Exploring Grade 12 Learners’ Understanding of Geometric Transformations Through the STAD Cooperative Learning Model

Abstract

Research has shown that both learners and teachers struggle to understand and teach geometric transformations meaningfully. This mixed-methods case study examined the efficacy of the Student Teams Achievement Division (STAD) cooperative learning model in fostering learners’ conceptual understanding of geometric transformations. This study involved 28 Grade 12 learners from one intact class. In addition to pre- and post-intervention tests, which measured learners’ conceptual understanding of geometric transformations, participants completed a feedback questionnaire at the end of the intervention. The results show that the STAD model significantly improved learners’ grasp of geometric transformations, as demonstrated by higher test scores in the post-test. Participants also highlighted the importance of well-crafted questions during group discussions and oral quizzes, teacher explanations during whole-class presentations, and the motivational impact of criteria for selecting and awarding top-performing groups. Based on Shapiro’s intervention evaluation criteria, the STAD model was found to be effective, with high levels of acceptability, integrity, and social validity. While this study confirms the STAD model’s effectiveness in enhancing conceptual understanding and social learning, it also emphasizes the importance of considering contextual factors, such as group dynamics and classroom resource availability, when implementing this cooperative learning model. Teachers are encouraged to tailor this learning strategy to their specific classroom environments and learners’ needs.

1. Introduction

Geometry is a critical component of mathematics education, widely recognized for its ability to foster mathematical reasoning, spatial visualization, and environmental awareness [1,2,3,4]. Geometry holds a significant place in school mathematics curricula worldwide [5,6,7,8,9], with a strong emphasis on visual and intuitive understanding [1,10]. Geometric transformation is one of the branches of geometry that is particularly important for offering learners the tools to comprehend how shapes change position, orientation, or size while maintaining certain properties.

Despite its importance in the school mathematics curriculum, both teachers and learners often struggle to understand and effectively teach geometric transformations. For example, a study conducted by Aktas and Ünlü [11] in Turkey found that eighth-grade learners faced challenges in distinguishing between similarity and congruence in reflections, identifying axes of symmetry, understanding intersection rules, and recognizing the relationship between symmetry axes and polygon sides. Additionally, learners had difficulties in determining and applying the correct angle of rotation. Similar challenges were observed among Zimbabwean secondary school learners, as reported by Mukamba and Makamure [12], who noted persistent difficulties in understanding and connecting various geometric transformation concepts.

It is also worth noting that challenges with geometric transformations are not limited to learners. Teachers also encounter difficulties in teaching these concepts. For instance, Niyukuri et al. [13] found that many Burundian teachers struggled with teaching geometric concepts like isometries, often opting to skip these topics with the intention of covering them later, though they were frequently not taught at all. Another study conducted in Indonesia [14] found that preservice teachers faced some difficulties with geometric transformations, particularly in applying the concepts, visualizing the shapes, understanding the problems, and constructing proofs.

In the Zambian school curriculum, geometric transformations are introduced in Grades 11 and 12, with particular importance given to the topic in national Grade 12 examinations [8,15,16,17,18]. This concept, which involves isometries like translations, reflections, and rotations and non-isometries like enlargements, shear, and stretch, often poses challenges for learners. Performance reports from Zambian national examinations repeatedly highlight the difficulties learners face with geometric transformations, a problem also reflected in studies across sub-Saharan Africa [7,12,13]. Researchers [6,7,13] agree that some of the challenges learners face are compounded by teachers’ struggles with the subject, leading to a reluctance to teach geometric transformations. As a result, there is a need to explore not only the challenges learners face but also strategies to support teachers in teaching this content effectively. The challenges learners encounter in mastering geometric transformations highlight the need for targeted interventions [14,19,20,21].

Various studies have proposed the use of cooperative learning as an effective strategy to address difficulties in the teaching and learning of mathematics [22,23,24,25,26]. Among cooperative learning models, the Student Teams Achievement Division (STAD) model has been widely used to foster collaboration and improve learning outcomes across various mathematics topics, including geometry [22,23]. However, there has been no study, particularly in the Zambian context, that has investigated the efficacy of any cooperative learning model in fostering learners’ understanding of geometric transformations.

This study, therefore, explores how the STAD model of cooperative learning can be used to enhance Grade 12 learners’ understanding of geometric transformations. The rationale for selecting STAD lies in its structured approach, which combines direct instruction with peer interaction. This allows learners to collaboratively address complex mathematical concepts. Furthermore, STAD emphasizes group accountability, which can be particularly beneficial in overcoming learners’ and teachers’ challenges with geometric transformations. STAD, as described by Slavin [27], is a structured model where students work in small, diverse groups. The teacher first provides direct instruction, after which learners collaborate on tasks designed to reinforce their learning. Individual and group assessments follow, and learners receive feedback based on their group’s collective performance. The following research questions guided this inquiry:

• How does the STAD cooperative learning model impact learners’ conceptual understanding of geometric transformations?
• What features of the STAD learning model are regarded as effective in fostering learners’ conceptual understanding of geometric transformations?

This paper builds upon a conference paper [28] that was previously presented as a short paper at the 2023 Southern African Association for Mathematics, Science, and Technology Education (SAARMSTE). Regarding the first research question, our aim was to investigate changes in learners’ understanding of geometric transformations before and after participating in STAD cooperative learning activities. This allowed us to explore whether the STAD model enhances comprehension of geometric transformations. The second research question sought to identify specific aspects of the STAD learning approach (e.g., team structure, peer interaction, teacher feedback, and teacher actions) that positively influence conceptual understanding. We achieved this by considering learners’ perceptions and experiences related to these features. Additionally, the second research question examined learners’ suggestions for improving the teaching of geometric transformations, a topic considered challenging for most secondary school learners in Zambia. By addressing these two research questions, it was anticipated that this study could contribute by offering practical implications for teachers using the STAD model, along with policy and theoretical recommendations.

2. Literature Review

2.1. Theoretical Perspectives on Cooperative Learning

Cooperative learning has been extensively studied for its positive impact on learners’ academic achievement. Several theoretical perspectives provide insights into how this instructional strategy enhances learning outcomes. According to Slavin [29], four major perspectives are key to understanding the effectiveness of cooperative learning: motivation, social cohesion, cognitive development, and cognitive elaboration.

The motivational perspective highlights the importance of rewards and incentives, emphasizing that individual learning within a group contributes to the overall success of the group [30]. In cooperative learning models like STAD, students are motivated by group rewards, which are contingent on the individual efforts of group members. This links directly to STAD, where students are accountable not only to themselves but also to their peers, creating a shared responsibility for learning.

The social cohesion perspective posits that the relationships and sense of belonging among group members are crucial to the success of cooperative learning. Johnson and Johnson [31] argue that students who feel connected to their group are more likely to assist one another. In STAD, this perspective is operationalized through the group structure, where students work collaboratively, fostering a supportive learning environment that enhances both academic and social outcomes.

From a cognitive developmental perspective, cooperative learning encourages cognitive growth through peer interactions, as explained by Piaget [32] and Vygotsky [33]. Vygotsky’s zone of proximal development (ZPD) is particularly relevant, as STAD allows students to work together, enabling them to solve problems with the guidance of more knowledgeable peers. This model promotes scaffolding, where learners internalize knowledge shared by their peers, aligning well with the goals of this study to enhance learners’ understanding of geometric transformations.

The cognitive elaboration perspective argues that explaining and discussing concepts within a group leads to deeper understanding [30]. In STAD, students must explain their reasoning to others during group discussions, thereby reinforcing their own learning while helping others. This aligns with this study’s goal of improving the conceptual understanding of geometric transformations through collaborative engagement.

By integrating these theoretical perspectives, STAD serves as a structured method to address the specific challenges learners face in understanding geometric transformations, such as difficulty with visualizing and applying concepts. The next section explores how these theories underpin the implementation of STAD in this study.

2.2. STAD Implementation in Mathematics Classrooms

STAD is one of the most widely researched structured team learning (STL) models within cooperative learning, particularly in the field of mathematics education [24,34,35]. It is designed to improve both academic achievement and social skills by structuring group work with individual accountability. Studies have consistently demonstrated that STAD enhances learners’ conceptual understanding, motivation, and self-confidence in mathematics [24,27,36,37].

The implementation of STAD in the current study was informed by the challenges learners and teachers face in teaching and learning geometric transformations. Research shows that students often struggle with visualizing transformations and understanding key concepts, such as congruence, similarity, and rotational symmetry [11,12]. STAD addresses these challenges by promoting peer-assisted learning, where students help each other in dealing with difficult concepts through structured group activities and individual assessments.

In the context of the current study, STAD was applied as an intervention to enhance Grade 12 learners’ understanding of geometric transformations, with specific attention to overcoming common obstacles in applying and visualizing these concepts. The structured nature of STAD, with teacher presentations followed by group work and individual quizzes, is well-suited to addressing these learning difficulties. Additionally, this model fosters the social interactions necessary for collaborative learning, as students are encouraged to explain their reasoning and challenge each other’s thinking.

However, it is important to acknowledge the challenges associated with implementing cooperative learning strategies such as STAD, particularly in environments with large class sizes, limited resources, or resistance from students [38,39,40,41]. This study considered these contextual challenges and examined how STAD can be adapted to the specific educational setting to maximize its effectiveness. In this study, these challenges were taken into account by ensuring that the cooperative learning activities were designed to fit the specific needs and constraints of the learners, such as their readiness to engage in group work and the teacher’s ability to facilitate group discussions effectively.

3. Materials and Methods

3.1. Research Design

This study employed an “embedded mixed methods case study design” to answer the stated research questions. According to Yin [42], this design integrates both qualitative and quantitative data within a single case study, involving multiple levels of analysis where different types of data collection and analysis are nested within a broader framework. The case boundaries were established by including participants from a single Grade 12 class at a private boarding school in Luanshya District, Copperbelt Province, Zambia. These learners were taught geometric transformations using the STAD cooperative learning model. Qualitative data, collected via a semi-structured participant feedback questionnaire, provided depth and context, while quantitative data from pre-test and post-test scores confirmed the effectiveness of the STAD cooperative learning model.

By embedding both quantitative and qualitative methods, we sought to address both exploratory and confirmatory research questions [43], ensuring a comprehensive understanding of the intervention’s impact. Specifically, this study assessed learner improvement in the conceptual understanding of geometric transformations using pre- and post-test scores, which aligned with a confirmatory research approach to evaluate the effectiveness of the STAD model. This allowed us to quantitatively confirm whether the intervention led to measurable gains in students’ learning outcomes. At the same time, an exploratory research question was addressed through qualitative feedback, capturing participants’ perceptions and experiences of the learning process. These insights provided a deeper understanding of how learners engaged with STAD and helped identify factors influencing its effectiveness. Despite some limitations in the intervention setup, combining these approaches enabled us to draw more robust conclusions and offer practical recommendations for future implementations.

3.2. Research Setting and Participants

This study involved 28 Grade 12 male learners from a private school in Luanshya district, Copperbelt Province, Zambia. The participants were aged between 15 and 18 years, with a mean age of 16.8 and a standard deviation of 0.819. In Zambia, secondary school learners may attend either private or public schools. Since the implementation of the educational policy in 2021 [44], public school learners have received free education, except for those attending boarding schools, who pay boarding fees to cover lodging and feeding costs. In contrast, private schools charge varying and generally higher fees. As a result, private schools in Zambia are attended mostly by learners from well-to-do families.

Zambian schools include gender-specific institutions (boys-only and girls-only) as well as co-educational schools. While this study took place in a boys-only private school, it is worth noting that the curriculum and teaching approaches remain the same across all types of schools, whether private or public. Teachers across both sectors hold similar qualifications, with junior secondary teachers (Grades 8 and 9) required to have a minimum of a three-year diploma and senior secondary teachers (Grades 10–12) needing at least a four-year bachelor’s degree [45]. Since this study was conducted at a private boys-only school, its findings should be interpreted with caution, as the context may vary with different school types. Factors such as group dynamics and the availability of classroom resources should be considered.

3.3. Data Collection

3.3.1. Instruments Used to Answer the Research Questions

Two key instruments were used to collect data that directly addressed the research questions: the Achievement Test (pre-test and post-test) and the Student Feedback Survey.

The Achievement Test (File S3), consisting of a pre-test and post-test, was administered to assess learners’ conceptual understanding of geometric transformations. The pre-test established a baseline of learners’ prior knowledge before the intervention, while the post-test evaluated the improvement in their understanding after implementing the STAD instructional approach. These tests were crucial in determining the effectiveness of the intervention. The test items were a mix of teacher-made questions and adapted items from past national examination papers based on the Zambian school certificate level. To ensure the validity of the assessment, all items were reviewed by three experienced mathematics teachers and one mathematics teacher educator. In line with the guidelines from our previous study [37], the validators were asked to provide feedback on the following indicators:

• Sufficiency: Whether the items were adequate for assessing learners’ understanding of geometric transformations in the context of the Zambian school curriculum.
• Clarity: Whether the items were well articulated and easily comprehensible by Grade 12 learners in Zambia.
• Coherence: Whether the items were logically connected to the concepts they were intended to assess.
• Relevance: Whether the items were essential and important for measuring learners’ reasoning abilities and conceptual understanding.

The feedback from these experts was used to refine the test questions, ensuring they were appropriate in terms of difficulty and alignment with curriculum standards. Nonetheless, both the adapted and teacher-made questions were framed in line with guidelines by previous studies conducted in similar contexts [20,46].

Learners’ marks/scores for achievement tests are provided in the Excel sheet labeled “group progress chart” of dataset 1. While both the pre-test and post-test measures provided very useful information, their focus was largely knowledge-based and provided little information about learners’ satisfaction with the implemented classroom activities regarding their understanding of transformation geometry. As such, a learner feedback survey was also administered.

The Student Feedback Survey (File S4) was administered to capture learners’ perceptions of the STAD model and how it contributed to their understanding of geometric transformations. The survey included both closed-ended and open-ended questions. The closed-ended questions required learners to rate their satisfaction with various classroom activities, while the open-ended questions invited learners to provide qualitative feedback on how specific activities supported their learning. The data from this survey were analyzed to gather insights into learners’ experiences and the perceived effectiveness of the intervention in enhancing their understanding of the topic.

Information obtained from closed-ended questionnaire items is available in the Excel sheet named “closed-ended responses” of dataset 1. On the other hand, open-ended questionnaire responses were extracted from all the completed questionnaires and typed into a Microsoft Word document, as illustrated in the qualitative dataset 2. It is worth noting that all the supplementary files and datasets described in this section are openly available in the Mendeley Data Repository at [https://data.mendeley.com/datasets/72946296mv/1, accessed on 3 August 2024].

3.3.2. Instruments Used to Inform Teachers and Pedagogical Adjustments

Other data sources, including class exercises, small group discussions, oral quizzes (File S2), and written quizzes (File S1), provided useful feedback for guiding instructional adjustments throughout the intervention. These instruments were not analyzed to answer the research questions but served an important role in identifying learners’ difficulties during the intervention. Based on the feedback from these sources, the teacher was able to make pedagogical shifts and provide remedial teaching where necessary. The marks from the written and oral quizzes were recorded in the “group progress chart” of dataset 1, helping to monitor learners’ progress, though the data were not formally analyzed for the purposes of this paper.

3.4. Intervention Description

The decision to adopt the STAD instructional approach was based on its demonstrated effectiveness in improving learners’ conceptual understanding of mathematics. The intervention spanned 4 weeks, with three 80 min lessons each week, during which, six geometric transformations—translation, reflection, rotation, enlargement, shear, and stretch—were taught. The intervention was implemented in a similar way as in an earlier study [37], which was also consistent with the recommended guidelines [27,47]. In the context of this study, the lead author acted as the teacher to assess the efficacy of the STAD model of cooperative learning in a private school setting. The lead author has over nine years of experience in mathematics instruction at the secondary school level. Although the author’s original teaching style has been predominantly lecture-based, this study involved a shift to the cooperative, learner-centered STAD approach to enhance engagement and the conceptual understanding of geometric transformations. The following steps were taken.

3.4.1. Step I: Whole Class Presentation

At this stage, each lesson was introduced through a short presentation, mainly focusing on explaining key concepts that were new to learners. Due to the nature of the topic, some demonstrations were performed by the teacher to enable learners to observe how a particular geometric shape was transformed from one position to the other following a specified set of rules. Time allocation for this phase varied between 15 and 30 min depending on the nature of the concept taught for a particular lesson. For instance, more time was dedicated to concepts like rotation, enlargement, shear, and stretch, compared to simpler ones like translation and reflection.

3.4.2. Step II: Small Group Discussions

The second step comprised small group discussions, where learners were divided into heterogeneous groups of four based on their mathematical abilities, assessed through pre-test test scores (referred to as base score in the group progress chart of dataset 1). Therefore, each group comprised one high-performing learner, two moderate-performing learners, and one low-performing learner. Within these groups, learners worked collaboratively to ensure that everyone understood the material, encouraging peer explanations, debates, and justifications to foster interdependence and accountability. After group discussions, each group presented their solutions to the class, with representatives chosen randomly to ensure active participation. The tasks for each group discussion varied depending on the specific transformation (reflection, translation, rotation, enlargement, shear, or stretch) being taught during that lesson. While the exact discussion questions varied across these topics, they followed a similar structure to the written and oral quiz questions provided in the supplementary files (Files S1 and S2). Approximately 40 min were allocated for group work and group presentations to the whole class. Depending on how much time was spent in step I, between 10 and 25 min were reserved for lesson evaluation, which comprised clarifications and homework administration.

3.4.3. Step III: Quiz Administration

Written and oral quizzes were each administered once. Quiz 1 (written) took place at the end of week 2, while Quiz 2 (oral) was conducted at the end of week 3. The written quiz, like class exercises and tests, was completed individually. During the oral quiz, questions were randomly assigned to groups. Seven questions of equal weight were formulated and numbered 1 to 7. These numbers were written on equally sized pieces of paper that were then placed in a box. Seven group representatives were invited to select a piece of paper from the box, with the number on the selected piece of paper corresponding to the question allocated to their group. Although all groups were encouraged to work on every question, only the designated group provided the answer or solution. Before presenting their response, the assigned group discussed and agreed on the answer, which was then delivered by one group member.

The quizzes were administered to offer alternative methods for monitoring learners’ progress. Although class exercises and/or homework were assigned for nearly every lesson (depending on time availability), it became evident that traditional assessments alone were insufficient for tracking learners’ progress. Therefore, both oral and written quizzes were introduced and conducted under controlled conditions to provide an alternative assessment of learners’ understanding of geometric transformations. As pointed out earlier, the insights gained from all forms of the administered assessments such as group work, class exercises, homework, and quizzes helped to identify learners’ levels of understanding and the challenges they faced. This information guided necessary pedagogical adjustments that eventually aided learners’ understanding of geometric transformations.

3.4.4. Step IV: Revisions and Test Administration

During week 4, the focus shifted to whole-class revisions, where the teacher revisited concepts that learners found particularly challenging. To further support learning, similar tasks as those discussed in Step II were administered for small group discussions, and students were encouraged to meet in their respective groups, even outside of class time to continue their discussions. At the end of the week, a post-test was administered. It is noteworthy that similar questions were administered before the intervention (pre-test), though with adjustments to the order and depth of the questions. In other words, pre-test questions were similar to post-test questions but tailored to the learners’ level of understanding.

After the post-test, learner performance was evaluated using a method adapted from an earlier study [37]. Each learner’s post-test score was compared to their pre-test score, and points were awarded to the group based on the improvement shown by each member. The degree of improvement was assessed using the criteria prescribed in

Table 1. Progress tracking score sheet.

Progress Description	Earned Group Points
Post-test score below the base score	−5
Post-test score equal to the base score	0
Post-test score above base score by 1 to 5	5
Post-test score above base score by 6 to 10	10
Post-test score above base score by 11 to 15	15
Post-test score above base score by 16 or more	20
Outstanding performance	20

. This table highlights how points were assigned based on how much a student’s new score exceeded their base score.

Although the criteria prescribed in Table 1 were inspired by the guidelines provided by Li and Lam [47], which also guided our earlier study [37], some adjustments were made to align with the current context. In reference to the last description in Table 1, any new score exceeding 85% was classified as outstanding, warranting an automatic 20-point allocation to the group. This means that even if a learner’s post-test score was below the pre-test score, they would still be considered outstanding, emphasizing sustained high-level performance despite the decrease in marks.

3.4.5. Step V: Group Recognition

Based on the points accumulated by members (as indicated in the Excel sheet labeled “group progress chart” of dataset 1), the group with the highest average points received a group award. Awards such as graph papers, ballpoint pens, notebooks, or the same score/mark for all the group members were given. To ensure that no one lagged, every member of the class was given a pad of graph papers and past examination papers to use for practice. In certain situations, material rewards can pose challenges, particularly when teachers or school administrations are unable to provide them. Therefore, the essential aspect here was that all group members received the same mark collectively. This approach encouraged each individual to actively contribute to the group’s success and avoid letting the team down.

3.5. Data Analysis

To evaluate the effectiveness of the STAD model in enhancing learners’ understanding of geometric transformations, we employed the intervention evaluation criteria proposed by Shapiro [48]. This approach focuses on four key strategies: intervention effectiveness, integrity, social validity, and acceptability. Although originally developed for school psychology, Shapiro’s criteria have been effectively applied in mathematics education research [49,50,51]. This makes the criteria useful and suitable for this study.

Intervention effectiveness pertains to the degree of behavioral change, its immediacy, and the maintenance and generalization of that change. In this study, the impact of the STAD model on learners’ understanding was assessed by comparing pre-test and post-test scores. Intervention integrity refers to the extent to which the STAD model was implemented as intended, following the guidelines established by its proponents [37] and prior research [24,27,47]. Participants’ feedback on the implementation of classroom activities (outlined later in Section 4.2 was also analyzed to assess this aspect. Social validity examines the social significance of treatment goals, the appropriateness of procedures, and the importance of outcomes. This was evaluated by analyzing participants’ perceptions of how the STAD model fostered peer learning, cooperation, mutual support, and individual accountability. Intervention acceptability, closely related to social validity, measures participants’ enjoyment and satisfaction with implemented activities. This was assessed through learners’ feedback on their overall satisfaction with the classroom activities implemented during the intervention.

Consistent with the pragmatic research paradigm within which this embedded mixed methods case study was framed, both quantitative and qualitative data analysis techniques were used to address the research questions. For intervention effectiveness (research question 1), a paired samples t-test was conducted to evaluate the significance of improvements in test scores. Intervention integrity, social validity, and acceptability (addressing research question 2) were assessed using descriptive statistics (e.g., mean and standard deviation) and qualitative analyses of participants’ open-ended responses.

In line with some existing studies in mathematics education [49,50,51], this approach allowed for a thorough examination of the intervention’s effectiveness, integrity, social validity, and acceptability. This could be attributed to the fact that an intervention may be effective but lacks social validity or acceptability. This clearly affirms the importance of evaluating these four dimensions to establish an intervention’s overall significance.

3.6. Ethical Considerations

Permission to collect data from learners was sought and granted by the school authorities. Before the commencement of data collection, all participants were briefed on the purpose of this study, after which informed consent was sought from every learner who participated. Participants were assured of confidentiality without any abrogation of privacy, as no participants’ full names are revealed in any publications associated with this study.

4. Results

As earlier stated, this study aimed to evaluate the impact of the STAD cooperative learning model regarding learners’ conceptual understanding of geometric transformations and identify the effective features that contribute to this understanding.

4.1. Impact of STAD on Learners’ Conceptual Understanding

To establish the effectiveness of the STAD cooperative learning model on learners’ understanding of geometric transformations, we used a paired samples t-test. Given the small sample size, we followed established guidelines [52] to ensure that the data met the necessary assumptions.

First, test scores for both the pre-test and post-test were continuous variables collected from the same individuals before and after the intervention. It was also important to determine whether the normality assumption was met before performing the statistical test. As recommended in the literature [52], we subjected the score differences between pre-test and post-test scores to the Kolmogorov–Smirnov (K-S) and Shapiro–Wilk (S-W) normality tests using SPSS version 27.

After conducting these normality tests for the dataset with all 28 participants, the results showed that the distribution of the score differences was significantly different from normal, as evident by p-values (sig.) below the 0.05 threshold in the first trial (

Table 2. Normality tests.

	Kolmogorov–Smirnov			Shapiro–Wilk
	Statistic	df	p-Value	Statistic	df	p-Value
Trial 1	0.170	28	0.037	0.861	28	0.002
Trial 2	0.08	26	0.200	0.982	26	0.909

). Upon establishing that the normality assumption had been violated, we explored outliers further. We identified two outliers: one participant improved from 31% in the pre-test to 88% in the post-test and another participant’s score declined from 83% in the pre-test to 64% in the post-test.

Considering the effect of outliers on the overall mean and standard deviation, we excluded these two cases and conducted a second normality check. Based on the results of the second trial (Table 2), the distribution of the score differences was not significantly different from normal, as the p-values for both the K-S and S-W normality tests were above the 0.05 threshold. This indicated that the distribution of the score differences was normal. Additionally, the skewness (0.115) and kurtosis (0.119) values were close to zero and within the recommended intervals for chance fluctuations.

After ensuring that the data adhered to the recommended guidelines, a paired samples t-test was undertaken with the exclusion of the two outliers.

Table 3. Descriptive statistics on test scores (n = 26).

Measure	Minimum	Maximum	Mean	Std. Deviation
Pre-test	31	83	60.3	13.3
Post-test	30	100	66.4	16.9

illustrates the descriptive statistics associated with learners’ performance on geometric transformations for both the pre-test and post-test.

The descriptive statistics displayed in Table 2 indicate an improvement in the mean test scores from the pre-test (60.3) to the post-test (66.4). The range of scores also expanded in the post-test, with a higher maximum score reaching 100 compared to the pre-test maximum of 83. The increase in the standard deviation from 13.3 in the pre-test to 16.9 in the post-test suggests greater variability in the post-test scores. The improvement in mean test scores from the pre-test to the post-test is also evident in the paired samples t-test results displayed in

Table 4. Paired samples t-test results.

Measure	Mean	SD	95% Confidence Interval		t-Value	df	p-Value
			Lower	Upper
Pre-test–post-test	−6.115	8.311	−9.472	−2.759	−3.752	25	0.001

.

The paired samples t-test results in Table 4 show a significant difference between the pre-test and post-test scores. The negative mean difference of −6.115 indicates that, on average, post-test scores were higher than pre-test scores.

The 95% confidence interval for the mean difference did not include zero (−9.472 to −2.759), which enforced the significance of the improvement. The t-value of −3.752 with 25 degrees of freedom corresponded to a p-value of 0.001, which was well below the conventional threshold of 0.05. This indicated that the observed improvement in scores was statistically significant. The effect size, represented by Cohen’s d, was 8.311, indicating a large effect. This suggested that the STAD model had a substantial practical impact on learners’ conceptual understanding of geometric transformations. Overall, these results highlight the intervention’s success in improving learners’ performance, thereby fulfilling Shapiro’s criterion for “intervention effectiveness”.

4.2. Perceived Effective Features of the STAD Model

After the intervention, learners were asked to indicate the classroom activities within the implemented STAD model that they viewed as being helpful in their conceptual understanding of geometric transformations. They were asked to rate each of the activities displayed in

Table 5. Participants’ perceptions on implemented classroom activities (n = 28).

Code	Classroom Activities	Min	Max	Mean	SD
CA1	Teacher explanations during whole-class presentations	3	5	4.00	0.609
CA2	Teacher guidance before group discussions	2	5	3.96	0.693
CA3	Teacher support during group discussions	2	5	3.93	0.979
CA4	Quality of questions for group discussions	3	5	4.25	0.645
CA5	Quality of questions for the oral quiz	2	5	4.18	0.863
CA6	Group member participation during discussions	2	5	3.75	0.799
CA7	Group member participation during the oral quiz	1	5	3.68	1.056
CA8	Individual accountability to group goals	2	5	3.36	0.870
CA9	Cooperation among group members during discussions	2	5	3.64	0.780
CA10	Individual contributions to group success	1	5	3.68	0.819
CA11	Criteria for selecting and awarding the best group(s)	1	5	4.14	1.008

on a scale ranging from 1 (not satisfied at all) to 5 (very satisfied).

The results displayed in Table 5 show that the average satisfaction levels among participants were on the higher side, considering that 3.36 was the lowest and 4.25 was the highest. This gave an indication that on average, learners were of the view that all the implemented activities, in a way, contributed to their improvement in the conceptual understanding of geometric transformations. This was also evident in participants’ open-ended responses. For instance, participant #23 stated the following:

I think all the activities helped me practice more to avoid bringing the group down. The group activities encouraged me to study and practice more since points for the group were earned based on each member’s improvement of the previous score. It also helped me because I was able to sharpen my understanding by explaining what I knew to my fellow group members.

The most satisfying aspects based on learners’ ratings included the quality of questions posed by the teacher for group discussions (CA4; M = 4.25, SD = 0.645), followed by the quality of questions for the oral quiz (CA5; M = 4.18, SD = 0.863), the criteria for selecting and awarding the best-performing groups (CA11; M = 4.14, SD = 1.01), and the teacher explanations during whole-class presentations or lectures (CA1; M = 4.00, SD = 0.609).

In terms of Shapiro’s intervention evaluation criteria, these results indicate that the implemented classroom activities adhered to “intervention integrity”; teacher quality in terms of explanation and the type of questions posed were regarded as particularly satisfactory by the majority of participants. The fact that learners rated CA11 highly also gave an indication of “intervention acceptability”, especially given that even those who belonged to groups that did not win also found the idea of group awards quite useful. These insights were also reflected in participants’ responses when asked to comment on the aspects of the classroom activities that they found most helpful. Below are some of the responses that justify this aspect:

Participant #1: My understanding of geometrical transformation was enhanced by teacher explanations during whole-class lesson presentations. Knowing that the best performing group would be recognized and rewarded also made us make sure that everyone in the group understood the topic.

Participant #2: Group awards and the criteria used to select the best performing group. This enhanced my understanding of the topic in the sense that it was rather more encouraging to be recognized as it helps me to put more effort not only to improve my grades but also to help my group. The quality of questions for small group discussions also helped me a lot as I saw the way questions would be asked in the exam giving me more reasoning as I need to put more effort to be able to answer those questions.

Participant #10: Teacher’s explanations, oral quiz, and the quality of groupwork questions motivated me a lot. My friends also helped me to understand the topic.

Participant #11: The questions given for group work helped me to get better because my friends in the group helped me to correct my mistakes and showed me what should be done. Oral quizzes helped when it came to our thinking, it sharpened our mind to think fast.

Participant #28: The oral quiz made me realize the problems I was facing, which made me consult widely not only with my group members but also with other class members who were more knowledgeable than me.

These learners’ responses indicate that the STAD cooperative learning model was both accepted and effectively implemented. The motivation provided by group recognition and rewards, coupled with the quality of teacher explanations and discussion questions, contributed to the intervention’s success in enhancing learners’ understanding of geometric transformations.

Group member participation during discussions (CA6; M = 3.75, SD = 0.799), individual contributions to group success (CA10; M = 3.68, SD = 0.819), and cooperation among group members during discussions (CA9; M = 3.64, SD = 0.780) were all moderately rated, a clear indication of learners’ satisfaction with these activities. This implies that the “social validity” of the intervention was also assured. The following quotes by research participants justify this:

Participant #14: Through groupwork, I learnt more from my friends because other group members understood things better than me. The idea of recognizing and rewarding hardworking groups encouraged me to work harder and not to let my group down.

Participant #17: With the help of group members when I missed the point or explanation in class, I was able to understand the topic. Amazing cooperation from group members making sure that at least everyone understands what they are doing which helped me learn a lot more.

Participant #21: Group work was very helpful in times when I did not understand fully from the teacher’s demonstrations. What I like about it is we have groups to refer to if we don’t understand through teacher’s presentation.

These responses from the participants indicate that the STAD cooperative learning model had high social validity. This reflects that the intervention’s design fostered an environment of peer learning, cooperation, and mutual support, which were highly valued by the learners. The recognition and rewards for group performance further motivated learners to engage actively and contribute to their group’s success. Overall, the learners perceived the intervention as effective and beneficial for their understanding of geometric transformations, which supports the social validity of the STAD model in this educational context.

Notwithstanding the high satisfaction levels expressed by the majority of the participants, we are also cognizant of the fact that some individuals found three activities (CA7, CA10, and CA11) not satisfying at all. This is attributed to the fact that the minimum ratings (Table 5) for these activities stood at 1. Insights into the reasons behind participants’ dissatisfaction with these activities are reflected in the following direct quotes:

Participant #4: Most of the class activities that took place were helpful. Only that transformation geometry is too involving for me.

Participant #6: While the teacher’s explanations and announcement of group rewards inspired me to work hard, I did not understand enough from my fellow group members.

Respondent #15: Participation in the group was not impressive, but an improvement can be seen.

Participant #16: Though group work is good, I feel my group was not very supportive. I could have done better if my group members were not too busy.

The responses from these participants provide insights into the varying levels of benefits and challenges experienced during the intervention. Despite overall high satisfaction, specific difficulties related to the complexity of geometric transformations and group work dynamics were evident. The complexity of the topic itself posed a significant barrier for some learners. It also became evident that the effectiveness of the STAD model relies heavily on peer interactions. As such, limited understanding from peers and inadequate support can reduce the approach’s effectiveness. Active participation and support within groups are vital, as poor group dynamics or less engaged members can negatively impact individual learning outcomes.

5. Discussion

Overall, the findings from this study demonstrate that the STAD cooperative learning model had a significant impact on learners’ conceptual understanding of geometric transformations. The improvement in test scores from the pre-test to the post-test suggests that the intervention was effective. This is consistent with previous studies conducted in other settings that have confirmed the effectiveness of STAD in fostering learners’ understanding of geometric transformations [23,53,54] and other mathematical concepts at various levels of education [24,37].

Furthermore, specific features of the STAD model were identified as particularly effective in contributing to this improvement. Participants highlighted the quality of questions posed during group discussions and oral quizzes, teacher explanations during whole-class presentations, and the criteria for selecting and awarding the best-performing groups as the most impactful. The high ratings associated with these activities suggest that the implemented instructional approach was not only effective in enhancing conceptual understanding but also executed with integrity. This resonates with previous studies that emphasize the importance of well-structured questions and teacher guidance in cooperative learning environments [37,39,55,56].

The intervention was also perceived to have high social validity, as participants valued the peer learning, cooperation, and mutual support fostered by the STAD model. These aspects are key indicators of social validity, which focuses on the acceptability and perceived usefulness of an intervention by its participants. Specifically, participants’ recognition of rewards for group performance as a motivating factor reflects not only the intervention’s success in terms of learning outcomes but also its overall acceptance by the learners. This finding aligns with broader research on cooperative learning environments, which have been shown to promote both social and academic benefits, such as group accountability and personal responsibility. For example, a recent systematic review [57] reported that cooperative learning is associated with improved self-reflection, reduced negative behavior, and increased cultural integration among learners. Similarly, in an Indonesian university study, students who engaged in the team-assisted individualization cooperative learning model showed significant gains in problem-solving skills, communication, and self-efficacy compared to those exposed to traditional teaching methods [58]. These findings further substantiate the role of cooperative learning in enhancing both social and academic dimensions, highlighting its effectiveness in diverse educational settings.

The high level of intervention acceptability in this study challenges the findings of an earlier study [39], where teachers reported that students were reluctant to participate in cooperative group discussions, preferring the expository teaching methods. However, it is also important to acknowledge the differences in context between the two studies. While the earlier study was conducted in public schools, the present study took place in a private school setting, characterized by smaller class sizes and better resources than most public schools in Zambia. This may lend credence to the idea that cooperative learning, particularly the STAD model, could be more effective in well-resourced environments with enough classroom space [39].

Despite the overall positive feedback, some participants encountered challenges, especially with the group work dynamics. A few learners reported difficulties in understanding the concepts due to the topic’s complex nature or insufficient support from group members. These challenges underline the need for careful attention to group formation and ensuring that all learners receive adequate support, consistent with findings from other studies [40,41,59,60].

Besides highlighting the challenges encountered during the intervention and their interactions with the taught concepts, participants made some suggestions on how the teaching and learning of geometric transformations could be improved upon. Prominent suggestions included regular revision sessions through oral and written quizzes, splitting the topic into smaller pieces, swapping group members rather than maintaining the same groupings, revisiting prior concepts such as matrices and analytic geometry (coordinate geometry), and encouraging group members to cooperate and contribute to group success. The provision of adequate learning time and materials such as past examination papers, graph papers, and other related equipment for practice was another prominent suggestion.

6. Conclusions

This study revealed that the STAD model significantly enhanced learners’ grasp of geometric concepts, as evident by improved test scores from pre-test to post-test. Participants noted the value of well-crafted questions for group discussions and oral quizzes, teacher explanations during whole-class presentations, and the motivation provided by the criteria for selecting and awarding top-performing groups. These factors suggest that not only was the STAD model implemented effectively but it was also received positively by the learners. This study also highlights the social benefits of the STAD model, with participants appreciating the peer learning, cooperation, and mutual support that it fostered. The recognition of rewards for group performance as a motivating factor further affirms the model’s success and acceptability.

However, this study does have limitations that need to be acknowledged. The absence of a control group and the relatively small sample size drawn from a private school may limit the generalizability of the findings. Additionally, conducting this study in a private school setting, with smaller class sizes and better resources, differs from the typical public school environment where most previous studies on this subject were conducted. While this setting introduced a new perspective, suggesting that STAD may be particularly effective in well-resourced environments, future research should explore its effectiveness across diverse educational contexts, including under-resourced public schools, to better understand the broader applicability of the model.

This study employed the Shapiro Intervention Criteria to ensure that the STAD model was appropriate, feasible, and aligned with this study’s objectives in the given context. Nonetheless, a fidelity assessment, which monitors the extent to which an intervention is implemented as designed, was not conducted. Without this measure, it is challenging to determine whether variations in the implementation might have influenced this study’s outcomes. Future research should incorporate fidelity assessments alongside frameworks like the Shapiro Intervention Criteria to provide a more comprehensive evaluation of the intervention, ensuring that both its design and execution are rigorously assessed.

Furthermore, this study highlights the importance of contextual factors in cooperative learning implementation. Policymakers and educators should consider these factors when adopting cooperative learning strategies, ensuring that they are tailored to the specific needs and constraints of different educational settings.