Abstract
Background: Ghana’s commitment to quality education has been reflected in its goal of providing equal access to highquality education, leading to the reforms of the New Common Core Mathematics Curriculum for Basic Schools. The study explored the integration of mathematical modelling using reflective thinking skills, which are not currently core competencies in Ghanaian basic school curriculum.
Aim: The study examined the modelling proficiency of preservice mathematics teachers by assessing their reflective thinking abilities in a modelling laboratory context.
Setting: The study focussed on preservice mathematics teachers from two education colleges in Ghana.
Methods: Using purposive sampling to select participants, a quasiexperimental design with pre and posttest interventions was employed. Data were analysed through content and inferential analysis, supplemented by interviews.
Findings: The findings indicated that the comparison group lacked prior knowledge of modelling problems and struggled with comprehension tasks. In contrast, the experimental group successfully translated realworld problems into mathematical models.
Conclusion: Providing preservice mathematics teachers access to a modelling laboratory and modellingeliciting activities was essential for developing future modellers. This approach would enhance their effectiveness in teaching foundational mathematics in Ghanaian education.
Contribution: This study advocated for reorienting the mathematics curriculum at both Basic schools and Colleges of Education in Ghana to include mathematical modelling and reflective thinking skills as core components.
Keywords: developing modelling; modelling competencies; reflective thinking skills; modelling laboratory; preservice mathematics teachers.
Introduction
This article describes a study that explored the impact of a modelling laboratory on the reflective thinking and modelling proficiency of preservice mathematics teachers in Ghana. It begins with a background on the importance of mathematical modelling in education, followed by research questions and the hypothesis, focussing on assessing how reflective thinking skills can enhance modelling abilities. The article then details the research methodology, including a quasiexperimental design with pre and posttests. It presents the key findings, highlighting differences in performance between the experimental and comparison groups. Finally, the article discusses the implications of these findings for mathematics education in Ghana and offers recommendations for integrating modelling into the curriculum.
In today’s mathematics education, preservice mathematics teachers play a crucial role in their ability to engage in mathematical modelling and develop reflective cognitive skills. Mathematical modelling serves as a creative strategy and a fundamental concept that enhances mathematical knowledge, giving the learning process a clear purpose (Salha & Qatanani 2021). It involves the process of translating a realworld situation into a mathematical model. However, the mathematics laboratory is more than just a space. It is a structured set of activities designed to help students construct meaningful mathematical concepts through handson learning, observation and communication with peers and experts. According to Maschietto and Trough (2010), the mathematics laboratory is integral to teacher education’s perspective and practice.
The concept of a mathematics laboratory did not originate from pedagogical research but from the reflections of mathematicians concerning the use of artefacts. Global studies on the mathematical modelling process emphasise the importance of reflective thinking (Thahir et al. 2019; Yasin et al. 2020). However, there is a notable gap in research focussed on preservice mathematics teachers and the tasks leading to creating a modelling laboratory to support the development of reflective thought processes and modelling proficiency. Establishing a learning environment that fosters reflective thinking and understanding the foundations of this skill are complex tasks (Davydov & Rubtsov 2018). Reflection is a central element in teacher education and the professional development of aspiring preservice mathematics teachers (Agustan, Junniati & Siswono 2017; Amidu, 2012; Lim 2011).
Strong contentoriented knowledge underpins preservice mathematics teachers’ proficiency in modelling (GochevaIlieva et al. 2018; Govender 2018). Besides broadening mathematical understanding and providing direction in the learning process, mathematical modelling is a tool for creativity (Salha & Qatanani 2021). According to Rellensmann, Schukajlow and Leopold (2017), preservice mathematics teachers’ proficiency in solving modelling tasks increases as they become skilled in creating and using diagrams to represent these tasks.
Mathematical modelling, which involves applying mathematics to realworld problems, benefits significantly from developing reflective thinking skills. These skills can be purposefully cultivated through experimental approaches, such as designing and facilitating problembased learning, simulations and collaborative learning activities. These approaches would help preservice mathematics teachers develop the ability to understand and tackle complex modelling tasks (Davydov & Rubtsov 2018; Yasa & Karatas 2018).
Analysis can be considered an experimental approach when it is used to explore how people think and solve mathematical problems. It helps uncover the relationships or principles needed to solve a problem and understand the factors involved (Davydov & Rubtsov 2018:303). Furthermore, analysis helps preservice mathematics teachers practise and improve their skills in theoretical and practical modelling activities (BorromeoFerri 2018), building reflective thinking skills to solve complex mathematical problems.
Modelling is not listed as a core competency in the newly implemented 2019 curriculum or explicitly mentioned in the mathematics curriculum. However, Ghanaian education colleges should include modelling laboratories because, as Durandt (2021) suggests, they foster a positive attitude towards learning. Preservice mathematics teachers need mentoring, which involves applying mathematics to simple and complex realworld problems, bridging the gap between everyday and academic discourse (Durandt 2021).
To achieve this, the pedagogical modelling cycle proposed by Mhakure and Jakobsen (2021) should be modified to incorporate reflection and innovative approaches (Lu & Kaiser 2022). An accurate interpretation obtained through reflection would allow the solution to be validated, and the outcome can be reexamined using the model. Preservice teachers must repeat the modelling cycle if the procedure does not accurately estimate the findings. They should use models to solve problems and carry out the necessary steps of the modelling process (Rellensmann, Schukajlow & Leopold 2020). However, preservice mathematics teachers may find modelling challenging because of a lack of experience or limited professional knowledge (Breiner et al. 2012; Corlu & Capraro 2014).
Asante and Mereku (2012) assert that preservice mathematics teachers do not have sufficient time to practice teaching at the foundational level of Ghanaian education. Moreover, pedagogical approaches are not given adequate consideration. This lack of practical teaching experience and emphasis on pedagogy extends to mathematical modelling, where teachers may struggle to apply theoretical concepts in realworld contexts. Without adequate practice in modelling, these teachers may find it challenging to develop the reflective thinking skills necessary to integrate modelling into their teaching, limiting their ability to prepare students for complex problemsolving tasks.
In the 2011 Trends in International Mathematics and Science Study (TIMSS) assessment, Ghanaian students consistently scored below international benchmarks. Therefore, another assessment, the Early Grade Mathematics Assessment (EGMA), was conducted in 2015 to diagnose the problems of earlygrade students concerning basic mathematical skills and competencies (Armah & Mereku 2018). The findings revealed that only 25% of students could accurately answer questions in the conceptual knowledge subtasks, compared to 46%–72% in the procedural knowledge subtasks. Furthermore, the Basic Education Certificate Examination (BECE) results showed that students’ performance was low (Akyeampong 2017; Asante & Mereku 2012).
Therefore, preservice teachers were recommended to take practical courses in mathematics pedagogy that offer ample opportunities to practice teaching at the basic level of Ghanaian education. Mathematical modelling is considered a challenging procedure, particularly when understanding, simplifying, synthesising and realising the problem within given constraints (Govender 2020; Govender & Machingura 2023). The study sought to investigate how preservice mathematics teachers could use their reflective thinking skills to solve problems related to mathematical modelling. In addition, the establishment of a modelling laboratory at education colleges was proposed, along with a restructuring of the mathematics curriculum to include modelling as one of the core competencies.
Research questions and hypothesis
This study explored the development and application of reflective thinking skills in preservice mathematics teachers, particularly within mathematical modelling. The following questions guided the research:
How do preservice mathematics teachers use their reflective thinking skills to solve reallife problems in a mathematical modelling context?
What is a modelling laboratory’s relevance in developing preservice mathematics teachers’ reflective thinking skills when solving modelling problems?
The following hypothesis ties together the research questions by testing whether a modelling laboratory significantly enhances reflective thinking skills in preservice mathematics teachers. It checks explicitly if there is a measurable difference between those with and without access to the laboratory:
H_{0}: The reflective thinking skills of preservice mathematics teachers in the experimental and comparison groups employing the modelling process technique do not differ significantly.
Research methods and design
A pretest and posttest quasiexperimental design was used in this study. According to Johnson and Christensen (2012), quasiexperimental research provides the most robust evidence for causeandeffect correlations from manipulating and controlling irrelevant variables. The quasiexperimental research methodology was employed in this study, involving two Colleges of Education: one providing the experimental group and the other the comparison group to evaluate the intervention’s impact on B.Ed. students.
Mathematics preservice teachers were sampled based on their willingness instead of random selection (Johnson & Christensen 2012). However, a quasiexperimental design may offer weak evidence of a causal relationship between variables because there is no random assignment to groups or manipulation of the independent variable (May 2017). Nevertheless, this design is essential in educational research because many research questions in education do not lend themselves to experiments (Creswell & Creswell 2018).
Using a purposive sample, 35 preservice mathematics teachers were selected from a Ghanaian College of Education for the comparison group and 38 preservice mathematics teachers were selected from another Ghanaian College of Education for the experimental group. Using semistructured interviews and tests as datacollecting instruments, the participants were divided into seven groups of five using the random number table. Mhakure and Jakobsen’s (2021) theoretical and pedagogical modelling frameworks were modified to address the research questions and the hypothesis (see Figure 1 and Table 1). The most important feature of the theoretical framework was the skill of reflective thinking, which is applied in all the stages of the modelling cycle. Moreover, applying innovative, flexible approaches to develop the mathematical model, followed by reflection and interpretation of the results, validated and refined the model, ensuring it reflected the realworld scenario (see Figure 1).

FIGURE 1: Modified pedagogical modelling cycle. 

TABLE 1: Pedagogical mathematical activity framework. 
A pedagogical modelling activity framework also guided the process that preservice mathematics teachers used to obtain accurate estimates of their learning outcomes. This was achieved by applying the pedagogical mathematical activity (PMAD) framework outlined in Table 1, categorising the activities involved in the modelling process.
These activities were successful based on an intervention modelling activity lesson plan detailing the critical aspects regarding mathematical activity and modelling activity (see Table 2). The intervention took 8 weeks of facetoface instruction for B.Ed. Mathematics Education preservice mathematics teachers for the 2021–2022 academic year at the Colleges of Education in Ghana. During the intervention, the comparison group solved the tasks using the conventional approach through reflective thinking, recalling concepts and theories to apply to the realworld scenario. However, the experimental group solved the task using theoretical and pedagogical activity frameworks. Furthermore, the experimental group worked in a wellplanned and conducive modelling laboratory. After the problemsolving, the groups did PowerPoint presentations of their findings on a whiteboard.
TABLE 2: Sample intervention modelling lesson activity. 
Table 2 presents a sample of what and how preservice mathematics teachers conducted the intervention through modelling.
Table 1 and Table 2 show the conceptual framework of the experimental group work, outlining reflective thinking and mathematical modelling. The experimental group members applied the concept of reflective thinking to review previous concepts, subject matter and theories to facilitate their comprehension of the task. Thereafter, they demonstrated knowledge by representing the reallife problem as a mathematical model through mathematisation and simplification. Their next task was to apply flexible methods or suitable theories to solving the mathematical problem embodied in the model. Finally, they applied reflection by looking back to learn, unlearn and relearn through interpreting, verifying, validating and communicating how they turned the reallife problem into a mathematical model, indicating the methods and theories that led to the result.
Ethical considerations
The Humanities and Social Sciences Research Ethics Committee for Ethical Research (HSSREC) at the University of the Western Cape was consulted, and an ethics clearance certificate was issued with the HSSREC reference number HS22/6/54 before the researcher began collecting data. Moreover, permission approval letters were obtained from the Director General of the Ghana Tertiary Education Commission, principals from the sampled Colleges of Education, the Heads of the Department of Mathematics/ICT and the tutor trained for the intervention. Students signed consent forms indicating their willingness to be part of the study. They were assured of confidentiality and informed that it was for research purposes only.
Results
Quantitative and content analyses were conducted on the pretest scores of the comparison and the experimental groups, and the quantitative results and qualitative findings are presented and discussed in detail.
Table 3 presents the quantitative results of Levene’s test for equality of variance for the pretest scores and the ttest for equality of means between the experimental and comparison groups. These statistical tests assessed whether there were significant differences in variance and mean scores between the two groups before the intervention.
TABLE 3: Levene’s test of equality of variance of pretest scores. 
Table 3 indicates that with t (0.246), df (12) and a pvalue of 0.81, greater than the alpha level of 0.05, there was no statistically significant difference in the pretest scores of the comparison and experimental groups. Furthermore, Levene’s test for equality of variance, with F (0.013) and a pvalue of 0.91, confirmed no significant differences in variance of the pretest results for the two groups.
Pretest answers were subjected to content analysis to determine the preservice mathematics teachers’ competency levels in applying reflective thinking skills to mathematical tasks. The comparison group was classified as groups A, B, and so on, while the experimental group was classified as groups 1, 2, and so on. This approach was based on the understanding that, through extensive reflection, the preservice teachers would start with simple realworld problems, applying foundational concepts. As their knowledge deepened, they would move on to complex tasks, applying increasingly advanced mathematical concepts, including those used in trigonometry (Tan & Ang 2012).
The first research question was:
How do preservice mathematics teachers use their reflective thinking skills to solve reallife problems in a mathematical modelling context?
To address this question, the study focussed on observing their problemsolving approaches and subsequent reflections. The research explored whether reflective thinking enabled these preservice teachers to connect theoretical knowledge with practical application, adapt their strategies and improve their modelling processes over time. The findings, detailed in the subsequent sections, reveal how reflective thinking facilitated their ability to break down complex problems, apply mathematical concepts effectively, and refine their approaches to achieve accurate and relevant solutions in realworld scenarios.
The findings revealed that every group refrained from attempting Task 3, except for those in Comparison Group G, who had an idea and could unearth pertinent data to solve every task based on assumptions from trigonometry concepts. However, while employing relevant mathematical approaches to tackle the challenges, this group did not effectively utilise reflective thinking skills. A sample of Group G’s solutions in Task 3 is illustrated in Figure 2. The scoring was carried out using the following criteria: B for Basic Concept or Best Accuracy, M for Method, A for Answer and NJ for Not Judicious (indicating that the answer was correct, but the basic concept or method applied was incorrect), as shown in the sample solutions in Figure 2.

FIGURE 2: Comparison Group G’s solution for Task 3. 

Task 3’s instructions were as follows (see Appendix 1):
3. A spotlight for a theatre production illuminates a triangular area on stage. Actors are to stand at the corners of the illuminated area at P, Q and R. The actors at P and R have to stand 5m and 4m away from the actors at Q, respectively. The angle of elevation of S from P is 45°, and the angle of S from R is 60°. If the spotlight is placed at a point vertically above PR.
Draw a diagram to illustrate the given information.
Determine how the actors at P and R must stand from each other.
The actor at P enters on stage by sliding down a wire from S to P. How long is the wire that the actor slides along?
Figure 2 presents Group C’s solution to Task 3. The diagram demonstrates Group C’s calculations for the actors’ placement based on the given distances and angles of elevation. Additionally, it shows their method for determining the distance between the actors at P and R, as well as the length of the wire along which the actor at P slides down from point S. The diagram effectively captures the geometric relationships and trigonometric calculations used by Group C to solve the task, providing a visual representation of the problem and their approach to the solution.
Group G of preservice mathematics teachers in the comparison group was able to conceptualise the real scenario simply and construct the trigonometric model, as shown in Figure 2. After calculating the angle at Q using the idea of cyclic quadrilateral production as a result of the spotlight, they employed trigonometry and discovered that the angle was 105°. They built a trigonometric diagram to mathematically represent the reallife situation using notation and measurements like 4 m and 5 m to illuminate a shadow from the theatre production. With this model’s aid, students could access and deconstruct the necessary mathematical knowledge to generate a solution.
They could determine the angle at Q by adding the opposite angles in a cyclic quadrilateral using the keyword ‘spotlight’ that was generated to the cycle. With the help of this result, they could use the Cosine rule to determine that the PR length was 7.17 m. This finding demonstrated that preservice mathematics teachers (1 out of 7 groups) could evaluate, check and reflect on the discovered answer to the extent that they could examine their trigonometric diagram and had the correct outcome from the spotlight on the theatre production.
Task 1 required the students to determine the value of the unknown variable in a triangle correct to one decimal place. However, because of their inability to apply the assumptions of the Sine and Cosine rules, Comparison Group C divided the given triangle in half to produce a rightangled triangle (see Figure 3 and Figure 4).

FIGURE 4: Comparison Group C’s solution to Task 1. 

Instead of using the Sine and Cosine rule, the members of Group C applied the Pythagoras theorem, which produced the correct answer and is also considered logically and mathematically accepted because the vertical line meets the base at 90°, creating a rightangled triangle. However, the group’s approach might have prevented them from completing the other tasks because Tasks 2 and 3 were beyond their comprehension because they could not apply the Sine and Cosine rules.
Because of a lack of concept application, two comparison groups could not do Tasks 2 and 3. Although they applied previously taught trigonometric functions, they failed to reflect on and integrate their prior knowledge and experience with these concepts. These groups correctly drew the diagram for Task 3 but did not find the solution because they could not locate the correct number of angles. However, one particular group correctly drew the diagram for Task 3 using a novel approach but could not find the solution. Most groups approached the different tasks in conventional ways and could not apply the mathematical modelling processes.
The experimental groups could do Task 1 by applying assumptions and mathematical skills. However, experimental groups 2, 4 and 6 could not do tasks 2 and 3, while experimental groups 3 and 5 did not even attempt them. Experimental Group 7 formulated prepositions, facilitating their completion of Task 1. However, as explained earlier in the intervention stage, they did not think critically or creatively when making the diagram for Task 3. They used the wrong concept in the wrong diagram, which led to a correct answer, which was not judicious. Even though this approach helped them answer other tasks correctly, they were only given marks for the method they used, not for the accuracy of the answer, as shown in Figure 5.

FIGURE 5: Experimental Group 7’s solution for Task 3. 

Experimental Group 7 created the Task 3 diagram using a novel drawing technique but could not show clearly how the spotlight casts a shadow to form the quadrilateral. A mark for the method was given, but they were denied the answer mark. Although the answer was correct, it was not judicious because concepts were missing from the diagram. They understood the new concepts, correctly applied the Cosine rule, and they justified their answer. However, they eventually had to acknowledge that the missing basic concept (B) and method (M) marks indicated that they had not solved the problem. In addition, a not judicious (NJ) mark was subtracted from the answer (A) mark, suggesting that while the answer was technically correct, the approach was inappropriate.
An analysis of the covariance (ANCOVA) test was performed to examine the differences in posttest scores between the groups while controlling for any initial differences in the pretest scores. This test would quantitatively address the hypothesis that the reflective thinking skills of preservice mathematics teachers in the experimental and comparison groups employing the modelling process technique do not differ significantly.
Table 4 presents the ANCOVA results for both the experimental and comparison groups. The table provides a detailed view of the observed differences’ statistical significance and effect size.
TABLE 4: The analysis of the covariance test results for experimental and comparison groups. 
Table 4 shows a statistically significant difference between the experimental and the comparison groups in how they invoked their reflective thinking abilities when solving problems using the modelling process because the pvalue of 0.031 was lower than the alpha value of 0.05. While both groups could think reflectively when responding to modelling tasks, the partial eta squared of 0.33 indicated a significant effect size on the modelling process on the reflective thinking ability of preservice teachers in the experimental group.
This result implied that the null hypothesis was not accepted and that there was no difference between the experimental and comparison groups in their reflective thinking when solving problems through mathematical modelling. However, juxtaposing the posttest results of the groups indicated that the experimental group used reflective thinking, producing accurate mathematical models in Tasks 1 and 2.
Figure 6 presents a sidebyside boxplot illustrating the impact of the 8week intervention on the posttest scores of both the comparison and experimental groups. This visual representation allows for a clear comparison of the score distributions between the two groups, highlighting the extent of the intervention’s effect.

FIGURE 6: A sidebyside boxplot of the groups’ test scores. 

Group 6 completed all tasks successfully by utilising relevant modelling skills. However, three other groups chose not to undertake Task 3 because of a lack of modelling proof. In other words, they did not have sufficient understanding, reasoning or evidence to attempt to solve the task. They were unsure how to justify their methods or solutions using the appropriate mathematical concepts and reasoning.
Both groups applied modelling techniques, but the experimental group demonstrated a deeper understanding and more accurate application of the concepts, leading to better results. The experimental group saw the distinction between solving word problems and modelling. They understood that solving word problems involves applying a straightforward, often pretaught formula to arrive at a solution. In contrast, modelling requires creating a mathematical representation of a realworld scenario, interpreting the situation, making assumptions and refining the model to find a solution. Modelling requires critical thinking, creativity and connecting mathematical concepts with realworld situations. It involves creatively combining and adapting multiple mathematical tools to develop a new, composite solution that accurately represents a reallife problem.
The experimental group extensively utilised reflective thinking, progressing from simple to complex approaches enhancing their proficiency in modelling. They successfully applied flexible methods alongside the modelling cycle and recognised that reflective thinking is integral to every aspect of the process. This realisation underscored the importance of establishing modelling laboratories in Colleges of Education in Ghana, addressing the second research question:
What is a modelling laboratory’s relevance in developing preservice mathematics teachers’ reflective thinking skills when solving modelling problems?
A modelling laboratory would enhance reflective thinking skills by providing a controlled environment where preservice teachers can engage in handson, practical activities that promote reflection, critical thinking and applying theoretical knowledge to realworld problems. This structured setting would help teachers refine their problemsolving approaches, leading to an understanding and mastery of mathematical modelling, which is essential for effective teaching in the classroom.
The first experimental group comprehended and analysed the essential details of all tasks. They organised their thoughts and transformed the realworld problem into mathematical models by creating appropriate diagrams. Although some concepts in Subtask 1.3 led to erroneous substitutions and incorrect answers, the students accurately derived the necessary relationships by methodically applying relevant ideas and measurements in their calculations. Additionally, the group utilised a trigonometric model to enhance their understanding of the modelling process. This approach resulted in accurate solutions for both Tasks 1 and 2. Group 1 particularly valued the modelling process, which helped them isolate essential data, apply facts and formulas, and understand the underlying concepts. For instance, in Task 3, they successfully used the Pythagorean theorem by identifying rightangled triangles within the figure.
Experimental Group 1 derived a model to solve the problem of the distance between two cliffs using trigonometric relationships. This is indicated here as:
The students first established the equation DE = ⇒ EF = DESinα. They deduced that BE = BF + EF but BF = y and EF = DESinα as Equation 1. They then revealed that tanθ = but BC = x ⇒ tanθ = ⇒ BE = x tanθ as Equation 2 equating their two modelling equations, which resulted in the following model: y + DESinα = x tan θ ⇒ DESinα = x tanθ – y. Thus, they formulated the required model relation as DE = . Substituting the correct parameters or measures, the distance between the cliffs was found to be 750 m.
Members of Group 2 approached Task 1 by leveraging reflective thinking to bridge the gap between a realworld scenario and a mathematical model. They successfully calculated the necessary measures and accurately represented them through mathematisation, creating an appropriate model for Task 1. Their creativity enabled them to identify rightangled triangles within the diagram, which they used to solve subtask 1.3 and obtain the correct answer. By reflecting on theorems and trigonometric models, as illustrated in Figure 5 (see Appendix 2), they effectively interpreted, verified and validated their results.
Figure 7 illustrates the stepbystep process followed by Group 2 in solving Task 1. The diagram on the left (1.1) shows the application of angle properties within triangles to determine an unknown angle in the triangle. In step 1.2, the Sine rule was employed to calculate the length of side AD in the same triangle. Diagram 1.3 demonstrates how Group 2 used trigonometric principles, precisely the Sine function, to find the size of the side CD in the rightangled triangle. The remarks of the variables in red indicate critical points in their solution, such as the correct application of theorems and the steps leading to the accurate calculation of the required measures.

FIGURE 7: Experimental Group 2 sample posttest solution. 

For Task 2, Group 2 created an appropriate diagram using the same conceptualisation techniques and reflective thinking. The group was able to apply trigonometric ratios in determining the length of AD regarding y and α as seen as = . Similarly, for the length of BD regarding y and Ɵ, they obtained BD = . For Task 3, group members successfully answered Subtask 3.1 and deduced the relationship by applying concepts to the diagram. However, they gave the incorrect response to Subtask 3.2 owing to substitution and computational errors, leading to the incorrect final answer.
Group 3 approached Task 1 by reflecting on the details learned during the initial intervention stage. They understood the task requirements and organised their thoughts to create an accurate diagram. They reached the correct conclusion for Task 1 by correctly applying trigonometric theorems. Furthermore, in Task 2, they applied their modelling skills, presenting a clear diagram and using it to calculate the lengths of AD and BD regarding y, α and Ɵ. However, when attempting Task 3, the group struggled to apply the correct solution method, resulting in an incorrect equation, as illustrated in the following examples (see Equation 3 and 4):
They considered another triangle from the figure and came out with the result of tan45 = ⇒ 250 tan 45 = y – 250 ⇒ y = 500m. The solution was presented inaccurately, and the trigonometric modelling principles and theorems were improperly exploited.
Despite using the correct theorem of the Sine rule, Group 4 had difficulty drawing the model. This affected their solution and made the final answer to Task 1 inaccurate. Group 4 attempted to draw the diagram for Task 2. However, they failed to identify the measurements in the diagram, which resulted in incorrect answers. Group 4 did not even attempt Task 3.
Group 5 thoroughly understood Task 1 and successfully transitioned the task’s context from a realworld scenario to a mathematical model through mathematisation. They accurately estimated the necessary measurements in the model to find the correct solution. The group members reflected on relevant mathematical concepts, facts, formulas, and theorems critical to solving the task. Specifically, the preservice mathematics teachers applied the concept that the sum of interior angles in a triangle is 180° to determine the 46° angle. They then used the Sine rule to arrive at the correct answers for subtasks 1.2 and 1.3, as illustrated in Figure 8.

FIGURE 8: Experimental Group 5 sample posttest solution. 

Comparable concepts and formulas learned earlier during the intervention were applied by Group 5 to construct the Task 2 model, which was then used to answer the subtask and obtain the correct answers. None of the group members could complete Task 3.
Group 6 utilised reflective thinking skills to analyse the tasks and deconstruct the data to create a model. After reviewing the task and recalling relevant facts, they developed a mathematical model based on an actual scenario. The group reconsidered mathematical theories, facts, formulas and concepts and applied the Sine rule to arrive at correct solutions. They completed subtasks 3.1 and 3.2 of Task 3 using the diagram. Additionally, they simplified their presentation of findings and solutions, making them accessible and relatable to concepts that Junior High School students might use.
Group 7 approached the task by reflecting on the intervention process and previously taught concepts, allowing them to understand the task thoroughly. They adopted a different approach to drawing the model, while partially accurate, was informed by their reading and comprehension of the task. Through extensive reflection, they calculated the length of AD as 61.08 m, although this approach was not entirely judicious because of the placement of two angles at angle CAD, as shown in Figure 9.

FIGURE 9: Experimental Group 7 sample posttest solution. 

Their reflections involved retrieving information from memory, reviewing the task from simple to complex, investigating ideas, and applying concept development using the modelling process, including applying the Sine rule. Group 7 clarified and comprehended Task 2, producing a sophisticated model compared with that of Task 1. In addition, Group 7 demonstrated a significant level of reflection throughout the process.
Group 7 drew the Task 2 diagram with two different angles, 78° and 65°, at point A, which was geometrically impossible. As a result, they were awarded marks for their method but received no credit for the final answers, even though the answers were correct, because of the flawed approach. This group did not attempt Task 3, as they felt they lacked sufficient time.
In contrast, Group 5 effectively utilised their reflective thinking skills by analysing Task 3 and applying various theories. They accurately interpreted the given diagram and arrived at the correct answers, as illustrated in Figure 10.

FIGURE 10: Experimental Group 5 sample posttest result. 

Group C demonstrated creative thinking by generating ideas applied to design the diagrams for Tasks 1 and 2, resulting in correct answers through a focus on conventional concepts. For each subtask in Task 3, the group completed the tasks by analysing the diagram given. However, instead of deriving the required relationship, they merely substituted the provided measurements into the equation and solved it. This approach was not ideal for addressing the subtasks. Moreover, they had difficulty recalling basic concepts, and simplification strategies hindered their ability to model the given relationship accurately and arrive at correct answers.
Figure 11 presents Group C’s posttest solutions for various tasks. The diagram showcases their approach to solving Tasks 1.1, 1.2, 1.3, 2.1, 2.2, 2.3 and 3.1. The annotations and calculations illustrate the group’s application of trigonometric concepts and their transition from theoretical understanding to practical problemsolving. Group C demonstrated their knowledge of concepts, such as the Sine rule, angle sum properties, and the relationships between angles and sides in rightangled triangles. The figure indicates the group’s ability to connect abstract mathematical principles to realworld scenarios.

FIGURE 11: Comparison Group C sample posttest solution. 

Figure 11 highlights the group’s systematic steps in deriving lengths and angles, emphasising the accuracy of their mathematical reasoning. However, the figure also shows areas where the group’s understanding or application of trigonometric concepts may have been inadequate.
These shortcomings may have stemmed from a misinterpretation of the task requirements, incorrect assumptions or minor errors in calculation that could have impacted the final results. These inadequacies highlight the importance of precision and the need to carefully verify each step in mathematical problemsolving.
Comparison Group G struggled to apply reflective thinking in recalling the necessary concepts and theories to analyse the given diagram and answer the subtask. Instead of systematically approaching the problem by leveraging their prior knowledge, they attempted to generate triangles from the overall diagram. However, this approach was insufficient, leading to an incorrect solution for the subtask, as illustrated in Figure 12.

FIGURE 12: Comparison Group G sample posttest solution. 

Their inability to apply relevant trigonometric concepts, such as identifying and using angles and sides in the diagram, suggested a gap in their understanding or a lack of confidence in their problemsolving ability. This shortfall resulted in a fragmented approach, with the group attempting to break down the problem without a clear strategy and failing to solve the problem.
Figure 12 highlights the challenges and emphasises the importance of reflective thinking and a foundational understanding of mathematical concepts in tackling complex problems. The figure underscores the need for students to connect theory with practice and to apply their knowledge to arrive at correct solutions methodically.
Figure 12 shows that Group G demonstrated its ability to produce correct diagrams derived from the central diagram and accurately applied trigonometric ratios. Specifically, they correctly identified and calculated tanα using the given angles and sides in the triangles. This indicated their understanding of basic trigonometric principles and their application to the problem. However, despite this correct initial step, the group encountered difficulties proceeding with the proper method to solve and prove the given relation. After calculating the tangent values, Group G struggled to connect these results with the required final solution. They could not utilise the trigonometric properties to advance their solution, leading to an incorrect final answer.
Interview findings: Comparison group
During interviews, the comparison group revealed insights into the effectiveness of the conventional approach following the intervention. These preservice mathematics teachers found applying the appropriate theorem and trigonometric ratios challenging. They acknowledged their prior knowledge but failed to reflect deeply enough to retain it for realworld application. Their justifications for their techniques were often ineffective, leading them to repeatedly acquire, unlearn and relearn the rules and theorems needed to complete the tasks.
While most preservice mathematics teachers could identify the concepts behind the Sine and Cosine rules, they also recognised the challenge of drawing accurate mathematical models or diagrams. Many students struggled to complete tasks because of insufficient understanding of the theory or its application.
The following sample from the interview between the modelling instructor and a preservice mathematics teacher highlights several challenges:
‘What kind of mathematical modelling experience do you have?’ (Modelling teacher educator)
‘I never realised that I was studying mathematical modelling, even though I did understand some of the basic concepts at one point.’ (Preservice mathematics teacher)
‘Do your tutors discuss or employ mathematical modelling in their teaching?’ (Modelling teacher educator)
‘No, the term ‘mathematical modelling’ has never been mentioned.’ (Preservice mathematics teacher)
At times, the modellling teacher educator had to explain the concept of mathematical modelling to respondents before they could answer the interview questions. For example, when the modelling instructor asked, ‘Do you believe that mathematical modelling will have a significant impact on Basic School pupils?’, one of the preservice teachers responded:
‘Yes, I believe it will have a significant impact due to the increased freedom for students to research and expand their knowledge. They won’t feel that there’s only one correct way to solve a problem or that maths is inherently difficult. Students might prefer reallife problems over traditional mathematical problems, as many tutors and preservice teachers use modelling without realising it’s mathematical modelling. With increased awareness, they would be better equipped to help basic school students understand mathematical concepts by teaching through mathematical modelling rather than relying on teachercentred approaches.’ (Preservice mathematics teacher)
The interview responses from the comparison group revealed preservice mathematics teachers’ challenges in applying the conventional approach after the intervention. Despite acknowledging their prior knowledge, these teachers struggled to apply the appropriate theorems and trigonometric ratios. A lack of deep reflection hindered their ability to retain and use this knowledge in realworld contexts, leading to repeated cycles of acquiring, unlearning and relearning the necessary rules and theorems.
While many could identify the concepts behind the Sine and Cosine rules, they found it difficult to accurately draw mathematical models or diagrams, impacting their ability to complete tasks. This reflected a gap in their understanding of both theory and its practical application.
One preservice teacher admitted to not realising they were studying mathematical modelling, indicating a disconnect between their learning experiences and recognising these experiences as part of a modelling framework. In addition, the term ‘mathematical modelling’ was reportedly never mentioned in their coursework, suggesting a lack of emphasis on this aspect of education.
The modelling teacher educator often had to explain mathematical modelling during the interview, showing a broader awareness issue. However, there was recognition of the impact of mathematical modelling on students, particularly in promoting independent thinking and a deeper understanding of concepts.
The responses suggested that the conventional approach may fail to foster deep reflection, practical application of theoretical knowledge and integration of mathematical modelling in teaching. This finding highlights the need for focussed training to prepare preservice teachers.
Interview findings: Experimental group
Preservice mathematics teachers could generally solve the tasks with relative ease and recognise the importance of the intervention. One group drew the model correctly but struggled to apply the appropriate theorems to arrive at the correct answers. However, most preservice teachers successfully interpreted and validated their results to ensure accuracy.
Although some group members mentioned the challenges they faced in drawing the diagrams, most effectively utilised the modelling framework to produce accurate results. Below is a sample of an interview:
‘What kind of mathematical modelling experience do you have?’ (Modelling teacher educator)
‘Yes, during our internal quiz, we were asked to estimate the number of hours and the amount of money that needed to be paid in Ghanaian cedis. For example, if someone was paid for working six days, how much would they be paid? Similarly, if someone worked seven days, how much would they be paid? When calculating the total, which option would be best?’ (Preservice mathematics teacher)
‘Do your tutors discuss or employ mathematical modelling in their teaching?’ (Modelling teacher educator)
‘Yes, in the Algebraic Thinking course, linear equations were modelled using algebraic tiles.’ (Preservice mathematics teacher)
‘Have you dealt with the New Common Core Mathematics Curriculum (NCCMC) statement’s aspects of mathematical models? If so, could you please explain how?’ (Modelling teacher educator)
‘Yes, I taught Junior High School mathematics during my macroteaching [offcampus] experience. Although many tutors focus on teaching for learning, which is not the best approach, I was not impressed with how the students were thinking. This modelling approach would help students think more deeply. Allowing students to conduct independent research, explore, and take ownership of their education would significantly improve their comprehension of mathematical concepts.’ (Preservice mathematics teacher)
The interview responses provided insights into the experiences of preservice mathematics teachers with mathematical modelling in their education and teaching. The preservice teacher recounted a quiz where they estimated payments for different working days, demonstrating some practical experience with mathematical modelling. However, it was more focussed on basic arithmetic or algebra rather than a deeper engagement with modelling. They also mentioned that mathematical modelling had been included in their coursework, particularly in an algebraic thinking course where linear equations were modelled using algebraic tiles. However, this exposure seemed limited to specific courses rather than a widespread approach to their education.
Reflecting on their experience with the New Common Core Mathematics Curriculum (NCCMC) during macroteaching, the interviewee expressed concern that many tutors emphasised ‘teaching for learning’, which the teacher felt was ineffective. The interviewee believed mathematical modelling could foster more profound, more critical thinking among students. The preservice teacher advocated for a studentcentred approach, where learners could explore and take ownership of their education through modelling.
The teacher also expressed dissatisfaction with students’ thinking during their teaching experience, suggesting that conventional methods were not promoting deep understanding. The interviewee believed integrating mathematical modelling into teaching could help students approach problems from multiple angles and think critically.
The responses suggested that while preservice teachers experienced little exposure to mathematical modelling, they saw its potential to enhance student learning and critical thinking. The interviewee advocated for the integrated use of modelling in teaching, moving beyond surfacelevel understanding, engaging students deeply and acknowledging that their experiences with modelling may have been limited and that there was room for further development in this area.
Discussion
The pretest findings revealed that the experimental group (Subgroups 1, 2, 3 and 5) struggled with Task 3, successfully completing only Tasks 1 and 2. However, Groups 4 and 7 could not translate the realworld model into a mathematical model, drawing an appropriate diagram and aligning with the theories proposed by Csíkos, Szitanyi and Kelemen (2012). After thoroughly examining, verifying and validating their results, they used the diagram from Task 3 to obtain correct results, even though these differed from the ideal model.
In the comparison group, Group C divided the task into rightangled triangles and solved it to obtain the correct answer. However, Groups A, B, C, D and F could not complete Tasks 2 and 3. Group G, on the other hand, applied their knowledge creatively but struggled with drawing or finalising diagrams, affecting their performance in the pretest tasks. As Rellensmann et al. (2017) suggest, failing to sketch or finalise diagrams hinders understanding mathematical modelling.
The pretest results indicated that the comparison group did not use reflective thinking to uncover concepts and complete tasks. Consequently, many group members found applying their skills to the modelling tasks challenging and failed to attempt most mathematical problems. Rellensmann et al. (2020) recommend that students repeat procedures by sketching diagrams if they fail to reach a solution. The limited professional knowledge or lack of understanding among preservice teachers became a significant barrier, consistent with findings by Breiner et al. (2012) and Corlu and Capraro (2014).
The findings highlighted that preservice mathematics teachers had limited knowledge of mathematical modelling, aligning with BorromeoFerri’s (2018) finding. Many solved the questions without utilising reflective thinking, which aligns with Galbraith’s (2012) finding that preservice teachers focussed on converting realworld problems into mathematical terms without fully engaging in deeper reflective thinking about the broader context of the problem. They were essentially applying mathematical techniques to realworld scenarios without critically reflecting on the underlying concepts or the significance of the problemsolving process.
Posttest results showed that the experimental group performed better in modelling problems than the comparison group, which is consistent with the findings of Yasa and Karatas (2018), who indicated that their experimental group outperformed the control group in mathematical modelling. Furthermore, in this study, the preservice mathematics teachers in the experimental group demonstrated competencies such as knowledge about modelling problems, classroom management during modelling activities, and the ability to interpret and respond to students’ thinking, consistent with Blum’s (2011) and Schmidt’s (2011) findings. Groups 1, 2, 6 and 7 in the experimental group validated their solutions through reflective thinking, while Group 5 struggled with Task 3, and Group 4 failed to complete any tasks despite their efforts.
The primary finding was that the experimental group used reflective thinking in posttest modelling. To address the tasks, they employed flexible approaches, such as problembased learning and inductivedeductive reasoning. In contrast, the comparison group fell short in these areas., to solve mathematical problems and reach conclusions. These approaches aligned with Lu and Kaiser’s (2022) assertion that adaptable strategies lead to successful mathematical outcomes.
Reflective thinking, often seen as a psychological phenomenon, is realistic when viewed as a pedagogical concept (Clarà 2015). Reflection plays a central role in teacher education and professional development, as noticed by Agustan et al. (2017), Amidu (2012) and Lim (2011). Interpreting a problem, unpacking information, modelling a real scenario, mathematising it into a mathematical model and using flexible techniques to arrive at solutions requires reflective thinking.
The experimental group demonstrated more reflective thinking than the comparison group, aligning with the findings of Thahir et al. (2019) and Agustan et al. (2017), who emphasise that reflective thinking helps identify ideas, concepts, formulas and theorems needed to solve mathematical problems using the modelling approach. The analysis of covariance indicated a statistically significant difference between the two groups, with a large effect size of 0.33, showing a significant impact of the mathematical modelling process on the reflective thinking ability of preservice mathematics teachers. This result aligned with Salha and Qatanani’s (2021) findings.
One of the essential skills or competencies preservice mathematics teachers should develop is a contentoriented approach. According to GochevaIlieva et al. (2018), a contentoriented approach involves focussing on the deep understanding of mathematical content and concepts, ensuring that teachers are not just proficient in procedural knowledge (how to solve problems) but also in conceptual understanding (why the solutions work and how they apply to various contexts).
In this study context, the experimental group benefited from an intervention that utilised a theoretical framework designed to enhance their modelling competencies. This framework included strategies to help preservice teachers understand mathematical concepts and apply them in realworld scenarios. By linking these modelling competencies with a pedagogical mathematics activity framework, the intervention ensured that preservice teachers were not just learning theory in isolation but also developing the ability to teach concepts.
This holistic approach of combining content knowledge with pedagogical skills led to the experimental group performing better than the comparison group. The superior performance is visually represented in the sidebyside boxplot in Figure 6, showing the distribution of scores or outcomes for both groups and highlighting the positive impact of the intervention on the experimental group. The boxplot results showed that the experimental group, outperformed the comparison group, which may not have received the same level of comprehensive training, thanks to the integrated approach of the intervention.
Conclusion
Preservice mathematics teachers in Ghana should be trained to move beyond solving word problems and engage with reallife situations or global events through mathematical modelling. As Tan and Ang (2012) and Durandt (2021) suggested, preservice teachers must be helped to tackle simple to complex authentic problems and bridge the gap between everyday mathematical language and educational discourse. To achieve this, mathematics pedagogy must be grounded in the preservice teachers’ prior knowledge and enhanced through reflective thinking skills. This approach involves incorporating horizontal and vertical mathematisation into mathematics lessons, improving the relevance and depth of the subject for preservice mathematics teachers.
Horizontal mathematisation helps to translate realworld problems into mathematical language, making the subject applicable to everyday situations. Vertical mathematisation deepens understanding by connecting and abstracting mathematical concepts. By integrating both, preservice teachers would develop a comprehensive skill set, enabling them to teach mathematics as both a practical tool and an interconnected field of knowledge. This dual focus makes mathematics meaningful and equips future teachers to approach problems from multiple perspectives.
Given the positive impact observed from the intervention using the modelling approach, the researchers propose the establishment of a mathematical modelling laboratory equipped with modellingeliciting materials at the Colleges of Education in Ghana. Additionally, it is recommended that mathematical modelling should become a core competency within the mathematics curriculum at Ghana’s Colleges and Basic Education levels.
Recommendation for future research
Future research can investigate how college tutors apply mathematical modelling to preservice teachers and implement these strategies at the Basic Education level in Ghana. In addition, studies could include interviews and observations of tutors to examine how mathematical modelling is being integrated into the curriculum at Colleges of Education in Ghana.
Longitudinal studies could track preservice teachers’ progress as they transition into inservice teachers, assessing how their training in mathematical modelling influences their teaching practices and their students’ learning outcomes over time.
In terms of curriculum development, future research could focus on designing and assessing specific curricular materials or teaching strategies that integrate mathematical modelling, creating new modules or resources tailored to the needs of preservice teachers in Ghana.
Technology integration is another crucial area for future research. Studies could examine how digital tools and technology can support the teaching and learning of mathematical modelling, enhancing engagement and understanding among preservice teachers.
By pursuing these research avenues, a comprehensive understanding of how mathematical modelling can be effectively integrated into teacher education could be developed, ultimately improving mathematics education in Ghana.
Acknowledgements
Competing interests
The authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article.
Authors’ contributions
H.I.B. contributed to the literature review, methodology, data collection, analysis and writing of the original draft of the article. R.G. contributed to the conceptualisation of the study, resources, writing, reviewing and editing the final draft of the article.
Funding information
This research received no specific grant from any funding agency in the public, commercial or notforprofit sectors.
Data availability
The data that support the findings of this study are available from the corresponding author, H.I.B. upon reasonable request.
Disclaimer
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.
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Appendix 1: Data collection instrument (pretest)
Determine the value of the unknown variable in each of the following triangles, correct to 1 decimal place:
In the diagram below, D is a point vertically above C. DC is ymetre long. The angle of elevation of D from B is θ. Angle DAB = α and DÉA = β.
 2.1 Determine the length of DB regarding y and θ.
 2.2 Show that
A spotlight for a theatre production illuminates a triangular area on stage. Actors are to stand at the corners of the illuminated area at P, Q and R. The actors at P and R have to stand 5 m and 4 m away from the actors at Q, respectively. The angle of elevation of S from P is 45°, and the angle of elevation of S from R is 60° if the spotlight is placed at a point vertically above PR.
 3.1 Draw a diagram to illustrate the above information
 3.2 Determine how the actors at P and R must stand from each other
 3.3 The actor at P enters on stage by sliding down a wire from S to P. How long is the wire that the actor slides along?
Appendix 2: Data collection instrument (posttest)
A, B and C are three points in the same horizontal plane, and AB is 53 m long. CD is a vertical tower, and the angle of elevation of D from A is 65°. < DAB = 78° and < DBA = 56°
 1.1 Draw a diagram to illustrate the above information
 1.2 Determine the length of AD
 1.3 Determine the height of the tower CD
In a diagram, C is a point vertically above D. CD is y metres long. The angle of elevation of C from B is θ, and the angle of elevation of C from A is α. Angle ADB = β
 2.1 Draw a diagram to illustrate the given information
 2.2 Determine the length of AD regarding y and α
 2.3 Determine the length of BD regarding y and θ
A telephone cable is to be erected between 2 Cliff sides, AD and BE. An engineer stands at point C in the same horizontal plane as the foot of the cliffs. He measures the angle of E from C and D to θ and α, respectively. Cliff DA is y meters in height, and C is x metres from the foot of cliff BE.
 3.1 Show that the length of the telephone cable is
 3.2 If x = 1000m, y = 250m and θ = α = 45°. What is the distance between the Cliffs?
