Project-based learning in computing-oriented applied mathematics: Methodology for designing effective driving questions

Introduction

Project-Based Learning (PBL) is an educational paradigm in which students acquire the necessary knowledge and skills through sustained teamwork on real-world, practically meaningful projects (Chen and Yang, 2019; Guo et al., 2020; Lavado-Anguera et al., 2024; Strobel and Van Barneveld, 2009). Over the past 2 years, the relevance of PBL has increased markedly: educational leaders now position it as a key post-pandemic engagement strategy, and analytical reviews in Forbes and Social Studies (December 2024 - January 2025) include it among the “critical trends”, highlighting the need for interdisciplinary STEAM solutions and the cultivation of competencies that cannot be automated by AI (Hendricks, 2024; Marr, 2024). Since PBL denotes a broad approach rather than a prescriptive methodology, its core principles can - and do - take various forms even within a single discipline, let alone across different fields (Guo et al., 2020).

The most challenging phase of PBL is formulating the project’s Driving Question (DQ) ¹ (Markula and Aksela, 2022; Mentzer et al., 2017; Ssali et al., 2025). Difficulties in crafting a well-defined project task frequently lead educators to abandon PBL implementation, despite its widely recognised effectiveness (Mӓkelӓ et al., 2023; Zhang and Ma, 2023). Surveys show that over 70% of educators experience difficulties in locating or developing well-structured tasks that align with curricular requirements (Baines et al., 2021).

A poorly formulated task is virtually impossible to transform into a successful project. The literature offers no guidelines for task formulation or clear criteria that such tasks should meet. A review of existing studies shows a lack of a systematic operational approach to developing a Driving Question in discipline-specific contexts. The analysis revealed that:

(1) In many cases, a clear Driving Question is not formulated, and only a general theme is provided.

It is evident, however, that each discipline requires its own set of criteria and methodology. Moreover, different methodologies may be developed even within a single discipline (Markula and Aksela, 2022; Rogers et al., 2011).

In applied mathematics, educators encounter specific challenges when formulating project tasks because of the inherently abstract nature of the subject. Such tasks must link mathematical abstractions with real-world practice, requiring educators to possess expert knowledge in relevant domains (e.g., physics, economics, biology). Furthermore, applied tasks often demand real data and/or access to necessary resources, and securing such access can itself pose a significant challenge.

Previously, we developed a PBL methodology for students enrolled in the “Intelligent Data Analysis” university programme majoring in Computational Applied Mathematics (Akhiiezer et al., 2023). It should be noted that the proposed methodology pertains to the major rather than its individual courses, which renders it unique. This approach enables more effective attainment of the programme’s educational objectives. In (Akhiiezer et al., 2023), we outlined the overarching project workflow and defined the roles of each participant in the process. In the present article, the term computing-oriented Applied Mathematics refers to Applied Mathematics programmes in which mathematical modelling is systematically combined with programming, data analysis, and the development of software or software-hardware solutions.

The present study aims to develop and substantiate a detailed methodology for selecting and formulating project tasks within PBL for computing-oriented Applied Mathematics, supporting the systematic design of Driving Questions that promote the achievement of the core educational objectives of project-based learning.

Analysis of typical mistakes in PBL problem statement and formulation

An analysis of numerous sources (e.g., Bender, 2012; Fisher et al., 2020; Wolf et al., 2013) presenting applied mathematics PBL problem statements revealed that nearly every example contains recurring methodological errors capable of undermining PBL effectiveness. The identified shortcomings were categorized into six principal groups (Table 1). Table 1 details the nature of each error, its practical manifestations, and a real-world example drawn from the literature. Each error prevents learners from developing the intended target skills; furthermore, a single task may exhibit multiple error types. The examples in Table 1 are intentionally presented in sufficient detail because the methodological error often becomes visible only through the concrete wording of the project formulation.

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Table 1.

Characteristic methodological errors in formulating applied mathematics problems in PBL.

№	Typical error	Practical manifestation	Example of formulation
1	No clearly formulated Driving Question	Students tend to simplify the task when given options	Children in third-world countries. This is a mathematically and thematically complex project (interdisciplinary!), which mainly involves tables, work with graphical representations, statistical reports, independent collection of ‘real’ data, image statistics, calculations with large numbers, etc. a (Wolf et al., 2013).
2	Overly trivial problem	Students complete it quickly with minimal engagement.	“In Search of a Bi-Dimensional Geometric Model”: Consider the object you were given as having negligible thickness. To sketch the graphs, use a support software such as Symbolab Math Solver (for integrals calculus) and the online services provided by Desmos, Inc. — “Desmos” (Graphing Calculator, San Diego, CA, USA). Consider the following questions: Q1. How should the object be placed in a bi-dimensional frame? Q2. What are the algebraic expressions of the functions involved (at the boundary of the object)? Q3. What is the object’s surface area (in square centimeters’)? Q4. A variation on the proposed challenge. Note: Please keep in mind that you can use a ruler.” b (Rézio et al., 2022).
3	Overly complex multi-component problem, resource inaccessibility, excessive workload	Excessive technical work unrelated to mathematics	“An example of a project-based activity is students examining the development of a new suburb and utilising statistics to justify the inclusion of a range of services such as retail, health, education, leisure and transport. Students utilise census data and examine the viability of their inquiry and demonstrate their learning through presentations which utilise graphs and statistical tools.” (Rijken and Fraser, 2024).
4	The project is not related to real life and lacks explicit interdisciplinarity	The academic project lacks a convincing connection to real professional tasks.	“Multiple proofs of the Pythagorean theorem: students research various proofs of the Pythagorean Theorem. c The end product can be a webpage showing the proofs or an oral presentation of one of the proofs.” (Stivers, 2010).
5	Task accompanied by detailed solution instructions	Eliminates the need for independent research, analysis of different solution methods, etc.	“A plantation home in Virginia, the Cedar Plantation. The family wants to allow a family-owned furniture company to harvest a selected allotment of white and red cedar trees each year for furniture production. On this plantation property, cedar trees occupy all of the 49 acres of woods, but the family is not sure how many trees are on the property, and they want to be assured that selected harvesting does not deplete their entire supply of cedar trees. The family wants to harvest no more than 50% of the cedar trees in any given year. Students will work in groups to accomplish several tasks: 1. Classify the types of cedar trees on the plantation. For the most part, white cedars grow in the lower-land swamps on the plantation, whereas red cedars grow on the higher ground. Verify the average number of trees on each acre and the total number owned by the plantation. These data on the extrapolated cedar tree count should be summarized on a spreadsheet. 2. Determine the life span for cedar trees in upstate Virginia. If possible, provide some guidance on “worst case” scenarios (e.g., a 2-year drought, late snowfall that kills new-growth seedlings in a given year) that might limit the number of trees that should be harvested yearly. Determine the number of trees available for harvest each year, based on average tree count on the four acres and other.” (Bender, 2012).
6	Disconnection between the problem and the target competencies	The problem does not require the use of mathematical models and methods.	“Design of a shopping mall: students work as architects and design a shopping mall. Students research the design of malls and the feasible sizes for various types of stores. The end product can be a brochure, poster, or web-page presenting the design to a panel of adults or students serving as a city-planning board.” (Stivers, 2010).

These errors indicate a fundamental methodological gap: the absence of sufficiently specific and verifiable guidance for formulating PBL problems in applied mathematics. Existing descriptions of authenticity, interdisciplinarity, openness, and complexity provide important general principles, but they do not offer a practical procedure for determining whether a particular project formulation will function effectively in a specific disciplinary context. A literature-based analysis is therefore an important starting point, but it needs to be complemented by practice-based evidence from actual project implementation. The following section examines such evidence from early project formulations and explains how it provided the empirical rationale for developing the Driving Question criteria.

Empirical rationale for developing the driving question criteria

In addition to analysing examples of PBL tasks presented in the literature, the methodology was developed through the examination of our own early project formulations. At the initial stage of PBL implementation, project tasks were designed mainly in accordance with general principles emphasized in the PBL literature: the task should be authentic, interdisciplinary, practically meaningful, and supportive of teamwork (Guo et al., 2020; Markula and Aksela, 2022). However, the first project cycles showed that these characteristics provided only a general direction and were not sufficiently specific and verifiable for designing tasks in computing-oriented Applied Mathematics. Features such as authenticity, interdisciplinarity, and teamwork did not, by themselves, reliably distinguish a methodologically sound Driving Question from an externally convincing but pedagogically weak formulation. More precise and checkable criteria were therefore needed for the preliminary evaluation of Driving Questions.

To analyse the early projects, the teaching team used materials systematically collected for each project: the initial problem statement received from the Customer in natural language, the final Driving Question, the project specification, the software product, the technical report, the presentation, the video recording of the public defence, brief Customer feedback, and students’ post-project reflections. These materials made it possible to compare the initial project idea with how the task functioned in the project process: whether it stimulated independent inquiry, required substantive mathematical analysis, supported the work of all team members, corresponded to students’ level of preparation, and led to a practically meaningful outcome.

Particular attention was paid to unsuccessful or substantially revised early formulations. This analysis showed that a task could appear applied or interdisciplinary while still failing to fulfil the educational function of PBL. Some tasks had a convincing applied context but, in practice, were reduced to plotting functions, finding extrema, calculating a volume, or applying a standard formula. In such cases, the applied narrative served mainly as an external framing of a conventional academic exercise rather than as a source of a genuine problem situation. Other tasks required the development of a software product but did not involve non-obvious mathematical modelling and could be completed by one strong student; as a result, teamwork became largely formal.

Another group of difficulties concerned misalignment with the programme’s intended learning outcomes. Some formulations shifted the focus toward engineering, design, or narrowly technical development, thereby losing a clear connection to the mathematical models and methods specified in the curriculum. Others, by contrast, contained a mathematical core that was too simple and did not create productive difficulty for students at the relevant stage of study.

After each project cycle, the teaching team compared the initial formulations with the process of project implementation and the final artefacts. The analysis focused on cases in which a project was completed too quickly, was carried out mainly by one participant, was broken down into sequential stages that forced part of the team to wait, did not require substantive mathematical modelling, was weakly connected to a real Customer need, or did not correspond to students’ current level of preparation. Students’ post-project reflections were used as an additional source of information about task difficulty, distribution of work, sufficiency of time, the need to master new methods independently, and students’ perception of the relationship between mathematics and real-world practice.

This analysis enabled moving from general pedagogical principles to a more rigorous procedure for the preliminary evaluation of project tasks. Unsuccessful formulations were treated not as isolated organizational failures, but as indicators that specific qualities were missing in the Driving Question itself. The next stage of the study, therefore, involved identifying a set of operational requirements for project tasks. These requirements are presented in the following section as a system of criteria for formulating PBL Driving Questions.

Core requirements for the PBL project problem

Based on an analysis of contemporary pedagogical and psychological methodologies (González-Pérez and Ramírez-Montoya, 2022; Graesser et al., 2022; Mohammed and Ozdamli, 2024), employer requirements for potential employees’ soft- and hard-skills (Poláková et al., 2023; Taguma et al., 2023; Yong and Ling, 2023), the rapid advancement of AI capabilities, and within the framework of the Applied Mathematics Specialty Standard (Ministry of Education and Science of Ukraine, 2018) and the learning outcomes of the “Intelligent Data Analysis” educational programme (Educational Programme, 2023), we have formulated the following criteria that a PBL project problem in computing-oriented Applied Mathematics programmes must satisfy:

Criterion I. Market-Ready Product: The result of the project should be a software or software-hardware product ready for a presentation on the market.

Criterion II. Applied relevance: The problem should have a relevant applied nature.

Criterion III. Interdisciplinarity: The solution to the problem should require applying knowledge from different domains.

Criterion IV. Implicit mathematical modelling: The problem must entail mathematical modelling that is not evident from the problem statement.

Criterion V. Decomposability: The problem can be decomposed into several separate, independently running subproblems (students within the team should be able to work in parallel).

Criterion VI. Team-based working: The impossibility of one person completing the task on time.

Criterion VII. Alignment to Curriculum: The models and methods used to solve the problem correspond to the curriculum.

Every problem submitted by a potential Customer is first evaluated against these criteria. In most cases, the raw problem will not meet all of them, since real-world Customers often lack awareness of academic curricula and student proficiency levels. A committee of educators reviews the Customer’s initial problem statement: if it can be revised to satisfy all criteria, the formulation is modified and confirmed with the Customer; otherwise, it is rejected. Once the Customer approves the revised version, the official PBL project Driving Question will be issued. The sequence of steps for this formulation process is depicted in Figure 1 as a BPMN diagram (International Organization for Standardization, 2022).OPEN IN VIEWER

Next, we justify each criterion and show how the absence of each criterion may weaken the pedagogical function of the project task and reduce the likelihood of achieving the intended educational objectives. To this end, consider the following example problem that meets all criteria:

Problem. Develop software for designing the cost-optimal layout of an electrical cable network for a village street lighting, given a specified map and a known power supply ingress point.

The remaining sections describe and substantiate each criterion. In each case, we also present a modified version of the original formulation that fails to satisfy the criterion under consideration.

Template for constructing the PBL problem statement

We have developed a problem-statement template that functions like a modular constructor, enabling the formulation of an effective PBL Driving Question that satisfies all of the criteria outlined above. According to this template, the Driving Question is expressed as a single sentence, the structure of which is shown in Figure 3.OPEN IN VIEWER

The formulation must include the following components: 1. Outcome, 2. Function, 3. Criterion, 4. Constraints, and 5. Additional Conditions. Each component corresponds to a segment of the sentence that conveys the following semantic content:

Outcome is an Authentic Achievement as a software product or a software-hardware system. Specifying a concrete outcome establishes a clear target for students, which is crucial for goal-directed learning. According to Locke and Latham’s goal-setting theory, specific and challenging goals yield higher motivation and performance than vague or simple ones (Chen and Yang, 2019; Locke and Latham, 1990). Clarity of the expected outcome reduces cognitive load and enables more efficient allocation of cognitive resources.

Function is the primary function the software (or software–hardware (SW/HW) system) must perform.

Criterion is a condition that a correct solution must satisfy.

Constraints are constraints on input data and the outcome. This component serves two essential purposes: first, constraints are intrinsic to any real-world problem; second, they allow for fine-tuning problem complexity. Working within set constraints forces students to think both creatively and critically. These constraints not only enhance their problem-solving skills but also cultivate the ability to operate under realistic limitations - an essential competence for their future professional practice.

Additional Conditions are secondary requirements - compliance with regulatory documents, Customer preferences, reference data, etc. - commonly encountered in real-world scenarios. These conditions should not substantially alter the underlying mathematical model or solution algorithms. As an optional element, this component increases the template’s adaptability and provides another lever for adjusting problem complexity. Incorporating additional conditions readies students for the reality that standards, national regulations, and legislative frameworks often govern professional tasks. It equips them with the vital skill of navigating such requirements in professional life.

It should be noted that a single Driving Question may include multiple functions and multiple criteria.

By specifying these five components, one obtains a ready-to-use Driving Question that contains all the information the project team needs to begin work. Figure 3 presents five example Driving Questions - each line corresponding to one component - that conform to this structure. These examples are drawn from actual student projects carried out between 2020 and 2023.

All Driving Questions are deliberately stripped of concrete data to develop students’ critical thinking, information-retrieval abilities, client-communication skills, and other competencies. According to the methodology in (Akhiiezer et al., 2023), once the Driving Question is received, the project team embarks on a domain exploration phase to prepare for the Customer meeting and to formulate all clarifying questions before drafting the technical specification.

To illustrate how a customer’s initial request can be evaluated against the proposed criteria and reformulated through the Driving Question template, we present a case from our implementation practice concerning an apple-sorting project. The Customer initially described the need as a system for separating sweeter apples intended for jam from more acidic apples intended for juice. This request was practically meaningful but methodologically vague, since sweetness and acidity were not directly specified as measurable input parameters, and the apple variety was not defined. During methodological reformulation, the task was translated into the following Driving Question: “Create an HW/SW system that sorts apples by degree of ripeness, given a specified apple variety, into separate containers in real-time mode.” The constraint concerning apple variety was essential, because visual indicators of ripeness differ substantially across varieties; without this constraint, the classification task would have exceeded the intended level of complexity for the student cohort. The final project output was a functioning HW/SW prototype that included a conveyor simulator assembled from available materials, a camera for real-time image acquisition, classification software, and a stepper motor that redirected apples according to the software output. The prototype was demonstrated during the public defence, and the project subsequently resulted in a peer-reviewed publication (Galuza et al., 2021). This case illustrates how the template helps transform an initially vague Customer request into a feasible, product-oriented, mathematically meaningful, and curriculum-aligned PBL task.

Recommendations for Driving Question formulation

This section provides practical recommendations for formulating Driving Questions and supporting the PBL process, derived from an analysis of common methodological errors and aimed at enhancing the effectiveness of project-based learning.

When issuing the Driving Question and throughout project work, educators should:

(1) Refrain from supplying additional informational resources when the Driving Question is issued (this would negate Criterion IV). In the traditional PBL approach, educators often hold a discussion immediately after presenting the Driving Question and distribute resources (links to websites, books, lecture notes, etc.) containing the information needed to solve the problem (Bender, 2012). This practice leads students to replicate prescribed procedures without conducting a comparative analysis of alternatives, thereby undermining the development of critical thinking and independent research strategies.

For educators integrating project-based learning (PBL) into the Applied Mathematics curriculum who face difficulties sourcing problems from real Customers, we recommend seeking collaboration with:

• Local enterprises (industrial manufacturers, IT companies, banks, insurance firms, etc.);

Additionally, we advise actively attending:

• Industry conferences and exhibitions, where you can meet company representatives and learn about the challenges they are trying to solve;

Systematic application of these recommendations helps projects maintain an appropriate level of cognitive challenge and authenticity, thereby enhancing their educational value and boosting graduates’ professional competitiveness.

Discussion

The contribution of this study should be understood in relation to existing PBL task-design frameworks. Concepts such as authenticity, interdisciplinarity, teamwork, and labour-market relevance are already well established in the PBL literature; however, they often remain too general to guide the formulation of concrete project tasks in a specific disciplinary context. The originality of the proposed methodology lies in translating these broad principles into a coherent set of specific and verifiable criteria and a structured template for designing Driving Questions in computing-oriented Applied Mathematics. In this sense, the methodology is not intended to redefine PBL or the concept of a Driving Question, but to provide a discipline-specific procedure for transforming customer problems stated in natural language into mathematically substantive, product-oriented, team-based, and curriculum-aligned project tasks.

The proposed methodology for selecting and formulating PBL Driving Questions is grounded in seven interrelated criteria, each designed to develop specific hard skills, soft skills, and self-skills as codified in the updated professional standards SFIA 9 ² (SFIA Foundation (2025)) and the competency forecasts (WEF (2025)). Table 2 presents the mapping between these criteria and the targeted skills, fulfilling two functions. First, it supports the internal coherence of the methodology by showing how each criterion is aligned with specific skills specified in the curriculum and demanded by the labour market. Second, this mapping can be used upon project completion as an assessment tool to evaluate the degree to which PBL objectives have been achieved. Such reflective practice is crucial for educators, as it enables them to appraise how accurately Driving Questions were formulated and to identify areas for refinement in future problem formulation.

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Table 2.

Correspondence between PBL Driving Question Criteria and the Skills they provide.

Criterion	Hard skills	Soft and self skills
1. The project result should be a software or a software-hardware product ready for a presentation on the market.	• Programming • Writing discipline-specific texts reports, etc. • Presenting the product to an audience • Clear and concise information delivery • Public speaking before unfamiliar audiences, etc.	• Customer orientation • Flexibility and adaptability when working with Customers, etc.
2. The problem should have a relevant applied nature.	• Collection and analysis of real data • Interpretation of results • Working with incomplete data, etc.	• Initiative • Ability to work with uncertainty, etc.
3. The solution to the problem should require applying knowledge from different domains.	• “Learn to learn” capability • Interdisciplinary thinking • Understanding the context of data sets • Critical thinking • Cognitive flexibility • Adaptability to new information, etc.	• Communication with domain experts • Openness to new experiences • Tolerance for ambiguity, etc.
4. The problem must entail mathematical modelling that is not evident from the problem statement.	• Mathematical and algorithmic formalization of the problem • Development of solution strategies • Modelling objects, processes, and decision-making procedures • Critical evaluation of AI recommendations	• Patience • Persistence • Attention to detail • Systems thinking, etc.
5. The problem can be decomposed into several separate, independently running subproblems	• Fundamentals of project management • Decomposition of mathematical problems • Decomposition of programming tasks, etc.	• Teamwork • Leadership • Conflict resolution, etc.
6. The impossibility of completing the task on time by one person	• Resource planning • Critical-path (network) scheduling, etc.	• Delegation • Time management • Responsibility • Team coordination, etc.
7. The models and methods used to solve the problem correspond to the curriculum	• Application of curriculum-based mathematical models and methods • Transfer of theoretical knowledge to applied problem solving	• Self-directed learning • Awareness of one’s own level of preparation • Confidence in applying course knowledge

Table 2 demonstrates that the proposed set of criteria collectively addresses a broad spectrum of skills essential for a Computational Applied Mathematics specialist. Criterion VII differs from the previous criteria in that it primarily functions as a curriculum-alignment and feasibility criterion. However, it is included in Table 2 because it also supports the transfer of course-based knowledge to applied problem solving and helps maintain the project within a productive level of difficulty.

In addition to the criteria, we have developed a unified, structured template to help instructors avoid errors in Driving Question formulation and provide direction for all subsequent project-based activities. The PBL Driving Question template significantly streamlines the formulation process by:

• Facilitating the standardization of both the problem-development process and the overall PBL instructional workflow;

Problem complexity can be substantially adjusted via the template’s two final components - Constraints and Additional Conditions. Moreover, according to Lev Vygotsky’s theory, maximal educational effectiveness is achieved when problem complexity is balanced at the edge of the Zone of Proximal Development (Van de Pol, Volman, and Beishuizen, 2010). Within this zone, skills first develop “between” learners and then internalize into individual cognitive processes. At the same time, Driving Questions should be pitched just above students’ independent capabilities while still remaining within achievable limits.

Thus, the combination of a clearly defined set of criteria and a unified template establishes the foundation for several methodological and practical advantages of the developed driving-question methodology:

• The criteria set - each explicitly linked to specific learning outcomes - serves as a “quality filter.” A poorly formulated Driving Question not only fails to develop the targeted competencies but also creates a false sense of goal completion: knowledge is retained only until the next assessment and then quickly dissipates from working memory. By leveraging Bloom’s cognitive pyramid hierarchy, the methodology intentionally guides students to the Create level, thereby transferring knowledge from short-term to durable long-term memory, unlike traditional “exam-driven rote learning,” in which superficially acquired material is often forgotten immediately after the test.

The methodology presented in this study is the result of long-term practice-based refinement within programme-level PBL implementation. The criteria and the Driving Question template were iteratively revised on the basis of actual project cycles, student reflections, educators’ observations, Customer feedback, and analysis of project artefacts. From 2018/2019 to 2025/2026, this implementation involved 196 students, 51 qualitatively distinct project problems, 12 educators, and 15 industry and research partners serving as Customers. Additional feedback was obtained from employers who hired graduates trained under this methodology; this feedback indicated strong preparedness and work-related skills among these alumni.

The caliber of these projects is such that several have yielded peer-reviewed publications (Galuza et al., 2019, 2021; Savchenko et al., 2021).

Student reflections indicated the high motivational value of the methodology: an overwhelming majority of respondents consider PBL in the proposed format the optimal way to master applied mathematics, as it provides a direct link between theory and practice.

The scope of the proposed methodology should also be specified. The criteria and template were developed for a programme in which applied mathematics is closely connected with programming, data analysis, and software development. Therefore, the methodology should not be interpreted as directly applicable to all Applied Mathematics programmes without adaptation. In programmes where project outcomes are not expected to take the form of software or software-hardware products, several criteria, especially those related to market-ready deliverables and software development, may require modification.

Thus, within computing-oriented Applied Mathematics programmes, the developed methodology combines a clear methodological framework, pedagogical adaptability, and alignment with real IT-industry demands. It offers a practical approach for supporting the systematic design of PBL Driving Questions under conditions of rapid technological change.

Conclusions

In this study, we have proposed a discipline-specific methodology for selecting and formulating PBL Driving Questions for university students in computing-oriented Applied Mathematics programmes. The key achievements of this study are:

(1) Typical errors commonly made in the formulation of mathematical PBL problems have been identified and categorized through the analysis of numerous existing cases. For each category, it is shown how such errors reduce the effectiveness of PBL.

Overall, the proposed methodology addresses a critical gap in PBL practice by offering a systematic and transferable approach to Driving Question development for computing-oriented Applied Mathematics. It provides educators with a practical framework for systematically and replicably designing PBL Driving Questions.