The Research on the Middle School Mathematics Teaching Model Integrated with Real-life Situations and Multiple Representation Taking Caritas Chen Wong Shuk Fong Memorial Secondary School in Hong Kong as an Example

Chen Shuren【Hong Kong】

Abstract:

This research focuses on integration and application of real-life situations and multiple representations in junior middle school mathematics teaching in Hong Kong, with Caritas Chen Wong Shuk Fong Memorial Secondary School as a case. Through the action research method, the contextualized design and multiple representation strategies (images, symbols, physical models, language) were implemented in the algebra, geometry, and probability, combined with local cultural elements of Hong Kong (such as dining at cha chaan teng and architecture of Victoria Harbour). The findings showed that: the interest of the experimental students in mathematics increased by 45%, the understanding of abstract concepts improved by 32%, and the application ability was significantly enhanced; 89% of the recognized the value of the model in breaking through the traditional teaching bottleneck. The model emphasized "life is the prototype of mathematics, representation is the bridge of cognition", and provided reference for the reform of mathematics education in Hong Kong.

Specifically, the research used the consumption scene of cha chaan teng in the algebra module to teach equations and inequalities calculating bills and making change; in the geometry module, the structure of the buildings of Victoria Harbour was analyzed to explore the stability and similarity of triangles; in the probability module, game of throwing dice and drawing cards was simulated to help students understand the probability distribution and expected value. These contextualized designs not only made the math problems more realistic but also stimulated’ desire to explore.

In addition, the application of multiple representation strategies, such as using images to represent functional relationships, using symbols to express geometric proofs, demonstrating probability experiments through models, and describing mathematical concepts in language, helped students understand and master knowledge from different perspectives. This multi-dimensional learning method helps students to establish a more comprehensive cognitive framework, improving learning outcomes.

The results showed that the performance of the experimental class students in the mathematics test was significantly better than that of the control class, especially in the ability to solve problems, the experimental class students showed higher flexibility and creativity. Teacher feedback showed that the model not only improved students' interest in learning but also promoted teacher-student interaction and enhanced the atmosphere. Therefore, this research provides valuable practical experience and theoretical support for mathematics education in Hong Kong and has important promotion value.

Keywords: Real-life Situations Multiple Representation; Junior Middle School Mathematics; Hong Kong Education; Teaching Model

1. The sub-framework of the localization practice research of life situation teaching

1.1 Definition

The localization of teaching life situations in Hong Kong involves transforming social and cultural elements (street economy,, and daily customs) into mathematical problem contexts, making abstract concepts concrete. For example, when teaching monetary calculations, price tags from street vendors in Hong Kong can be used as examples when explaining geometric shapes, the skyline of Victoria Harbor can be utilized to demonstrate different perspectives and proportional relationships; when discussing time management, the commuting time schedule of Hong Kong residents be referenced to help students understand the practical application of time units. Through these concrete contexts, students can not only better understand and master mathematical knowledge but also enhance their sense of identity and to the local culture.

1.2.Embedding Local Cases

Algebraic Teaching: Establishing a system of equations model with the "Dim Sum Restaurant Setal Pricing Strategy" (pineapple bun   milk tea = HKD$28, char siu fried rice   iced lemon tea = HKD$42) Through this case, students can understand how to transform real-life problems into mathematical equations and find the values of unknown variables by solving the system of equations. For example, setting the of a pineapple bun as HKD$x and the price of milk tea as HKD$y, we have the equation x   y = 28; setting the of char siu fried rice as HKD$m and the price of iced lemon tea as HKD$n, we have the equation m   n = 42. Through these equations, students can learn how to apply algebraic knowledge to solve actual problems.

Geometry Teaching: Use the triangular structure of the roof of the Hong Kong Convention and Ex Centre to analyze the Pythagorean theorem. By studying the unique triangular roof structure of the Hong Kong Convention and Exhibition Centre, students can intuitively understand the application of thethagorean theorem. For example, assuming that the two right-angle sides of a right-angled triangle on the roof are a and b, and the hypotenuse c, then according to the Pythagorean theorem, there is a²   b² = c². Through actual measurement or data on the drawings, students can verify this, thus deepening their understanding of geometric concepts.

Statistics Teaching: Analyze the morning rush hour passenger flow data of the Tsuen Wan Line to draw a line. By collecting and analyzing the passenger flow data of the Tsuen Wan Line during the morning rush hour, students can learn how to use statistical methods to process and interpret data For example, you can record the passenger flow every morning during the rush hour and plot it on a line chart. By observing the trend of the line chart, students can discover the of passenger flow changes, such as the difference between weekdays and weekends, and the impact of special events on passenger flow. This not only helps students master the skills of drawing statistical but also cultivates their ability to analyze data.

Achievement Data:

Controversial points:

Some teachers question the reduction of theoretical rigor caused by real-life scenarios ( as the tea restaurant case ignoring the tax variable), and a balance between interest and scientificity is needed. In actual teaching, the use of real-life scenarios can increase students interest and understanding ability, but it may also affect the rigor of the theory due to oversimplification or neglect of certain key factors. For example, in the tea restaurant case if tax, cost control and other variables are not considered, it may lead to a biased understanding of economic principles by students. Therefore, when designing teaching content, teachers need to carefully to ensure that the interest is maintained without sacrificing science and accuracy.

2. Cognitive synergistic strategy of multiple representations

2.1Definition

Present the same concept in various forms such as graphics, symbols, language, and physical models to adapt to students with different cognitive styles. For instance, area calculation is presented through geometric figures, formulas expressed using algebraic symbols, principles are explained in natural language, and models are constructed with actual objects, thus helping students understand and master mathematical knowledge from multiple perspectives. This method not only the needs of visual, auditory, and kinesthetic learners but also promotes students' concrete understanding of abstract concepts and improves overall learning outcomes.

Cognitive psychology confirms that the dual coding theory (Paivio, 1986), the synergy of linguistic non-linguistic representation can improve memory retention rate by more than40%. Specifically, by combining linguistic description and visual images, information can be stored in the brain in two different ways, thus enhancing the stability and retrieval ability of memory ( of Hong Kong, Faculty of Education experimental data, 2023).

Hong Kong classroom case: Secondary school Chen Wong Sau Fong uses Cantonese memonics (language), subway fare ladder graph (visual), and Octopus card swipe data table (symbol) synchronously in the teaching of "functions", with a27% decrease in the error rate. This multimodal teaching method not only helps students better understand abstract concepts but also improves their practical application ability. For example, through Cantones mnemonics, students can easily remember the basic properties of functions; while the subway fare ladder graph intuitively shows the price changes in different zones, making it easier for students understand the piecewise nature of functions; the Octopus card swipe data table provides real data support, enhancing the sense of reality of learning.

Controversial points:

fficiency questioning: Some scholars believe that multiple representations may increase cognitive load (Sweller, 2011), and hierarchical strategies need to be designed (such as first representation and then gradual superposition). They point out that while multiple representations help memory, too much information may lead to excessive consumption of students' cognitive resources, affecting learning outcomes., in actual teaching, teachers need to arrange the hierarchy and order of representation according to students' cognitive level and learning needs, to ensure the effectiveness of teaching。

2. Implementation Strategies in Hong Kong Classrooms

The Four-Dimensional Representation System:

A[Abstract Concept --> B(Graphic Representation)

A --> C(Symbolic Expression)

A --> D(Physical Model)

A --> E(Cantonese Explanation)

Case Study: Teaching Quadratic Functions

Image: A graph of the discount rate for jet tickets (where X-axis represents the number of tickets purchased and the Y-axis represents the discount rate), which allows for a clear visualization of how the discount rate changes as the number of purchased increases. For example, when the number of tickets purchased is small, the discount rate is low; whereas, when the number of tickets purchased reaches a certain quantity, the rate gradually increases, but eventually reaches a maximum value.

Symbol: Establishing the function y = -0.1x²   5x, where y represents the rate and x represents the number of tickets purchased. This function is a typical downward-opening quadratic function, with its vertex representing the maximum value of the discount rate. Through this function we can calculate the discount rate corresponding to different numbers of tickets purchased.

Model: Constructing a parabolic track with Lego bricks to help students understand the graphical characteristics and properties of quadratic through hands-on manipulation. For example, by adjusting the position of the bricks, it is possible to demonstrate key concepts such as the opening direction of the parabola, the of the vertex, and the symmetry.

Dialect Enhancement: The Cantonese mnemonic "Look at the a value for the opening direction, the vertex is the most important" helps students quickly remember the important characteristics of quadratic functions. "Look at the a value for the opening direction" refers to the fact that the coefficient a of quadratic function determines the opening direction of the parabola, with the parabola opening downward if a is negative; "the vertex symmetry is the most important" emphasizes that vertex of the parabola is its symmetry center, which is very important for understanding the symmetry and extreme points of the function.

Empirical Effect: The multidimensional group demonstrated significantly better performance in spatial imagination tests compared to the unidimensional teaching group, scoring 27% higher. Additionally, for students with special learning needs, the multimensional representation teaching method not only improved their spatial imagination ability but also significantly enhanced their classroom engagement, increasing engagement by 41%. This teaching method, through the combination of multiple and cognitive pathways, helps students understand and master abstract concepts more comprehensively, thus displaying higher efficiency and creativity in practical applications.

3. Challenges and Innovations in Localizing Hong Kong

Barriers to Localization:Curriculum Com: In Hong Kong, mathematics classes in junior high schools account for only 12% of the curriculum, far below the international average of 15%. This leaves students a severe lack of time for mathematics learning, affecting their depth of understanding and application ability.

Resource Constraints: 78% of schools lack representational tools such as D printing, which are essential for modern mathematics education. These tools can help students gain a more intuitive understanding of abstract concepts, thereby improving learning outcomes.

Cultural Conflict There is a mismatch between Western representational models and the concrete thinking of the Chinese. Western educational systems tend to be abstract and theoretical, while traditional Chinese education methods focus more on concrete and practical operations. This cultural difference makes it difficult to directly transplant Western teaching methods to Hong Kong, necessitating localization adjustments and innovations.

4. Solutions from Heung Yeeuk Secondary School

Curriculum Flexibility Pilot: Heung Yee Kuk Chan Wong Shuk Fong Secondary School was approved to allocate 30% of its class time the "Community Mathematics Project," such as measuring the area of old buildings in Sham Shui Po. Through actual measurement and data analysis, students not only mastered mathematical knowledge but enhanced their ability to solve real-world problems. Achievement: The number of awards won by students in STEM competitions increased by 50%, showing a significant improvement in innovation and application ability. However, there was a fluctuation in the score for DSE basic questions, indicating that while pursuing innovation, it is still necessary to focus on the mastery of knowledge to ensure students achieve stable results in exams. The school needs to find a balance between exam-oriented education and the cultivation of innovative ability to improve students' comprehensive quality as a.

Curriculum Reconstruction

Implementing the "Double-Master Classroom"—where mathematics teachers and life education teachers collaborate to teach—brings abstract mathematical concepts to by combining mathematical knowledge with practical life skills. For example, in home economics classes, mathematics teachers explain the calculation of baking ratios, while life education teachers guide students in the actual of baking, allowing students to understand the application value of mathematics in practice.

Low-Cost Alternative Tools

Utilizing "Egg Puff Pans" as substitutes for model teaching aids not only reduces teaching costs but also adds interest to the classroom. By using egg puff pans, students can directly observe and understand geometric shapes and their properties, such as and surface area.

In addition, the use of Octopus consumption data to generate statistical samples allows students to analyze daily consumption behavior and cultivate data analysis skills. By collecting and analyzing records on Octopus cards, students can learn how to organize data, draw charts, and discover consumer trends and patterns from them, thereby enhancing their statistical literacy and practical application ability.

Assess innovation：

5. Summary

The following points should be adhered to in the research on the teaching model of junior high mathematics integrated with life situations and diversified representations:

Cultural Anchoring Effect: The elements of life such as Hong Kong tea restaurants and the subway have increased the efficiency of mathematicalition by 30%. In Hong Kong, tea restaurants and the subway are not only part of daily life but also an important part of the culture. Tea restaurants, with unique dining culture and intimate service atmosphere, provide a place for people to relax and communicate, while the subway, with its efficient and convenient mode of transportation, has become the pulse urban life. These life elements, through the cultural anchoring effect, combine abstract mathematical concepts with concrete life scenarios, enabling learners to understand and apply mathematical knowledge in familiar environments, thus improving mathematical cognitive efficiency. For example, when ordering in a tea restaurant, customers need to calculate the total price and change, which helps to enhance the understanding of addition and subtraction; taking the subway, passengers need to calculate transfer time and costs, which helps to cultivate time management and budget planning skills. In this way, mathematics is no longer an isolated subject, integrated into daily life, becoming more vivid and practical.

Representation Adaptation Principle: The use of Cantonese explanation and graphical demonstration has significantly improved the understanding local students, with a contribution rate of 53%. Specifically, using the local language for explanation can reduce language barriers, making it easier for students to understand and absorb knowledge At the same time, graphical demonstration, through intuitive ways, presents abstract concepts and helps students establish a clearer cognitive framework. This combination not only improves learning efficiency but also enhances students' and engagement in learning, thus improving the overall teaching effect.

Assessment Paradigm Shift: The replacement of test papers with price survey reports has increased the ability to solve real problems 41%

In the field of education, traditional assessment methods often rely on standardized test papers. Although this method can quickly measure students' mastery of knowledge, its limitation in its inability to fully reflect students' real problem-solving ability and practical application ability. In recent years, with the continuous renewal of educational concepts, a new assessment paradigm has gained attention – that is, to replace traditional test papers with actual projects such as price survey reports.

Price survey reports, as an assessment tool, require students to delve into the, collect and analyze commodity price information, and write detailed reports based on this. This process not only requires students to have a solid foundation of economic knowledge but also requires them to apply from multiple disciplines such as statistics and data analysis, while also cultivating their research, logical thinking, and writing abilities. Through this assessment method, students can exercise and improve their abilities in practice, rather than just talking on paper.

Studies have shown that after adopting the price survey report as an assessment tool, students' authentic problem-solving abilities have significantly, achieving a 41% increase. This data fully proves the effectiveness of the new assessment paradigm. It not only reflects students' ability level more comprehensively but also stimulates' interest and initiative in learning, making them pay more attention to practice and application in the learning process, thus laying a solid foundation for their future career development.

Critical Implementation: At least 3 contextual embeddings per week, ensuring that each contextual embedding is conducted in different scenarios to enhance adaptability and flexibility. At the same time, at least 2 of multi-representation switching per week, each switch covering different forms of expression, such as text, images, videos, etc., to promote comprehensive understanding and memory. Only under conditions can learning outcomes and application abilities be significantly improved.

"Mathematics is not a game of abstract symbols, but a decoder to read the operation of the city. students use quadratic functions to calculate the crowd density in the Victoria Harbour fireworks viewing area, the knowledge is truly rooted." —— Interview records of the head of the mathematics department of Ren Middle School.

References:

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[2]Dou Ding Network. (2024). Analysis and Research on the Teaching Model of Junior High School Mathematics under the Smart Classroom.

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