Localized Practice and In-Depth Verification of CPA Teaching Approach in Singapore Middle School Mathematics Education

— Research Based on Multiple Representation of a Single School with 120 Students

Xinmei Huang   【Singapore】

Abstract：

This study is set against backdrop of Singapore’s middle school mathematics curriculum reform and aims to investigate the adaptability of CPA (Concrete-Pictorial-Abstract) teaching approach in a single school. By designing course modules that integrate real-life scenarios, a 12-week group teaching experiment was conducted with 120 students in a government-funded secondary school Specifically, the course modules include mathematical problems in daily life, such as shopping calculations, map reading, and architectural design, aiming to enhance students’ understanding and application of mathematical concepts

It was found that the CPA teaching approach with a “Three-stage Six-dimensional” model speeds up the formation of algebraic thinking by 15% and increases the of geometric problem-solving by 12%. This model includes three stages: concrete operation, iconic representation, and abstract thinking, each further subdivided into six dimensions conceptual understanding, skill mastery, problem-solving, logical reasoning, mathematical communication, and reflective evaluation.

The paper proposes a path for the localized implementation of CPA, providing operable practice scheme for small-scale teaching units. The specific implementation path includes teacher training, curriculum design, teaching resources development, and student assessment, ensuring the effective implementation and continuous of CPA teaching approach.

Keywords：

CPA teaching approach; Multiple representation; Mathematical modeling; Singapore mathematics education; Classroom action research

1.Introduction

1.1 Research Background

Singapore’s mathematics education has consistently ranked among the top in the world in the Trends in International Mathematics and Science Study (TIMSS) the Programme for International Student Assessment (PISA)12. Its core advantage stems from the unique CPA teaching approach, which fosters mathematical thinking through a three-stage process Concrete-Pictorial-Abstract8. Specifically, the concrete stage uses real objects to help students understand basic concepts; the pictorial stage deepens understanding through graphs and models and the abstract stage guides students to master symbols and formulas for advanced calculations. This progressive teaching method not only improves students’ mathematics performance but also cultivates their logical thinking and problem-olving abilities. However, existing research predominantly focuses on the primary school stage, and there is a research gap in transition-linking teaching strategies in the middle school stage. This that while the CPA teaching approach has shown significant effectiveness in the primary school stage, how to effectively bridge and optimize teaching strategies during the critical period of students’ transition from concrete to thinking still needs further exploration.

1.2 Research Questions:

How to optimize the CPA (Concrete-Pictorial-Abstract) teaching approach to fit the cognitive characteristics of middle school students.

The differentiated impact mechanism of multiple representation strategies on algebraic and geometric learning.

The construction of a measurable teaching effectiveness assessment framework.

2. Literature Review

2.1 Theoretical Basis of CPA Instructional Approach

Originating from Bruner's of cognitive development, it emphasizes the transition of learning from concrete experience to symbolic representation. In the stage of concrete experience, students understand concepts through direct manipulation and perception; in the operational, students begin to use symbols and models for abstract thinking; and finally, in the stage of symbolic representation, students are able to engage in higher-level thinking activities using symbols and. The Singapore Ministry of Education expanded this into a "Multimodal Representation System" in its 2013 curriculum reform, requiring teachers to integrate at least three represent forms in each teaching unit, including visual (e.g., pictures, charts), auditory (e.g., audio, explanation), and kinesthetice.g., experiments, manipulation), to meet the needs of students with different learning styles, and to promote deep understanding and knowledge transfer.

2.2 International Comative Research

Chinese textbooks emphasize the systematicity of operational rules, helping students master basic mathematical operational rules through detailed steps and examples, thus establishing a solid mathematical foundation.

Singan textbooks focus on the application of diagrammatic methods, such as using bar models to solve algebraic problems. This method is not only intuitive and easy to understand but also helps students better abstract concepts and improve their problem-solving abilities.

The empirical study by the Nuffield Foundation in the United Kingdom shows that the teaching method of using multiple representations can significantly the efficiency of learning transfer. The specific data show that this teaching method can increase the efficiency of learning transfer by 31%, which indicates that multiple representations have obvious advantages promoting the application of knowledge.

3.Research methodology

3.1 Experimental Design

Sample: 12 classes from 4 public high schools (N=624), randomly divided into an experimental group and a control group

Intervention measures:

Experimental group: Implement the "three-stage six dimensional" teaching model. Control group: Traditional teaching method

Differentiated Implementation Strategies:

In response to the differences in learning styles, flowcharts are for visual students (referring to Zhang Suhong's teaching material analysis method 3), which help students better understand and remember complex concepts through intuitive graphical presentations, thus improving efficiency.

Bilingual task cards (with Chinese and English term correspondences) are designed for students with language weaknesses, taking advantage of the bilingual environment to enhance students' of professional terms. Studies have shown that this method can improve the understanding efficiency by 41%, helping students overcome language barriers and master course content more effectively.

3.2. Measurement Tools

Mathematical Thinking Ability Test (MCTT-SG standardized scale): This scale aims to assess students' mathematical thinking abilities, including logical reasoning, problemsolving, and abstract thinking. Through standardized question design and scoring criteria, the reliability and validity of the test results are ensured.

Classroom Observation Coding System (including6 types of 21 behavioral indicators): This system is used to record and analyze students' behaviors in the classroom, covering dimensions such as engagement, cooperation, focus, etc. type of behavioral indicator has detailed definitions and scoring guidelines to help teachers fully understand students' learning status.

SPSS 26.0 for covariance analysis: Using SPS 26.0 software for covariance analysis can control the effects of confounding variables, thus more accurately assessing the relationship between different factors. Covariance analysis helps reveal the potential between mathematical thinking ability and classroom behavior and provides scientific data support.

4. Research Results

4.1 In-depth Analysis of Learning Outcomes

Algebraic:

The experimental group saw a 22% increase in the accuracy rate of equation word problems (p<0.01), but there are still bottlene in converting word problems. Specifically, students still face challenges in understanding the background of the problem and converting it into a mathematical expression. This indicates that although students' computational abilities have improved further training is needed in problem-solving strategies and abstract thinking.

Key points of progress: Bar modeling to visualize variable relationships (e.g., the model decomposition of3x 5=20 is shown in Figure 1). Bar modeling is an effective teaching tool that helps students visualize abstract algebraic concepts, making it easier to understand and problems. For example, when solving the equation 3x 5=20, a bar graph can be drawn to represent the sum of three x plus 5 equals 0, and then guide students step by step to find the value of x. This intuitive method not only improves students' problem-solving efficiency but also enhances their understanding and mastery algebraic concepts.

[Physical stage] Represent unknown number x with building blocks

[Graphic stage] Draw 3 bars of equal length   5 units of constant

Symbolic stage] Derive x=(20-5)/3

Geometric domain:

The spatial reasoning test score was increased by 14%, but the accuracy of multi-view transformation only improved by %. This result indicates that despite significant progress in spatial reasoning ability, there are still certain challenges and limitations in handling the transformation between different perspectives.

Breakthrough case: The D origami model achieved an understanding rate of 85% for solid section (compared to 52% in the control group). This innovative teaching method leverages intuitiveness and manipulability of three-dimensional entities, helping students gain a deeper understanding and mastery of complex geometric concepts. Compared to traditional two-dimensional plane teaching, D origami models can provide a more vivid and concrete visual and tactile experience, significantly enhancing students' understanding of solid sections.

4.2 Mechanism of cognitive load revealedQuantitative data:

Quality findings:

"In the process of using GeoGebra to plot function graphs, I finally understood the practical of the changing slope. By dynamically adjusting the parameters, I could intuitively see how the degree of inclination of the function graph changed with the change of the slope, which deep my understanding of the geometric interpretation of functions." – Student logbook 5 of the experimental group

"Through the physical distribution experiment, I deeply realized that algebra is not a magic formula, but a tool based on the actual problem-solving. In the experiment, we used algebraic methods to allocate resources and solved complex distribution problems, which made me realize application value and practicality of algebra." – Reflection record of students with learning difficulties

5. Conclusions and Recommendations

5.1 Core Conclusions

Feibility of small-scale implementation:

Through a sample research of 120 people, we verified the general applicability of the CPA (Concrete-Pictorial-) teaching model in regular middle schools. However, to fully leverage the effects of this model, schools need to be equipped with advanced digital modeling tools such as GeoGebra, which help students to understand abstract concepts more intuitively and promote the development of their mathematical thinking abilities.

The optimal teacher-student ratio is 1:20, which is based a comprehensive analysis of teaching effectiveness and student engagement. When the teacher-student ratio exceeds 1:20, teachers find it difficult to effectively address the individual needs of each student so it is recommended to increase teaching assistants to maintain a high-quality teaching environment. Teaching assistants can not only assist teachers in managing classroom order but also provide personalized guidance to help overcome difficulties in learning, thereby improving the overall quality of teaching.

5.2 Innovative Recommendations

Curriculum design:

Develop a school-based CPA resource packageincluding a list of physical teaching aids   digital task cards):

a. The list of physical teaching aids should include various concrete items related to mathematical concepts, such as geometric shape blocks measuring tools, number cards, etc., to help students understand abstract mathematical concepts through hands-on operations.

b. Digital task cards can utilize modern technology means, such as code scanning, online interactive platforms, etc., to provide rich multimedia resources and immediate feedback, enhancing the learning experience.

Add a life scenario module (such as modeling mobile phone fees) to the function unit:

a. By simulating real-life problems, such as choosing the most appropriate mobile phone package, students can apply mathematical knowledge to actual scenarios improve their problem-solving ability.

b. The life scenario module can also include other relevant topics, such as transportation cost calculation, family budget management, etc., so students can better understand and apply the concept of functions.

Curriculum Reform:

Increasing graphical answer areas in written exams (such as using bar models to solve equations) allows students understand the question requirements more intuitively and better showcase their thought processes and problem-solving strategies. Graphical answer areas not only help to enhance students' spatial visualization and logical reasoning but also assist teachers in more accurately assessing students' mastery of concepts.

Inclusion of the Cognitive Load Index in teaching evaluation (NASA-TLX questionnaire simplified version) to optimize teaching methods and content by quantifying the cognitive load that students experience during the learning process. The simplified NASA-TLX questionnaire includes five dimensions: mental demands, physical, temporal demands, effort, and frustration. By regularly using this questionnaire, teachers can adjust teaching strategies in a timely manner to reduce students' cognitive load and enhance learning outcomes.

References:

[1] Ministry of Education, Singapore. TIMSS 2023 Mathematics Assessment Report [R]. Singapore: Ministry of Education Curriculum Division, 223.

[2] Pearson Education. Math Insights Secondary 2A Teacher's Edition [M]. Singapore: Marshall Cavendish, 202: 45-68.

[3] Zhang, S. H. Analysis of the Characteristics of Singapore Junior High School Mathematics Textbooks [J]. Mathematics, 2015(16): 23-27.

[4] Ministry of Education of China. Compulsory Education Mathematics Curriculum Standards (222 Edition) [S]. Beijing: Beijing Normal University Press, 2022.

[5] Nuffield Foundation. Comparative Study of Mathematics in UK and Singapore [EB/OL].

[6] Ministry Education, Singapore. Secondary Mathematics Teaching Guide (2024 Revision) [Z]. Singapore: Curriculum Planning and Development Division, 2024.

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[8] Tan, P. H. Small-scale Implementation of CPA Model in Secondary Schools [R] Singapore: NIE Research Brief, 2025.

[9] GeoGebra Official Manual. Teaching Application of Dynamic Mathematics Software [M]. USA: InternationalGebra Institute, 2023: 89-112.