A qualitative examination of the mediating void: teacher PCK, classroom management, and mathematical achievement

1 Introduction

1.1 Background of the study

In the study of mathematics education, identifying classroom factors that enhance students' academic achievement is a major concern for educators and researchers. Within Senior High Schools in Ghana, there is constant pressure to improve student outcomes in mathematics. Traditional research often looks at these classroom variables through a structural lens. Many researchers operate under the assumption that a direct relationship exists between what a teacher knows, how a teacher manages the class, and how the students perform.

Central to this relationship is Pedagogical Content Knowledge (Shulman, 1986), which represents the unique way teachers combine their knowledge of mathematics with effective teaching methods. In mathematics education, Pedagogical Content Knowledge is seen as a necessary requirement for good teaching. It allows teachers to break down difficult mathematical topics and proofs into simpler explanations that students can easily grasp.

Along with Pedagogical Content Knowledge, classroom management is also considered a pillar of effective teaching. It is widely believed that good classroom management provides a stable and orderly environment where learning can take place. In many quantitative studies, classroom management is modeled as a mediator. This means it is supposed to help transfer the teacher's knowledge into better performance for the students. The common expectation is that when a teacher has good knowledge, they will plan lessons well, which leads to a well-managed classroom, and this orderly environment will automatically result in high mathematics achievement.

1.2 Statement of the problem

Although this model sounds logical, recent structural equation modeling analysis by Owusu et al. (2022) involving senior high school students in Kumasi, Ghana, revealed a different result. Their study showed that classroom management and mastery-oriented instructional practices do not mediate the relationship between teachers' Pedagogical Content Knowledge and students' mathematics achievement. This finding means a teacher can have an excellent understanding of mathematics and keep a very orderly classroom, but this does not automatically lead to high achievement for the students.

Quantitative studies are effective at showing that these relationships are not statistically significant, but they cannot explain why this happens. By using Likert scale questionnaires, quantitative survey's view classroom interactions as a black box. When a student chooses a number on a questionnaire about classroom order, it does not reveal the internal tension, confusion, or anxiety they might feel during a difficult mathematics lesson.

There is a clear gap in the literature regarding how students experience and interpret classroom management styles in high-pressure mathematics environments. A classroom that looks quiet and well-managed to an outside observer might actually be an environment where students feel disconnected or anxious about the subject matter. When researchers treat classroom management as just a tool for order rather than a psychological relationship, they miss the hidden factors that prevent teacher knowledge from reaching the students. Without qualitative studies to explain this disconnect; professional development programs will continue to focus only on strict behavioral rules that do not help students understand mathematics.

1.3 Purpose of the study

The purpose of this qualitative multi-case study is to investigate the lived experiences of Senior High School students in the Kumasi Metropolis of Ghana. Specifically, the study explores the underlying psychological and pedagogical reasons why orderly classroom management and high teacher Pedagogical Content Knowledge fail to translate into deep conceptual mathematics achievement.

1.4 Research questions

To address this problem and understand the reasons behind the statistical findings, this study answers three research questions:

• RQ1: How do Senior High School students in the Kumasi Metropolis perceive and experience their teachers' classroom management styles during mathematics lessons?
• RQ2: In what ways do these perceived management styles encourage or hinder students' emotional and cognitive engagement with difficult mathematical concepts?
• RQ3: How do students make sense of the psychological disconnection and low performance in mathematics classrooms that feature highly qualified teachers and strict classroom management?

1.5 Significance of the study

This study contributes to research on teacher knowledge and classroom environment. Theoretically, it challenges the idea that a quiet and compliant classroom is the same as a learning classroom, providing a deeper look at secondary school dynamics. Practically, it gives school heads and mathematics teachers in Ghana descriptive insights into how strict management styles affect students. These findings can help reshape teacher preparation programs by moving the focus from mere classroom control toward creating supportive learning environments.

2 Literature review and theoretical framework

2.1 Theoretical framework: self-determination theory (SDT)

To understand the underlying reasons why classroom management and instructional patterns may fail to mediate mathematics achievement, this study adopts Self-Determination Theory as its theoretical framework. Self-Determination Theory, as articulated by Deci and Ryan (2000) and expanded in recent educational contexts by Howard et al. (2021) and Ryan and Deci (2020), posits that student learning is not merely a product of behavioral compliance. Instead, deep conceptual learning depends on the satisfaction of three basic psychological needs, which are autonomy, competence, and relatedness.

Autonomy refers to the student's need to feel a sense of voice, choice, and personal agency during the learning process. Competence involves the student's internal feeling of capability in mastering difficult concepts, while relatedness refers to the feeling of being securely connected to and valued by the teacher and peers.

In many senior high school mathematics classrooms, traditional classroom management is characterized by a controlling motivating style where teachers prioritize absolute quiet, rigid seating arrangements, and strict pacing to cover the examination syllabus. According to Ryan and Deci (2020), when a classroom environment focuses heavily on external behavioral compliance, it actively thwarts the students' need for autonomy. Even though a teacher may possess high Pedagogical Content Knowledge and keep the room orderly, the controlling nature of the management style creates internal anxiety and emotional disconnection.

As a result, students may appear attentive on the outside, but they lack the autonomous motivation required to process abstract mathematical proofs. This framework helps explain the statistical findings of Owusu et al. (2022) by showing that structural order can sometimes co-exist with psychological alienation, thereby blocking the path to mathematics achievement.

2.2 Conceptualizing teacher PCK and student achievement

It is widely accepted in mathematics education literature that teacher Pedagogical Content Knowledge is an essential predictor of student academic outcomes. Recent studies by Dunekacke et al. (2022) and Gasteiger et al. (2021) confirm that when teachers have a deep understanding of mathematical structures alongside the correct methods to represent them, student performance improves. This is because high-PCK teachers can readily anticipate student misconceptions and provide clearer illustrations of complex topics like algebra, calculus, and geometry.

However, recent research also indicates that high teacher knowledge does not automatically guarantee student understanding if the classroom dynamic is unfavorable. For instance, a study by Depaepe et al. (2020) noted that the positive effect of PCK on achievement is highly dependent on how teachers enact that knowledge during live classroom discourse. If a teacher possesses excellent subject matter understanding but lacks the relational capacity to deliver it in an engaging, pressure-free manner, the cognitive benefit of their expertise is lost. Therefore, while PCK establishes the potential for effective teaching, it requires a supportive classroom environment to successfully influence student achievement metrics.

2.3 The role of classroom management in mathematics education

Classroom management has traditionally been conceptualized as a logistical mechanism designed to maximize time-on-task and eliminate student disruptions. Quantitative studies by Korpershoek et al. (2020) and Kwok (2021) have often shown that well-managed classrooms create a sound environment that is conducive to studying complex subjects like mathematics. When behavioral rules are clearly established, instructional delivery can proceed without constant interruptions, which is expected to result in an increase in student performance.

Despite these traditional assumptions, contemporary researchers have found that classroom management alone cannot influence student performance if it relies purely on intimidation or rigid control. Recent investigations by Aldrup et al. (2020) and Mitchell et al. (2023) reveal that strict behavioral management often leads to surface-level compliance rather than genuine cognitive engagement. In mathematics classrooms specifically, an over-emphasis on behavioral silence can suppress student questions and discourage collaborative problem-solving. This creates a classroom paradox where the learning environment appears perfect to an outside observer, but the students are internally experiencing high levels of math anxiety and conceptual stagnation.

2.4 Mastery-oriented instruction versus performance pressures

Mastery-oriented instruction focuses on helping students deeply understand mathematical concepts at their own pace, rather than simply memorizing procedures to pass examinations. Research by Bardach et al. (2020) and Lazarides et al. (2021) shows that when teachers emphasize mastery, students develop stronger problem-solving skills and higher self-efficacy. Mastery-oriented instructions encourage students to view mistakes as natural parts of the learning process, which is critical when dealing with abstract mathematics.

Nevertheless, the implementation of mastery-oriented instruction in secondary schools is frequently hindered by external performance pressures. Studies within Sub-Saharan African contexts by Anyagh et al. (2024) and Mensah and Kioko (2023) observe that the highly competitive nature of national exit examinations forces teachers to abandon genuine mastery approaches. Instead, teachers often resort to rote memorization and exam-drilling to meet systemic targets. Consequently, even when a teacher attempts to use mastery-oriented instructions, the rigid classroom structures and time constraints often render these efforts ineffective, which explains why its mediating effect on mathematics achievement is frequently found to be non-significant.

2.5 The mediation disconnect in recent literature

The breakdown in the mediating paths of classroom management and mastery-oriented instructions points to a larger gap in educational research. Quantitative methodologies are skilled at identifying that these variables do not statistically mediate the link between teacher PCK and student achievement, but they leave the operational details unexplained (Owusu et al., 2022). Recent qualitative insights by Boston et al. (2023) and Russo et al. (2024) suggest that this mediation disconnect occurs because statistical models cannot capture the moment-to-moment instructional friction that happens in a classroom.

For example, when a high-knowledge teacher uses strict classroom management to maintain focus, the students may feel too intimidated to speak up when they do not understand a step in an algebraic equation. The survey questionnaires used in large-scale studies are completely blind to these psychological barriers. Therefore, to truly understand why classroom management and instructional styles fail to act as effective pathways for teacher knowledge, researchers must move away from closed-ended numbers and instead use qualitative methods to investigate the actual lived experiences of the students within those spaces.

3 Methodology architecture

3.1 Research design

To provide an in-depth explanation of the reasons why classroom management and instructional practices fail to mediate the relationship between teacher knowledge and achievement, this study adopts a qualitative interpretive multi-case study design. Qualitative case study methodology is particularly appropriate when a researcher seeks to answer how and why questions about a contemporary phenomenon within its real-life context, especially when the boundaries between the phenomenon and the context are not clearly evident (Yin, 2018).

By utilizing a multi-case design, each selected Senior High School mathematics classroom serves as a distinct, bounded case. This approach allows for a cross-case analysis to discover patterns that are common across different schools within the Kumasi Metropolis. This qualitative approach moves away from the statistical models used by Owusu et al. (2022) and instead focuses on gathering detailed descriptions of classroom life.

3.2 Target population and context

The context of this study is public Senior High Schools located within the Kumasi Metropolis of the Ashanti Region, Ghana. This urban educational environment is characterized by large class sizes, high-stakes academic expectations, and strict adherence to the national mathematics syllabus. The target population consists of Form 2 and Form 3 elective and core mathematics students, as well as their respective mathematics teachers. Form 2 and Form 3 students are selected because they have spent considerable time within the secondary school system and have developed clear perceptions of various teaching and management styles in mathematics classrooms.

3.3 Sampling techniques and participant profiles

To be selected, a classroom must meet specific criteria that reflect the statistical paradox identified in recent literature (Owusu et al., 2022). Because formal, standardized psychometric testing of PCK was outside the scope of this qualitative inquiry, “high PCK” was operationalized using three established proxy indicators: (a) holding a specialized Bachelor's or Master's degree in Mathematics Education, (b) possessing a minimum of five years of continuous teaching experience at the Senior High School level, and (c) receiving direct recommendations from the school's Head of Mathematics Department as demonstrating strong subject matter expertise. Furthermore, the teacher must be recognized by school heads as an effective classroom manager, yet the students must demonstrate uneven or stagnant academic performance in internal mathematics assessments. The final participant group across the four cases includes four mathematics teachers for individual interviews, and 32 students distributed across four focus group discussions, with eight students participating in each group. A sample of four cases was judged adequate for this qualitative inquiry because it provided sufficient cross-case variability to observe different management styles while allowing the researcher to reach thematic saturation within the practical and temporal constraints of the two-month data collection period. This is shown in Table 1 below.

Table 1. Participant profiles.

3.4 Data collection instruments

To ensure data trustworthiness and qualitative rigor, three distinct data collection instruments are used to achieve data triangulation:

• Non-Participant Classroom Observations: The researcher uses a semi-structured observation protocol to record field notes during live mathematics lessons over a four-week period for each case. The observations focus on the interaction patterns between teachers and students, the methods used to handle classroom disruptions, student body language during difficult math topics, and how the teacher responds to student mistakes.
• Student Focus Group Discussions: Focus groups are conducted using an open-ended interview guide. Focus groups are highly effective for encouraging students to share their shared experiences and reveal underlying feelings, such as math anxiety or alienation, which they might be hesitant to share in an individual setting (Morgan, 2019).
• Semi-Structured Teacher Interviews: Individual interviews are conducted with each of the four case teachers. These interviews allow the researcher to explore the teachers' professional beliefs, their rationale for specific classroom management choices, and their views on student engagement.

3.5 Data collection procedures

Before starting data collection, official ethical clearance is obtained, and introductory letters are submitted to the heads of the selected institutions. Informed consent is obtained from all participating teachers, and parental consent alongside student assent is secured for all participating minors.

The data collection is carried out sequentially over a period of two months. The classroom observations are conducted first to allow the researcher to become familiar with the classroom dynamics and to identify specific management incidents to discuss during the interviews.

The student focus group discussions and teacher interviews are held in quiet, private rooms within the respective schools. All interviews and focus groups are audio-recorded with the permission of the participants, and each session lasts between 45 and 60 min.

3.6 Data analysis plan

The qualitative data gathered from field notes and verbatim interview transcripts were analyzed using thematic analysis, following the systematic guidelines established by Braun and Clarke (2022). To ensure the findings were authentically grounded in the participants lived experiences rather than predetermined by theory, an inductive-first approach was employed.

During the initial phase, the researcher engaged in line-by-line open coding of the transcripts, remaining strictly close to the participants' own words (in vivo coding). Raw data generated inductive codes such as “fear of answering wrongly,” “rushing for the syllabus,” and “forced to copy.” These initial codes were then clustered into sub-themes reflecting the actual classroom dynamics. Only after these grounded themes were firmly established did the researcher introduce Self-Determination Theory (Ryan & Deci, 2020) in a deductive phase, utilizing it as an analytical lens to conceptually categorize the students' experiences of autonomy, competence, and relatedness. This dual inductive-deductive approach ensured that the theoretical framework illuminated the data rather than forcing it into predetermined categories. Thematic saturation was monitored iteratively throughout the analysis. Saturation was deemed to have been reached after the analysis of the third case, as no new substantive inductive codes regarding management styles and student apprehension emerged. The fourth case was subsequently coded to confirm these established patterns and ensure no minority perspectives or disconfirming data were missed. This is shown in Table 2 below.

Table 2. Thematic coding framework (inductive to deductive progression).

3.7 Trustworthiness and rigor

To ensure the rigor of this qualitative investigation, the evaluative criteria proposed by Lincoln and Guba (1985) were strictly applied. Credibility was established through methodological triangulation and formal member checking. Two weeks after the initial data analysis, the researcher provided a one-page summary of the preliminary inductive themes to the four participating teachers and a representative subset of the focus group students. Participants were asked to verify if the summaries accurately reflected their classroom realities, and all confirmed the interpretations without requiring substantive alterations. Transferability was facilitated by providing thick, rich descriptions of the Kumasi Metropolis educational context and the high-stakes nature of the mathematics classrooms. Dependability was maintained through a detailed audit trail documenting all methodological shifts and reflexive notes (Nowell et al., 2017). Finally, confirmability was ensured through peer debriefing; an independent mathematics education researcher, who was not involved in data collection, reviewed 20% of the raw transcripts alongside the initial codebook. This debriefing session was specifically utilized to deliberately challenge the primary researcher's assumptions and ensure the raw data was not being prematurely forced into the deductive Self-Determination Theory framework.

4 Findings and discussion

4.1 Perceptions of classroom management styles (RQ1)

The first research question sought to understand how Senior High School students in the Kumasi Metropolis perceive and experience their teachers' classroom management styles during mathematics lessons. The inductive thematic analysis of the focus group transcripts revealed that students predominantly describe classroom management as an instrument of rigid behavioral containment rather than an instructional support system. Across all four cases, students consistently characterized their environments as spaces where absolute silence and strict compliance are prioritized.

Students indicated that teachers appear to equate an orderly classroom with a quiet classroom, an interpretation that aligns with traditional, teacher-centered management frameworks. For example, a student from Focus Group 2 remarked:

“Our teacher knows the mathematics very well, but his main focus during lessons is for everyone to sit quietly and look at the board. If you talk or try to explain something to your friend who is confused, you will be punished for making noise. So, most of us just keep quiet and pretend to follow the lesson even when we are completely lost.”

This excerpt illustrates how students experience the management style as an external pressure that demands surface-level behavioral compliance. The individual interviews with the mathematics teachers provided context for this dynamic. The teachers expressed that due to large class sizes and the heavy burden of completing the West African Senior School Certificate Examination (WASSCE) syllabus, strict behavioral enforcement feels like the only viable option to maintain order. Consequently, what an external observer or a closed-ended quantitative survey captures as a “well-managed” learning environment is experienced internally by students as a restrictive space that discourages peer communication.

4.2 Management styles and student engagement (RQ2)

The second research question investigated the ways in which these perceived management styles encourage or hinder students' emotional and cognitive engagement with difficult mathematical concepts. The qualitative data indicated that while strict classroom management successfully secures behavioral compliance, it acts as a significant barrier to emotional and cognitive engagement. Students explained that the pressure to maintain absolute quietness fosters an atmosphere of apprehension. When viewed through the lens of Self-Determination Theory (Ryan & Deci, 2020), this environment appears to thwart students' basic psychological needs for autonomy and relatedness.

During the focus groups, students revealed a strong reluctance to raise their hands and ask questions when they do not understand a mathematical step. They expressed concern that asking a question might disrupt the teacher's pacing or invite negative sanctions. A student from Focus Group 4 stated:

“When the teacher is solving an algebraic equation and you miss a step, you cannot ask for clarification because the environment is too tense. The teacher wants to move quickly to cover the topic, and if you interrupt, it looks like you are causing trouble. It is safer to remain silent and copy down whatever is on the board.”

This finding highlights how strict behavioral control contributes to a psychological disconnect between the teacher and the students. Cognitive engagement in mathematics inherently requires students to actively manipulate ideas, ask questions, and make mistakes. However, when the management style treats mistakes and inquiries as disruptions to classroom order, students resort to passive copying.

The non-participant classroom observations supported this finding; while students-maintained eye contact with the board and remained quiet, their actual involvement in problem-solving was minimal. Uniform, choral responses of compliance were often followed by independent work that revealed deep conceptual misunderstandings. Therefore, the rigid management styles appear to suppress the cognitive vulnerability necessary for mastering abstract mathematics.

While the dominant narrative highlighted psychological alienation, a minority perspective did exist. A few highly achieving students noted that the strict silence helped them concentrate. One student from Focus Group 3 stated, “I actually like it when the teacher is very strict because the class is too large. If people talk, I get distracted. But even then, when the math gets very hard, the silence doesn't help me understand the formula any better, it just means the room is quiet.” This disconfirming evidence is vital; it demonstrates that while strict management may prevent logistical distractions for a few, it still universally fails to facilitate the interactive discourse required for conceptual mastery.

4.3 Unpacking the mediation disconnect (RQ3)

The third research question explored the phenomenon of student disconnection in secondary mathematics classrooms characterized by highly qualified teachers and strict classroom management. This question addresses the statistical paradox found by Owusu et al. (2022). Rather than establishing definitive causality, the qualitative findings of this study offer a plausible, experience-based interpretation for this “black box.” The data illuminates a fundamental psychological disconnect between the delivery of knowledge and the student's internalization of that knowledge, providing contextual insight into why statistical mediation might fail to materialize.

The data suggests that high teacher Pedagogical Content Knowledge (PCK) remains isolated within a teacher-centered framework when combined with rigid classroom management. Because the teachers are highly knowledgeable, they present mathematical proofs flawlessly. However, because the management style values compliance over dialogue, the feedback loop required to assess student processing is missing. A student from Focus Group 1 summarized this disconnect clearly:

“We all know our teacher is very brilliant and understands the math perfectly. But she teaches at her own pace and uses strict rules to keep the class quiet. You cannot tell her that her speed is too fast for you. The high knowledge she has stays on the blackboard because the strict environment makes it hard for us to interact with her or explain where we are struggling.”

This indicates that a “well-managed” classroom creates a silent background but does not necessarily facilitate the cognitive processing of teacher expertise. Under Self-Determination Theory, when students' needs for autonomy and competence are thwarted by an over-controlling management approach, their autonomous motivation to learn diminishes (Howard et al., 2021).

Furthermore, the teachers' interviews revealed that efforts toward mastery-oriented instruction are often reduced to structural checklists, such as writing lesson objectives on the board, rather than cultivating environments where students are permitted to struggle with concepts collaboratively. The qualitative evidence suggests that the statistical non-significance observed in previous mediation models may be contextually understood by viewing rigid classroom management as a perceived psychological barrier, one that isolates the teacher's PCK from the student's cognitive reach.

5 Conclusion and recommendations

5.1 Conclusion

This study provides a necessary qualitative exploration of the statistical findings identified by Owusu et al. (2022), which showed that classroom management and mastery-oriented instructional practices do not mediate the relationship between teachers' Pedagogical Content Knowledge and students' mathematics achievement. By peering inside the classroom “black box” through an interpretive multi-case design, this study offers a plausible interpretive framework: the breakdown in this statistical mediation path closely parallels a lived, psychological disconnect between behavioral compliance and cognitive engagement.

While senior high school mathematics teachers utilize rigid classroom management styles to secure a quiet and orderly environment, this surface-level order frequently comes at the expense of students' psychological needs. Analyzed through Self-Determination Theory, the findings illustrate how an over-controlling management style limits students' autonomy and relatedness (Ryan & Deci, 2020). Students are often driven by examination pressure and apprehension to remain silent, a dynamic that external quantitative surveys can easily misinterpret as a highly effective learning environment.

In reality, this enforced quietness establishes a psychological environment that hinders the transfer of the teacher's Pedagogical Content Knowledge to the students. High teacher knowledge remains isolated at the front of the classroom because students feel too intimidated to voice confusion or display the cognitive vulnerability required to master complex mathematics. Therefore, this study posits that classroom management may fail to act as a statistical mediator because, in practice, it frequently prioritizes structural containment over the interactive, communicative discourse necessary for deep mathematical understanding.

While this study is deeply contextualized within secondary mathematics classrooms, it is highly probable that these psychological dynamics are not entirely unique to mathematics. The intense pressure to cover the WASSCE syllabus and manage overcrowded classrooms affects multiple core subjects. Therefore, the finding that rigid behavioral containment suppresses cognitive engagement would likely manifest in other cognitively demanding subjects, such as physics or chemistry, taught under identical high-stakes structural constraints.

6 Recommendations

Based on the qualitative findings and conclusions of this study, the following recommendations are made for policy and practice in mathematics education:

• Shift Focus in Teacher Professional Development: The Ministry of Education, through the Ghana Education Service, should restructure professional development workshops for secondary mathematics teachers. Training programs should move away from traditional, teacher-centered management models that emphasize absolute silence and rigid behavioral control. Instead, emphasis should be placed on autonomy-supportive classroom management strategies that encourage dialogue, peer collaboration, and active student participation during mathematics lessons (Lazarides et al., 2021).
• Cultivate Safe Relational Spaces for Error: Mathematics teachers within Senior High Schools should consciously build a classroom culture where mistakes are treated as natural diagnostic tools for learning rather than disruptions to classroom order. By reducing pedagogical pacing and creating a pressure-free environment, teachers can fulfill students' psychological needs for competence and relatedness, thereby allowing high Pedagogical Content Knowledge to be effectively internalized by the students (Boston et al., 2023).
• Revise Institutional Supervision Metrics: School heads and internal supervisors should change their evaluation criteria for assessing instructional quality. Rather than judging the success of a mathematics lesson based on how quiet and compliant the classroom is, supervisors should look for evidence of cognitive engagement, student-initiated questioning, and active problem-solving discourse.
• Future Research Frameworks: Future studies should follow the recommendations of Owusu et al. (2022) to combine qualitative methodologies with standardized achievement tests across a wider geographic sample. Longitudinal qualitative designs should also be conducted to track how a shift from controlling management to autonomy-supportive teaching affects student self-efficacy and mathematical performance over an extended academic cycle.

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Footnotes

Edited by: Kamilah B. Legette, Rutgers, The State University of New Jersey, United States

Reviewed by: Dejene Girma Denbel, Dilla University, Ethiopia

Helen Emeka, Afe Babalola University, Nigeria