Original Research

An Action–Process–Object–Schema study on first-year undergraduate students solving related rates problems

Fitzgerald Abakah, Deonarain Brijlall, Pholoho Morojele

Received: 03 Nov. 2025; Accepted: 11 May 2026; Published: 02 Oct. 2026

Copyright: © 2026. The Authors. Licensee: AOSIS.
This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0) license (https://creativecommons.org/licenses/by/4.0/).

Abstract

This qualitative case study investigated first-year undergraduate students’ challenges with calculus and how those challenges can be addressed with a focus on related rates problems. This study was undertaken at a University of Technology in South Africa, involving two groups: control (n = 145) and experimental (n = 136). In the control group lessons were conducted via the traditional approach. Lessons were conducted in a collaborative classroom setting for the experimental group during which participants’ muddiest points at the end of each lesson were investigated and analysed using thematic content analysis. Action–Process–Object–Schema (APOS) theory was applied as theoretical framework. Newman’s error analysis model and APOS theory were employed as data analysis tools in identifying students’ challenges. Whereas the former was applied in identifying students’ problem-solving challenges, the latter was utilised to monitor and assess students’ mental constructions when solving related rates problems. Group activities, classroom observations and a standardised test were modes for data collection. The researchers established from this study that: (1) participants who encountered comprehension error and transformation error were unable to operate at the Action level; and (2) finding participants’ muddiest points at the end of each lesson is an appropriate intervention strategy for uncovering students’ challenges in solving related rates problems. The researchers, therefore, recommend finding participants’ muddiest points at the end of related rates lessons to identify, address and minimise students’ challenges.

Contribution: This study offers practical insights into formative assessment for improving conceptual understanding in undergraduate calculus.

Keywords: APOS theory; Challenges; Learning difficulties; Muddiest point; Related rates problems

Introduction

The underperformance of first-year undergraduate calculus students in universities in South Africa is a cause of concern and there is a pressing need to ascertain the root cause of this conundrum (Brijlall, 2020). Calculus is indispensable as many undergraduate courses in mathematics, science, technology and commerce rely on its contents. However, despite its relevance in academia, undergraduate students’ challenges with this mathematical content persist, resulting in their underperformance in examinations (Brijlall, 2020). Related rates problems (RRPs) are one of the integral calculus concepts. Solving RRPs requires students to make a diagrammatical representation of the given problem, construct a functional relationship, relate the rates, solve for the unknown rate, and check the answer for reasonability (Engelke, 2009). Many students, however, encounter challenges in each of these five mentioned procedures; this incapacitates them in solving those problems (Engelke, 2009).

Habibullah and Hartono (2019) argue that the word-problem nature of RRPs contributes significantly to students’ difficulties in solving them. Solving RRPs requires students to read the given problem, identify critical information, plan for a solution, and then put their plan into action by solving the problem (Powell, 2011). However, many students are unable to logically reason around relevant mathematical concepts, to understand the mathematical language of RRPs (Habibullah & Hartono, 2019). In addition, students are unable to link relevant mathematical ideas and their chains of reasoning so as to effectively solve RRPs, which demands advanced mathematical thinking (Habibullah & Hartono, 2019). Solving RRPs demands creative, critical and systematic thinking; however, students often resort to memorisation, which poses significant challenges for their conceptual understanding. In addition, students’ learning difficulties and misconceptions in relation to understanding variables, functions and their misunderstanding and misapplication of the chain rule contribute to their challenges in solving RRPs. Maharaj (2013) maintains that undergraduate students encounter challenges in understanding derivatives and their applications, such as RRPs, and that undergraduate students’ ineptitude in applying appropriate rules of differentiation were due to them not possessing appropriate mental structures at the Process, Object, and Schema levels of the Action–Process–Object–Schema (APOS) theory.

One of the most relevant calculus concepts required for solving RRPs is the chain rule; this is described as differentiation of a function of a function (Rodríguez et al., 2010). That is, if y is a function of u and u is a function of x, then y is called a function of a function of x. This is differentiated using the chain rule, in Leibniz’s notation: . For example, if y = (2x + 5)⁸, let u = (2x + 5). So . Hence, . Therefore, . The chain rule, a foundational concept in calculus, plays a central role in solving RRPs, yet many students misapply it due to conceptual gaps.

In this study, the researcher investigated participants’ challenges with RRPs in accordance with Newman’s error analysis model and APOS theory’s mental conception levels. Also, the researchers investigated a pedagogical approach known as ‘muddiest points’. Cappetta (2018) explains that the ‘muddiest point is a classroom assessment technique that gives students opportunities to point out what they are most confused about’ (p. 4). Finding participants’ muddiest points at the end of a lesson on RRPs, therefore, is an innovative intervention approach for uncovering and addressing students’ challenges and this assertion was investigated in this study.

Despite recognition of students’ challenges with RRPs, few studies have examined these difficulties using both cognitive and error analysis frameworks while testing practical pedagogical interventions. Thus, this is what the researchers sought to investigate in this study by developing a rationale for identifying and addressing students’ muddiest points, as a reliable classroom assessment practice during RRPs lessons. In fulfilling this aim, the following critical research questions were formulated:

• What challenges did participants encounter when solving RRPs in relation to muddiest points, Newman’s error analysis and APOS analysis?
• How did the muddiest point technique impact participants’ performance in solving RRPs?

Literature review

Students’ challenges with related rates problems

Calculus has tremendous relevance in academia; however, Tall (1992) asserts that in whatever manner calculus is taught, there are embedded difficult concepts that pose challenges to students – one such calculus concept is RRPs. Cappetta (2018) asserts that calculus instructional challenges are so prevalent that ‘about 25% of Calculus 1 students successfully complete their course’. This further speaks to the pedagogical challenges pertaining to calculus resulting in students’ underperformance and students’ attrition from calculus-related courses.

RRPs is a crucial calculus concept; it goes beyond simple arithmetic and algebra, and involves infinite processes requiring indirect arguments embedded with advanced mathematical thinking. However, most first-year undergraduate students experience extensive cognitive challenges in solving RRPs (Li et al., 2017). Probably, this is because those students are unable to develop appropriate mental structures required for solving complex calculus concepts such as RRPs. Students are mostly unable to interpret RRPs appropriately and they are unable to represent RRPs diagrammatically using correct mathematical notations (Li et al., 2017). This results in students’ inability to attain conceptual understanding, relevant problem-solving skills and the expected level of cognition required for solving RRPs. Students are also unable to find derivatives of given functions, are unable to differentiate equations with respect to time, as well as being unable to eliminate one of the variables by substitution (Rodríguez et al., 2010). In addition, students’ solutions to RRPs are inundated with misapplications of the chain rule (Rodríguez et al., 2010). This limits students’ ability to be able to put their thoughts together to form a meaningful coherent framework, making them unable to link various variables of RRPs correctly as required for solving RRPs (Stewart, 2015).

According to Cappetta (2018) and Hashemi et al. (2019), ineffective problem-solving instructional approaches may contribute to undergraduate students’ challenges with RRPs. Hashemi et al. (2019) assert that university students are unable to solve RRPs because they lack the relevant prior knowledge to conceptually understand this calculus concept. In addition, Maharaj et al. (2015) note that students are unable to master basic skills and knowledge in algebra – functions, reasoning, using symbols and connectives – required for understanding RRPs. The researchers assert that if students’ challenges with RRPs are uncovered, then the necessary pedagogical remediation strategies can be employed to minimise those challenges. This will guide students to develop the required skills for them to be able to attain proficiency in solving RRPs.

Students’ challenges in solving RRPs can be attributed to students’ and teachers’ overdependence on ‘memorisation and mimicry’; these impede conceptual understanding and rather promote rote learning, according to Cappetta (2018). This author, in addition, cites lack of motivation and preparedness as another contributing factor of students’ challenges. This situation highlights that students’ challenges with calculus are enormous, hence further impeding their learning of advanced calculus concepts, such as RRPs.

Tziritas (2011), however, found APOS theory’s mental conception levels as not a wholly appropriate framework for nurturing students’ problem-solving skills and reasoning needed for them to develop mastery in solving RRPs. The author, thus, recommended the incorporation of a combination of frameworks for teaching RRPs. In this study, therefore, in addition to the APOS theory, the researchers implemented the five stages of Newman’s error analysis model – reading error, comprehension error, transformation error, process skill error and encoding error – for guiding participants on how they can appropriately solve RRPs. The incorporation of Newman’s error analysis model also guided the researchers to correctly identify participants’ challenges when solving RRPs.

How students’ challenges with related rates problems can be uncovered utilising the muddiest point technique

Soliciting students’ muddiest points as an assessment technique was first conceived by F. Mosteller at Harvard in 1988 for teaching undergraduate statistics modules (Mosteller, 1989). For this process, students were requested to write their areas of concerns, anonymously highlighting what they did not understand at the end points of a lecture, a classroom presentation, or other forms of instruction, on a piece of paper or on a card. In the context of this study, in a small class setting, each muddiest point on RRPs can be addressed individually; however, for a large class, students’ muddiest points with RRPs can be put into categories. For example, students who are unable to interpret RRPs can be labelled as category 1, students who misapply the chain rule can be labelled as category 2, students who are unable to differentiate functions or equations can be labelled as category 3, et cetera. The instructor, after ascertaining students’ concerns with RRPs then addresses the areas of the articulated challenges during the normal class time or during extra classes.

According to Majumdar and Iyer (2013) in large classes the muddiest point procedure can be partitioned into three phases: collection, addressal and closure. Hall et al. (2002) argue that the muddiest point technique promotes active and differentiated learning and instant feedback to the questions raised is guaranteed. King (2011) adds that the muddiest point technique can be implemented in large classes – the justification for implementing this technique on 136 participants in the experimental group in this current study. The aforementioned three phases of employing the muddiest point technique in a large class by Majumdar and Iyer (2013) informed how participants’ challenges with RRPs were uncovered and remedied in this study.

The traditional practice of identifying and addressing students’ learning difficulties during lessons has been to encourage students to ask questions in class or to go to teachers after the lesson (Waters et al., 2016). Many students, however, shy away from this practice; thus, anonymously finding students’ muddiest points serves as an innovative approach for identifying and addressing students’ challenges without anyone knowing the student who asked that question (Cappetta, 2018). This approach is able to identify students’ conceptual misunderstandings; in this way, students are accorded leeway to ask any question that hampers their understanding without feeling ridiculed by their contemporaries.

According to Waters et al. (2016), identifying and addressing students’ muddiest points uncovers their misconceptions and instructor’s blind spots. Hence, areas of RRPs that students could not conceptually understand are identified and areas of RRPs omitted or inadequately explained by the teacher are brought to the teacher’s attention. Therefore, muddiest points can be an effective formative assessment technique in the context of RRPs pedagogy. Angelo and Cross (1993) labelled muddiest points as an effective classroom assessment technique applicable in mathematics classrooms; it is remarkably efficient, time saving, inexpensive, and easy to implement, and responding to them requires some higher-order thinking. These authors further maintain that ‘the Muddiest Point technique is not only quick, simple and easy to administer; it also requires very little preparation’.

The limitations of previous research studies discussed under this literature review include small samples and lack of longitudinal tracking. Also, gaps in how previous studies evaluated the effectiveness of formative assessments like ‘muddiest points’ were identified, justifying the conducting of this study. In addition to employing the muddiest point technique, this study further adopted Newman’s error analysis in identifying students problem-solving challenges associated with RRPs. Also, this study utilised the four APOS mental conception levels to assess and monitor students’ mental constructions when solving RRPs making the conducting of this study unique. This is because questions raised but not addressed by past research studies such as how to remediate errors at the schema level were some of the identified gaps in literature the conducting of this study hopes to fill.

Theoretical framework

In this study, APOS theory was adopted as the theoretical framework. This theory is characterised by four distinct components, namely: Action, Process, Object and Schema (Dubinsky, 2001). Its unique theoretical approach, methodology and outcome from its implementation make it stand out from other mathematics education theories (Dubinsky, 2001; Maharaj, 2013). The premise for the adoption of APOS theory as a theoretical framework is its employment as an analytical lens for interpreting students’ mental constructions and reasoning processes when solving RRPs. An exemplar of RRPs and the characterisations of the expected mental constructions students are expected to demonstrate at each of the four levels of the APOS theory is delineated next. The RRP discussed herein (see Box 1) is a typical standard high-order RRP at the first-year undergraduate level. Thus, any other standard high-order RRP could have being selected for discussion.

APOS theory

Action level

Students operating at this mental conception level are expected to demonstrate the ability to be able to read, interpret and understand the given rate of change problem, in the format of a word problem, by representing the given problem diagrammatically and mathematically as illustrated in Figure 1. Students are also expected to be able to recall from memory appropriate trigonometric concepts such as trigonometric ratios, measurement formulae, such as , and S = πrl, et cetera. These concepts are then applied in solving the given problem to the extent that their reaction to stimuli (perceived as external) is first conceived as action – by abiding by concise step-by-step procedures showing ‘what’ and ‘why’ of each step of the transformation required of them (Abakah & Brijlall, 2024).

FIGURE 1: Diagrammatical representation of the given rate of change problem.

Process level

Students operating at the process mental conception level should be able to demonstrate direct applications of trigonometric ratios and finding derivatives of an equation. With reference to the given rate of change problem, from Figure 1: ; students are expected to make r the subject to obtain r = h tan 30°, . They then substitute this value of r into the volume formula: , resulting in . Also, to find the wet surface area, S = πrl, with reference to Figure 1: ; making l the subject, we get resulting in l = 2r. This can be plugged into the appropriate formula S = πrl by reflecting on the action conceptions of applying correct formula, making a variable the subject of an equation and substituting it into relevant equations, thereby describing or reversing steps of internalised actions without performing those steps explicitly. These action conceptions are iterated and interiorised forming a mental process imaginatively, devoid of step-by-step procedures, which gives S = πr(2r), S = 2πr²; therefore, . This is simplified to obtain (Abakah & Brijlall, 2024).

Object level

At this mental construction level, students are expected to demonstrate high-order reasoning and creative thinking around the relevant parameters of the given rate of change problem to find the rate of rise of the water level when the depth is 6 cm. This requires the student to logically and creatively know that they are expected to find which entails finding and . This can happen after the student is able to encapsulate the internalised processes into a cognitive object, by reflecting on them repeatedly as they become totally aware of correct procedures, by means of a process termed ‘encapsulation’ and their associated reversal termed ‘de-encapsulation’. The student totally realises, constructs, and mentally imagines and automatically performs required transformations correctly (Abakah & Brijlall, 2024; Maharaj, 2013).

Schema level

At this mental conception level, students are expected to construct, interconnect, organise and link actions, processes and objects into a coherent framework – the schema. This demands critical, logical and creative thinking around relevant concepts to form a productive thought, forming the chain rule: so we get resulting in . Given that V = 81π cm³, this implies that ; this is simplified to obtain h³ = 729 resulting in h = 9. Therefore, ; this is simplified to obtain . This means that the surface area is decreasing at (Abakah & Brijlall, 2024; Maharaj, 2013).

Research methods and design

Sample and data collection

This study was undertaken within a duration of three months at a university of technology in South Africa which was purposively sampled to serve as the research field. According to Tajik et al. (2024), in purposive sampling, the researcher makes specific decisions to ascertain who ought to be included in the study. In addition, Cohen et al. (2007) assert that purposive sampling can be employed to access those who have in-depth knowledge about a particular issue. Due to the peculiar nature of this research, a research field with some desired characteristics motivated the need to employ purposive sampling. Constraining selection factors such as obtaining consent from the university, financial considerations, suitability of the university for the research – a university offering undergraduate mathematics modules, easy accessibility to the university, et cetera – were considered. In addition, the researchers did not have the wherewithal to conduct this study at other universities which were far away from their locations.

In all, 18 lessons on RRPs were conducted. Each lesson lasted for a duration of one hour, in the second semester of the 2025 academic year involving first-year undergraduate calculus students which consisted of two classes. One class formed the control group while the other class formed the experimental group. Three classroom observation sessions were conducted – before, during and after the intervention (Abakah, 2023). Before the intervention, inter alia, the researchers sought to familiarise themselves with the research field, students who were the target for the study, and how lessons were conducted at the research field. During classroom observation sessions, the researchers primarily focused on how the participants learnt and solved RRPs in both groups. Also, group activities were administered during conducted lessons, and a standardised test was administered in both groups after the intervention; these were the avenues for data collection.

Data validity and trustworthiness

The standardised test on RRPs was validated by three mathematics experts; adjustments based on the structure and content of the test were effected on the first draft after two of the three mathematics experts raised some concerns. The final draft of the test was re-evaluated by those experts and a unanimous affirmative response from all of them was received before the final draft was admitted as the standardised test instrument for this study. Also, three different sources of data, namely group activities, classroom observations and a standardised test were utilised in this study to validate the research findings during triangulation of data (McMillan & Schumacher, 2014).

Research paradigm

According to Creswell and Poth (2018) a research paradigm is a philosophical framework guiding the conducting of research, embodying the ideas and beliefs of that specific philosophical practice. Constructivism was implemented as the research paradigm of this study. According to Burns et al. (2022), constructivism as a paradigm positions students at the centre of learning experiences. This guides students to become active knowledge constructors by placing a focus on the ‘why’ of human experiences and actions in lieu of being knowledge absorbers. The constructivist paradigm views reality to be subjective, multiple, and socially constructed through one’s own experiences including interactions with the environment (Burns et al., 2022; Cobb, 2006).

Constructivism guided the researchers to facilitate and assess participants’ learning and mathematical errors utilising Newman’s error analysis model and students’ mental constructions utilising the APOS levels. Also, how participants’ level of errors or thinking influenced their performance as they solved RRPs was guided by constructivism. In addition, constructivism guided the researchers to appropriately position participants at the centre of the learning and problem-solving sessions on RRPs. Specific focus was placed on assessing the efficacy of the muddiest point technique. Error analysis, nurturing students’ mental constructions and the muddiest point technique are germane to this study and they are all constructivist practices, justifying its utilisation as the research paradigm of this study.

Research approach

This study primarily followed a qualitative research approach. According to Creswell and Poth (2018) in qualitative research, the researcher’s priority is ascertaining the why and how of human behaviour and experiences via in-depth description, understanding, and analysis of a case or multiple cases. This gives the researcher the leverage to gain deeper insights into a problem and collate participants’ experiences, perceptions and behaviour to answer open-ended questions to solve real-world problems (Tenny et al., 2022).

The two analytical tools employed in this study, namely Newman’s error analysis model and APOS, were undertaken after a comprehensive content analysis of each participants’ paper-and-pencil or oral responses. Content analysis of participants’ written responses to the standardised test was the researchers’ main focus. In addition, participants’ oral responses captured during classroom observation sessions as they explained how they arrived at their conjectured solutions either individually or as a group served as the supplementary data. Content analysis, integral to this study, is a qualitative data analysis procedure. Therefore, the implementation of qualitative research approach in this study is justified.

Research design

To obtain adequate, concise and unabridged first-hand information at the research field, a case study research design was considered appropriate for this study. According to Creswell and Poth (2018), a case study design accords the researcher the leeway to adequately observe and monitor participants in their natural classroom setting, over an extended period. This design guided the researchers to extensively monitor and assess participants as they went about their learning and solving of RRPs.

Participants

In this study, two participating groups were involved: the control group (n = 145) and the experimental group (n = 136). First-year undergraduate students at a university of technology in South Africa were purposively sampled to serve as participants for this study. They were from separate pre-existing intact groups of classes: Class A formed the control group and Class B formed the experimental group. These two classes were constituted from students who had passed an admission test. The same teacher (one of the researchers) taught both groups; potential bias of the dual role of the researcher was mitigated using triangulation. These participants availed themselves voluntarily.

How participants’ challenges with RRPs were uncovered utilising the muddiest point technique

In the control group, the traditional mode of instruction was employed; participants critically observed the teacher as he explained to them how to solve RRPs and they were expected to regurgitate the information. In the experimental group, finding participants’ muddiest points, that is, their learning difficulties at the end of each conducted lesson, was tested in this study which lasted for a duration of three months. Students were anonymously asked to write on a piece of paper their ‘muddiest point’ (most difficult learning barrier) from the day’s lesson; these were submitted in a box placed in front of the class. Before the next day’s lesson began, the researcher reported back to participants as he responded to the most mentioned points and any other relevant points cited by participants. In doing so, relevant explanations, activities, investigative tasks or RRPs that could be used to address participants’ stated points were utilised (Majumdar & Iyer, 2013). In both the control and experimental groups, group activities and classroom observations were avenues for data collection during lessons. After all lessons were conducted, a standardised test on RRPs was administered to the two participating groups.

Data analysis

The unique nature of rate of change problems – word problems embedded with advanced reasoning and application of relevant mathematical ideas – prompted the researchers to adopt both Newman’s error analysis model and APOS as analytical tools. Although Newman’s error analysis model provides an appropriate problem-solving path to identify participants’ challenges, it does not address how students’ advanced mathematical reasoning can be nurtured explicitly. This is embedded in rate of change problems which can be addressed by APOS.

Newman’s error analysis model is an in-depth diagnostic assessment procedure, specifically identifying the root causes of students’ consistent problem-solving limitations by systematically adhering to the five stepwise process when solving RRPs (Seng, 2020). However, Newman’s error analysis model does not identify errors in the development of students’ mental structures and mathematical reasoning required for solving high cognitive concepts such as RRPs. Newman’s error analysis model primarily targets students’ errors from written tasks and does not specifically assess in detail students’ mental constructions and mathematical reasoning – the focus of APOS. This weakness in the utilisation of Newman’s error analysis model resulted in its integration with APOS (Inglis, 2015; Makgakga, 2023; Seng, 2020).

In word problems such as RRPs, understanding precedes mathematical reasoning (Jäder & Johansson, 2025; Polya, 1945). This is to say students do mathematics after understanding mathematics and that deviations in understanding – interpretations and contexts – leads to deviations in approach and reasoning (Stein & Smith, 1998). Therefore, in this study, Newman’s error analysis which tends to assess participants’ reading, understanding and problem-solving abilities was undertaken before assessing students’ mental constructions and mathematical reasoning utilising APOS.

Data analysis was carried out qualitatively in five parts (1) Newman’s error analysis of students’ written responses on RRPs; (2) presentation of participant’s written responses; (3) APOS analysis of students’ written responses on RRPs; (4) juxtaposition of the results from Newman’s error analysis and APOS analysis; and (5) thematic analysis of participants’ muddiest points data. From these analyses, concise and empirically proven findings, conclusions and recommendations were made.

Stage 1: Newman’s error analysis

During this analysis, the number of participants who encountered reading error, comprehension error, transformation error, process skill error and encoding error were ascertained after content analysis of their written responses to the standardised test on RRPs were carried out. The results after conducting this analysis are displayed in Table 1.

Table 2 shows that in the control group, at the Action level, 64 (44%) participants could operate well while 81 (56%) participants could not operate well; at the Process level, 41 (28%) participants could operate well while 104 (72%) could not operate well. At the Object level, 23 (16%) participants could operate well while 122 (84%) participants could not operate well; at the Schema level, 12 (8%) participants could operate well while 133 (92%) participants could not operate well. In the experimental group, at the Action level, 102 (75%) participants could operate well while 34 (25%) participants could not operate well; at the Process level, 88 (65%) participants could operate well while 48 (35%) could not operate well. At the Object level, 42 (31%) participants could operate well while 94 (69%) participants could not operate well; at the Schema level, 28 (21%) participants could operate well while 108 (79%) participants could not operate well.

The data in Table 2 suggest that more participants in the experimental group could operate at each APOS level compared to the control group. More experimental group participants could recall relevant calculus concepts, they could internalise relevant calculus concepts and procedures and encapsulate those internalised processes forming a cognitive object (Abakah, 2023). This indicates that more experimental group participants could master the relevant calculus knowledge, they could appropriately reason around and apply the concepts to form the appropriate schema required for solving RRPs (Abakah, 2023; Martin et al., 2020).

According to Yunus et al. (2019), ‘the mathematical reasoning ability of students taught using a contextual approach is better than those taught using direct instruction’ (p. 24). Therefore, the ‘muddiest point’ technique which was investigated in the context of RRPs as an intervention might have enhanced the experimental group participants’ learning. This was made possible as those participants’ relevant reflective abstractions required for solving RRPs were developed compared to the control group participants who were taught RRPs via the traditional approach. This indicates improved conceptual understanding and mathematical reasoning skills among students in the experimental group (Abakah, 2023). Perhaps, the ‘muddiest point’ technique contributed in the construction of relevant calculus knowledge and mental structures in the experimental group participants resulting in a better performance (Arnon et al., 2014).

According to Haeruman et al. (2024) and Khardita and Agoestanto (2023), students’ level of mathematical reasoning has an impact on their level of mathematical proficiency. The ‘muddiest point’ technique, which identified and addressed participants’ challenges, served as a medium through which participants’ misconceptions with RRPs could be minimised, thereby enhancing their conceptual understanding and adaptive reasoning skills (Khardita & Agoestanto, 2023). This assisted participants in the experimental group to learn RRPs well, and to develop their conceptual understanding and problem-solving skills well (Hansen, 2022; Sholihah & Listanti, 2022).

According to Liang and She (2023), high-order mathematical reasoning corresponds to good mathematics learning, improved problem-solving skills and good performance in mathematics. In addition, Akuba et al. (2020) assert that ‘students’ mathematical reasoning ability positively influences their mastery of mathematical concepts, self-efficacy and problem-solving abilities’ (p. 1). These assertions by Liang and She, and Akuba et al. were verified in this study as the ‘muddiest point’ technique resulted in improved classroom dynamics; this was recorded in the classroom observation notes. Also, more participants in the experimental group who could operate at the high APOS levels – object and schema – could solve high-order RRPs, demonstrated advanced problem-solving skills and produced a better performance than the participants in the control group.

Stage 4: Comparing the results from Newman’s error analysis and APOS analysis

In addition, this analysis shows that participants who encountered comprehension error and transformation error were unable to operate at the Action level of the APOS theory, participants who encountered process skill error were unable to operate at the Process level, and participants who encountered encoding errors were unable to operate at the Object and Schema levels of the APOS theory.

Stage 5: Analysis of participants’ muddiest points

Thematic coding was applied for analysing participants’ muddiest points data. Student responses were transcribed, coded and categorised leading to the formulation of relevant themes. Five themes emerged, namely language comprehension difficulties, conceptual confusion, difficulty with variable relationships, challenges with chain rule, and inability to follow a logical thinking path. In this section, samples of common participants’ learning difficulties they encountered at the end of each conducted lesson in this study are presented. The collected samples of participants’ muddiest points that are typical of others are presented herein.

Figures 2–6 highlight the various degrees of errors expounded by Newman’s error analysis model. These errors, if not uncovered and addressed, would have resulted in participants not developing conceptual understanding of solving RRPs. Also, these errors could have limited participants’ mathematical reasoning skills. Therefore, these errors could have impeded participants’ development of the relevant reflective abstractions required for solving RRPs as expounded by APOS theory (Majumdar & Iyer, 2013).

FIGURE 2: Muddiest point note from participant E12.

FIGURE 3: Muddiest point note from participant E58.

FIGURE 4: Muddiest point note from participant E117.

FIGURE 5: Muddiest point note from participant E45.

FIGURE 6: Muddiest point note from participant E92.

Ethical considerations

Prior to the commencement of this study all ethical procedures were followed: informed consent, offering confidentiality to participants and voluntary participation. First, ethical approval was granted by the Durban University of Technology Institutional Research Ethics Committee (IREC) (no. IREC 001/25). Also, permission was obtained in writing from prospective participants and their parents or guardians. Second, anonymity and confidentiality were ensured by not requesting any form of identification from the study participants. Third, only students who were willing to partake in this study were admitted as participants.

Results and discussion

The research findings established in this study are presented and delineated with justifications in this section with reference to the research questions investigated in this study.

Research question 1

Research question one asked: What challenges did participants encounter when solving RRPs in relation to muddiest points, Newman’s error analysis and APOS analysis?

Participants’ challenges with RRPs identified in this study are presented in three parts, namely challenges in view of muddiest points, challenges in view of Newman’s error analysis and APOS analysis, as well as deductions from comparing Newman’s error analysis and APOS analysis.

Challenges in view of participants’ muddiest points

In Figures 2–6 typical muddiest points, which are representations of a broader set of narrations by participants, are presented. First, reference is made to Figures 2–4: In Figure 2: ‘I am unable to interpret rate of change questions’; Figure 3: ‘I don’t understand rate of change questions so I find it difficult to solve them’; and Figure 4: ‘Rate of change questions are difficult for me. It has too much grammar that I find difficult to interpret’. Figures 2–4 attest that participants incorrectly interpreted RRPs and they were unable to represent RRPs on appropriate diagrams using correct mathematical notations (Li et al., 2017). Furthermore, participants’ misconceptions were uncovered, showing barriers in their understanding and mathematical reasoning pertaining to solving RRPs (Waters et al., 2016).

In addition Figures 2–4 support the assertion that students cannot find derivatives of given functions and equations (Rodríguez et al., 2010). Figures 2–4 also confirm students’ lack of prior knowledge they may require for solving RRPs such as finding derivatives of functions and equations (Hashemi et al., 2019). This caused participants to develop misconceptions which detrimentally impacted their ability to solve RRPs correctly (Waters et al., 2016).

Reference can also be made to Figures 5–6. In Figure 5: ‘How to apply the variables in a rate of change question to form the chain rule’; and Figure 6: ‘I can interpret the given question and express it mathematically. I can also work out the main variables using trig ratios and differentiation but I find putting the variables together for the chain rule confusing’. Figures 5–6 support the assertion that students misapply the chain rule (Rodríguez et al., 2010) resulting from their inability to link variables of RRPs appropriately (Stewart, 2015). This hampered participants’ proficiency in conjuring appropriate solutions to the given RRPs.

Challenges in view of Newman’s error analysis and APOS analysis

The results after content analysis of the control group’s written responses (Newman’s error analysis in Table 1 and APOS analysis in Table 2) answers this research question. Newman’s error analysis revealed that no participant encountered reading errors, 54% of participants encountered comprehension error, 70% of participants encountered transformation errors, 81% of participants encountered process skill errors, and 86% of participants encountered encoding errors when solving RRPs. On the other hand, the APOS theory analysis resulted that 56% of participants could not operate at the Action level, 72% of participants could not operate at the Process level, 84% of participants could not operate at the Object level, and 92% of participants could not operate at the Schema level when solving RRPs. The result obtained from Newman’s error analysis (see Table 1) and APOS analysis (see Table 2) testify that students experience challenges in solving RRPs (Cappetta, 2018; Habibullah & Hartono, 2019; Hashemi et al., 2019; Klymchuk et al., 2010; Sofronas et al., 2011; Thieken, 2012; Tziritas, 2011). Newman’s error analysis and APOS analysis carried out on the control group participants’ written responses to the standardised test are presented next.

Newman’s error analysis of the control group’s written responses

Figure 2 is a sample of typical written responses of the control group participants demonstrating comprehension error when solving RRPs because they mostly provided irrelevant responses. This indicates that these participants were unable to interpret and understand the given problem; therefore, they could not represent the word problem incorporated in RRPs mathematically and diagrammatically. Participants could not also interpret and understand the relevant parameters of the given problem, leading to them writing , demonstrating wrong application of trigonometric ratios (Hashemi et al., 2019; Tziritas, 2011).

In Figure 3 and Figure 4, participants could correctly read the given RRPs with understanding. Participants were able to represent the given RRPs mathematically and diagrammatically. They were also able to apply the correct formula for finding the volume of a cone and they applied the appropriate trigonometric ratio () and rightly made r the subject (see Figure 3 and Figure 4). In addition, they correctly substituted the obtained equation for r = h tan30° into the formula for volume of a cone (see Figure 4). However, those participants could not go beyond this point; they could not recognise and apply correct mathematical processes: trigonometric ratios and finding derivatives of a function to form a logical thought. Participants’ written responses in Figure 3 and Figure 4 are evidence to testify that participants demonstrated transformation errors. Also, participants’ written responses in Figure 3 and Figure 4 are evidence to assert that they demonstrated process skill errors. This is because they were unable to identify and apply mathematical processes appropriately: they could not work out the relevant parameters required to apply the chain rule. This highlights that participants could not apply relevant chains of reasoning needed to solve RRPs because they were unable to apply appropriate mathematical procedures demonstrating encoding errors (Hashemi et al., 2019).

Newman’s error analysis of experimental group’s common written responses

Figure 5 and Figure 6 show common written responses of the experimental group participants. It is obvious in these figures that these participants could interpret, understand and represent RRPs mathematically on a diagram. These participants could also apply appropriate mathematical processes to constitute , made r the subject to get r = h tan30° and . Thereafter, they correctly substituted into the formula for finding the volume of a cone: and . Also, these participants recognised and applied the appropriate formula for finding the surface area of a cone, that is, S = πrl and, from Figure 1, , leading to . They made l the subject to get l = 2r and made correct substitutions to obtain S = πr(2r), ∴ S = 2πr2; thus, which was simplified to obtain . They differentiated S with respect to h, that is, , and they could interpret from the given problem that . However, these participants could not strategically reason and so interconnect their understanding and mathematical ideas illustrated in Figure 5 and Figure 6 required for applying the chain rule. Therefore, misapplication of the chain rule is evident in Figure 6 while no attempt was made in Figure 5 to apply the chain rule (Seng, 2020). Thus, the required answer to the given problem could not be reached and verified. These are evidence to assert that these participants demonstrated encoding errors (Hashemi et al., 2019; Seng, 2020; Tziritas, 2011).

APOS analysis of control group’s written responses (Figures 2–4)

Figure 2 shows the written response of control group participants who could not operate at the Action level. This is because they were unable to recall from memory appropriate mathematical concepts and procedures, causing them to write irrelevant responses, wrongly apply trigonometric ratios (e.g. ) and produce incorrect diagrammatical representation of RRPs. Also, it can be observed in Figure 2 that these participants could recall from memory the correct formula for finding the volume of a cone: . However, they were unable to apply appropriate mathematical concepts in solving the problem at hand. This is evidence to assert that these participants were unable to operate at the Process level (Abakah & Brijlall, 2024). In Figure 3 and Figure 4, participants operated at the Action level of the APOS model, but failed to transition to Process or Object level due to an inability to apply derivatives or link concepts meaningfully. This resulted in the misapplication of the chain rule. Newman’s error analysis and APOS analysis carried out on the control group participants’ written responses suggest that comprehension errors are not only linked to a failure in initiating mathematical procedures (Action level). Comprehension errors may also impair deeper conceptualisation, especially when compounded by poor process skills when solving RRPs.

APOS analysis of experimental group’s written responses (Figures 5–6)

Experimental group participants, mainly, could operate at the Action and Process levels. An exemplar of their written response is illustrated in Figure 5 and Figure 6: they could recall from memory and apply relevant mathematical concepts and procedures resulting in them obtaining and . However, many experimental group participants were unable to operate at the Object and Schema levels. This assertion originates from the evidence that these participants were unable to reflect repetitively and encapsulate internalised processes into a cognitive object, as they performed those processes automatically. Thus, they were unable to logically put their thoughts together to conjecture the required chain rule: . From this point, these participants were unable to meaningfully interconnect and link Actions, Processes and Objects to form the required Schema. Hence, they could not systematically apply their encapsulated internalised processes to derive the expected answer that the surface area is decreasing at the rate of (Abakah & Brijlall, 2024).

Deductions from comparing Newman’s error analysis and APOS analysis

The researchers further deduced from the two analyses that participants who encountered comprehension errors and transformation errors were unable to operate at the Action level of the APOS theory. This implies that participants who could not understand the given rate in an RRP and could not apply correct mathematical procedures in solving RRPs could not, as well, recall from memory appropriate mathematical concepts and procedures in solving RRPs (see Figure 2). Also, participants who encountered process skill errors were unable to operate at the Process level. This indicates that participants who could not perform mathematical procedures correctly in solving RRPs could not, as well, apply relevant mathematical procedures in solving RRPs (see Figure 3 and Figure 4). In addition, participants who encountered encoding errors were unable to operate at the Object and Schema levels of the APOS theory. This depicts that participants who could not brainstorm and inductively apply relevant trigonometry and differentiation procedures to constitute the required Schema involving high-order critical and reflective thinking in solving RRPs could not, as well, justify their conjectured solutions (see Figure 3 and Figure 4).

In addition, it was noted in Table 1 and Table 2 that more participants in the control group encountered challenges at each of the five levels of Newman’s error analysis and also at each of the four APOS levels in comparison with their peers in the experimental group. This analysis indicates that the pedagogical technique of finding participants’ muddiest point at the end of each conducted lesson on RRPs proved to be an effective intervention approach. This is because this approach guided the teacher to uncover the experimental group participants’ challenges in solving RRPs and to address them accordingly (Cappetta, 2018). This resulted in better performance in participants in the experimental group compared to participants in the control group (see Table 2). This is because minimal challenges were noted from the experimental group participants which nurtured them to be able to develop appropriate and relevant mental structures required for solving RRPs. Thus, more participants in the experimental group compared to those in the control group could internalise and encapsulate relevant concepts integral for solving RRPs. This observation originates from the premise that more participants in the experimental group compared to those in the control group could operate at the higher APOS levels, namely Object and Schema levels (Abakah & Brijlall, 2024).

Research question 2

Research question two asked: How did the muddiest point technique impact participants’ performance in solving RRPs?

It can be deduced from Newman’s error analysis in Table 1 that more participants in the control group were faced with challenges when solving RRPs compared to their compatriots in the experimental group. This is because the experimental group participants’ challenges with RRPs were identified and their misconceptions were uncovered and addressed which minimised the occurrence of those challenges (Waters et al., 2016). The APOS analysis in Table 2 indicates that more participants in the experimental group could operate at each of the four APOS levels compared to participants in the control group. Probably, the introduction of muddiest points as assessment for teaching and learning RRPs proved to be effective resulting in better problem-solving abilities of participants in the experimental group (Angelo & Cross, 1993). According to Ncube and Luneta (2025) concept-based instructions promote conceptual understanding of mathematical concepts thereby improving students’ performance. Hence, the utilisation of muddiest points for RRPs lessons promoted students’ conceptual understanding of RRPs resulting in better performances of the experimental group participants.

More participants in the control group experiencing challenges compared to participants in the experimental group reinforces the assertion that inappropriate pedagogies for RRPs contribute to students’ inability to solve RRPs (Cappetta, 2018; Hashemi et al., 2019). These inappropriate pedagogies for RRPs such as the traditional approach investigated in this study, are dominated by ‘memorisation and mimicry’ as asserted by Cappetta (2018). In addition, investigating participants’ muddiest points in this study has ascertained that the traditional practice of asking questions in class or going to teachers after the lesson is counter-intuitive (Waters et al., 2016). This is because the researchers noted in their classroom observation notes that this practice does not promote active and differentiated learning owing to fear of ridicule from peers (Hall et al., 2002).

Conclusion

First-year undergraduate students’ challenges with RRPs, mainly attributed to ineffective pedagogies, necessitated the conducting of this study. Therefore, the researchers sought to find an alternative mode of pedagogy for RRPs, other than the traditional approach. In realisation of this aim, participants’ challenges with RRPs were investigated in view of Newman’s error analysis and APOS analysis. In addition, the muddiest point technique was investigated as a formative assessment strategy for RRPs. In this study the researchers established that: (1) participants who encountered comprehension errors and transformation errors were unable to operate at the Action level of the APOS theory; (2) participants who encountered process skill errors were unable to operate at the Process level; and (3) participants who encountered encoding errors were unable to operate at the Object and Schema levels of the APOS theory. The researchers noted that misapplication of the chain rule was participants’ main challenge when solving RRPs. In addition, the researchers found ‘finding participants’ muddiest point’ to be effective for uncovering and addressing participants’ challenges with RRPs. Participants’ written responses to the standardised test indicated that most of the participants’ learning difficulties raised were addressed; that is why they could operate at the APOS levels. This implies that the muddiest point intervention strategy has been proven to be an effective tool for remedying students’ challenges with RRPs during conducted lessons. Therefore, we recommend the ‘muddiest point’ technique to be incorporated in RRPs lessons.

Acknowledgements

The authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article.

Fitzgerald Abakah: Conceptualisation, Data curation, Formal analysis, Methodology, Writing – original draft. Deonarain Brijlall: Formal analysis, Methodology, Project administration, Supervision, Writing – review & editing. Pholoho Morojele: Supervision. All authors reviewed the article, contributed to the discussion of results, approved the final version for submission and publication, and take responsibility for the integrity of its findings.

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

The authors declare that all data that support this research article and findings are available in the article and its references.

The views and opinions expressed in this article are those of the authors and are the product of professional research. It does not necessarily reflect the official policy or position of any affiliated institution, funder, agency, or that of the publisher. The authors are responsible for this article’s results, findings, and content.

References