A Practice-Based Study on the Cultivation of Visual Thinking in Singapore Primary Mathematics Education—Localized Based on the CPA Approach

Li Maocheng   【Singapore】

Abstract:

This paper, through an analysis of the current situation of mathematics education in Singapore creatively proposes a “Visual Thinking Training Framework”, thus providing a new idea for mathematics education in Singapore’s primary schools.

Keywords: Primary Mathematics; Visualization; Training; Localization

Ⅰ. Introduction

Singapore’s mathematics education has attracted international attention for its outstanding performance in TIMSS and PISA tests. Its core lies in systematic implementation of the CPA (Concrete-Pictorial-Abstract) approach to teaching1. The CPA approach helps students gradually understand mathematical concepts through three stages: concrete stage (Concrete), the pictorial stage (Pictorial), and the abstract stage (Abstract). The concrete stage uses actual objects such as building blocks and to help students intuitively perceive mathematical problems; the pictorial stage uses pictures or models to transform concrete objects into visual representations, helping students transition to abstract thinking; and the abstract stage guides students to master symbols and formulas for pure mathematical operations.

However, there are three major challenges in current teaching practice:

1. Over-reliance on standardized teaching aids the concrete stage, neglecting the development of life-oriented materials: Although standardized teaching aids help to unify teaching standards, excessive reliance on these teaching aids may result in students lacking an of mathematical phenomena in daily life. For example, when teaching addition and subtraction, teachers can use life-oriented materials such as the number of desks and chairs in the classroom, number of students on the playground, etc., to make it easier for students to understand and apply mathematical knowledge.

2. Discontinuity in the transition from concrete objects toorial representation, making it difficult for students to establish a connection between the two on their own: The transition from concrete objects to pictorial representations is a key link in the CPA. However, many students have difficulty in this process and cannot establish a connection between the two on their own. Teachers can enhance students’ understanding ability by designing more interactive activities, such having students draw the corresponding pictures of concrete objects themselves, or using dynamic software to show the transition process from objects to pictures.

3. Early maturity of abstract symbols, causing students to give up on drawing strategies too early: As students progress to higher grades, they gradually tend to directly use abstract symbols for calculations, neglecting the importance of the pictorial strategy This may cause some students to become confused when facing complex problems. Teachers should encourage senior students to continue using the pictorial strategy when solving complex problems and show how to combine abstract symbols the pictorial strategy to solve problems through examples, thus improving students’ comprehensive mathematical abilities.

This study proposes a "Visual Thinking Training Framework" aimed at achieving a deep reform of the CPA (Concrete-ictorial-Abstract) teaching method by reconstructing the design of textbook exercises and developing localized teaching tools. Specifically, the framework first emphasizes the introduction of concrete physical operations in the teaching to help students establish intuitive understanding. Secondly, abstract concepts are visualized through visual aids such as images and charts, making them easier for students to understand and remember. Finally, are gradually guided to transition to abstract thinking, mastering complex mathematical concepts and problem-solving skills. This process not only helps improve students' mathematical abilities but also cultivates their thinking and innovative abilities.

Ⅱ. Theoretical Basis and Singapore's Characteristics

(a) The Evolution of Localized CPA Instructional Approach

The2006 Singapore Primary Mathematics Syllabus established a five-dimensional framework (Concepts/Skills/Attitudes/Metacognition/Processes) that centered onproblem-solving." This framework aims to help students gradually master mathematical knowledge and cultivate problem-solving abilities through three stages: concrete operations, iconic representation, and abstract thinking The latest revision emphasizes:

Mediating role of visual representation: Listing "Model Drawing" as a required skill. Model drawing is a method of representing mathematical problems through the drawing figures, which can help students understand problems more intuitively, thus finding solutions. This method is widely used in Singapore's education system because it not only helps improve students' performance but also enhances their logical thinking and creative abilities.

Cultural adaptability: Incorporating multiracial living contexts (such as public housing distribution, hawker center stalls) Singapore is a multicultural country, and its education system emphasizes integrating mathematics learning with the local culture and social environment. For example, when teaching geometry, teachers might use the layout of housing as an example to help students understand the practical application of different shapes and structures. In teaching statistics, teachers might use data from hawker centers, asking students to analyze and interpret data to better understand the concepts and methods of statistics. This culturally adaptive teaching method not only makes mathematics learning more closely related to students' lives but also promotes their understanding and respect multiculturalism.

Critical innovation point: Existing research focuses too much on problem-solving results (such as the "Singapore Mathematical Drawing Method") and neglects the visualization process of thinking. In fact, the process of visualizing thinking is crucial for understanding the essence of problems, developing solutions, and cultivating logical thinking abilities. Through visual thinking, students can see the structure and relationships of problems more clearly, thereby better analyzing and solving them. For example, when using the Singapore Mathematical Drawing Method, not only should attention be paid to the final graphical result, but also emphasis should be placed on how to gradually draw the graph to reveal the key information and logical relationships in the problem. This process oriented learning method helps improve students' mathematical literacy and comprehensive abilities.

（b）Breakthrough from the perspective of international comparison

Comparing the teaching of the "Ten Point Method" in the Chinese Beijing Normal University textbook (below) and the Singaporean workbook (above):

Ⅲ. Innovative Teaching Framework Design

(a) Development of Three-dimensional Visualization Tools

Concrete level: Modular Currency ModelsCompatible with Singapore Dollar Design)

Features: Transparent Layering (Coins can be separated by denomination), Supports Fraction Concept Construction, Different colors and levels differentiate denominations, facilitating students' intuitive understanding of the relationship between monetary units. Additionally, the models feature a rotation function, allowing students to observe coin structures from various perspectives.

Teaching Case: P3 "Fraction Addition" uses 1/2 SGD coin   1/4 SGD coin = 3 pieces of 25 cents. By breaking down half-dollar and quarter-dollar coins into their corresponding 25-cent coins, students can understand the practical application of fraction addition while enhancing their understanding of conversion.

Diagrammatic level: Use of dynamic thinking recording charts.

Abstract level: Symbol migration bridge question

“If □ represents a 1 yuan coin and △ represents a 25 cents coin then □   3△ = ______ yuan”

(b) Exercise System Reconstruction Principles

Reverse interference design: 30% of exercises provide incorrect, requiring students to diagnose (e.g., overlapping parts are drawn repeatedly in the perimeter calculation of P4). This design aims to cultivate students' critical thinking and problemsolving ability by identifying and correcting errors, thus deepening their understanding of concepts. For example, in a question about perimeter calculation, there might be a case where the overlapping part drawn repeatedly, and students need to analyze carefully and point out where the error lies.

Interdisciplinary integration: Converting plant growth data from science class into materials for drawing statistical. This principle emphasizes the integration of knowledge from different disciplines to enhance learning effectiveness. For instance, students collect plant growth data in science class, which can be used as materials for drawing graphs in math class. In this way, students not only consolidate their science knowledge but also learn how to convert actual data into charts, improving their data analysis and visualization skills.IV. Empirical Research

Ⅳ. Research Methods

1.Research Methodology

Participants: 12 classes from 3 neighboring primary schools (P3P5), comprising approximately 360 students, were involved in this study. The schools were selected to ensure a diverse range of socio-economic backgrounds and cultural diversity, for the generalizability of the findings.

Duration: The research spanned a period of six months from January to June 2024, including a one-month preparation phase a three-month intervention phase, and a one-month evaluation phase. Pre- and post-tests were conducted at the beginning and end of the study, respectively, to measure effects of the intervention.

Instruments: MOE Standard Diagnostic Test: Developed by the Ministry of Education, this test is designed to assess students' academic performance in core such as mathematics, language, and science.

Adapted Visualization Thinking Scale: This scale evaluates students' thinking processes and problem-solving skills through various visual, including charts, images, and symbols. It aims to capture cognitive skills and creativity that traditional tests may not measure.

2.Key Findings

A significant and improvement in the use of the drawing strategy was observed (N=326), particularly among the experimental group, where students were more inclined to actively use drawing as a tool problem-solving. Specifically, the proportion of students in the conventional teaching group who used the drawing strategy at the P5 stage was only 18%, whereas it was high as 73% in the experimental group.

There was a 200% increase in the complexity of the drawings produced by high-ability students, who were only able to produce complex two-dimensional diagrams but also began to experiment with three-dimensional coordinate systems, indicating significant progress in their spatial thinking and graphical representation abilities.

Mal students demonstrated the most significant improvement in problem-solving confidence, with an average increase of 34.7 points, compared to only 8.2 points in the control, suggesting that the teaching method used in the experimental group had a notable effect in enhancing confidence among students from specific ethnic groups.

Typical Teaching Episode:

Chen, a ChineseMalaysian student, exhibited exceptional innovative thinking when solving the "Juice Mixing Problem." He innovatively used a tiered color bottle diagram method, representing the heights different juices using different color tiers, such as a red tier for the height of strawberry juice, a green tier for apple juice, and a yellow tier for orange juice,. This method not only visually presented the proportion of each juice in the mixture but also broke through the limitations of traditional bar graphs, making the visualization of data more vivid and concrete., this tiered color bottle diagram also helped students gain a deeper understanding of how differences in liquid densities affect the mixing outcome, thus sparking their interest in and spirit of exploration for experiments.

V. Conclusions and Recommendations

1. Curriculum Design Innovation

The development of the "Visual Thinking Growth Portfolio" and integration of the EPMS teacher evaluation system aims to enhance students' thinking abilities and learning outcomes comprehensively through intuitive visual tools and systematic assessment mechanisms. This portfolio not only records students learning journey but also showcases their thinking processes and achievements through various forms such as charts, images, and mind maps, making abstract concepts concrete and tangible.

The concept of "matic Bank" is introduced in the P1-P2 stage, encouraging students to establish their personal schema library. This innovative measure can stimulate students' creativity and autonomous learning abilities. Schematic Bank is a dynamic knowledge base where students can add, modify, and share their own schematics at any time. These schematics can be hand-drawn drafts, digital, or images collected from the internet. In this way, students can not only consolidate their knowledge but also cultivate their information organization and analysis skills.

Moreover, the Sche Bank can also serve as an important resource for classroom discussions and group collaborations. Students can promote mutual communication and understanding by presenting and interpreting their schematics, thus forming an interactive learning. Teachers, through the EPMS system, can monitor and assess students' performance in real-time, providing personalized feedback and support to ensure that each student can progress at their own.

In conclusion, the combination of the "Visual Thinking Growth Portfolio" and the "Schematic Bank" not only enriches teaching methods but also enhances students' experiences and outcomes. This innovative curriculum design provides new ideas and directions for the development of future education.

2. Teacher Development Pathway

Novice Teacher: Basic model demonstration skills, including how to use multimedia tools and whiteboards for teaching, and how to design effective classroom activities. Dynamic thinking recording template, which helps teachers record the students' thinking process in the classroom, so as to better understand students' thinking patterns and provide targeted guidance.

Backbone Teacher: Error diagram diagnosis ability, including identifying and analyzing common types of errors that students make during the learning process, and corresponding corrective strategies. Cross-disciplinary integration case library, containing integrated teaching cases between multiple disciplines, to help teachers design more comprehensive and innovative teaching plans, and enhance students' quality.

3. Policy Recommendations

The "Thinking Visualization" specific ability index was added in the 2027 version of the syllabus, aiming to students understand and express complex concepts more intuitively through means such as graphics, charts, and models. The cultivation of this ability not only helps to improve students' logical thinking and problemsolving abilities but also enhances their comprehensive application abilities in cross-disciplinary fields.

The figure blank rate of the textbook was increased from 15% to 35% and the drawing space was reserved compulsorily, in order to encourage students to actively engage in visual thinking and creation during the learning process. This not only stimulates students' creativity imagination but also promotes their in-depth understanding and memory of knowledge. In this way, students can directly record their thinking process in the textbooks, form personalized learning notes, and better master the content they have learned.

References:

[1] Ministry of Education, Singapore. (2020). Primary Mathematics Teaching Syllabus.E Press.

[2] Pu Li Fang. (2015). An Analysis of the Singapore Primary Mathematics Syllabus. World Education Information, 34(4), 28-31. 2

[3] Fan, L. (2018). Problem Solving in the Singapore Mathematics Curriculum. NIE Research Paper Series.

[4] Kaur, B. (2019). Visualisation in Singapore Mathematics Education. Springer

[5] MOE. (2023). PISA 2022: Singapore Students’ Mathematical Literacy. Ministry of Education Report.

[6] Ng, S. F. (2020). Model Method: A Visual Tool for Mathematics Problem Solving. Singapore Teachers' Academy