Abstract
Background: Students’ prior mathematical knowledge plays a critical role in their success in university mathematics, yet it is often treated diagnostically rather than instructionally.
Aim: This study examines students’ prior mathematical knowledge at the start of a university calculus course, with the purpose of identifying conceptual and procedural strengths and weaknesses that are relevant for subsequent learning and for a mathematical modelling intervention.
Setting: The study was conducted in South Africa and involved three cohorts of first-year science and engineering students (1874 participants) across the years 2020, 2021 and 2022.
Methods: A mixed-methods sequential explanatory design was employed. Quantitative data were collected using a diagnostic test. Welch’s one-way analyses of variance (ANOVAs) and Games–Howell post hoc tests were used to compare cohort performance across content areas. Qualitative analyses of written solutions from a stratified random sample of 90 students were conducted to examine errors and misconceptions.
Results: Although overall test performance showed modest improvement across cohorts, students’ results remained low across most content areas. Algebra emerged as a relative strength, indicating procedural fluency, whilst weaknesses were evident in Analytical geometry, Functions and Modelling. Qualitative analyses revealed persistent conceptual and representational difficulties, in interpreting inequalities, graphing functions and constructing and using linear models.
Conclusion: The findings suggest that apparent improvements in preparedness may cover structural weaknesses. The study highlights the importance of using diagnostic evidence not merely to identify at-risk students, but to inform responsive teaching that addresses conceptual gaps.
Contribution: By integrating large-scale diagnostic data with qualitative error analysis, students’ prior knowledge can be leveraged as an instructional resource to support modelling-oriented teaching in university mathematics.
Keywords: constructivist perspective; mathematical errors; mathematics students; mathematical modelling; prior knowledge; teaching intervention; tertiary level; university teachers.
Introduction and research question
Teaching interventions in mathematics are typically designed with the intention of advancing student learning. Such interventions involve the interaction of several expert roles, including teachers, students and mathematical content. Since the emergence of cognitive perspectives on learning, increasing attention has been given to the expertise of the learner, particularly the role of students’ existing knowledge in shaping learning outcomes. Early work in mathematics education highlighted that students’ mathematical errors are not random but arise from coherent conceptual frameworks informed by previously acquired knowledge (Nesher 1987). From this perspective, errors and misconceptions provide valuable insight into students’ thinking and can be used productively to inform instruction.
More generally, prior knowledge is understood to encompass not only content knowledge and skills, but also beliefs and ways of reasoning that influence how new information is interpreted (Hattie & Yates 2013). A substantial body of research has demonstrated strong associations between students’ prior knowledge and subsequent learning across educational contexts (Du Plessis & Gerber 2012; Hailikari, Nevgi & Lindblom-Ylänne 2007; Kosiol, Rach & Ufer 2019; Thompson & Zamboanga 2003). In mathematics, this relationship is particularly pronounced due to the abstract nature of the subject and the cumulative way in which new concepts build on foundational ideas. Recent research further shows that strengthening topic-specific prior knowledge in mathematics can have a significant positive impact on learning outcomes (Alreshidi 2023). These findings are especially relevant when teachers plan targeted teaching interventions aimed at advancing students’ mathematical competencies.
Despite this, mathematics students, particularly those enrolled in science and engineering programmes, often experience substantial difficulties at the start of university mathematics courses, with many of these challenges emerging in the first year of tertiary study. Research consistently points to the transition from school mathematics to university mathematics as a critical and demanding phase (Di Martino, Gregorio & Iannone 2023; Van Appel & Durandt 2019). Students’ difficulties during this transition have been attributed to several factors, including gaps in prior knowledge, the abstract and formal nature of advanced mathematics, changes in learning environments and teaching approaches that may differ markedly from those encountered at school (Rach & Ufer 2020). In courses that involve more demanding mathematical activity, such as applied problem solving, these challenges may be further intensified, as students are required to draw flexibly on prior knowledge and problem-solving strategies.
In response to these challenges, university mathematics teachers have increasingly explored tools to identify students’ strengths and weaknesses early in the learning process. One common approach is the use of diagnostic tests at the start of a course to identify knowledge gaps, detect students at risk of failure and inform targeted support (Greefrath, Koepf & Neugebauer 2017). Other studies have focused on predictive models to identify at-risk first-year students at different points during the semester (Van Appel & Durandt 2019; Van Appel, Pretorius & Durandt 2024), or on analysing student profile data to better understand performance in calculus (Padayachee, Mudavanhu & Campbell 2021). In addition, meta-analytic work has examined the level of prior mathematical knowledge required to predict study success at the start of tertiary education (Rach & Ufer 2020).
Whilst these studies collectively underscore the central role of prior mathematical knowledge, they raise an important question: to what extent do such approaches help students address gaps in their understanding, rather than merely identifying or classifying those at risk of failure? At the primary and secondary school levels, learner errors and misconceptions are routinely analysed to diagnose underlying conceptual difficulties and to inform instruction (Makonye & Fakude 2016; Makonye & Luneta 2014; Pournara et al. 2016). However, similar practices are less common and less well documented at the tertiary level, often due to large class sizes and time constraints. Nevertheless, research in higher mathematics education emphasises the importance of analysing students’ mathematical thinking and conceptual understanding as a basis for instructional design (Centre for Higher Mathematics Education [Kompetenzzentrum Hochschuldidaktik Mathematik (KHDM)], University of Paderborn n.d.).
In science and engineering mathematics courses, students are frequently required to engage with mathematically demanding application problems, commonly referred to as mathematical modelling tasks. Such tasks aim both to support understanding of real-world situations (modelling as content) and to promote mathematical learning (modelling as a vehicle). Mathematical modelling involves interpreting a real-world problem, translating it into mathematics, working within a mathematical model and interpreting and validating the results in context. Modelling competency thus refers to the ability to construct, use, analyse and evaluate mathematical models in a meaningful way (Niss & Blum 2020). Empirical research has shown that each step of the modelling process poses significant cognitive challenges for students and can act as a barrier to successful problem solving, particularly when students work independently (Durandt 2018; Durandt & Jacobs 2017; Durandt & Lautenbach 2020; Kaiser 2017; Stillman 2019). From a constructivist perspective, successful engagement with such tasks depends heavily on students’ prior conceptual knowledge, which supports the development and application of procedures in new contexts.
Despite extensive research on diagnostic testing and predictive modelling, relatively little is known about how specific conceptual and procedural weaknesses in tertiary students’ prior mathematical knowledge can be systematically leveraged to inform modelling-oriented teaching interventions. This study addresses this gap by focusing on students’ prior mathematical knowledge in precalculus content, which forms a necessary foundation for learning calculus and engaging with applied problem solving. The aim of the study is to identify strengths, weaknesses, errors and misconceptions in first-year students’ prior mathematical knowledge at the start of a university calculus course, in preparation for a modelling-oriented teaching intervention.
The research question guiding the study is:
- Which conceptual and procedural components of students’ prior mathematical knowledge emerge as strengths and weaknesses across cohorts, and how do these differences inform instructional opportunities for first-year calculus and modelling?
The remainder of the article presents the conceptual framework underpinning the study (‘Conceptual framework’ section), followed by a description of the modelling intervention and the design and implementation of the diagnostic assessment (‘Research methods and design’ section). The ‘Results’ section reports the quantitative results and qualitative analyses of students’ mathematical work, and the ‘Discussion’ and ‘Conclusion’ sections discuss the findings and their implications for teaching and learning in tertiary mathematics.
Conceptual framework
This studay is grounded in a constructivist view of learning, which posits that students actively construct mathematical knowledge by building on their existing understandings. Within this perspective, students’ prior mathematical knowledge – comprising both conceptual and procedural components – plays a central role in shaping their engagement with university mathematics. Errors and misconceptions are therefore not interpreted as signs of failure, but rather as meaningful indicators of students’ underlying knowledge structures. Mathematical modelling is conceptualised as a cognitively demanding domain that makes these structures particularly visible, especially in situations where conceptual understanding and representational fluency are weak. When diagnostic assessment is used in conjunction with systematic error analysis, it offers an instructional lens through which university mathematics teachers can design responsive teaching practices that better support students’ transition into tertiary mathematics.
Prior knowledge in tertiary mathematics learning
Prior knowledge is widely recognised as a key determinant of learning outcomes in university mathematics. In this study, prior knowledge refers to the mathematical knowledge students bring with them at the start of a first-year calculus course, shaped by their schooling experiences and earlier encounters with mathematical concepts and procedures. The literature commonly distinguishes between two interrelated forms of prior knowledge (see, for example, Alexander, Schallert & Hare 1991; Ningsih & Retnowati 2020): conceptual knowledge, which concerns understanding mathematical concepts, relationships and meanings (the what), and procedural knowledge, which involves knowing how and when to apply mathematical procedures (the how and when). Importantly, possession of one form of knowledge does not necessarily imply mastery of the other, and the relationship between these components is complex (Alreshidi 2023).
Empirical findings are inconclusive regarding which component of prior knowledge contributes more strongly to student achievement in mathematics. Some studies emphasise the role of conceptual understanding (e.g. Richland, Stigler & Holyoak 2012), whilst others highlight procedural fluency (e.g. Hailikari et al. 2007). This ambiguity resembles a classic ‘chicken-and-egg’ dilemma, in which it remains unclear which component precedes or exerts greater influence on learning. From a constructivist perspective, however, such a dichotomy may be less productive, as new knowledge is constructed through the interaction of existing concepts and procedures within an interconnected conceptual framework (Nesher 1987).
From this perspective, learners construct new mathematical knowledge individually by connecting new information to existing ideas and experiences (Rach & Ufer 2020; Wilson 1996). This process is particularly salient in mathematics, and even more so in contexts that require the application of mathematics, such as mathematical modelling. When solving unfamiliar problems, students draw on prior conceptual knowledge to generate new procedures and to adapt familiar procedures to new situations. However, students may demonstrate procedural fluency in routine contexts whilst lacking the conceptual understanding required to interpret problems, coordinate representations or transfer knowledge to novel situations (Alreshidi 2023; Hailikari et al. 2007). This distinction becomes increasingly important in tertiary mathematics, where learning demands flexible reasoning, representational competence and conceptual coherence rather than routine execution alone. When prior knowledge is fragmented or incomplete, students’ reasoning may result in systematic and persistent errors across tasks and contexts. Such errors should therefore be understood as manifestations of underlying knowledge structures rather than as isolated mistakes.
Errors and misconceptions as indicators of prior knowledge, and readiness for mathematical modelling
Mathematical proficiency is widely conceptualised as comprising multiple, interrelated components (Baroody & Dowker 2013; Schoenfeld & Kilpatrick 2008), including conceptual understanding, procedural fluency, strategic competence, adaptive reasoning and productive disposition. Learning mathematics involves the integration of these components, and working mathematically ‘correctly’ entails not only the accurate application of procedures, but also the ability to justify reasoning, interpret representations and communicate understanding. The integration of these components becomes particularly important in mathematical modelling, where problems originate in real-world contexts, and students must identify, select and apply appropriate mathematical tools to construct and analyse models (see Durandt & Jacobs 2017, for an example at the tertiary level; Niss & Blum 2020, for an overview of the modelling process).
Mathematical modelling places distinctive demands on students’ prior knowledge. When students’ knowledge is predominantly procedural, they may struggle to engage meaningfully with modelling tasks, even if they can perform isolated calculations. Difficulties in interpreting graphs, understanding functional relationships, or reasoning geometrically may lead to breakdowns early in the modelling process, thereby limiting students’ ability to progress beyond surface-level engagement (Durandt & Jacobs 2017; Kaiser 2017). Readiness for modelling should therefore be understood not merely in terms of content coverage, but in terms of the conceptual coherence, representational fluency and flexibility of students’ prior knowledge.
Errors are a natural and ubiquitous feature of mathematical activity, ranging from minor slips to deeply rooted misconceptions. Within mathematics education research, errors and misconceptions are recognised as valuable sources of insight into students’ thinking. Early constructivist work on misconceptions (e.g. Nesher 1987; Smith III, DiSessa & Roschelle 1994) characterises them as incorrect conclusions arising from faulty reasoning or inaccurate underlying beliefs. In line with this view, Makonye (2012) defines misconceptions as entrenched but incorrect principles that give rise to recurring errors. Such misconceptions often originate in prior learning, may be widespread, and yet simultaneously offer important opportunities for instruction (Brodie 2014; Makonye & Fakude, 2016; Pournara et al. 2016).
Analysing students’ written work enables researchers and lecturers to distinguish between difficulties arising from procedural breakdowns (such as incorrect execution of algorithms) and those rooted in deeper conceptual misunderstandings (such as misinterpretations of inequalities, functions or geometric constraints). This distinction is particularly relevant at the tertiary level, where surface-level performance may mask underlying conceptual fragility. Whilst error analysis is well established in school mathematics research (e.g. Makonye & Luneta, 2014; Pournara et al. 2016), it remains less frequently documented in university mathematics contexts, despite its considerable instructional potential. Recent research in tertiary mathematics education underscores the importance of conceptual understanding, modelling competence and the pedagogical use of assessment (Advances in Mathematics Education series, Springer n.d.).
In this study, students’ errors are treated as indicators of the nature and structure of their prior mathematical knowledge. Identifying recurring error patterns provides insight into which concepts, representations and forms of reasoning require explicit instructional attention, particularly within a modelling-oriented approach to teaching first-year university mathematics.
Research methods and design
Strategy of inquiry
The study followed a mixed-methods sequential explanatory strategy, as described by Creswell and Clark (2018), where priority was given to the collection and analysis of quantitative data, followed by qualitative data to explain, elaborate and contextualise the results.
In the context of first-year mathematics, this strategy is particularly useful for investigating student learning and performance. For example, quantitative data from the diagnostic test (see later section for details regarding the test used in this study), identify trends in achievement, common error patterns and areas of conceptual difficulty. Subsequently, qualitative data – such as students’ written solutions – are used to explain why these difficulties occur by examining students’ reasoning, misconceptions and problem-solving strategies. This approach allows the author to move beyond performance outcomes to a richer understanding of mathematical thinking in the transition to university mathematics and in considerations for university mathematics teachers.
The mathematical modelling intervention
The mathematical modelling intervention intended to advance tertiary students’ mathematical and modelling competency forms part of the project CoSTAMM.1 The unit and teaching designs were developed in 2019 for South African first-year engineering students and are similar to the designs developed in the German DISUM2 project. For more details on the CoSTAMM design, see Durandt, Blum and Lindl (2022a), and Lindl, Durandt and Blum (2025), for details on the DISUM project, see Schukajlow et al. (2012). The tasks were conceived to consider the students’ mathematical prior knowledge from school and the demands of the first-year engineering mathematics curriculum (which focuses on the topic area of calculus and thus deals extensively with functions). Thus, all tasks were developed to require the construction of models involving elementary functions known from school.
Sample
The sample comprised science and engineering students from a large public university in Johannesburg, allocated to specific class groups through the university’s official registration system. Exposure to the study was limited to selected class groups, and comprehensive population-level data were not available due to feasibility and administrative constraints. In 2020, a sample of 555 first-year engineering mathematics students was exposed to the diagnostic test. In 2021, another sample of 989 first-year engineering mathematics students wrote the test. Then in 2022, a sample of 330 students wrote the test. The sample in 2022 consisted of both first-year engineering mathematics students and a small group of science students from a diploma programme in Analytical chemistry. Altogether, a total of 1874 mathematics students were exposed to the test that provided the quantitative data for the study. For the qualitative analysis, a stratified random sample of 90 student scripts was selected from the full dataset of 1874 scripts, with 30 scripts randomly drawn from each cohort. This sampling approach ensured representation across year groups whilst remaining feasible within the constraints of time and resources.
Diagnostic test instrument
The diagnostic test was designed based on guidelines from Stewart (2016) and included the content areas Algebra (the branch of mathematics that uses symbols, such as letters, to represent unknown numbers and express mathematical relationships in equations and formulas), Analytical geometry (the study of geometry using algebraic methods on a Cartesian coordinate system), Functions (it describes a relationship between an input [domain] and an output [range]), Trigonometry (the branch of mathematics that studies the relationships between the sides and angles of triangles) and elementary Calculus (focusing on the fundamental concepts of limits and derivatives). The idea to expose the students to a mathematical modelling intervention later in the semester led to the inclusion of a sixth area in the test: Modelling (focusing on creating a mathematical representation of a real-world problem to analyse, understand and make predictions about it). The test consisted of 25 tasks (with altogether 32 items, and 38 marks as a maximum), of which the format and level of difficulty ought to be mostly familiar to grade 12 students in South Africa.3 The only unfamiliar task was the second of two modelling tasks in the final section of the test (see detailed information about the Hot-air Balloon task in Durandt, Blum & Lindl 2022b). Table 1 provides an overview of the key aspects of the test with example items.
| TABLE 1: Features of the diagnostic test with example items. |
Data collection and analysis
The diagnostic test was administered in the first week of the university mathematics course during an official lecture period. In 2020 and 2022, the test was paper-based, and in 2021, the test was online due to coronavirus disease 2019 (COVID-19) restrictions. Both 2020 and 2021 samples could not be exposed to the modelling unit after the data on the diagnostic test were collected, as had been planned beforehand, also due to COVID-19 restrictions. Participants were not informed of the test ahead of time, so they could not prepare for the test, and were only allowed to use scientific calculators in the final test section on Modelling. On test day, participants were informed about the aim of the test that they were part of a research study and participation is voluntary (also see the later section regarding ethical considerations), and that test results will not form part of the formal course assessment structure.
All quantitative data processing and analyses were conducted using the statistical software R (R Core Team 2023). The raw values of the single items were combined to sum scores for each content area and to a total sum score. The internal consistencies of the various sections as well as the overall test, which were estimated using the reliability indicator McDonald’s Omega, are acceptable overall (partially low values in Trigonometry and Calculus are due to the low numbers of items). The differences in reliability between the years are not substantial.
The 90 documents for qualitative data processing were analysed to search for mathematical errors and misconceptions per content area, as informed by results from the quantitative analysis, and these were classified according to either conceptual or procedural errors (aligned with the concepts in the literature, see former sub-section). Qualitative data analysis was checked by a subject specialist and postdoctoral researcher to improve the trustworthiness of findings. A Cohen’s kappa value (a statistical measure of inter-rater agreement, assessing how consistently raters agree on categorical classifications, beyond what would be expected by chance) confirms a strong level of agreement (κ = 0.81) (compare Dettori & Norvell 2020).
Ethical considerations
Standard university ethical procedures were followed, and clearance to conduct this study was obtained from the University of Johannesburg, Faculty of Science Research Ethics Committee (Ethics reference number: 2020-09-04/Durandt). Participants gave written informed consent, and they could withdraw from the research at any time without any consequences.
Results
Quantitative results from the diagnostic test
An overview of the descriptive test results in all 3 years for each individual section and overall is given in Table 2. Comparing the test results, the 2022 participants perform best in the first three categories (Algebra, Analytical Geometry and Functions) and overall, even if their overall score differs only marginally from that of the 2021 group. The 2021 students achieved the highest scores in Trigonometry, Calculus and Modelling. The lowest scores overall and in four of the six test sections (Analytical geometry, Functions, Trigonometry and Modelling) were achieved by the participants in 2020. Welch’s one-way analyses of variance were conducted to compare performance across three cohorts (2020, 2021 and 2022) for each test section, given unequal variances and sample sizes (see Table 3). No significant year effects were found for Algebra or Calculus. Significant differences across years were observed for Analytical geometry, Functions, Trigonometry and Modelling. These differences are surprising and might be related to the entrance requirements of the different groups and the characteristics of each sample (more data on the specific samples are not available, which is a limitation in interpreting the results). However, Welch’s analyses of variance (ANOVA) are regarded as robust and accurate for real-world data where variances are often unequal (Delacre, Lakens & Leys 2017; Welch 1951; Zimmerman & Zumbo 1993).
| TABLE 2: Internal consistencies (McDonald’s Omega), means (M) and standard deviations (s.d.) per test section. |
| TABLE 3: Welch’s one-way ANOVA comparing test sections. |
In view of the maximum number of marks that can be achieved, the mean scores achieved in each individual section and overall are quite low (see Table 2 and Table 3). On average, the participants achieve only about one-third of all possible marks in the overall test. In 2 years (2020 and 2022), no one was even able to solve a test task (Hot-air Balloon task) in the modelling section, and in 2021, only one person was able to solve it – as expected, the item was therefore removed from the scale. One reason might be that students were completely unfamiliar with this modelling task and had no idea how to start the solution process. Other reasons for the poor performance overall might be that the time allocation was too strict or that students did not take the test seriously. One might argue that students could have performed much better if they were informed of the test beforehand and specifically prepared for it. Another reason might be that students felt ‘lost’ without the use of a calculator, in five of the six sections of the test, as South African students (at the secondary level) are used to writing mathematics tests with a calculator.
Overall, relatively the best results were achieved, in all years, in the Algebra section, which can be seen as a strength, referring to strong procedural mathematical knowledge. All other sections showed weaknesses in students’ prior mathematical knowledge, and students were almost completely unfamiliar with advanced problem-solving. Games–Howell post hoc comparisons (see Table 4) indicate that mean scores in Analytical Geometry and Functions increased significantly across cohorts, with 2022 outperforming both 2020 and 2021. For Trigonometry and Modelling, performance in 2021 was significantly higher than in 2020 and 2022, indicating a temporary cohort effect. Total test scores were significantly higher in both 2021 and 2022 compared to 2020, with no significant difference between the latter two cohorts. Games–Howell post hoc tests were used following significant Welch’s ANOVAs because they provide robust pairwise comparisons under conditions of unequal variances and unequal sample sizes across groups (see Games & Howell 1976). The overall test performance improved from 2020 to later years, but there was no difference between 2021 and 2022. Also, no substantial differences can be identified between the samples that completed the test on paper (2020 & 2022) and the sample that completed the test online (2021).
| TABLE 4: Games–Howell post hoc comparisons between years (significant Welch tests only). |
Qualitative findings from studying students’ mathematical work
Whilst the overall low performance in the diagnostic test is concerning, a closer examination of students’ written solutions provides important insight into how students reason mathematically and where their difficulties originate. In line with the sequential explanatory design of the study, the qualitative analysis was guided by the weakest-performing content areas identified in the quantitative results and focused on questions that were particularly relevant for the planned mathematical modelling intervention. These included tasks from Analytical geometry (Q12 & Q13), Functions (Q18) and Modelling (Q24). Across the analysed scripts, evidence of both conceptual and procedural weaknesses was observed, often intertwined within the same solution attempts. Importantly, students’ errors were not random but reflected consistent reasoning patterns, indicating gaps in underlying conceptual understanding rather than isolated mistakes. The following examples illustrate typical difficulties identified in the qualitative analysis and highlight how these difficulties may constrain students’ engagement with modelling-oriented tasks.
Difficulties in analytical geometry
Questions 12 and 13, both from the Analytical geometry section, revealed substantial conceptual difficulties. In Question 12, students were asked to sketch the region in the Cartesian plane defined by two inequalities. Approximately 93% of the analysed sample were unable to produce a correct representation. Whilst many students attempted a sketch, they struggled to interpret simultaneous restrictions on both variables, often treating the inequalities independently or ignoring one constraint entirely. This suggests a limited conceptual understanding of inequalities as defining regions in the plane, rather than as isolated algebraic statements.
Similarly, Question 13 required students to construct a rectangle given three vertices and to determine the coordinates of the fourth vertex. Approximately 75% of students were unable to complete this task correctly. Although the given points were correctly plotted in many cases, students frequently placed the fourth point incorrectly, often avoiding the use of negative coordinates and producing a skewed quadrilateral instead of a rectangle. These responses indicate difficulties with spatial reasoning, coordinate geometry, and the interpretation of geometric constraints – competencies that are essential for constructing and interpreting mathematical models.
Difficulties with functions and representations
Question 18, which required students to sketch the graphs of an exponential and a rational function, revealed both conceptual and procedural weaknesses. Approximately 75% of the students in the qualitative sample were unable to produce acceptable sketches. Many students attempted to construct value tables for the exponential function but were unable to calculate sufficient points or to use the calculated values to generate a meaningful graph. Others calculated points correctly but struggled to translate numerical information into a visual representation. In the case of the rational function, several students produced graphs that bore little resemblance to the expected shape, indicating a lack of understanding of asymptotic behaviour and functional structure. These findings suggest that students’ difficulties extend beyond procedural calculation to include representational fluency – the ability to connect symbolic expressions, numerical values and graphical forms. The reliance on calculators for graphing at the school level may have contributed to these difficulties, as students were required in the diagnostic test to reason about function behaviour without technological support.
Difficulties in mathematical modelling
The modelling task (Question 24) revealed the most pronounced difficulties. Although most students were able to plot the given data points on a scatter plot, approximately 88% were unable to complete the task successfully. In particular, students struggled to identify and construct an appropriate linear model, even when the use of a calculator was permitted. Several students attempted to proceed with estimation despite having an incorrect or poorly constructed model, resulting in further compounding errors. These responses suggest that students possess fragmented procedural knowledge, such as plotting data, but lack a coherent conceptual understanding of modelling as a process that involves identifying relationships, constructing models and interpreting results in context. This finding is particularly significant given the central role of modelling in science and engineering mathematics and underscores the dependence of modelling competency on strong prior knowledge of functions and representations.
Overall, the qualitative findings reveal persistent conceptual and representational gaps in students’ prior mathematical knowledge, particularly in areas foundational to calculus and modelling. Whilst some procedural competence was evident, especially in algebraic manipulation, students frequently struggled to interpret constraints, reason geometrically, visualise functions and construct meaningful models. From a constructivist perspective, these errors reflect coherent but incomplete knowledge structures formed through prior learning experiences. The qualitative analysis should be further expanded by analysing more questions and a larger sample.
Discussion
This study sets out to examine first-year science and engineering students’ prior mathematical knowledge at the start of a university calculus course, with the explicit aim of informing modelling-oriented teaching. By combining large-scale diagnostic data with qualitative analyses of students’ written work, the findings provide a layered understanding of not only how students performed, but why particular difficulties persist during the transition to tertiary mathematics.
Prior knowledge as uneven and structurally fragile
Across all three cohorts, students’ overall performance in the diagnostic test was low, with mean scores indicating that, on average, students achieved only about one-third of the possible marks. Whilst modest improvements were observed over time, particularly in Analytical Geometry and Functions, these gains did not translate into strong overall preparedness. Algebra emerged as a relative strength across cohorts, suggesting a degree of procedural fluency. However, this fluency was not accompanied by corresponding conceptual understanding in other content areas that are central to calculus and modelling, such as functions, geometry and representation. These findings align with earlier research highlighting the cumulative and hierarchical nature of mathematical learning (Hailikari et al. 2007; Rach & Ufer 2020), but they also extend this work by showing that apparent procedural competence may co-exist with substantial conceptual fragility. From a constructivist perspective, this unevenness in prior knowledge is consequential: when foundational concepts are weak or poorly connected, students struggle to construct new knowledge, particularly in contexts that require flexible application and interpretation.
Explaining quantitative trends through qualitative evidence
The qualitative analysis provides critical insight into the quantitative patterns observed across cohorts. For example, the low performance in Analytical Geometry and Functions is illuminated by students’ difficulties in interpreting inequalities, reasoning about regions in the Cartesian plane and visualising functions without technological aids. These difficulties point to conceptual gaps rather than isolated errors, particularly in students’ understanding of relationships between algebraic expressions, graphical representations and geometric meaning. Similarly, the consistently weak performance in the modelling section (despite minor cohort-related fluctuations) reflects students’ limited experience with constructing and using mathematical models. Although many students could plot data points, they struggled to identify appropriate functional relationships and to interpret models meaningfully. This suggests that modelling challenges are not merely due to unfamiliar task formats but are rooted in insufficient conceptual understanding of functions and representations, which are prerequisites for successful modelling (for more details on learning modelling see Niss & Blum 2020).
The integration of quantitative and qualitative findings thus underscores the value of mixed methods approaches in tertiary mathematics education research. Performance data alone may indicate where students struggle, but qualitative analyses are essential for understanding how students reason and which misconceptions shape their mathematical activity.
Implications for modelling-oriented teaching
The findings have important implications for modelling-oriented teaching in first-year university mathematics. Mathematical modelling places high cognitive demands on students, requiring them to coordinate conceptual understanding, procedural fluency and representational competence across multiple stages of problem solving. When students’ prior knowledge is fragmented or overly procedural, modelling tasks can become overwhelming, leading to breakdowns early in the solution process. The results of this study suggest that modelling interventions cannot be effective if they assume a level of conceptual readiness that many students do not yet possess. Instead, modelling-oriented teaching needs to be deliberately scaffolded to address common conceptual gaps – particularly in functions, geometry and interpretation – before or alongside engagement with complex real-world problems. Diagnostic evidence, when analysed at the level of specific errors and misconceptions, provides a valuable foundation for designing such scaffolding. Importantly, the study challenges deficit-oriented interpretations of students’ prior knowledge. Whilst students’ performance was undeniably weak in several areas, the qualitative findings reveal coherent, if flawed, reasoning patterns. From a constructivist standpoint, these patterns represent starting points for instruction rather than obstacles to learning. Errors, in this sense, are not signs of failure but indicators of how students are making sense of mathematical ideas.
Rethinking diagnostic testing at the tertiary level
A further implication of this study concerns the role of diagnostic testing in university mathematics. Much of the existing literature emphasises diagnostic tests as tools for identifying at-risk students or predicting academic success (e.g. Greefrath et al. 2017; Van Appel et al. 2024). Whilst such uses are valuable, the present findings suggest that diagnostic tests have greater potential when they are used as instructional resources. By linking diagnostic results to qualitative analyses of students’ work, this study demonstrates how diagnostic testing can inform teaching decisions, curriculum design and the sequencing of learning activities. In this sense, diagnostic assessment becomes a bridge between students’ prior knowledge and instructional design, rather than a mechanism for classification alone. This shift is particularly important in large first-year classes, where opportunities for individualised feedback are limited.
Cohort differences and contextual considerations
The observed differences between cohorts (such as the temporary peak in Trigonometry and Modelling performance in 2021) should be interpreted cautiously. These variations may reflect cohort-specific characteristics, differences in schooling experiences or contextual factors related to the COVID-19 pandemic. Nevertheless, the persistence of low performance across content areas and years suggests that the challenges identified are structural rather than incidental. This finding reinforces concerns raised in the literature about the alignment between school and university mathematics curricula (Di Martino et al. 2023). Whilst secondary schooling may support procedural competence, it appears less successful in developing the conceptual and representational understanding required for tertiary mathematics and modelling. Addressing this misalignment remains a systemic challenge that extends beyond individual courses or institutions.
Finally, the findings of this study highlight the complexity of students’ prior mathematical knowledge at the start of university mathematics and the limitations of viewing preparedness through aggregate performance measures alone. The integration of quantitative trends and qualitative error analysis reveals persistent conceptual weaknesses that have direct implications for calculus learning and modelling-oriented teaching. By treating students’ errors as sources of instructional insight, rather than as indicators of deficit, university mathematics teachers can design more responsive and effective learning opportunities that support students’ transition into tertiary mathematics.
Conclusion
This study examined first-year science and engineering students’ prior mathematical knowledge at the start of a university calculus course, with a focus on its implications for modelling-oriented teaching. Across three cohorts, the findings show that whilst students display some procedural fluency in Algebra, persistent conceptual and representational difficulties remain in areas such as Analytical geometry, Functions and Mathematical modelling. These weaknesses were evident despite modest improvements in overall performance.
The integration of quantitative diagnostic data with qualitative analyses of students’ written work revealed that many errors were systematic and rooted in incomplete conceptual understanding rather than isolated procedural mistakes. Such difficulties might constrain students’ engagement with calculus and modelling tasks that require flexible reasoning and coordination of representations. The findings highlight the value of using diagnostic assessment as an instructional resource to make students’ prior knowledge visible early in the course. For university mathematics teachers, this underscores the importance of explicitly addressing conceptual gaps and supporting students’ transition into tertiary mathematical learning through targeted, modelling-oriented instruction.
Acknowledgements
This article is based on research originally conducted as part of the CoSTAMM studies. The author gratefully acknowledge the contribution of the CoSTAMM research project in providing the foundation for this work.
Competing interests
The author declares that no financial or personal relationships inappropriately influenced the writing of this article.
CRediT authorship contribution
Rina Durandt: Conceptualisation, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Validation, Writing – original draft and Writing – review & editing. The author confirms that this work is entirely her own, has reviewed the article, approved the final version for submission and publication, and takes full responsibility for the integrity of its findings.
Funding information
This work was supported by the National Research Foundation (NRF) of South Africa (grant number: 121969).
Data availability
The data that support the findings of this study are not publicly available due to ethical and confidentiality restrictions related to student participation. The datasets are available from the corresponding author, Rina Durandt, upon reasonable request and with permission from the institutional ethics committee.
Disclaimer
The views and opinions expressed in this article are those of the author and are the product of professional research. They do not necessarily reflect the official policy or position of any affiliated institution, funder, agency or that of the publisher. The author is responsible for this article’s results, findings and content.
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Footnotes
1. Comparative Studies into Teaching Approaches for Mathematical Modelling is a South African/German project launched in 2018. The project is directed by R. Durandt (mathematics education, University of the Witwatersrand [formerly University of Johannesburg]), W. Blum (mathematics education, University of Kassel), and A. Lindl (mathematics education, University of Regensburg).
2. Didaktische Interventionsformen für einen selbständigkeitsorientierten aufgabengesteuerten Unterricht am Beispiel Mathematik – in English: Didactical intervention modes for mathematics teaching oriented towards students’ self-regulation and guided by tasks. The project was directed by W. Blum (mathematics education), R. Messner (pedagogy, both University of Kassel), and R. Pekrun (pedagogical psychology, University of München) and was carried out 2002–2013.
3. The South African Grade 12 Mathematics curriculum (CAPS) focuses on preparing students for higher education, covering key topics including algebraic manipulation, functions and graphs, differential calculus, patterns and sequences, financial mathematics, probability, Euclidean and analytical geometry, trigonometry and statistics.
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