Toward a Design Framework for Mathematical Modeling Activities: An Analysis of Official Exemplars in Hong Kong Mathematics Education

Abstract

Mathematical modeling is considered a bridge to STEM education and has been incorporated into K–12 mathematics curriculums in various countries. However, it has a relatively short history in Hong Kong schools. The lack of high-quality, relevant exemplars of mathematical modeling activities is a challenge to teacher practice in this area. Hence, this study aims to establish a design framework for mathematical modeling activities suitable for teachers and students in Hong Kong. We explore the design and content of the official mathematical modeling exemplars published by the Hong Kong Education Bureau using a document analysis approach. The findings provide the basis for developing a framework to be used in the future design of mathematical modeling activities. Four exemplars were found and analyzed in terms of their structural components, level of learning experience in mathematical modeling, and design characteristics. Based on our findings, we discussed various strategies to enhance the design of a mathematical modeling activity, including setting diversified learning objectives, cross-subject collaboration when formulating the problem context and instructions, designing more activities suitable for average and underperforming students, emphasizing the evaluation of modeling outcomes, and providing relevant supporting materials. Our study thus lays the groundwork for advancing the teaching and learning of mathematical modeling in school contexts.

1. Introduction

Mathematical modeling and its applications have been receiving increasing attention worldwide. As Kertil and Gurel [1] described, mathematical modeling is a process of mathematizing, interpreting, verifying, revising, and generalizing real-life situations. It is a cyclical process (i.e., a modeling cycle) in which a real-world model is transformed into a mathematical model, and then the formulated model is validated and iteratively adjusted based on the real-life situation [2]. Mathematical modeling is widely applied in the field of sustainable development across STEM (Science, Technology, Engineering, and Technology) disciplines, such as monitoring water quality [3], waste treatment infrastructure planning [4], sustainable energy systems [5], financial risk assessment [6], and pandemic control strategy [7]. Accordingly, mathematical modeling is considered a bridge to STEM education [1].

Although training in mathematical modeling is most commonly offered at the university level, it has been the foundation for the mathematics framework of the Programme for International Student Assessment (which measures 15-year-olds’ abilities) since 2015 [8]. As the interest in STEM education has increased, mathematical modeling has been incorporated into K–12 mathematics curriculums in various countries, such as Singapore [9] and Australia [10]. Sokolowski [11] conducted a meta-analysis which involved 1670 participants in mathematical modeling research across countries (e.g., USA, Germany, and Turkey). His results suggested that engaging students in mathematical modeling activities significantly promoted their understanding and application of mathematics concepts compared to traditional teaching methods. In particular, the interventions in high school settings showed an overall large effect on student achievement (g = 0.94, 95% CI [0.79, 1.08]) in favor of the mathematical modeling classrooms over the traditional classrooms [11]. This result thus provided the ground for practicing mathematical modeling activities in schools to enhance student learning of mathematics.

So, why do we need to investigate the way of designing mathematical modeling activities? In Hong Kong, there is a relatively short history of teaching and learning mathematical modeling in primary and secondary schools. The 2017 curriculum guide [12] was probably the first to explicitly introduce mathematical modeling into Hong Kong mathematics education. To promote the teaching and learning of mathematical modeling, Ang [9] pointed out that teachers need a set of good, ready-made exemplars of modeling problems and relevant resources. At the time of writing, however, we could find only four official (i.e., published by the Hong Kong Education Bureau) exemplars [12,13,14,15] of mathematical modeling activities that were accessible to teachers. Further effort is required to develop high-quality exemplars and resources relevant to the government’s direction.

Therefore, we synthesized the design characteristics of the official mathematical modeling exemplars in Hong Kong using a document analysis approach, aiming at establishing a design framework for mathematical modeling activities suitable for our teachers and students. Using the established framework, researchers can have a more focused agenda for future research to examine the effect of mathematical modeling activities on students’ ability and interest in mathematics in Hong Kong school contexts. Furthermore, this study can help researchers from other regions to understand how official mathematical modeling activities are designed in Hong Kong, laying the groundwork for further comparative studies involving different educational contexts.

2. Conceptual Background

The conceptual background of this study is threefold. First, we provide an overview of the mathematics education and mathematical modeling in Hong Kong schools. This can help readers understand the curriculum arrangements regarding the mathematics topics involved in this study and the government’s effort in promoting mathematical modeling in recent years. Second, we explicate Ang’s [9] three levels of learning experience in mathematical modeling. Third, we discuss recent design principles for mathematical modeling activities. These are the useful resources to establish the design framework in this study.

2.1. Mathematics Education and Mathematical Modeling in Hong Kong School Contexts

In Hong Kong, mathematics is a compulsory subject in both primary and secondary schools. The mathematics curriculum covers various learning units across different content areas (e.g., algebra and data handling) [12]. Nevertheless, the Education Bureau allows teachers to cater to the needs and abilities of their students through flexibility and diversification. For example, the secondary mathematics curriculum includes both foundation topics (i.e., essential concepts and knowledge) and non-foundation topics (i.e., content beyond the foundation topics). For the non-foundation topics, teachers can evaluate their suitability for and relevance to their students. In addition to the compulsory part of the curriculum, there is an optional extended part in the senior secondary school mathematics curriculum [12]. The extended part consists of two modules: Module 1 (Calculus and Statistics) and Module 2 (Algebra and Calculus). Students who wish to learn more advanced mathematics can take either Module 1 or Module 2. To illustrate how mathematics topics are arranged in the curriculum,

Table 1. Mathematics topics of three learning units in Hong Kong mathematics curriculum [16,17].

	Compulsory Part
Learning Unit	Foundation Topics	Non-Foundation Topics	Extended Part
Probability	The concept of probability; the calculation of probabilities of events by listing the sample space and counting; etc.	The concept of expectation; the addition law and multiplication law of probability; the concept and notation of conditional probability; etc.	The concept of discrete probability distribution; the concepts of expectation E[X]; etc.
Exponential and logarithmic functions	None.	The definition and properties of logarithms; the properties of exponential functions and logarithmic functions; etc.	The definition of e and the exponential series; using exponential functions and logarithmic functions to solve problems; etc.
Differentiation	None.	None.	The addition rule, product rule, quotient rule, and chain rule of differentiation; solving the problems relating to rate of change, maximum, and minimum; etc.

provides examples of several topics in three learning units (probability, exponential and logarithmic functions, and differentiation) [16,17]. These units are involved in the official exemplars retrieved in this study, and thus can help readers understand the analysis of the exemplars.

The teaching and learning of mathematical modeling are at an initial stage in Hong Kong. Mathematical modeling was not explicitly mentioned in the government’s mathematics curriculum documents until 2017. In response to the evolving changes in the contemporary world, the most recent mathematics curriculum guide for Primary 1 to Secondary 6 [12] has various updates, including an emphasis on STEM education and an introduction to mathematical modeling. It states that “STEM education could be strengthened through creating opportunities for students to apply the mathematical knowledge and skills in analysing and modelling real-life problems” [12] (p. 43). To this end, the curriculum guide recommends organizing STEM-related activities to provide students with “more opportunities to participate in mathematical modelling through identifying, formulating and solving the problem” [12] (p. 44). Therefore, topic- and project-based approaches to integrating learning elements from different key learning areas (e.g., science education and technology education) are proposed (see [12] (pp. 70–75) for a review). In other words, mathematical modeling can be (1) incorporated into the teaching and learning of specific topics inside the classroom and/or (2) conducted outside the classroom as a project-based learning activity.

In each of the last three academic years (i.e., 2019/20, 2020/21, and 2021/22), the Education Bureau implemented a seed project to promote STEM education by infusing mathematical modeling into secondary school mathematics education [18,19]. The implementation of these three consecutive seed projects reflects not only the Education Bureau’s determination to promote STEM education through mathematics but also the substantial demand for professional development opportunities and school support. However, Hong Kong lacks established teacher professional development programs on mathematical modeling, unlike nearby regions. For example, the Singapore Ministry of Education and mathematics educators from the National Institute of Education in Nanyang Technological University have collaborated to provide professional development in designing and implementing mathematical modeling activities in schools since 2010 [20]. To enhance teachers’ professional capacity in promoting STEM education through mathematics, The Education University of Hong Kong has launched a new professional development program on mathematical modeling (to be offered in 2023) for in-service teachers. Hence, further effort is required to investigate how to advance the teaching and learning of mathematical modeling in Hong Kong schools.

2.2. Three Levels of Learning Experience in Mathematical Modeling

To help teachers plan and set goals for a mathematical modeling activity, Ang [9] divided the learning experience in mathematical modeling into three levels of cognitive demands and expectations. Using this taxonomy, teachers have a clear trajectory to build their students’ modeling capacities by moving from one level to the next. The three levels of learning experience are summarized below [9] (pp. 60–69):

• Level 1 focuses on acquiring skills in a modeling context. The skills are either purely mathematical skills or some specific skills that are used in mathematical modeling. Ang [9] observed that in Singapore, Level 1 modeling activities are trim enough to fit into a typical one- or two-period mathematics lesson. In his “Mountain climbing” activity, students were required to find an exponential function (i.e., a mathematical skill) that fitted real data on atmospheric pressures at various altitudes above sea level. In addition, the students were introduced to the use of Excel’s Solver Tool (i.e., an IT skill) in finding the value of each parameter in their model.
• Level 2 focuses on developing students’ modeling competencies in applying knowledge specific to mathematical modeling. The activity objectives at this level are to help students learn to make assumptions that simplify a problem, identify the factors that influence a variable, and interpret a mathematical solution in real-world terms, among other modeling skills. Ang [9] cautioned that the Level 2 modeling activities are more advanced than the Level 1 activities and thus require more instructional time. In his “Water warming” activity, a cup of ice-water was left to warm up, and its temperature was recorded every 5 s. Students developed a model that used their knowledge of differential equations to describe how the water temperature changed over time. Through this activity, teachers helped their students learn to state the factors that can affect water temperature and to make relevant assumptions about the warming process.
• Unlike Level 2 (which develops students’ modeling competencies), the overarching objective of Level 3 is to tackle a mathematical modeling problem. Students are required to work in groups and apply various modeling skills, such as developing a model, solving the model, and making a presentation. The activities at this level further develop the students’ modeling competencies, and they may take a few days to complete. In his “Accident at the MRT [Mass Rapid Transit] station” activity, Ang’s [9] students were given the scenario of a girl who accidentally fell onto the tracks after walking in a random manner on the platform. The students were expected to communicate their ideas and construct a simulation model to study her random walk by (1) listing the variables in the model, (2) making assumptions about the situation and simplifying the problem, and (3) designing and carrying out the simulation.

2.3. Design Principles for Mathematical Modeling Activities

In addition to students’ levels of learning experience in mathematical modeling, some researchers have considered the instructional dimension when designing mathematical modeling activities [21,22]. Geiger et al. [21] generated a set of task design principles based on both the seminal work of Galbraith [22] and their collaboration with frontline teachers. The researchers adopted and enriched five of Galbraith’s [22] six principles, namely relevance and motivation, accessibility, feasibility of approach, feasibility of outcome, and didactical flexibility principles (Principles 2 to 5 and 7,

Table 2. Design principles for mathematical modeling activities based on Geiger et al. [21] (p. 324) and Galbraith [22] (p. 235).

Principle	Description
Principle 1—Nature of problem	Problems must be open-ended and involve both intra- and extra-mathematical information. The degree of open-endedness depends on students’ previous experience with modeling.
Principle 2—Relevance and motivation	There are some genuine links with students’ everyday lives. Therefore, the problem context must be a part of their everyday experience or related to their personal circumstances.
Principle 3—Accessibility	It is possible to identify and specify mathematically tractable questions from a general statement. Suitable sub-questions can be implied by the general problem.
Principle 4—Feasibility of approach	The formulation of a solution process is feasible and involves (a) the use of mathematics available to students, (b) the making of necessary assumptions, and (c) the assembly of necessary data.
Principle 5—Feasibility of outcome	It is possible for students to solve the mathematics for a basic problem and interpret the results.
Principle 6—Feasibility of evaluation	An evaluation procedure is available that enables students to check for mathematical accuracy and the appropriateness of the solution in the contextual setting.
Principle 7—Didactical flexibility	The problem is structured into sequential questions that retain the integrity of the real-world situation. These questions are either given as occasional hints or used to provide organized assistance by scaffolding a line of investigation.

). Based on the teachers’ feedback, Geiger et al. [21] revealed the need to emphasize the nature and authenticity of the problems (i.e., open-ended real-world problems). Therefore, they formulated the nature of problem principle (Principle 1, Table 2) which emphasizes the open-endedness of a problem and the involvement of both intra-mathematical information (i.e., pure mathematics) and extra-mathematical information (i.e., information outside of mathematics). For brevity, Niss and Blum [23] noted that a situation in the extra-mathematical domain may simply be referred to as “context.” In contrast, intra-mathematical problems are pure mathematical tasks that are not connected to reality [24].

It is worth nothing that one of Galbraith’s [22] principles of model evaluation was not included in Geiger et al. [21]. That principle emphasizes the feasibility of checking the mathematical accuracy and appropriateness of students’ solutions in real-world settings. In line with Galbraith [22], we regard model evaluation as an essential procedure because mathematical modeling is a cyclical process in which a formulated model is validated and iteratively adjusted based on the real-world situation [2]. Therefore, we include Galbraith’s [22] feasibility of evaluation principle (Principle 6, Table 2) in our analytical framework.

After synthesizing Geiger et al. [21] and Galbraith [22], Table 2 shows seven design principles which we drew as the foundation for developing a design framework relevant to Hong Kong school contexts. For example, if we wish to adopt the “Accident at the MRT station” activity [9], the relevance and motivation principle (Principle 2) suggests that we should use similar incidents in our own region to contextualize the problem for our students. Furthermore, the feasibility of approach principle (Principle 4) emphasizes the consideration of our students’ mathematics backgrounds. Taking the “Mountain climbing” activity [9] as an example, if the modeling process involves only linear functions (a foundation topic in Hong Kong) instead of exponential functions (a non-foundation topic in Hong Kong), more students can manage and formulate a solution.

3. Methods

Although Hong Kong teachers lack established guidelines for designing mathematical modeling activities, the Education Bureau has published several exemplars which can inform our design. Accordingly, a document analysis approach is suitable for identifying the structural components (i.e., the parts or sections that the exemplars comprise) and design characteristics underpinning those exemplars. With the above conceptual background, the following research questions (RQ1 to RQ3) guided our study:

• RQ1: What structural components do the official exemplars comprise?
• RQ2: Which levels of learning experience in mathematical modeling do the official exemplars provide?
• RQ3: How can the design principles for mathematical modeling activities be enacted in the context of Hong Kong mathematics education?

The outcomes solicited from RQ1 can be used to formulate a template for teachers’ activity plan, whereas the findings of RQ2 and RQ3 can inform the way of setting learning objectives and offering instructions of a mathematical modeling activity, respectively.

3.1. Document Analysis

Document analysis is a qualitative research method used for reviewing and evaluating documents to elicit meaning, gain understanding, and develop empirical knowledge [25,26]. This research approach has been widely applied in the field of sustainable development [27,28,29,30,31,32] and mathematics education [33]. Depending on researchers’ objectives, their documents can take a variety of forms, such as policy documents [27,28,29], curriculum documents [30], community sustainability plans [31], and media articles [32]. For example, Yang [33] analyzed three textbook chapters on the Pythagorean theorem. She identified their common patterns to develop an understanding, compared their instructional approaches, and made appropriate recommendations. Through document analysis, researchers can further generate theories to explain phenomena and guide their actions [25].

According to Bowen [25], document analysis procedures entail first retrieving and selecting relevant documents and then appraising (i.e., making sense of) and synthesizing the data contained in those documents. These procedures are detailed in the following sub-sections.

3.2. Retrieval and Selection of Documents

To retrieve relevant documents, we searched the official website of the Hong Kong Education Bureau and examined the curriculum documents for mathematics education. As Mhlanga [32] noted, an advantage of this approach is that the documents are available in the public domain, making it easier for researchers and readers to obtain information without the need to ask permission from the original authors. The following four factors were considered when selecting exemplars of mathematical modeling activities [26] (pp. 71–72):

• Authenticity: the extent to which a document is genuine.
• Credibility: the extent to which a document is free from errors.
• Representativeness: the extent to which a document is typical.
• Meaning: whether the evidence in a document is clear and comprehensible.

As of 20 June 2022 (the date on which we finalized our search), we had found four exemplars.

Table A1. Documents reviewed in this study.

Documents/Sources * (Year of Publication, If Any)	Exemplars	Included Exemplars
Mathematics Education Key Learning Area Curriculum Guide (Primary 1—Secondary 3) (2002)	N = 13	N = 0
Mathematics Education Key Learning Area Curriculum Guide (Primary 1—Secondary 6) (2017)	N = 22 **	N = 1
Mathematics Key Learning Area—Pure Mathematics Curriculum and Assessment Guide (Advanced Level) (2004)	N = 3	N = 0
Target Oriented Curriculum Programme of Study for Mathematics—Key Stage 1 (primary 1–3) (1995)	N = 5	N = 0
Target Oriented Curriculum Programme of Study for Mathematics—Key Stage 2 (Primary 4–6) (1995)	N = 7	N = 0
(Website) Resources—STEM Examples: Examples on STEM Learning and Teaching Activities	N = 29 ***	N = 3

Note. * Other reviewed documents without exemplars included: Checklist of major updates on Mathematics Curriculum and Assessment Guide (S4–6) (November 2015); Comparison between Syllabuses for Secondary Schools—Mathematics (Secondary 1–5) (1999) and Syllabuses for Secondary Schools—Syllabus for Mathematics (Forms I–V) (1985); Comparison of Syllabuses for Primary Schools—Mathematics (1983) and Mathematics Education Key Learning Area—Mathematics Curriculum Guide (P1–P6) (2000); Comparison of Target Oriented Curriculum Programme of Study for Mathematics (1995) and Mathematics Education Key Learning Area—Mathematics Curriculum Guide (P1–P6) (2000); Comparison of the Content of the Revised Mathematics Curriculum and Current Mathematics Curriculum; Consultation on the Revised Mathematics Curriculum (P1–S6) (held from March to May 2017); Explanatory Notes to Junior Secondary Mathematics Curriculum (2020); Explanatory Notes to Primary Mathematics Curriculum—Key Stage 1 (2018); Explanatory Notes to Primary Mathematics Curriculum—Key Stage 2 (2020); Explanatory Notes to Senior Secondary Mathematics Curriculum—Module 1 (with updates in August 2018); Explanatory Notes to Senior Secondary Mathematics Curriculum—Module 2 (with updates in August 2018); Explanatory Notes to Senior Secondary Mathematics Curriculum—Module 1; Explanatory Notes to Senior Secondary Mathematics Curriculum—Module 2; Explanatory Notes to Senior Secondary Mathematics Curriculum—The Compulsory Part (with updates in December 2021); Explanatory Notes to Senior Secondary Mathematics Curriculum—The Compulsory Part; Guidelines on Catering for Learner Diversity and Creating Space in Senior Secondary Mathematics (2021); Information Sheet “Summary of Changes to the Contents of Syllabuses for Secondary Schools—AL Applied Mathematics (1992)”; Mathematics Curriculum and Assessment Guide (Secondary 4–6) (2007); Mathematics Curriculum and Assessment Guide (Secondary 4–6) (with updates in December 2017); Mathematics Curriculum and Assessment Guide (Secondary 4–6) (with updates in November 2015); Mathematics Education Key Learning Area—Additional Mathematics Curriculum Guide (S4–S5) (2001); Mathematics Education Key Learning Area—Mathematics Curriculum Guide (P1–P6) (2000); Recommended implementation timeline for the revised Mathematics curriculum; Recommended implementation timeline for the revised senior secondary Mathematics curriculum; Supplement to Mathematics Education Key Learning Area Curriculum Guide: Learning Content of Primary Mathematics (2017); Supplement to Mathematics Education Key Learning Area Curriculum Guide: Learning Content of Junior Secondary Mathematics (2017); Supplement to Mathematics Education Key Learning Area Curriculum Guide: Learning Content of Senior Secondary Mathematics (2017); Supplementary Notes on Teaching of Advanced Supplementary Level—Mathematics and Statistics (1998); Supplementary Notes to Senior Secondary Mathematics Curriculum; Syllabuses for Primary Schools: Mathematics (1973); Syllabuses for Primary Schools: Mathematics (1983); Syllabuses for Secondary Schools—Additional Mathematics (Secondary 4–5) (1992); Syllabuses for Secondary Schools—Applied Mathematics (Advanced Supplementary Level) (1998); Syllabuses for Secondary Schools—Applied Mathematics (Advanced Level) (1992); Syllabuses for Secondary Schools—Mathematics & Statistics (Advanced Supplementary Level) (1991); Syllabuses for Secondary Schools—Mathematics (Secondary 1–5) (1999); Syllabuses for Secondary Schools—Mathematics (Secondary 1–5) (1999); Syllabuses for Secondary Schools—Pure Mathematics (Advanced Level) (1992); Syllabuses for Secondary Schools—Syllabus for Mathematics (Forms I–V) (1985). ** The Exemplar of “Flippable Measure Spoons” remarks that it involves “the ideas of mathematical modelling” [12] (p. 181). However, it appears that the primary focus of this exemplar is not highly related to mathematical modeling. Therefore, the exemplar was not included in our analysis. *** A slightly revised version of Exemplar 1 (“Modelling the spread of a disease,” as published in the curriculum guide for mathematics education [12]) is also published on this website [34].

in Appendix A shows the documents reviewed and the number of exemplars in each document. First, these exemplars are authentic because they are primary sources retrieved from the Education Bureau. As

Table 3. The official exemplars analyzed in this study.

ID	Title	Source
Exemplar 1	“Modelling the spread of a disease”	Curriculum guide [12]
Exemplar 2	“Mathematical modelling on the accommodation demand of visitors to Hong Kong”	Official website [13]
Exemplar 3	“Mathematical modelling on decision-making: Probabilistic model”	Official website [14]
Exemplar 4	“Investigation on the relation between the maximum walking velocity and the length of legs”	Official website [15]

shows, Exemplar 1 is published in the curriculum guide for mathematics education [12] (pp. 189–194), whereas Exemplars 2 to 4 [13,14,15] are published on the “Resources—STEM Examples: Examples on STEM Learning and Teaching Activities” website of the Education Bureau [34]. Notice that a slightly revised version of Exemplar 1 is also published on that website [35]. Second, these exemplars are credible because they were produced by the Education Bureau, and we retrieved the original documents. Third, these exemplars are representative because at the time of writing, they were the only available exemplars of mathematics modeling activities designed and published by the Education Bureau. Fourth, these exemplars are clearly written and understandable. Therefore, the four exemplars that we found were suitable documents for our analysis.

3.3. Data Analysis

We addressed our research questions through content analysis of the exemplars. To answer RQ1, we identified and summarized the structural components of the exemplars. To answer RQ2, we used the three levels of learning experience in mathematical modeling [9] presented in Section 2.2 as our framework for analysis. To answer RQ3, we used the design principles for the mathematical modeling activities [21,22] detailed in Section 2.3 as our lens through which to analyze the design characteristics of each exemplar. Although these frameworks provided a basis for our content analysis, we were open to refining or adding to them if new levels of learning experience or design principles emerged. To enhance the reliability of our analysis, all of the exemplars were double-coded by the first two authors and reviewed by the third author. In the event of disagreement, we re-examined the exemplars to come to a consensus. Multiple reviews were conducted until we reached perfect agreement.

4. Findings

4.1. RQ1: What Structural Components Do the Official Exemplars Comprise?

Table 4. Structural components of the official exemplars.

Component	Description	Exemplar 1	Exemplar 2	Exemplar 3	Exemplar 4
Title	A title containing the problem context.	√	√	√	√
Key stage	The grade levels of the targeted students.	√	√	√	√
Strand	The content area (e.g., data handling).	None.	√	√	√
Learning unit	The learning unit(s) in which the exemplar is situated.	√	√	√	√
Objective	The objective(s) to be achieved.	√	√	√	√
Prerequisite knowledge	The knowledge required for the mathematical modeling activities.	√	√	√	√
Relationship with other learning areas	The related topic(s) in other subjects.	None.	√	√	√
Background information/scenario	An introduction to the problem context.	√	√	√	√
Number of sub-activities	The number of sub-activities in the exemplar.	3	4	3	3
Description of activities	The descriptions of each sub-activity, including teaching instructions, questions for discussion, and notes for teachers (teaching recommendations, suggested answers/solutions, and further information).	√	√	√	√
Generic skills involved	The generic skills (e.g., critical thinking) that the exemplar requires.	√	None.	√	None.
References	The materials (e.g., articles and websites) related to the modeling problem.	√	√	√	√
Others (appeared once)			List of resources; worksheets.	Information sheet.

summarizes the structural components of the official exemplars. Most of the components were found in all of the four exemplars, including their title, key stage (i.e., grade level), learning unit, objective, prerequisite knowledge, background information or scenario, description of activities, and references. However, only Exemplars 2 to 4 explain the exemplars’ relationships with other key learning areas in Hong Kong secondary education. Exemplar 2 relates to a module in technology education (“Business Environments, Operations & Organisations”) in junior secondary education [36], whereas Exemplars 3 and 4 relate to other subjects (“Business, Accounting and Financial Studies” [37] and “Physics” [38], respectively) in senior secondary education. In addition, only Exemplars 1 and 3 describe the generic skills involved in the activities, such as critical thinking and problem-solving skills. In terms of the number of sub-activities, Exemplar 1 consists of three different parts that discuss a simple epidemic model, a counter plague model, and a challenging problem that requires the application of calculus. By contrast, Exemplars 2 to 4 are divided into a series of three to four interrelated activities. Finally, Exemplars 2 and 3 further provide useful teaching and learning resources, including a list of resources required in the activities, an annex of the information necessary for the modeling, and student worksheets.

4.2. RQ2: Which Levels of Learning Experience in Mathematical Modeling Do the Official Exemplars Provide?

We identified the levels of learning experience in mathematical modeling that the official exemplars provide by making sense of their expectations and instructions. We distinguished between Level 1 (Exemplar 1), Level 2 (Exemplars 2 and 3), and Level 3 (Exemplar 4).

Exemplar 1 is a modeling activity related to the spread of a disease. There are three objectives: (1) “To help students relate STEM education with the real life,” (2) “To let students recognise the mathematics in everyday life and apply information technology to solve problems,” and (3) “To let students recognise mathematics as a powerful tool for planning.” These objectives mainly focus on providing students with Level 1 learning experience in mathematical modeling. Specifically, in the three parts of this exemplar, the students are expected to learn to apply exponential functions (Model 1: a simple epidemic model), probability and expectation (Model 2: a counter plague model), and calculus (a challenging problem) in the context of modeling the spread of a disease. Although assumptions are involved in the modeling process, the methods of making those assumptions are given, along with the initial conditions. For example, “it is assumed that no one dies in 10 months” and “Initially there are 8 healthy people and 2 infected people.” In other words, Exemplar 1 does not emphasize developing students’ modeling competencies (e.g., making assumptions to simplify a problem).

Exemplar 2 is a modeling activity related to the accommodation demand of visitors to Hong Kong. There are two objectives: (1) “To enrich students’ experience in applying mathematics in handling daily life problems” and (2) “To enhance students’ abilities in applying the concepts of percentage in modelling real-life situations.” In a series of four sub-activities, the students are guided to formulate a basic model (i.e., involving only a growth factor in the number of visitors) and then progressively refine the model by considering other factors (e.g., whether or not the visitors would stay overnight). Although the objectives of Exemplar 2 focus on Level 1 learning experience in mathematical modeling, its activities attempt to develop students’ modeling competencies, such as recognizing the assumptions involved in a model (e.g., “What assumptions are made in your model?”) and interpreting a mathematical solution in real-world terms (e.g., “Which assumptions may be vulnerable to hold in real-life scenarios?”). Therefore, Exemplar 2 provides students with Level 2 learning experience in mathematical modeling.

Exemplar 3 is a modeling activity related to the decision-making process of buying a new smartphone. The following objective is stated: “To allow students to understand the applications of probability in modelling real-life scenarios such as decision-making process to make reasonable forecast and nurture students’ entrepreneurial spirit.” In a series of three sub-activities, the students are guided to formulate a basic model (i.e., involving two brands of smartphones) and then progressively refine the model by considering other factors (e.g., whether users are likely to buy a new smartphone) and incorporating additional information (e.g., the probability of a user replacing their smartphone in the next 12 months). Although its objective focuses on the Level 1 learning experience in mathematical modeling, its activities attempt to develop students’ modeling competencies, such as recognizing the assumptions involved in a model (e.g., “the new smartphones of both Brand I, Brand S and other brands are available at roughly the same time”) and identifying factors that influence a variable (e.g., “the users of Brand I smartphone using earlier generations might have different considerations from those using the latest available generation”). Therefore, Exemplar 3 provides students with Level 2 learning experience in mathematical modeling.

Exemplar 4 is a modeling activity related to the relationship between the maximum walking velocity and the length of legs. The following objective is stated: “To explain some phenomena in real-life situations through mathematical modelling.” A more specific direction of investigation is provided in its background information section: “to explore a suitable simple model to express the relation between the maximum walking velocity and the length of legs.” Therefore, this exemplar has some characteristics of Level 3 mathematical modeling activities in which students are expected to tackle the modeling problem. This exemplar includes three sub-activities. In Activity 1, the students are divided into groups and collect data on the lengths of their legs and their corresponding walking velocities. In Activity 2, the students formulate a model (e.g., a linear function or quadratic function) that fits their data. However, we identified several instructions (e.g., “The teacher guides students to extract the data” and the “students in each group examine whether the linear model of function… is suitable to describe the relation”) that facilitate the students’ investigation but might limit the level of their learning experience. Finally, Activity 3 is designed only for students who study senior secondary school physics. The students are guided to use their physics knowledge to formulate a theoretical model of the modeling problem. Their theoretical model is then compared with the modeling outcomes in Activity 2.

4.3. RQ3: How Can the Design Principles for Mathematical Modeling Activities Be Enacted in the Context of Hong Kong Mathematics Education?

We examined the design characteristics underpinning the official exemplars using the framework presented in Table 2.

Table 5. Similar enactments of the design principles in the official exemplars.

Principle (Description)	Exemplar	Representative Quotes
Principle 2—Relevance and motivation. (The problem context is related to students’ everyday lives.)	1	“Bird flu, SARS and Ebola are examples of fatal epidemics that have emerged in a large scale in the past two decades.”
	2	“Tourism industry is a mainstay of Hong Kong’s economy.”
	3	“The activities to be introduced are based on real life scenario on the decision-making process of buying a new smartphone.”
	4	“When you are hurry to somewhere, you may walk very fast.”
Principle 3—Accessibility. (Sub-questions can be implied by the general problem.)	1	“What is the difference if 3 persons are infected at each stage?”
	2	“Please suggest some ways to estimate the number of visitor arrivals to Hong Kong in a whole year.”
	3	“the teacher may… ask students to represent the scenario with a tree diagram with suitably defined events.”
	4	“students in each group examine whether the linear model of function… is suitable to describe the relation between V (their walking velocities) and L (lengths of their legs).”
Principle 5—Feasibility of outcome. (The solution can be addressed by students with relevant knowledge. Suggested questions are provided to guide their interpretation.)	1	“How many steps will it take to infect all the people in the classroom? How about the whole school?”
	2	“When will the room supply be inadequate if all growth rates are unchanged?”
	3	“by using the tree diagram, … Predict the future market share of Brand I and Brand S after their release of new models.”
	4	“According to students’ mathematical knowledge involving functions and indices, students may try the following functions for explorations: V = aL2 + bL + c…; V = abL…”
Principle 7—Didactical flexibility. (Procedures of class activities, notes for teachers, and scaffolded questions are provided.)	1	“Does the epidemic take off or die out in each case?”
	2	“What information is needed for constructing the model?”
	3	“Which group of smartphone users has greater brand loyalty, users of Brand I or Brand S?”
	4	“1. The class is divided into several groups… 2. In each group, every student walks as fast as possible… 3. In each group, the length of the legs of each group member is calculated…”

and

Table 6. Different enactments of the design principles in the official exemplars.

Principle	Exemplar	Description
Principle 1—Nature of problem	1 and 4	The problem is open-ended in terms of mathematical approaches to modeling.
	2 and 3	The problem is open-ended in terms of approaches to making assumptions.
Principle 4—Feasibility of approach	1	The formation of a solution requires knowledge of the concept of probability and expectation (JS; F), exponential functions (SS; NF), and calculus (SS; EP).
	2	The formation of a solution requires knowledge of the concepts of percentage changes and growth rates (JS; F). The websites with the necessary data are provided in the teacher notes and student worksheets.
	3	The formation of a solution requires the concept of probability (JS; F) and the concept and notation of conditional probability (SS; NF). The necessary data are provided in the annex of this exemplar.
	4	The formation of a solution requires knowledge of the concepts of linear (SS; F), quadratic (SS; F), and exponential and logarithmic (SS; NF) functions. Activity 3 also requires knowledge of physics (SS).
Principle 6—Feasibility of evaluation	1 and 3	None.
	2	Evaluation is feasible using the provided data on government websites, such as the websites of the Hong Kong Tourism Commission and the Hong Kong Census and Statistics Department.
	4	The students’ modeling outcomes in Activity 2 are compared with the theoretical model formulated in Activity 3.

Note. JS = junior secondary; SS = senior secondary. EP = topics in the extended part; F = foundation topics; NF = non-foundation topics.

summarize how the seven design principles of mathematical modeling activities are enacted in the exemplars. Table 5 shows that their design characteristics are similar in terms of Principle 2 (relevance and motivation), Principle 3 (accessibility), Principle 5 (feasibility of outcome), and Principle 7 (didactical flexibility). Taking the didactical flexibility principle (Principle 7) as an example, all of the four exemplars provide suggested procedures for class activities, notes for teachers, and scaffolded questions, enabling teachers to assist students’ learning and investigation.

Despite the similarity of certain design characteristics, Table 6 shows that the exemplars are different in their enactments of Principle 1 (nature of problem), Principle 4 (feasibility of approach), and Principle 6 (feasibility of evaluation). First, the nature of problem principle (Principle 1) emphasizes the open-endedness of the problem in a mathematical modeling activity. Although the problems in all four exemplars are open-ended, we found that Exemplars 1 and 4 are different from Exemplars 2 and 3. The former are open-ended in terms of the mathematical approach to modeling. Taking Exemplar 1 as an example, it discusses the following approaches to modeling the spread of a disease: (1) a simple epidemic model using knowledge of exponential functions; (2) a counter plague model using knowledge of probability and expectation; and (3) a theoretical solution using knowledge of calculus. In contrast, only one major mathematical approach is adopted in Exemplar 2 (growth rate) and Exemplar 3 (probability and relative frequency). Nevertheless, these two exemplars are open-ended in terms of their approaches to making assumptions in the modeling process. In a series of activities, the students are guided to consider new variables (e.g., Exemplar 3: “Consider a new smartphone?”) and refine their assumptions (e.g., Exemplar 2: “Which assumptions may be vulnerable to hold in real-life scenarios? Why? How can it be refined so as to improve the accuracy of the model?”) to improve their models.

Second, the feasibility of approach principle (Principle 4) includes three aspects: (a) the use of mathematics available to students, (b) the making of necessary assumptions, and (c) the assembly of necessary data. The major difference between the four exemplars concerns the use of mathematics available to students. Accordingly, the four exemplars target different students. Only Exemplar 2 is suitable for all of the students in Hong Kong because it involves only the foundation topics (e.g., concepts of percentage changes and growth rates) in the mathematics curriculum. Exemplar 3 is suitable for more capable students because it involves non-foundation topics (e.g., the concept and notation of conditional probability). Only some specific groups of students have adequate knowledge to complete all of the sub-activities in Exemplars 1 and 4 (Exemplar 1: students taking the extended part of the mathematics curriculum; Exemplar 4: students taking senior secondary school physics).

Third, the feasibility of evaluation principle (Principle 6) emphasizes that the students should be allowed to evaluate their model in their real-world setting. In Exemplar 2, this evaluation is feasible using the data on government websites, such as the Hong Kong Tourism Commission and the Hong Kong Census and Statistics Department. These websites are provided in both the teacher notes and student worksheets. An evaluation is stated in the teacher notes: “The model [formulated in Activity 1] does not fit the data in 2015, 2016 and 2017.” In Exemplar 4, the students collect data and formulate their models in Activities 1 and 2, respectively. Their modeling outcomes are then compared with the theoretical model formulated in Activity 3. However, Activity 3 requires physics knowledge (e.g., circular motion and centripetal acceleration). Therefore, not all of the students can formulate the theoretical model and make the required comparison. In Exemplars 1 and 3, no explicit statements about model evaluation were identified.

5. Discussion

The findings and implications pertaining to each research question are discussed in the following sub-sections. After that, the limitations of this study are acknowledged with recommendations for future research.

5.1. Structural Components of a Mathematical Modeling Activity Plan (RQ1)

In this study, we first analyzed the structural components of the official mathematical modeling exemplars (see Table 4). By and large, they can provide some basic information (e.g., content area and objective) and the details of their activities (e.g., descriptions of each sub-activity and questions for discussion). However, we found that their expected number of lessons and target students were missing. For example, students in Exemplar 4 have to collect data in an open area (where students can measure the time of walking along a 10 m straight path) and then analyze the data. The activity may require more instructional time. In addition, one of its sub-activities is only suitable for students taking senior secondary school physics, as it requires physics knowledge (e.g., circular motion and centripetal acceleration) to formulate a theoretical model. These are some practical issues that teachers may consider when adopting the mathematical modeling activity in their classrooms. Therefore, duration and target students of the activity should be mentioned in the future. To summarize,

Table 7. A template for a mathematical modeling activity plan.

Section	Component
Basic information	Title
	Key stage (or grade level)
	Strand (or content area)
	Learning unit
	Objective
	Prerequisite knowledge
	Relationship with other learning areas
	Duration (or number of lessons)
	Target students
	Background information (or scenario)
Activities (3 to 4 sub-activities)	Descriptions of each sub-activity
	Questions for discussion
	Note for teachers (including teaching recommendations, suggested answers/solutions, and further information)
Other information	Generic skills involved
	References
Annex	List of resources required
	Information sheets
	Sample worksheets

proposes a template for a mathematical modeling activity plan. Educators can use the template to guide them in designing future activities.

5.2. Setting Diversified Learning Objectives of a Mathematical Modeling Activity (RQ2)

Our findings suggested that the official exemplars are situated at Levels 1 to 3 of learning experience in mathematical modeling, as detailed in Section 4.2. This indicated the feasibility of using all three levels of mathematical modeling activities in Hong Kong schools. However, Ang [9] argued that teachers should progressively build students’ mathematical modeling capacity by moving from one level to the next over time. Therefore, future activity plans should be flexible in terms of their objectives and learning tasks. Using the taxonomy of Ang [9], we suggested that the objectives of future mathematical modeling activities can be threefold (i.e., Levels 1 to 3), with corresponding recommended learning tasks. Teachers can conduct the activities based on their students’ needs and abilities. By doing so, they can better enact the didactical flexibility principle (Principle 7) [21,22]. Using the problem context of Exemplar 1 (originally Level 1),

Table 8. Diversified objectives and learning tasks in Exemplar 1 by level of learning experience in mathematical modeling.

Level	Objective	Learning Task
1	To enhance the students’ skills in applying their knowledge of differential equations in modeling the spread of a disease.	With an infection rate and recovery rate along with other necessary assumptions and initial conditions, the students apply their knowledge to formulate the model.
2	To develop the students’ modeling competencies in simplifying the problem of the spread of a disease.	The students are directed to identify the variables in the model and to make assumptions to simplify the problem.
3	To explore a suitable model to express the spread of a disease.	The students formulate the model based on their knowledge.

demonstrates a possible method of designing a set of diversified objectives and learning tasks that provide different levels of learning experience in mathematical modeling.

5.3. Enactment of Design Principles for Mathematical Modeling Activities (RQ3)

Our findings suggested that the design principles of Geiger et al. [21] and Galbraith [22] were enacted, albeit with some variations, as shown in Table 5 and Table 6. These findings can provide teachers with insights into the design of mathematical modeling activities. Several implications are discussed in the following sub-sections.

5.3.1. Principles 1 to 3: The Need for Cross-Subject Collaboration

In terms of their problem contexts, all of the exemplars involve extra-mathematical information and relate to students’ everyday lives, indicating the enactment of the nature of problem principle (Principle 1) [21] and the relevance and motivation principle (Principle 2) [21,22]. Exemplar 1 is related to mathematical biology, whereas Exemplars 2 to 4 explicitly state their relationship with other learning areas in STEM, including tourism, business, and physics. Accordingly, it is unreasonable to expect that mathematics teachers alone can design the mathematical modeling activities that involve extra-mathematical knowledge. For example, they may not be able to explain the relationship between the established mathematical models and the tourism industry (Exemplar 2) and the real benefits of mathematical modeling in the global smartphone market (Exemplar 3).

As suggested in previous studies, cross-subject collaboration can facilitate the implementation of STEM education in both classroom and extracurricular activity settings [39]. Therefore, the development of mathematical modeling activities requires the involvement of teachers in different subject areas. Teachers in other subjects can help mathematics teachers to both formulate the problem context and design subject-specific instructions. However, it is worth noticing that not all problems can be addressed using the mathematics available to secondary school students. Therefore, mathematics teachers should ensure the enactment of the accessibility principle (Principle 3) [21,22] by evaluating the possibility of identifying and specifying mathematically tractable questions from the formulated problems.

5.3.2. Principles 4 and 5: The Need for More Mathematical Modeling Activities in Foundation Topics and Supporting Data in Modeling

According to the feasibility of approach principle (Principle 4) [21,22] and the feasibility of outcome principle (Principle 5) [21,22], students in a mathematical modeling activity should have adequate knowledge to formulate a solution process and address the solution. However, we found that only Exemplar 2 is suitable for all of the students in Hong Kong (see Table 6). The mathematical modeling activities in other exemplars involve some non-foundation topics (Exemplars 1, 3, and 4) and topics in the extended part of the mathematics curriculum (Exemplar 1). The average and underperforming students may not have adequate knowledge to handle the problems in these exemplars. There is thus a need for designing more mathematical modeling activities in the foundation topics (see Table 1).

Besides mathematics knowledge, the feasibility of approach principle (Principle 4) [21,22] emphasizes the importance of assembling the necessary data in the modeling process. Teachers should be aware that students’ modeling processes can be hindered by the fact that not all real-world data and research reports are publicly available. For example, Exemplar 3 requires the use of marketing research data which may not be accessible to students. Therefore, an information sheet with the data on the global smartphone market is provided for students to formulate their model in the annex of this exemplar. In Exemplar 2, unfortunately, some hyperlinks to necessary data are no longer working. In future mathematical modeling activity plans, teachers can consider appending the essential data required in the modeling process to ensure the enactment of Principle 4 [21,22].

5.3.3. Principle 6: The Need for Strengthening the Evaluation of Modeling Outcomes

Our findings revealed that the enactment of the feasibility of evaluation principle (Principle 6) [22] requires reinforcement in future mathematical modeling activities. As Galbraith [22] noted, students’ modeling outcomes should be tested in their real-world settings. However, we found that Exemplars 1 and 3 do not explicitly emphasize the evaluation of students’ modeling outcomes (see Table 6). For Exemplar 1, such an evaluation is feasible if the students obtain real-world data on the spread of a disease, such as its infection rate and recovery rate, whereas for Exemplar 3, the evaluation is feasible if the students obtain recent data from a marketing research report. Taking the spread of the COVID-19 pandemic as an example, real-world global data are available on the Web. As of 24 June 2022, for example, the recovery rate in Hong Kong was 97.4% [40]. In addition, teachers can ask students to compare the new model to the previously formulated models (i.e., the simple epidemic model and the counter plague model) in terms of their reliability and precision. Using real-world data and relevant instructions, future mathematical modeling activities can emphasize the feasibility of testing, refining, and comparing students’ modeling outcomes.

5.3.4. Principle 7: The Need for Supporting Materials

Finally, we found that some official exemplars can better support the enactment of the didactical flexibility principle (Principle 7) [21,22]. First, Exemplar 2 provides sample worksheets (see Table 4) in which teachers can understand how the modeling problem is structured into sequential activities and questions. The use of such ready-made resources not only decreases frontline teachers’ workloads developing instructional materials but also facilitates their teaching practice [41]. Second, although all of the exemplars list their references, those provided in Exemplar 4 are worth mentioning. Its references [42,43,44] are directly related to the modeling problem in this exemplar (i.e., the relation between maximum walking velocity and the length of legs), which can help teachers prepare for the activity and support their students’ independent exploration of the problem. Therefore, we recommend that these supporting materials be provided in future activity plans for mathematical modeling.

5.4. Limitations and Recommendations for Future Research

There are some limitations of this study. We acknowledge that the number of available official exemplars was limited. Therefore, the proposed set of design principles is preliminary and merits further research for improvement. We recommend that empirical studies be conducted in teacher education and professional development settings to solicit pre-service teachers’ and in-service teachers’ suggestions for enhancing the design framework.

Second, this study employed a document analysis approach. Therefore, our analysis focused only on the content of the official exemplars. Further authentic classroom studies are required to examine whether the mathematical modeling activities can develop students’ abilities and increase their interest in mathematics. Besides the official exemplars, teachers and researchers can use our framework to design more mathematical modeling activities for teaching and research purposes. Future research can thus have a more focused agenda to examine the efficacy of mathematical modeling activities and determine whether the design framework can be applied in other regions at similar grade levels.

Finally, the COVID-19 pandemic has a profound impact on educational systems, where some instructional activities have been transitioned fully online. Future research can examine the teaching and learning of mathematical modeling in different learning environments (e.g., face-to-face, online, and hybrid) and propose updates on the curriculum for sustainable practices [45,46,47].

6. Conclusions

This study analyzed four official mathematical modeling exemplars published by the Hong Kong Education Bureau. Our analysis was underpinned by Ang’s [9] taxonomy of learning experience in mathematical modeling and some existing principles for designing mathematical modeling activities [21,22]. Our findings indicated the feasibility of setting diversified learning objectives in future mathematical modeling activity plans, which enables teachers to conduct the activities based on their students’ needs and abilities. With reference to the official exemplars, we discussed various strategies to ensure a better enactment of the design principles in future mathematical modeling activities, including cross-subject collaboration, designing mathematical modeling activities in the foundation topics, emphasizing the evaluation of modeling outcomes, and providing more supporting materials (e.g., student worksheets and references about the modeling problem in concern). Hence, the design framework proposed is useful for teachers and researchers to design mathematical modeling activities that are suitable for both teachers and students in Hong Kong. This study also serves as a reference for teachers and researchers from other regions, informing both their development of mathematical modeling activities and the methodology of building their design framework.