educators and peers, and perceived capability, which incorporates individual competencies and environmental support. These components exhibit theoretical alignment with the core constructs of the Theory of Planned Behavior (attitude, subjective norms, and perceived behavioral control).
Within the TPB framework, perceived behavioral control (PBC) represents an individual’s evaluation of past experiences and expectations of future obstacles. Due to the abstract nature of PBC, which poses challenges for actionable measurement, the study operationalized it through three problem-solving reflection dimensions: self-efficacy, habit, and perceived usefulness. Furthermore, the framework included knowledge and value of problem-solving reflection as exogenous latent variables by integrating secondary mathematics teachers’ instructional experiences and accounting for reflection-specific intentions and behaviors. This operationalization maintains consistency with the TPB’s theoretical structure while improving contextual relevance.
Subsequently, new model assumptions were developed to analyze secondary school students’ intentions to engage in reflective problem-solving, as depicted in Fig. 1.
[See PDF for image]
Fig. 1
The initial model for analyzing behavioral intentions for problem-solving reflection.
Knowledge of problem-solving reflection
In Polya’s four-stage problem-solving model, the Looking Back phase involves reviewing the complete solution, reevaluating it, analyzing the results and their derivation pathways, and facilitating appropriate knowledge transfers. Polya (2004) observes that even high-performing students tend to close their books and move on to other tasks after obtaining an answer and documenting the solution process, thereby omitting a critical and beneficial stage of problem-solving. Effective problem solving requires the solver to verify the solution process and results, assess the problem-solving approach, and seek optimization. Problem-solving reflection knowledge refers to understanding its components and mechanisms. This reflective practice can be guided by prompt questions such as: Can the results be verified? Can the arguments be validated? Are there alternative solutions? Is the solution immediately apparent? Can the results or methods be applied to other problems? Some researchers (Cai and Brook, 2006) have proposed three approaches to foster student reflection: (1) generating, analyzing, and comparing alternative solutions, (2) formulating new questions, and (3) participating in generalization discussions. Despite these insights, Polya’s reflective approach to problem solving remains understudied, and the review process is frequently neglected (Sowder, 1986). This study proposes that knowledge of problem-solving reflection influences the behavioral intention to engage in reflection.
Value of problem-solving reflection
The value of problem-solving reflection refers to the judgment regarding the worth of engaging in problem-solving reflection. This reflective practice consolidates knowledge and enhance cognitive skills (Krulik and Rudnick, 1994). Research has demonstrated that strong reflective skills can boost students’ motivation, comprehension, and academic performance when acquiring new knowledge or skills (Paris and Ayres, 1994). Consequently, effective educators should recognize this principle and emphasize to their students that no problem should be considered complete, as further investigation and insight can lead to refinement of solutions and a deeper understanding. This study proposes that the value of problem-solving reflection shapes behavioral intentions toward problem-solving reflection.
Self-efficacy of problem solving
Self-efficacy, also termed optimistic confidence, is defined as an individual’s belief in their capacity to successfully complete tasks and achieve desired outcomes successfully (Akhtar, 2008). Research has shown that initial self-efficacy perceptions significantly predict persistence in problem solving (Voica, Singer, and Stan, 2020). Problem-solving efficacy represents one’s self-assessed competence in addressing problems. The perceived behavioral control over problem-solving increases proportionally with two factors: (1) the breadth of one’s problem-solving knowledge and strategies an individual possesses, and (2) the frequency of one’s successful problem-solving experiences. This study postulates that perceptions of problem-solving efficacy affect the behavioral intention to engage in problem-solving reflection.
Attitude toward problem-solving reflection
Attitude represents an individual’s evaluative disposition, either positive or negative, toward a specific behavior (Ajzen, 2005). As a domain of the affective sciences, attitude is distinguished from emotion by its greater cognitive component and stability; similarly, it differs from belief in being less purely cognitive (Katrancı and Şengül, 2019). Students’ mathematical attitudes emerge from their cumulative mathematical experiences (Goldin et al., 2016), with their problem-solving performance being significantly associated with these attitudes. Conceptually, attitude is a stable psychological disposition toward specific ideas, objects, or entities, characterized by positive or negative valence. Regarding problem-solving reflection, attitude encompasses an individual’s overall assessment of this practice. In this study, problem-solving reflection attitude denotes an individual’s predisposition to favor or disfavor reflection, which affects their behavioral intention to reflect.
Perceived usefulness
Perceived usefulness is generally defined as the degree to which a person believes using a particular system will enhance their performance at work (Davis, 1989). Empirical evidence suggests that perceived usefulness significantly influences learning motivation; specifically, the more useful students perceive the content to be, the more likely they are to learn successfully (Huang, 2021). In this study, perceived usefulness specifically denotes the belief that problem-solving reflection facilitates problem-solving competence and learning outcomes. Furthermore, the more strongly students believe in the usefulness of problem-solving reflection, the greater their behavioral intention to reflect after solving problems.
Habit of problem solving
Habits, which are formed gradually over time and are closely associated with the development of human conditioned reflexes, constitute a significant factor influencing behavioral intentions (Gefen, 2003). In mathematics education, cultivating students’ effective mathematical thinking habits represents an important objective (Aslan and Özmusul, 2022). A problem-solving habit refers to an automated approach to addressing problems that individuals develop through experience. Well-developed problem-solving habits show strong correlations with the intention to engage in problem-solving reflection. Specifically, when a problem solver has developed the habit of reflecting after solving problems, they are more likely to demonstrate an intention to reflect on problem-solving.
Subjective norms related to reflection of problem solving
Subjective norm represents the social pressure perceived by an individual to perform a particular behavior, based on the anticipated behavior of others (Ajzen, 1985). This construct captures how individuals or groups influencing behavioral decisions affect the choice to adopt specific behaviors. Parents, mathematics teachers, experts, and peers collectively shape students’ learning behaviors in educational settings. In this study, subjective norms are operationalized as the environmental pressures that motivate students to engage in problem-solving reflection. Research demonstrates that teacher and parent involvement positively contributes to developing students’ problem-solving skills (Shulman and Shulman, 2004), thereby facilitating problem-solving reflection. Mukminin et al. (2020) found that subjective norms exert a significant positive influence on behavioral intentions in education, indicating that individuals develop stronger behavioral intentions when they perceive approval from significant others.
Behavioral intention to reflect problem solving
Behavioral intention constitutes an individual’s self-assessed likelihood of performing a particular behavior, reflecting their willingness to engage in that behavior. According to Ajzen (1991), behavioral intention is shaped not just by attitudes and subjective norms, but also by perceived behavioral control. Research indicates that attitudes partially predict behavioral intentions and serve as the most significant determinant of behavior (Gärling and Fujii, 2002). Furthermore, behavioral intention directly influences behavior, with stronger intentions leading to more accurate behavioral predictions. This study defines behavioral intention as a student’s propensity to engage in problem-solving reflection. The strength of an individual’s intention to reflect on problem solving is positively associated with the likelihood of performing such reflection.
The initial theoretical hypothesis of this study proposes that behavioral intention toward problem-solving reflection (Be) is determined by seven variables: knowledge of problem-solving reflection (Kn), value of problem-solving reflection (Va), self-efficacy in problem-solving (Se), attitude toward problem-solving reflection (At), perceived usefulness (Pe), problem-solving habit (Ha), and subjective norms regarding problem-solving reflection (Su). Building upon the Theory of Planned Behavior, we developed a model to examine students’ intention to engage in problem-solving reflection, with dual objectives: (1) identifying factors that exert significant positive effects on secondary school students’ reflection intentions, and (2) determining the strongest predictive factors. The model was validated using partial least squares structural equation modeling (PLS-SEM) implemented in SmartPLS 4.
Hypothesis 1. Knowledge of problem-solving reflection is positively related to self-efficacy in problem-solving.
Hypothesis 2. Knowledge of problem-solving reflection is positively related to attitude toward problem-solving reflection.
Hypothesis 3. Knowledge of problem-solving reflection is positively related to the perceived usefulness of problem-solving reflection.
Hypothesis 4. Knowledge of problem-solving reflection is positively related to the habit of problem-solving.
Hypothesis 5. Knowledge of problem-solving reflection is positively related to subjective norms of problem-solving reflection.
Hypothesis 6. The value of problem-solving reflection is positively related to self-efficacy of problem solving.
Hypothesis 7. The value of problem-solving reflection is positively related to attitude toward problem-solving reflection.
Hypothesis 8. The value of problem-solving reflection is positively related to the perceived usefulness of problem-solving reflection.
Hypothesis 9. The value of problem-solving reflection is positively related to the habit of problem solving.
Hypothesis 10. The value of problem-solving reflection is positively related to subjective norms of problem-solving reflection.
Hypothesis 11. Self-efficacy of problem-solving is positively related to attitude toward problem-solving reflection.
Hypothesis 12. The perceived usefulness of problem-solving reflection is positively related to attitude toward problem-solving reflection.
Hypothesis 13. Subjective norms of problem-solving reflection are positively related to the habit of problem solving.
Hypothesis 14. Self-efficacy of problem solving is positively related to behavioral intention toward problem-solving reflection.
Hypothesis 15. Attitude toward problem-solving reflection is positively related to behavioral intention toward problem-solving reflection.
Hypothesis 16. The perceived usefulness of problem-solving reflection is positively related to behavioral intention toward problem-solving reflection.
Hypothesis 17. The habit of problem solving is positively related to behavioral intention toward problem-solving reflection.
Hypothesis 18. Subjective norms of problem-solving reflection are positively related to behavioral intention toward problem-solving reflection.
Hypothesis 19. Knowledge of problem-solving reflection is positively related to behavioral intention toward problem-solving reflection.
Hypothesis 20. The value of problem-solving reflection is positively related to behavioral intention toward problem-solving reflection.
Method
Questionnaire design
This study developed a research instrument that was grounded in the Theory of Planned Behavior (TPB) and incorporated characteristics of secondary school students’ mathematical problem-solving reflection. The questionnaire was constructed following a rigorous scientific process:
Scale Development: The core observed variables were derived from Silver’s competency framework for mathematical reflection (Silver, 1994), Schoenfeld’s metacognitive framework for mathematical reflection (2016), and classical TPB scale structures, with item formulation adapted to align with the Chinese secondary school mathematics curriculum standards.
Preliminary Validation: Three mathematics education experts conducted two rounds of expert reviews to verify construct alignment, followed by a pilot study that eliminated four items with factor loadings below 0.6, resulting in 27 retained items.
Cognitive Adaptation: To accommodate the cognitive characteristics of middle school students, bilingual translation was performed under the guidance of both an English education specialist and a mathematics education specialist. Item readability was assessed through focus group interviews with three middle school students and three high school students.
The final questionnaire had two sections. The first section collected demographic information, such as gender and grade level. The second section measured eight latent variables related to behavioral intention toward problem-solving reflection using 24 observed items. All observed variables were assessed by a five-point Likert scale, with 1 to 5 corresponding to strongly disagree, disagree, neutral, agree, and strongly agree, respectively.
Data collection
This study employed a sample drawn from central and western China, specifically targeting Hunan Province and Guangxi Zhuang Autonomous Region, with participants spanning six grade levels from Grade 7 to Grade 12. The decision to investigate these students as a unified group was based on two key considerations. First, extant research has demonstrated the TPB model’s stability across adolescent age groups. For instance, McEachan et al. (2011) established that the TPB maintains consistent explanatory power for intention-behavior relationships among adolescents (aged 12–18), with nonsignificant age moderation effects. Second, the TPB’s core constructs exhibit structural invariance and are consistently applicable throughout adolescent cognitive development, as evidenced by Ajzen’s (2020) research findings.
The survey results indicated that 8.26% of participants were Grade 7 students, 9.67% were in Grade 8, and 25.31% were in Grade 9. Among senior high school participants, 26.54% were Grade 10 students, 14.59% in Grade 11, and 15.64% in Grade 12. Male students accounted for 56.41% of the sample, compared to 43.59% female students. Additionally, 19.33% of respondents were identified as mathematically gifted.
The study administered anonymous self-report questionnaires through the online platform ‘WENJUANXING’, collecting 579 responses in total. After implementing rigorous screening procedures to exclude incomplete or invalid submissions, 479 valid questionnaires were retained, yielding a validity rate of 82.73%. All collected data were processed under strict anonymization protocols and are stored securely to ensure participant confidentiality, with guarantees that no individual-level information would be disclosed throughout the research process.
Data analysis
This study adopted a quantitative survey design. It investigated the relationships among variables and assessed the model fit using structural equation modeling techniques. The analysis also evaluated untested predictors and theoretical frameworks. To address both predictive and exploratory objectives, the researchers employed partial least squares structural equation modeling (PLS-SEM) using SmartPLS 4, a specialized statistical software. PLS-SEM, conducted through SmartPLS 4, is particularly appropriate for measurement modeling and structural model analysis, especially when dealing with models comprising multiple latent variables. This method facilitates the examination of complex structural equation models featuring numerous latent variables and their indicators.
Results
The analysis revealed that the path coefficient for measurable variable Q12 (within the perceived usefulness construct) was statistically non-significant. As a result, it was considered for deletion. The final model is presented in Fig. 2.
[See PDF for image]
Fig. 2
The final model for analyzing behavioral intentions for problem-solving reflection.
Normality testing
The study rigorously evaluated data normality prior to conducting structural equation modeling analysis. Consistent with classical criteria established by West et al. (1995), all measurement items were assessed for skewness and kurtosis, with acceptable thresholds defined as absolute skewness values < 2 and absolute kurtosis values < 7. These criteria represent widely accepted benchmarks for ensuring robust parameter estimation in structural equation modeling contexts. In Table 1, statistical examination revealed that the absolute values of skewness for all questionnaire items remained below 0.973 (range: |0.065| to |0.760 | ), while kurtosis values uniformly stayed below 1.983 (range: |0.027| to |1.004 | ), thereby fully satisfying the normality requirements for subsequent multivariate analyses.
Table 1. Descriptive statistics for data normality testing.
Construct | Items | Min | Max | Mean | SD | Skewness | Kurtosis |
|---|---|---|---|---|---|---|---|
Knowledge of Problem-Solving Reflection | Q1 | 1 | 5 | 3.890 | 0.857 | −0.640 | 0.528 |
Q2 | 1 | 5 | 3.670 | 0.844 | −0.422 | 0.352 | |
Q3 | 1 | 5 | 4.100 | 0.762 | −0.619 | 0.327 | |
Value of Problem-Solving Reflection | Q4 | 1 | 5 | 4.270 | 0.720 | −0.719 | 0.371 |
Q5 | 2 | 5 | 4.400 | 0.638 | −0.724 | 0.206 | |
Q6 | 2 | 5 | 4.390 | 0.616 | −0.529 | −0.333 | |
Self-efficacy of Problem Solving | Q7 | 1 | 5 | 3.180 | 0.870 | −0.065 | 0.169 |
Q8 | 2 | 5 | 3.940 | 0.786 | −0.328 | −0.382 | |
Q9 | 1 | 5 | 3.750 | 0.884 | −0.515 | 0.225 | |
Perceived Usefulness | Q10 | 1 | 5 | 4.310 | 0.675 | −0.747 | 0.835 |
Q11 | 1 | 5 | 4.310 | 0.679 | −0.754 | 0.795 | |
Attitude toward Problem-Solving Reflection | Q13 | 1 | 5 | 4.080 | 0.804 | −0.573 | 0.058 |
Q14 | 1 | 5 | 4.250 | 0.706 | −0.760 | 1.004 | |
Q15 | 1 | 5 | 4.130 | 0.777 | −0.624 | 0.276 | |
Habit of Problem Solving | Q16 | 2 | 5 | 4.120 | 0.713 | −0.424 | −0.166 |
Q17 | 1 | 5 | 4.020 | 0.820 | −0.564 | 0.167 | |
Q18 | 1 | 5 | 3.870 | 0.803 | −0.377 | −0.027 | |
Subjective Norms | Q19 | 2 | 5 | 4.290 | 0.701 | −0.585 | −0.388 |
Q20 | 1 | 5 | 4.120 | 0.780 | −0.719 | 0.703 | |
Q21 | 1 | 5 | 4.280 | 0.694 | −0.702 | 0.535 | |
Behavioral Intention toward Problem-Solving Reflection | Q22 | 1 | 5 | 3.850 | 0.797 | −0.254 | −0.317 |
Q23 | 1 | 5 | 3.600 | 0.867 | −0.250 | 0.140 | |
Q24 | 1 | 5 | 3.830 | 0.881 | −0.437 | 0.045 |
Reliability and validity of the measurement model
The evaluation of the measurement model comprises reliability and validity analyses. This study assessed the model’s internal consistency and stability. According to Henseler et al. (2015), a higher Rho-A value corresponds to greater model reliability, with the recommended threshold being 0.70-0.95. Values exceeding 0.95 suggest redundant terms in the measurement model, which may compromise construct validity. As shown in Table 2, all Rho-A estimates in the research model fall within this range. Furthermore, the Cronbach’s alpha coefficients for all latent variables exceed 0.70. Both the Rho-A values and the Cronbach’s alpha coefficients meet the established criteria, demonstrating a high level of internal consistency in the model.
Table 2. Construct reliability and validity.
Construct | Cronbach’s Alpha | Rho-A | CR | AVE |
|---|---|---|---|---|
Subjective norms | 0.814 | 0.816 | 0.890 | 0.729 |
Perceived usefulness | 0.898 | 0.902 | 0.951 | 0.907 |
Habit | 0.807 | 0.807 | 0.886 | 0.722 |
Attitude | 0.845 | 0.847 | 0.906 | 0.763 |
Knowledge | 0.807 | 0.807 | 0.887 | 0.725 |
Behavioral Intention | 0.838 | 0.849 | 0.903 | 0.757 |
Value | 0.905 | 0.906 | 0.941 | 0.841 |
Self-efficacy | 0.769 | 0.769 | 0.866 | 0.683 |
The reliability assessment of the measurement model necessitates further examination of the composite reliability (CR) and the average variance extracted (AVE) values. The results indicate that all CR values for the latent variable constructs approximate 0.9, substantially exceeding the recommended threshold of 0.7. Similarly, the AVE values demonstrating the explanatory power of latent variables on observed variables all surpass the 0.5 benchmark, with higher values indicating stronger convergent validity. These findings provide evidence for the measurement model’s robust internal consistency and reliability (Hair et al., 2021).
As noted by Hair et al. (2019), a factor loading exceeding 0.708 indicates higher model reliability. The PLS algorithm analysis of external model load values revealed a range from 0.762 (minimum) to 0.956 (maximum), with most observed variables demonstrating factor loadings above 0.8. These results confirm the constructs’ high reliability.
Subsequently, discriminant validity was evaluated using three methods: the Fornell-Larcker criterion, the heterotrait-monotrait ratio (HTMT), and cross-loading analysis. When all three methods’ thresholds are satisfied, the discriminant validity analysis can be considered highly reliable (Hair et al., 2021). Specifically, the Fornell-Larcker criterion requires that the square root of each latent variable’s AVE must exceed its correlation coefficients with other latent variables. As shown in Table 3, the diagonal AVE square roots are consistently greater than the off-diagonal correlation coefficients, confirming strong discriminant validity among the latent variables in the structural model.
Table 3. Fornell-Larcker criterion value.
Su | Pe | Ha | At | Kn | Be | Va | Se | |
|---|---|---|---|---|---|---|---|---|
Su | 0.854 | |||||||
Pe | 0.627 | 0.953 | ||||||
Ha | 0.515 | 0.630 | 0.850 | |||||
At | 0.661 | 0.763 | 0.682 | 0.874 | ||||
Kn | 0.407 | 0.481 | 0.580 | 0.480 | 0.852 | |||
Be | 0.538 | 0.524 | 0.674 | 0.627 | 0.611 | 0.870 | ||
Va | 0.573 | 0.646 | 0.493 | 0.680 | 0.478 | 0.484 | 0.917 | |
Se | 0.315 | 0.483 | 0.578 | 0.474 | 0.526 | 0.547 | 0.275 | 0.827 |
The HTMT analysis yielded a maximum value of 0.872, which is below the threshold of 0.900, confirming adequate discriminant validity among the constructs. Thus, both the Fornell-Larcker criterion and HTMT values satisfy the established criteria, demonstrating the strong discriminant validity of the measurement model.
A third method for assessing discriminant validity involves cross-loading analysis, which requires that each indicator’s loading on its assigned construct should be higher than its correlations with other constructs. The results indicate that all indicators exhibit stronger associations with their intended constructs compared to other constructs, thereby validating their dimensional assignments.
In conclusion, this study systematically evaluated the measurement model by examining factor loadings, internal consistency reliability, convergent validity, and discriminant validity through PLS-SEM analysis. All statistical indicators met the required standards, confirming that the measurement model possesses satisfactory psychometric properties.
Evaluation of the measurement model
Structural model evaluation encompasses covariance analysis, goodness-of-fit, explanatory power, effect sizes, predictive power, as well as the significance and relevance of path coefficients. As Hair et al. (2019) suggest, structural model assessment typically commences with covariance analysis. Our calculations revealed that all variance inflation factor (VIF) values, which measure multicollinearity among observed variables, were below 5, with most below 3.0. This confirms the absence of multicollinearity issues in the structural model.
The coefficient of determination (R²), the primary metric for evaluating structural models, quantifies the relationship between dependent and independent variables while representing the variance explained by endogenous constructs, thereby measuring the model’s explanatory power (Gana and Broc, 2019). R² values range from 0 to 1, with 0.190–0.333, 0.333–0.670, and >0.670 indicating weak, moderate, and strong explanatory power, respectively, while 1 denotes perfect explanatory validity (Urbach and Ahlemann, 2010). The results of the PLS analysis demonstrated that most R² values fell within the moderate range. Although the problem-solving reflection self-efficacy construct showed weak explanatory power, the model nevertheless encompassed the primary predictors of secondary school students’ mathematical problem-solving reflection behavioral intention, exhibiting satisfactory predictive validity (see Fig. 2).
Beyond R² analysis, Hair et al. (2012) recommend calculating cross-validated redundancy (Q²) to evaluate predictive validity. Q² values > 0 that exceed 0.02, 0.15, and 0.35 signify weak, moderate, and strong predictive relevance, respectively (Wong, 2013). The blindfolding analysis via the PLS-Predict algorithm revealed Q² values for the eight endogenous latent variables ranging from 0.360 to 0.641, confirming the structural model’s strong predictive validity.
The fit of a structural model can be assessed by various metrics such as standardized root mean square (SRMR), canonical fit index (NFI), and goodness of fit (GOF = ). According to Dash and Paul (2021), assessing the fit of a PLS-SEM requires the SRMR value to be below 0.08 and the NFI value to exceed 0.80. Regarding the GOF index, the established thresholds suggest that values between 0.1–0.25, 0.26–0.36, and above 0.36 represent weak, moderate, and good global fit, respectively (Wetzels, Odekerken-Schröder, and Van Oppen, 2009). In the current study, the obtained values (SRMR = 0.064, NFI = 0.804, GOF = 0.596) satisfy all evaluation criteria, demonstrating the structural model’s strong fit and robustness.
Path coefficients (β-values), t-values, and p-values for all hypothesized relationships were computed using the bootstrapping method with 5000 resamples. Following Huang’s criteria (2021), path significance was determined as follows: 1.96 < |t | < 2.58 indicates p < 0.05; 2.58 < |t | < 3.29 indicates p < 0.01; and |t | > 3.29 indicates p < 0.001.
As presented in Table 4, several constructs demonstrated significant positive effects on problem-solving reflection behavioral intention: subjective norms (β = 0.172, p < 0.01), habit (β = 0.275, p < 0.001), attitude (β = 0.214, p < 0.001), knowledge (β = 0.244, p < 0.001), and self-efficacy (β = 0.161, p < 0.001). Conversely, perceived usefulness showed a significant negative relationship with behavioral intention (β = -0.135, p < 0.05), suggesting that behavioral intention may inversely affect perceived usefulness. However, the value of problem-solving reflection (β = 0.030, p > 0.05) did not exert a statistically significant effect on problem-solving reflective behavior.
Table 4. Significance and correlation of path coefficients.
Path | β | t | p | Significance |
|---|---|---|---|---|
At → Be | 0.214 | 3.685 | 0.000 | YES |
Ha → Be | 0.275 | 5.384 | 0.000 | YES |
Kn → At | 0.021 | 0.543 | 0.587 | NO |
Kn → Be | 0.244 | 4.907 | 0.000 | YES |
Kn → Ha | 0.400 | 8.592 | 0.000 | YES |
Kn → Pe | 0.223 | 4.535 | 0.000 | YES |
Kn → Se | 0.511 | 10.387 | 0.000 | YES |
Kn → Su | 0.172 | 3.576 | 0.000 | YES |
Pe → At | 0.473 | 8.761 | 0.000 | YES |
Pe → Be | −0.135 | 2.394 | 0.017 | YES |
Se → At | 0.146 | 4.050 | 0.000 | YES |
Se → Be | 0.161 | 3.696 | 0.000 | YES |
Su → Be | 0.172 | 3.171 | 0.002 | YES |
Su → Ha | 0.267 | 4.552 | 0.000 | YES |
Va → At | 0.325 | 6.457 | 0.000 | YES |
Va → Be | 0.030 | 0.465 | 0.642 | NO |
Va → Ha | 0.149 | 2.781 | 0.005 | YES |
Va → Pe | 0.539 | 10.624 | 0.000 | YES |
Va → Se | 0.031 | 0.634 | 0.526 | NO |
Va → Su | 0.490 | 10.193 | 0.000 | YES |
→ indicates the direction of the path.
Analysis of indirect effects revealed that among the 25 mediated pathways in the model, four relationships were statistically non-significant (p > 0.05): “Va→Se→At”, “Kn→At→Be”, “Va→Se→At→Be”, and “Va→Se→Be”, which indicates that the mediating effects of problem-solving reflection’s attitude and self-efficacy were either negligible or non-existent in these pathways (Gaskin, Ogbeibu, and Lowry, 2023). Regarding total effects, all indirect effects were significant at the 5% level except for the “Pe→Be” and “Va→Se” pathways.
Variance accounted for (VAF) values, calculated from indirect and total effects (Table 5), quantify mediation effects: VAF > 0.8 indicates full mediation; 0.2 < VAF < 0.8 suggests partial mediation; and VAF < 0.2 denotes no mediation (Hair et al., 2019). Notably, mediation analysis requires significant path coefficients as a prerequisite. While the VAF of problem-solving reflective value on behavioral intention was 0.876, its non-significant path precludes meaningful mediation interpretation.
Table 5. VAF value.
Path | Total indirect effect | Total effects | VAF |
|---|---|---|---|
Kn → At | 0.180 | 0.200 | 0.900 |
Kn → Be | 0.248 | 0.492 | 0.504 |
Kn → Ha | 0.046 | 0.446 | 0.103 |
Se → Be | 0.031 | 0.193 | 0.161 |
Su → Be | 0.073 | 0.245 | 0.298 |
Va → Be | 0.218 | 0.249 | 0.876 |
Va → Ha | 0.131 | 0.280 | 0.468 |
Va → At | 0.259 | 0.585 | 0.443 |
→ indicates the direction of the path.
The study observed significant mediation effects in the mathematical problem-solving reflection: knowledge on attitude (VAF = 0.900), value on attitude (VAF = 0.443), knowledge on behavioral intention (VAF = 0.504), subjective norms on reflection intention (VAF = 0.298), and value on reflection habit (VAF = 0.468). Conversely, no mediation effects were found for problem-solving reflection knowledge on habit (VAF = 0.103), and problem-solving reflection efficacy on behavioral intention (VAF = 0.161).
Relying exclusively on p-values to determine significance is methodologically limited; while p-values indicate statistical significance, effect sizes reflect practical significance. Importantly, statistically significant results may correspond to negligible effect sizes. Cohen’s f² effect size was therefore evaluated, with values below 0.02 considered negligible, and 0.02, 0.15, and 0.35 representing small, medium, and large effects, respectively (Hair et al., 2021).
As shown in Table 6, most predictors, including problem-solving habit, problem-solving reflection attitude, problem-solving reflection knowledge, problem-solving reflection self-efficacy, and problem-solving reflection subjective norms, exhibited small influence (0.02 < f² < 0.15) on behavioral intention. By contrast, neither the value of problem-solving reflection nor perceived usefulness demonstrated any significant effect on behavioral intention.
Table 6. Effect f²value.
Path | F-square | Effect size |
|---|---|---|
At → Be | 0.032 | Small |
Ha → Be | 0.075 | Small |
Kn → At | 0.001 | No effect |
Kn → Be | 0.081 | Small |
Kn → Ha | 0.214 | Medium |
Kn → Pe | 0.070 | Small |
Kn → Se | 0.279 | Medium |
Kn → Su | 0.035 | Small |
Pe → At | 0.315 | Medium |
Pe → Be | 0.016 | No effect |
Se → At | 0.040 | Small |
Se → Be | 0.036 | Small |
Su → Be | 0.037 | Small |
Su → Ha | 0.083 | Small |
Va → At | 0.165 | Medium |
Va → Be | 0.001 | No effect |
Va → Ha | 0.024 | Small |
Va → Pe | 0.412 | Large |
Va → Se | 0.001 | No effect |
Va → Su | 0.285 | Medium |
→ indicates the direction of the path.
Discussion and conclusions
Mathematical problem-solving reflection significantly contributes to the development of students’ problem-solving abilities and the enhancement of mathematical literacy. Employing partial least squares structural equation modeling (PLS-SEM) with SmartPLS 4, this study identified key factors influencing secondary school students’ intention to engage in problem-solving reflection. The analysis revealed the following statistically significant positive effects: (1) problem-solving self-efficacy, (2) reflection attitude, (3) problem-solving habits, (4) subjective norms, and (5) reflection knowledge on behavioral intention. Conversely, perceived usefulness demonstrated a negative relationship, suggesting that behavioral intention may influence perceived usefulness rather than vice versa, while the value of problem-solving reflection showed no significant effect.
Effect size analysis (f²) highlighted problem-solving habits and reflection knowledge as particularly influential predictors. Notably, neither the value of problem-solving reflection nor perceived usefulness affected behavioral intention. Synthesizing significance tests and effect sizes, three factors emerged as particularly impactful: problem-solving habits, subjective norms, and reflection knowledge.
A key finding concerns the substantial influence of reflection knowledge. Students possessing systematic reflection skills and successful experience with challenging problems are more likely to: (1) integrate reflection into their problem-solving process, and (2) regulate emotional fluctuations during mathematical tasks. This corroborates McCallie’s (2016) finding regarding the critical role of teacher-provided theoretical explanations and practical training. The internalization mechanism operates such that procedural knowledge (e.g., error analysis strategies) and conditional knowledge (e.g., reflection timing) acquired through systematic training enable students to: (1) internalize reflection as an inherent problem-solving component, and (2) sustain reflective practice under stress (Gross, 2015). This demonstrates a synergistic interaction between knowledge, behavior, and emotion.
Problem-solving habits also directly influence reflection intention. This aligns with the work of Martin-Requejo et al. (2023) on the correlation between habit and problem solving and with the work of Martins and Martinho (2021) on enhanced problem representation through reflection. As Güner and Erbay (2021) established, these crystallized metacognitive strategies enhance reflection depth through sustained problem representation. Our study advances this research by quantifying habit strength and overcoming qualitative limitations in measuring long-term behavioral tendencies. Consistent with Chen and Lin’s (2020) cross-cultural findings, we recommend that mathematics teachers: (1) cultivate strong problem-solving habits, and (2) provide strategic guidance through methods like pre-class study, peer error correction, and mathematical journaling.
Contrary to theoretical expectations, subjective norms exerted a relatively weak effect on behavioral intention toward problem-solving reflection despite achieving statistical significance. This finding suggests heightened student self-awareness in the contemporary digital economy, wherein problem-solving reflection behaviors appear more strongly influenced by intrinsic self-perception than external pressures. These results align with the core tenets of Self-Determination Theory (SDT; Ryan and Deci, 2000), particularly with regard to the autonomy of the digital-native generation in mathematics learning contexts.
This phenomenon corresponds to broader trends in digitalization of global education. The proliferation of ubiquitous learning resources (e.g., AI-powered problem-solving tools) has diminished the traditional authority of external norms, fostering a student preference for self-directed learning modalities (Bennett, Maton, and Kervin, 2008; Hwang et al., 2020; Zhao and Watterston, 2021). Crucially, external pressures fail to engender genuine internalization of reflective practices; rather, authentic adoption occurs only when students intrinsically value reflection as an essential problem-solving component.
These insights justify the integration of core elements from SDT, particularly autonomy needs and perceived competence, into planned behavior frameworks, thereby addressing theoretical limitations in digital-era behavior prediction.
Limitations and suggestions
This study established an effective prediction model to examine factors influencing students’ intention to engage in mathematical problem-solving reflection. While demonstrating both theoretical and practical contributions, several limitations should be noted.
First, the survey was limited to samples from central and western China, which may affect the generalizability of findings to other cultural contexts. Cross-cultural validation considering group differences would be necessary for broader application.
Second, although the structural model demonstrates theoretical robustness and predictive validity regarding factors affecting secondary students’ reflective behaviors in mathematical problem-solving, its reliability is bounded by predefined thresholds, as with all predictive models.
Third, the current model does not incorporate all potential influencing factors, as unmeasured variables may also play significant roles, suggesting opportunities for future research.
Finally, while focusing on general behavioral patterns among middle school students, the study did not examine grade-level variations, representing another limitation. Future studies should conduct more granular analyses of developmental differences across grades.
Acknowledgements
The research funding for this study was provided by the Scientific Research Project of the Education Department of Hunan Province—“Research on the Characteristics of Reflective Behavior in Problem-Solving and Intervention Strategies for Mathematically Gifted Students” (24A0042).
Author contributions
Conceptualization, investigation, JP; methodology, software, formal analysis, JP and LW; writing—original draft preparation, JP, LW, RX, ZJ; writing—review and editing, JP, RX, XR; visualization, all author; supervision, project administration and funding acquisition, JP. All authors have read and agreed to the published version of the manuscript.
Data availability
The data from professional journal articles are public data that anyone can access online (provided they subscribe to the journal). The datasets generated during the study are not publicly available due to them containing personal information that could compromise research participant privacy. But the data are available from the corresponding author on reasonable request.
Competing interests
The authors declare no competing interests.
Ethical approval
This study was conducted in accordance with the principles of the Declaration of Helsinki. Ethical approval was granted by the Ethics Committee of Hunan Normal University (Approval No. 2024321; Date: March 13, 2024).
Informed consent
Prior to participation, verbal informed consent was obtained from all participants and their legal guardians in March 31, 2024. Researchers explained the study’s purpose, procedures, and confidentiality measures during individual interviews, emphasizing voluntary participation and the right to withdraw at any time without penalty. Explicit verbal consent was recorded for data use in publications.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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