Analyzing Students' Critical Thinking Skills in Geometry Learning through Augmented Reality: An Evaluation Using the Technology Acceptance Model.

Technology acceptance model (TAM)

Technology Acceptance Model (TAM), which is the result of the Theory of Reasoned Action (TRA), describes the impact of individual motivation on the process of technology adoption (Al-Adwan et al., 2023; Hidayat et al., 2024; ). It is among the most powerful theories of explaining the behavioural intentions and attitudes of users towards technology, according to their perceptions of a system (Marian et al., 2025). TAM is used in the analysis of user-technology interaction widely in education, business, and healthcare. The model consists of five connected constructs, which are Perceived Usefulness (PU), Perceived Ease of Use (PEU), Attitude Toward Use (ATU), Behavioural Intention (BI), and Actual System Use (ASU) (Purwadi et al., 2023). PU represents the extent to which technology enhances performance, while PEU reflects how effortless it is to use. Both PU and PEU shape users’ positive attitudes and behavioural intentions toward adopting digital technologies.

One of the strengths of the TAM model is that it can be easily used to simplify user experiences and enhance efficiency in the processes of engaging a system (Al-Shorman et al., 2025; Purwadi et al., 2023). Users who think that a certain technology is useful and is readily available to them would likely have positive attitudes and intentions to carry on with the usage of the technology. ATU shows the feelings one has towards the system, and BI shows the feelings that one has towards using the same technology again (Li & Jiang, 2023; Trieu, 2023). Also, ASU can be used as a sign of an effective technology implementation regarding the experiences of the users that trigger the desire to be satisfied with it and use it constantly (Marian et al., 2025).

PU and PEU are two different constructs which always determine the selection decisions made by the users toward adopting a technology. Despite the relationship of these constructs, PU and PEU have an independent effect on attitude and behavioural intention of the users. Awareness of PU and PEU is a privilege in technology-based learning systems design, as it helps in generating user engagement, besides promoting technology acceptance (Dalle et al., 2024; Othman et al., 2024). TAM is a model that may be expanded with more aspects like learning motivation, digital experience, and social context.

The thesis of the study is the three primary variables of TAM, PU (perceived usefulness), PEU (perceived ease of use), and ASU (actual system use), due to their relevance in the context of learning with the use of the Augmented Reality (AR) technology (Nugroho & Sukirman, 2023; Putu et al., 2023; Wen et al., 2023; Wibowo, 2023). Moreover, the critical thinking abilities of the students are determined as the possible external variables that can influence the perception of AR technology and attitude towards learning among students (Koumpouros, 2024; Nikou, 2024; Rossano et al., 2020). The connection of TAM to the cognitive variables (analytical and problem-solving skills) substantiates the relevance of the model to technology-based learning in higher education (Al-Shorman et al., 2025). This is where a student learning technology framework proposed by Fred Davis (1989) is presented.

Figure 1 presents Technology Acceptance Model (TAM) diagram illustrating the relationships between variables in system acceptance. This figure shows how Perceived Usefulness (PU) and Perceived Ease of Use (PEU) influence Attitude Toward Using. External variables influence both Perceived Usefulness and Perceived Ease of Use, which in turn affect Behavioral Intention and ultimately Actual System Use (ASU).

Figure 1. Technology acceptance model (TAM).

Source: Davis & Venkatesh (2004)

Augmented reality

Augmented Reality (AR) is a technology according to which virtual elements can be superimposed on the visual perception of another user in the real world at the same time, which gives them some form of authentic learning, which is immersive and contextualised (Jang et al., 2021; Lismaya et al., 2022; Nusroh et al., 2022; Yanto et al., 2024). The students can immerse directly into the digital representations that become interacting with the real-world environment, and it is considered experience-based learning and active inquiry (Chiliquinga et al., 2024). AR has three signature features in the education industry, namely interactive visualisation, spatial tracking, and sense integration (Oktafiani et al., 2024; Putu et al., 2023). Students can view 3D objects in a dynamic environment thanks to interactive visualisation. To preserve the original object’s context, the system spatially monitors the objects, a process known as spatial tracking (Singh et al., 2024). Sensory integration produces additional multisensory reactions that enhance emotional, cognitive, and educational experiences (Singh et al., 2024). By utilising the AR interface to manipulate shapes and spaces, students can engage with geometry while also honing their spatial perception and critical thinking abilities (Farhah et al., 2024; Pratiwi & Nugraheni, 2024; Saphira et al., 2022).

Data collection

The entire research is composed of a number of themes of background, population, sample, data collection and research procedures. The subjects were 234 learners of 13 schools in the Special Region of Yogyakarta (DIY). The students were requested to use a specific Android application named GeoMater, based on an Augmented Reality (AR), in order to learn geometry concepts. The research was undertaken by visiting the schools and helping the students fill out questionnaires, and the subject teacher has control over the study. The general overview of the GeoMater application was collected in the form of descriptions/features, modes of use and geometry material shown.

Figure 2 presents the main menu of the GeoMaster application, which allows users to explore the world of geometry. The menu includes options for Instructions, Subject, Games, Cards, Augmented Reality, and Evaluations. The application is designed to provide an interactive and engaging learning experience for geometry.

Figure 2. Application menu.

Source: Author, 2026

Figure 3 presents the cover page of the GeoMaster geometry learning material. The image features two children, one girl and one boy, happily exploring the world of geometry. This material is designed to engage young learners with interactive content and visually appealing elements to enhance their learning experience.

Figure 3. Learning subject menu.

Source: Author, 2026

Figure 4 presents Augmented Reality (AR) interaction in the GeoMaster application, showing a 3D cube being displayed on a smartphone screen. The cube is visualized using Vuforia AR technology, with the dimension labeled as d = 6². This demonstrates how the application integrates AR to help students visualize geometric concepts interactively.

Figure 4. 3D AR display.

Source: Author, 2026

Measurement

A Likert-type scale measuring various statements from one to five was employed in this study. The measurement tool was developed from previous studies and was then adjusted to be appropriate for the current study. In particular, five items were used to measure the variable of critical thinking skills. Items developed from appropriate studies and validated in regard to technology-enhanced learning were adjusted to refer to students’ skills to utilise the GeoMaster application appropriately. Specifically, the students demonstrated their skills by analysing geometric objects, judging a visual representation, and drawing conclusions based on their activity using the virtual application.

The skills mentioned above (analysis, judgment, and conclusion) are the preliminary components of critical thinking constructed in a technology-enhanced, project-based, interactive learning environment. The students were interacting and manipulating the virtual elements provided by the GeoMaster application as an Augmented Reality-based learning medium to construct and demonstrate their critical thinking skills while engaging and observing virtual objects in a physical space.

Sample

The sample for this study consisted of 234 students from 13 junior high schools in Yogyakarta. Non-probability sampling with random sample selection was used for this purpose.

According to Table 1, out of the total number of 234 respondents, 116 (49.6) were male and 118 (50.4) were female. Students were stratified according to their classes with 90 students (38.5%), 34 students (14.5%), 41 students (17.5%), 16 students (6.8%), and 53 students (22.7) belonging to class 7A, 7B, 7C, 7D, and 7E respectively. According to age, 153 (65.4) were aged between 11–12 years, 64 (27.3) aged between 13–14 years and 17 (7.3) above 15 years.

Informed consent

In this study, written informed consent was obtained from the parents/guardians of the minor participants (ages 11–15). The parents/guardians were fully informed of the study’s purpose, procedures, potential risks, and their rights. In addition, the minor participants (students) provided verbal assent after being explained the study details in an age-appropriate manner. The participants were informed that their participation was voluntary and they could withdraw from the study at any time without any negative consequences. The data collected from the participants was kept confidential, anonymized, and solely used for academic purposes.

Structural equation modelling (SEM)

Figure 5 presents Structural Equation Model (SEM) illustrating the relationships between Critical Thinking Skills (CTS), Perceived Usefulness (PU), Perceived Ease of Use (PEU), and Actual System Use (ASU) in the context of the study. The diagram shows how Critical Thinking Skills (CTS) influence both Perceived Usefulness (PU) and Perceived Ease of Use (PEU), which in turn affect Actual System Use (ASU). The path coefficients represent the strength and significance of these relationships. The values near the arrows indicate factor loadings, showing how strongly each observed variable (such as CTS1, CTS2, PU1, ASU1, etc.) correlates with its respective latent variable (CTS, PU, PEU, ASU). Notably, Critical Thinking Skills (CTS) has a strong impact on Perceived Usefulness (0.819), while Perceived Ease of Use shows a moderate impact on Actual System Use (0.786). The Perceived Usefulness variable is directly influenced by CTS and has a significant effect on ASU. The overall model represents how these key constructs interact to explain users’ acceptance and usage of the system in educational contexts.

Figure 5. Outer loading.

Source: Author, 2026

The model illustrates the connections between Actual System Use (ASU), Perceived Utility (PU), Perceived Ease of Use (PEU), and Critical Thinking Skills (CTS). CTS has a moderate impact on PU (β = 0.345) and a strong influence on PEU (β = 0.693). Additionally, PEU has a significant effect on PU (β = 0.631), suggesting that perceived usefulness is increased by ease of use. ASU is positively impacted by both PU (β = 0.479) and PEU (β = 0.375), indicating that users are more engaged when systems are helpful and straightforward. Strong measurement reliability is confirmed by high indicator loadings (≥ 0.70). Overall, the model shows how perceived ease and utility in critical thinking indirectly motivate actual use.

Table 2 presents the measurement model results for the constructs in the study, including Critical Thinking Skills (CTS), Perceived Usefulness (PU), Perceived Ease of Use (PEU), and Actual System Use (ASU). The table shows the Outer Loadings, Cronbach’s Alpha, Composite Reliability (CR), and Average Variance Extracted (AVE) for each variable and its corresponding items. For Critical Thinking Skills (CTS), all items (CTS1 to CTS9) exhibit high outer loadings, ranging from 0.954 to 0.994, with excellent internal consistency (Cronbach’s alpha = 0.993) and strong reliability (CR = 0.994), along with a high AVE of 0.948. In the case of Perceived Usefulness (PU), the items (PU1 to PU9) show moderate to high outer loadings, ranging from 0.737 to 0.793, with Cronbach’s alpha of 0.914, CR of 0.929, and an AVE of 0.592, demonstrating good internal consistency and convergent validity. For Perceived Ease of Use (PEU), the items (PEU1 to PEU9) show strong outer loadings, ranging from 0.730 to 0.876, with excellent internal consistency (Cronbach’s alpha = 0.938), high reliability (CR = 0.948), and an AVE of 0.669. Finally, for Actual System Use (ASU), the items (ASU1 to ASU9) exhibit high outer loadings, ranging from 0.725 to 0.854, with strong internal consistency (Cronbach’s alpha = 0.922), high reliability (CR = 0.935), and an AVE of 0.616. These results indicate that all constructs in the study show strong reliability and validity, confirming the robustness of the measurement model.

Table 3 displays the outcomes of the discrimination validity assessment through the Fornell-Larcker criteria, suggesting that the root value of the AVE for each of the constructs exceeds the correlation between constructs, satisfying the criteria for discriminant validity. Table 4 shows the results of the discriminant validity test using the HTMT Ratio of Correlations with scores ranging from 0.715 to 0.936. Based on the accepted threshold of below 0.90. The values that meet the criteria are 0.715, 0.743, and 0.809, while the other values are close to the threshold and need to be considered. Furthermore, the results of the hypothesis testing are shown in Table 5.

Table 5 shows that five hypotheses were accepted because the T-statistic value exceeded 1.96 and the P value was less than 0.05 (Hair et al., 2021). Five hypotheses are declared significant and accepted because they have T values >1.96 and P values <0.05, namely CTS affects PEU (β = 0.693; T = 12.896; P = 0.000) and PU (β = 0.345; T = 6.895; P = 0.000), PEU affects ASU (β = 0.375; T = 4.683; P = 0.000) and PU (β = 0.631; T = 13.910; P = 0.000), and PU affects ASU (β = 0.479; T = 5.646; P = 0.000), while one hypothesis was rejected, namely CTS on ASU (β = 0.076; T = 1.103; P = 0.271) because it was not statistically significant.

Fuzzy C-means (FCM) clustering

The items in this study were analysed using FCM, a clustering approach that allows data elements to be assigned to more than one group with measurable membership degrees (Sawu et al., 2023). In the implementation, a degree of membership is assigned to each data point for each cluster, allowing the same data point to have different levels of membership. This study proposed an improved FCM based on t-SNE, focusing on decreasing the dimension of the t-SNE clustering samples and on classifying the sample points based on feature distribution as the initial cluster centre for FCM clustering analysis.

Determining the number of clusters

The Bayesian information criterion (BIC) is another statistical criterion that is similar to the Akaike information criterion (AIC). Statistical criteria, such as AIC and BIC, are frequently utilised in academic research both for model selection and as a statistical tool for assessing, in a timely manner, whether models are suited to the data while considering model complexity (Kasali & Adeyemi, 2022; Zhang et al., 2023). The lower the AIC or BIC, the better the model fits the data (Kasali & Adeyemi, 2022; Zhang & Meng, 2023). Researchers use AIC and BIC with many types of statistical analyses, including regression and time series. Similarly, AIC and BIC are also utilised in clustering analysis, namely the k-means algorithm (Cohen & Berchenko, 2021; Zhang et al., 2023). In these analytic approaches, AIC or BIC is used to determine the best number of clusters as well as WSS. The WSS, together with AIC or BIC, are all incorporated to determine cohesion or clustering of results for each statistical model.

Figure 6 presents three evaluation metrics: WSS (Within-Cluster Sum of Squares), AIC (Akaike Information Criterion), and BIC (Bayesian Information Criterion). According to the plot, the most evident elbow point appears at 4 clusters, marked by a red dot indicating the lowest BIC value. This suggests that at 4 clusters, the model achieves the best trade-off between the number of clusters and efficiency of modelling, without unnecessarily complicating the model. Thus, the most reasonable number of clusters to choose is 4.

Figure 6. Elbow Method Plot.

Source: Author, 2026

Figure 6 presents Elbow Method Plot used to determine the optimal number of clusters for the data. The plot shows the values of AIC (Akaike Information Criterion), BIC (Bayesian Information Criterion), and WSS (Within-Cluster Sum of Squares) for different numbers of clusters, ranging from 2 to 10. The red dot indicates the lowest BIC value, suggesting the optimal number of clusters. The AIC and BIC values help assess the model fit, while the WSS indicates the compactness of the clusters. The Elbow Method identifies the point where increasing the number of clusters no longer significantly improves the model, with the lowest BIC guiding the selection of the ideal number of clusters.

Table 6 presents the results of the clustering analysis, including the number of clusters (N), the R-squared (R²) value, Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and Silhouette Score for the optimal number of clusters (4 clusters). The number of observations in the 4 clusters is 234. The R² value of 0.552 indicates that the model explains 55.2% of the variance in the data. The AIC and BIC values are 4194.090 and 4691.660, respectively, which help assess the model fit and determine the optimal number of clusters. The Silhouette Score of 0.290 indicates a moderate level of clustering quality, with values closer to 1 suggesting well-separated clusters and values closer to 0 indicating overlapping clusters.

Table 7 demonstrates more evaluation metrics for cluster solutions. Person’s γ is 0.550, signifying a correlation between the distances of data within and between clusters. The Dunn Index (0.147) indicates the separateness of clusters and distances between data within a cluster and the distance between clusters. The entropy is 1.123, which suggests diversity of data within clusters. A smaller entropy value indicates higher data homogeneity. The Calinski-Harabasz index value is 87.969, indicating the effectiveness of data separation into clusters. A higher Calinski-Harabasz value indicates better separation between clusters. Based on these results, the identified clusters are of high quality.

Figure 7 presents t-SNE Cluster Plot showing the visual representation of clusters formed based on the data. Each point represents an observation, and the colors indicate different clusters. Cluster 1 is shown in pink, Cluster 2 in green, Cluster 3 in cyan, and Cluster 4 in purple. The plot demonstrates how the t-SNE (t-distributed Stochastic Neighbor Embedding) algorithm has grouped the data into distinct clusters, highlighting the separation between different clusters in the reduced-dimensional space.

Figure 7. t-NSE cluster plot.

Source: Author, 2026

Figure 7 presents t-SNE Cluster Plot showing the visual representation of clusters formed based on the data. Each point represents an observation, and the colors indicate different clusters. Cluster 1 is shown in pink, Cluster 2 in green, Cluster 3 in cyan, and Cluster 4 in purple. The plot demonstrates how the t-SNE (t-distributed Stochastic Neighbor Embedding) algorithm has grouped the data into distinct clusters, highlighting the separation between different clusters in the reduced-dimensional space.

All the clusters in this t-SNE Cluster Plot image are coloured in different colours: purple (Cluster 4), bluish green (Cluster 3), greenish yellow (Cluster 2), and pink (Cluster 1). The colour differences are the reflection of the sets of data with similar qualities in the original space prior to the dimension reduction. The points of a plot are very close to each other indicating that the data are related or similar to each other in high-dimensional space. To ensure that the closeness of colours in this visualisation should indicate similar patterns in the underlying data, t-SNE procedure is designed such that it preserves relative distances between data points as well as their local structures.

In the process of defining critical thinking abilities as an external cognitive construct that influences perceived usefulness (PU) and perceived ease of use (PEU), this study expands on the Technology Acceptance Model (TAM). Without specifically addressing the role of higher-order cognitive skills that mediate learning engagement, prior research has primarily used TAM to measure behavioural intention or technology acceptance (Al-Adwan et al., 2023). The present study contributes a new theoretical extension to the TAM framework by adding the dimension of critical thinking to the framework.