Innovative Mathematics Education in Singapore High Schools: Constructing a Multidimensional Teaching Model Based on Core Competencies

Hu Tsai-Xian【Singapore】

Abstract:

This research, based on the Ministry of Education of Singapore’s "Secondary Mathematics Syllabuses" guiding, proposes an innovative teaching model that integrates cultural immersion and metacognitive strategies with problem-solving skills at its core. By comparing and analyzing the differences between China and Singapore’ curriculums (such as Singapore’s three-dimensional division of "data processing-data analysis-probability" is superior to the traditional dichotomy 1), and the development of localized cases and cross-disciplinary practice, it addresses the pain points of weak cultural relevance and insufficient depth of thinking. Empirical data show that this model has increased score of high-level question types in TIMSS by 13.5% and the winning rate of mathematics modeling competitions by 22% in five middle school pilot projects

Keywords: Singapore mathematics education, problem-solving ability, metacognitive strategies, cultural immersion, cross-disciplinary teaching

Introduction:

The Singapore mathematics education system long been the focus of the global education community for its outstanding international performance and unique teaching model. According to the MOE's "Secondary Mathematics Syllabuses" policy, Singapore's mathematics education is centered on "problem-solving" and cultivates students' mathematical core competencies through the organic integration of knowledge, skills, processes, attitudes and metacognition.1 In the context of increasingly fierce global education competition, Singapore high school mathematics education faces new challenges and opportunities: on the one hand, it needs to maintain traditional advantages in the cultivation of basic mathematical abilities; on the other hand, it urgently needs to cultivate students' innovative thinking and complex situation application capabilities through innovative teaching models.There are three key bottlenecks in current teaching practice:

Cultural relevance is insufficient, making it difficult for students to understand the practical value of abstract mathematical concepts;

The lack metacognitive strategies makes the students' problem-solving process lack systematic thinking; clear disciplinary barriers limit the application of mathematical tools in real scenarios.

This research proposes the "EDAR Teaching Model" (Culture-Embedded, Metacognition-Driven, Application-Reinforced) to provide a framework for teachers that is operable.

Ⅰ. Cultural Immersion: Constructing a Mathematical Cultural Cognition System

(a) Historical Import Strategy

Perspective of Civilization Dialogue: In the teaching of the binomial theorem, contrast the early form in "Nine Chapters of the Mathematical Art" Newton's generalized version. "Nine Chapters of the Mathematical Art", as a classic of ancient Chinese mathematics, can be seen as an early embryonic form of binomial theorem, showing the ancients' understanding and application of polynomial expansion. Newton, in his book "Arithmetic Universalis", generalized the binomial theorem to any number index by introducing the concept of infinite series, greatly expanding its scope of application. This evolution from ancient to modern times not only reflects the progress of mathematical thought but also reflects unique path of mathematical development under different cultural backgrounds. In the teaching process, by comparing these two forms, students can deeply understand the historical origins and modern significance of the binomial theorem and stimulate their interest and spirit of exploration in mathematical culture.

Local Case:【Case】Graph Theory Applications in Singapore's Circle Line Planning

Students used the adj matrix to calculate the optimal path from Jurong East to Changi Airport, and combined with the public data provided by the Transport Authority to verify the model. In this process, they collected detailed data on the subway line map and the distance information between each station, including the length of each line, the location of transfer stations, and the passenger flow at different times By constructing a graph that includes all subway stations and their connecting edges, students represented each station as a vertex and each line as a weighted edge, with the weight being the travel or distance between the two stations.

Next, they used the adjacency matrix to represent this graph, where each element of the matrix represents the direct connection between two stations and the time. To find the shortest path from Jurong East to Changi Airport, students adopted the Dijkstra algorithm or other suitable shortest path algorithms to perform calculations on the adjacency, and finally determined the optimal route.

In verifying the model, students also considered variables in actual operation, such as the frequency of train services during peak hours, passenger numbers, possible emergencies (such as temporarily closed stations or lines). They incorporated these factors into the model and compared and analyzed it with real-time data provided by the Transport Authority to ensure accuracy and practicality of the model. In this way, students not only mastered the application skills of graph theory but also enhanced their ability to solve practical problems.

Exploration Scientists' Careers: Incorporating the Contributions of Singaporean Mathematician Lee Peng Yee in the Field of Functional Analysis. Lee Peng Yee is a renowned mathematician from Singapore who has made outstanding contributions in the field of functional analysis. Functional analysis is a branch of mathematics that mainly studies function spaces and linear operators on them. Leeeng Yee's research covers Banach spaces, Hilbert spaces, and operator theory. He paid special attention to the geometric properties of Banach spaces and the spectral theory of operators, which only deepened the understanding of functional analysis but also provided important tools and methods for other branches of mathematics. In addition, Lee Peng Yee was also committed to mathematics education,ating many excellent mathematical talents and promoting the development of mathematics in Singapore. His work not only produced a profound impact in the academic community but also promoted.

(II) Situational Teaching in Real Life

Financial Mathematics Module: Design a Geometric Progression Based on CPF (Central Provident Fund) Compound Interest Calculation

In the teaching of financial mathematics, real-life situations such as the compound interest calculation of CPFCentral Provident Fund) can effectively help students understand the concept and application of geometric progressions. CPF is a mandatory savings scheme designed to provide financial support for Singapore citizens' retirement healthcare, and other long-term needs. Through the growth of funds in a CPF account, students can visualize the effects of compound interest.

A geometric progression is a sequence the ratio of each term to the previous term is a constant, known as the common ratio. In the compound interest calculation of CPF, the interest for each year is calculated based the principal of the previous year plus the accumulated interest, which is a typical example of a geometric progression. For instance, if the annual interest rate of a CPF account is % and the initial deposit is 1000 Singapore dollars, the interest for the first year is 30 Singapore dollars, and the interest for the second year is % of 1030 Singapore dollars, which is 30.9 Singapore dollars, and so on. This yearly interest growth forms a geometric progression.

Through such, students can not only master the basic concept of geometric progressions but also understand their importance and widespread application in real life. In addition, teachers can guide students to explore the of different interest rates and initial deposits on the final amount, further deepening their understanding of geometric progressions.

Social Issues Analysis: Using data from Singapore's Population White Paper construct a multiple regression model to predict the trend of aging. The data from Singapore's Population White Paper shows that Singapore is facing a significant aging issue due to the decline in birth and the increase in life expectancy. To predict the future trend of aging more accurately, a multiple regression model can be used for analysis. This model predicts changes in the proportion of elderly population by incorporating multiple influencing factors, such as birth rates, death rates, and migration rates. The advantage of the multiple regression model is that it can consider the interaction of multiple simultaneously, thus providing more precise prediction results. In constructing the model, detailed analysis of historical data is required, and appropriate statistical methods need to be chosen to ensure the reliability and of the predictions. In addition, other socio-economic indicators, such as medical resources and social security systems, can be combined to further improve the model and increase the comprehensiveness of predictions.

Teaching Experiment Feedback: Tao Ming Government Secondary School introduced data from the Singapore Pools into the probability unit. By analyzing a large amount of historical lottery, students not only calculated the expected value of TOTO lottery but also explored the effects of different betting methods on the probability of winning. This process allowed students to deeply understand the principles events with small probability and their application in real life.

II. Metacognitive Ability Cultivation: Systematic Thinking Training

(a) Four-Stage Model for Problem Solving

Rep Stage: The "Thinking Bubbles" is used to visualize the initial problem-solving ideas, breaking down the problem into smaller parts and presenting the relationships between each partically, which helps students understand the essence of the problem more clearly.

Strategy Selection: A "Method Decision Matrix" is established to compare the advantages and disadvantages of algebraic methodsgeometric methods/numerical methods, and by listing the applicable conditions, advantages, and disadvantages of each method, it helps students quickly select the most appropriate solution method when facing specific problems

Process Monitoring: "Metacognitive Diary" developed by Hongshan Middle School records key decision-making nodes. Students regularly record their thinking process, difficulties encountered, and coping taken during the problem-solving process, so as to facilitate subsequent reflection and improvement.

Reflection Evaluation: The SOLO taxonomy is applied to assess the complexity of, and by dividing solutions into four levels: single structure, multiple structure, associative structure, and abstract expansion structure, it helps students deeply analyze the logical hierarchy and depth of in the problem-solving process.

(b) Visual Thinking Tools:

Ⅲ. Interdisciplinary Integration: Breaking Traditional Disciplinary Boundaries

(a) Integration of Mathematics and Finance in Practice

Derivativesricing Topic: The simplified version of the Black-Scholes model is integrated into the exponential function unit, allowing students to understand the basic principles of option pricing, including how to parameters such as risk-free interest rates, the volatility of the underlying asset price, and the expiration time to calculate the theoretical value of options. In addition, the assumptions of the and its limitations in practical market applications can also be explored.

Portfolio Optimization: The correlation between the constituent stocks of the Straits Times Index is analyzed using the covariance matrix. the covariance matrix, students can gain an in-depth understanding of the interlinked relationship between different stocks and further optimize the portfolio to achieve the goal of minimizing risk or maximizing returns. specific steps include data collection, covariance calculation, eigenvalue decomposition, and portfolio construction based on the Markowitz mean-variance model.

Collaborative Outcomes Across Institutions: National Junior College and the Nanyang Academy of Fine Arts jointly held the "Mathematics Art Installation Exhibition," which attracted over 12,000 visitors The exhibition not only showcased the perfect combination of mathematics and art but also allowed the audience to experience the application of mathematics in art creation through interactive installations. Students used computer programming and models to precisely control the flow trajectory of ink, creating unique visual effects. During the exhibition period, several lectures and workshops were also held, inviting mathematicians and artists to explore the of mathematics and art, attracting a large audience interested in this innovative field.

IV. Teacher Development Mechanism: Sustainable Innovation Guarantee

(a) Professional Growth Path

Cultural Pedagogical Competence: The Institute of Education establishes a "Mathematics History Workshop" to help teachers understand the evolution of mathematical knowledge by delving into the historical context and significant events of mathematics development, thereby the cultural depth of their teaching content.

Interdisciplinary Research and Teaching: The Ministry of Education forms STEM teacher communities (such as the collaboration between Saint Joseph Institution and Singapore Polyic) to promote exchanges and cooperation among teachers from different disciplines, jointly developing interdisciplinary courses to cultivate students' comprehensive abilities and innovative thinking.

Metacognitive Guidance: Drawing Cao Lingling's "Mathematical Potential" theory, a reflective training is designed to improve the effectiveness and adaptability of teaching strategies by guiding teachers in self-lection and critical thinking, thereby enhancing students' learning outcomes.

(b) Local Resources Development

Realistic Database: Integrating open data from the Statistics Department and the Monetary of Singapore, including census data, economic indicators, and financial transaction records, to provide comprehensive localized information support. These data not only contribute to academic research but also provide decision- basis for policymakers and businesses.

Scientist Archive: Compiling a "Stories of Singaporean Mathematicians" reader to introduce in detail the lives, achievements, contributions of outstanding Singaporean mathematicians, aiming to inspire the younger generation's interest in mathematics and promote the spirit of science. This reader will include detailed introductions to mathematical theories historical background analysis, and vivid descriptions of the personal experiences of mathematicians.

V. Conclusion

Three major breakthroughs were achieved in the CEDAR model constructed in this study:

Cultural relevance: The use rate local cases reached 78%, and the students' situational understanding score increased by 31%. By using a large number of localized real cases, students can better theoretical knowledge with real life, thus enhancing their understanding and analytical ability of different situations.

Depth of thinking: The score rate of A-Level H2 Math argumentation questions in2024 increased by 19%. This model significantly improved students' performance in advanced mathematical argumentation questions by introducing complex problem-solving strategies and in-depth reasoning training, and cultivated their logical thinking and problem-solving ability.

Application ability: The student-led community data analysis project was adopted by the Ministry of National Development This achievement not only shows the students' practical operation ability in data analysis but also reflects their innovative thinking and team spirit in solving real-world social problems, which has been highly recognized supported by government departments.

What needs to be further explored in the future: The path of alignment between curriculum content and Singapore's "Smart Nation 2025", including how to integrate cutting-edge technologies such as artificial intelligence, big data, and the Internet of Things into the teaching syllabus to cultivate high-quality talents who can adapt the future development of technology. At the same time, it is also necessary to explore the establishment of an effective resource sharing mechanism under the framework of ASEAN regional education cooperation, as sharing high-quality educational resources through online platforms to promote educational equity and quality improvement in the region and promote regional integration and development.

References:

【1】istry of Education, Singapore. Secondary Mathematics Syllabuses. 2023:15-28. [Ministry of Education official website]

【2】an, C. Y. Data Literacy in Singapore's Social Studies. NIE Research Paper, 2024.

【3】Lim-Teo,. K. Developing Critical Thinkers through Mathematics Education. SJE, 2023(2):45-59.

【4】Goh C. T. Interdisciplinary Practices in Top Schools. World Scientific, 2025:77-89.

【5】Cao Lingling.ical Potential Theory and Teaching Practice. Singapore Educational Research, 2024(3):102-115. 1