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

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## Abstract

**:**

## 1. Introduction

## 2. Conceptual Background

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

#### 2.2. Three Levels of Learning Experience in Mathematical Modeling

- 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

## 3. Methods

- 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?

#### 3.1. Document Analysis

#### 3.2. Retrieval and Selection of Documents

- 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.

#### 3.3. Data Analysis

## 4. Findings

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

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

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

## 5. Discussion

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

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

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

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

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

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

#### 5.3.4. Principle 7: The Need for Supporting Materials

#### 5.4. Limitations and Recommendations for Future Research

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

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 |

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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. |

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. |

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] |

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. |

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 = aL^{2} + bL + c…; V = ab^{L}…” | |

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…” |

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. |

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 |

**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. |

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## Share and Cite

**MDPI and ACS Style**

Lo, C.-K.; Huang, X.; Cheung, K.-L.
Toward a Design Framework for Mathematical Modeling Activities: An Analysis of Official Exemplars in Hong Kong Mathematics Education. *Sustainability* **2022**, *14*, 9757.
https://doi.org/10.3390/su14159757

**AMA Style**

Lo C-K, Huang X, Cheung K-L.
Toward a Design Framework for Mathematical Modeling Activities: An Analysis of Official Exemplars in Hong Kong Mathematics Education. *Sustainability*. 2022; 14(15):9757.
https://doi.org/10.3390/su14159757

**Chicago/Turabian Style**

Lo, Chung-Kwan, Xiaowei Huang, and Ka-Luen Cheung.
2022. "Toward a Design Framework for Mathematical Modeling Activities: An Analysis of Official Exemplars in Hong Kong Mathematics Education" *Sustainability* 14, no. 15: 9757.
https://doi.org/10.3390/su14159757