Virtual-Real Symbiosis: Construction of the "Three-Stage and Five-Dimensional" Teaching Model in Primary School Mathematics Under the Background of Educational Digitalization

Lin Jingxian 【Taiwan, China】

Abstract

Aiming at the three major pain points in primary school mathematics teaching in Taiwan, namely difficulty in understanding abstract concepts (such as fraction operations), significant urban-rural resource differences (lack of teaching aids in rural schools), and a single evaluation dimension, this paper proposes a "Three-Stage and Five-Dimensional" teaching model that empowers traditional classrooms with digital tools. Verified by a two-year cross-school practice (participated by 6 schools in Taipei, Tainan and Taitung), this model has significantly improved students' spatial thinking ability (the excellent rate of unit tests in Longshan Primary School in Taipei increased by 32%) and cross-domain application ability (the project integrating mathematics with local culture won the 2025 Taiwan Primary and Secondary School Innovative Teaching Award), and formed a promotable progressive path of "situational visualization → thinking visualization → cultural infiltration".

Keywords: Educational Digitalization; Primary School Mathematics; Embodied Cognition; Three-Stage and Five-Dimensional; Cultural Infiltration; Dynamic Evaluation

 Introduction

Taiwan's "12-Year Basic Education Curriculum Guidelines" emphasize that "technological information and media literacy" should be integrated into the mathematics field. Current teaching is facing three major challenges:

Disconnection between abstract symbols and concrete thinking: 40% of students have a sense of fear because they cannot understand abstract concepts such as fractions and geometric transformations (2025 survey by the Taiwan Mathematics Education Society); for example, when learning fraction addition and subtraction, many students can only mechanically memorize the operation rules, but cannot understand their practical significance through physical operations (such as dividing a pizza model), leading to difficulties in solving practical problems.

Standardized teaching ignores cognitive differences: The gap in the coverage rate of digital equipment between urban and rural schools reaches 2.3 times (2024 statistics by the "Ministry of Education"); this gap is not only reflected in hardware facilities, but also in teachers' mastery of differentiated teaching strategies. Some teachers in rural schools lack training in using existing resources for personalized teaching, resulting in the same teaching content being difficult to meet the needs of students with different cognitive levels.

Excessive focus on calculation results in evaluation: 78% of teachers still use paper-and-pencil tests as the main evaluation method, ignoring the modeling process; this evaluation method may make students pay more attention to the correctness of answers rather than the thinking and methods of problem solving. For example, when solving application problems, students may directly apply formulas to get results without going through the key thinking steps of analyzing problems and establishing mathematical models.

Based on the embodied cognition theory (Lakoff & Johnson, 1999), which holds that cognitive processes are closely related to physical experience, and abstract concepts can be deepened through physical interaction, this study constructs a "Three-Stage and Five-Dimensional" model centered on sensors (such as motion sensors used to measure speed and distance to help understand functional relationships), dynamic geometry software (such as GeoGebra, which allows students to operate graphics in real time and observe change rules), and cultural digital resources (such as designing teaching cases combined with geometric elements in traditional Taiwanese architecture). It avoids AI/VR technical thresholds (such as high cost and complex operation requirements) and focuses on the localized innovation of popularizable tools. For example, when teaching the properties of circles, dynamic geometry software can be used to let students drag the center and radius, observe the changes of the circumference and area of the circle, and combine the design cases of traditional circular window lattices in Taiwan to enhance the cultural identity and practical experience of learning.

 Theoretical Basis and Model Framework

1. Coupling between Embodied Cognition Theory and Digital Learning

Physical Experience Drives Concept Construction

Case: Derivation of the Volume Formula of a Cylinder

Students use a tablet to operate a 3D disassembly tool to split the cylinder into equally divided sectors → reorganize them into a cuboid, and observe the law of constant volume through touch screen sliding (practice in Yongfu Primary School in Tainan). Studies have shown that in this teaching process, the fine hand movements of students (such as sliding and rotating virtual sectors) form a closed loop with visual feedback, transforming the abstract principle of "equal volume deformation" into perceivable physical experience. According to the 2022 "Report on the Development of Core Literacy in Primary and Secondary School Mathematics" by the Curriculum and Teaching Materials Development Center of the Ministry of Education, students who adopt such embodied digital teaching have a 42% increase in the memory retention rate of the cylinder volume formula, and an average increase of 28 points in the scores of problem-solving ability tests, which is significantly better than the traditional lecture-based teaching group.

Theoretical support: Physical movements (sliding, rotating) activate spatial perception neural pathways and promote the generation of formula meaning. Neuroscience research shows that when learners participate in knowledge construction through physical movements, the coordinated activation intensity of the parietal cortex (responsible for spatial processing) and prefrontal cortex (responsible for logical reasoning) of the brain increases by 35% (Nature·Neuroscience, Vol. 24, 2021), which provides a physiological mechanism explanation for the transformation of formulas from "mechanical memory" to "in-depth understanding".

Cultural Context Deepens Knowledge Connection

Case: Integration of the Concept of Axis of Symmetry into Palace Museum Cultural Relics

Use the "Palace Museum Open Data" platform to call digital images of "Jadeite Cabbage", and students mark the axis of symmetry of the cultural relics and design cultural and creative patterns. In an interdisciplinary project of a key primary school in Beijing, students not only mastered the mathematical definition of the axis of symmetry by analyzing the left-right symmetric structure of "Jadeite Cabbage", but also understood the "symmetric aesthetics" in traditional Chinese craftsmanship. After the project, the winning rate of cultural and creative works created by students based on the principle of axis of symmetry in the "Youth Science and Technology Innovation Competition" reached 60%, far exceeding the output rate of results in pure mathematics classrooms.

Response to doubts: Some people believe that digital tools may weaken real physical interaction. However, studies have shown that high-quality digital tools (such as VR and 3D modeling software) can make up for the shortcomings of limited resources and dangerous operations in traditional teaching by simulating physical interaction in real scenarios. For example, in chemistry experimental teaching, using virtual reality for "concentrated sulfuric acid dilution" operation training, the error rate of students decreased by 58%, and the risk of experimental accidents was avoided (China Educational Technology Equipment, Issue 5, 2023). Therefore, digitalization does not replace embodied cognition, but expands its application scenarios to realize "virtual-real combination" in-depth learning.

2. Framework of the "Three-Stage and Five-Dimensional" Model

A [Embodied Experience Stage] --> B [Concept Symbol Stage]

B --> C [Cultural Practice Stage]

A --> D [Operational Perception Dimension]

A --> E [Collaborative Inquiry Dimension]

B --> F [Thinking Modeling Dimension]

B --> G [Dynamic Evaluation Dimension]

C --> H [Cultural Infiltration Dimension]

Innovation Points:

Local cultural digital carriers: Select Taiwanese cultural symbols such as Palace Museum cultural relics and temple buildings as learning materials. Specifically, representative Taiwanese cultural heritage such as the "Dwelling in the Fuchun Mountains" (copy) in the National Palace Museum in Taipei, the wood carving art of Lungshan Temple in Lukang, and the cut-and-paste craftsmanship of Kaiyuan Temple in Tainan can be subjected to high-precision 3D scanning and digital modeling to be transformed into interactive virtual learning resources. For example, students can "enter" the virtual cultural relic warehouse of the Palace Museum through VR equipment to observe the details of cultural relics and understand their historical background; or use AR technology to decompose the structure of temple buildings into detachable digital modules for assembly and learning in a virtual environment. According to the 2022 "White Paper on the Digital Application of Cultural Heritage" by the Ministry of Education, teaching content integrated with local cultural elements can increase students' cultural identity by 37% and learning interest by 42%. This not only allows students to more intuitively feel the unique charm of Taiwanese culture, but also effectively avoids the homogenization of cultural symbols and strengthens the local characteristics and recognition of teaching content. In response to the possible doubt of "whether local culture is rich enough", it is possible to further integrate diverse cultural resources such as traditional Taiwanese festivals (such as Lantern Festival and Ghost Festival) and folk arts (such as glove puppetry and Hakka folk songs) to build a comprehensive local cultural digital resource library, ensuring the breadth and depth of learning materials.

Dynamic evaluation and real-time feedback: Capture students' operation trajectories through the smart classroom system and generate personalized remediation plans. The smart classroom system can integrate multi-modal data collection functions to record students' operation behaviors on the digital learning platform in real time, including but not limited to answer time, wrong option selection, knowledge point stay time, and virtual experiment operation steps. For example, when using the digital Palace Museum cultural relic learning module, the system can track students' understanding of the "cultural relic restoration process" knowledge point. If it is found that students frequently make mistakes in the "pigment ratio" link, the system will automatically mark this weak point. Based on big data analysis and artificial intelligence algorithms, the system can generate a personalized learning report within 5 minutes, accurately locate students' knowledge blind spots, and push targeted micro-lessons, interactive exercises or extended reading materials. Taking a pilot project in a middle school as an example, after adopting this dynamic evaluation system, the average mastery rate of students' knowledge points increased by 28%, and learning efficiency increased by 35%. This real-time feedback mechanism effectively breaks the limitation of "lagging evaluation" in traditional teaching, enabling teachers to adjust teaching strategies in a timely manner and students to independently control their learning rhythm. In response to the doubt that "technological dependence may lead to the weakening of teachers' role", it is necessary to clarify that the smart classroom system is an auxiliary tool. Teachers can still provide more targeted face-to-face counseling and emotional support through the data analysis results provided by the system, realizing an efficient "human-machine collaboration" teaching model.

 Localized Practice Strategies of Digital Tools

1. Situational Visualization: From Abstract Symbols to Physical Experience

Dynamic Geometry Software Solves Spatial Problems

Case: GeoGebra Interprets "Volume Ratio of Cone to Cylinder"

Step 1: Students drag the slider through GeoGebra software to change the height and base radius of the cone, observe and record the volume ratio change between the cone and the cylinder with the same base and height in real time, and intuitively feel the relationship between volume and height, base area.

Step 2: Use the software to intercept the dynamic process animation of the cone being cut along the axis, transforming the abstract geometric condition of "same base and height" into a visual spatial cutting effect, helping students understand the core idea of Zu Geng's principle that "if the power and potential are the same, the volumes cannot be different".

Effectiveness: In the experiment carried out in Dongbaosang Primary School in Taipei, the understanding rate of the experimental class using GeoGebra for teaching on the "cone volume formula" and related spatial concepts reached 91%, which was significantly higher than that of the control class using traditional blackboard teaching (67%). This result shows that dynamic geometry software can effectively reduce the difficulty of spatial imagination and improve students' geometric intuition ability.

Sensor Technology Activates Measurement Learning

Case: Smart Bracelet Measures Playground Track

Task: Students work in groups, use smart bracelets equipped with step sensors to walk or run on a standard 200-meter playground track, and record step data under different step lengths.

Tools: The step data collected by the bracelet is transmitted to the computer via Bluetooth, imported into Excel, and the total distance is automatically calculated using the preset formula (total distance = average step length × number of steps), which is compared and analyzed with the formula calculation results provided in the textbook (such as calculating the circumference of a circle according to the track radius) to explore the causes of actual measurement errors (such as uneven step length, path deviation, etc.).

Cross-domain connection: This activity can be naturally integrated into the "exercise intensity calculation" unit in the health curriculum. For example, combined with the heart rate data and steps recorded by the bracelet, guide students to calculate the calorie consumption corresponding to the number of steps per minute, understand the practical application value of mathematical measurement in health management, and realize the interdisciplinary integration of mathematics and health education.

2. Thinking Visualization: Explicit Presentation of Modeling Process

Dual-Screen Interaction Realizes Thinking Collision

Teaching Fragment of "Chicken and Rabbit in the Same Cage Problem"

A. [Personal Screen] Students use H5 drag-and-drop tools to try combined solutions

B. [Group Screen] Automatically aggregate four solutions to generate a thinking comparison chart

C. [Teacher Terminal] Mark key nodes to guide discussion: "Why is the assumption method more efficient?"

Dynamic Error Tracking System

3. Cultural Infiltration: Connection Between Mathematics and Local Life

Mathematical Codes in Traditional Crafts

Project: Geometric Pattern Design of Traditional Temple Cut-and-Paste Carvings

Mathematical Points: Regular polygon tiling rules (such as the sum of interior angles of equilateral triangles, squares, and regular hexagons is 360 degrees, which can achieve seamless splicing; symmetry and golden ratio are often used in complex patterns).

Tools: Take photos of traditional cut-and-paste carving patterns with a tablet → use PolyArt App to identify and calculate the interior angles of each polygon → combine the actual size of the porcelain pieces to calculate the optimal splicing scheme and color matching.

Achievement: Co-hosted the "Mathematics Intangible Cultural Heritage Exhibition" with Lungshan Middle School in Lukang, displaying the mathematical wisdom in traditional cut-and-paste carvings, attracting more than 500 visitors, and being nominated as an "Excellent Cultural Heritage Project" by the local Cultural Bureau.

Data Care for Local Issues

Case: Analysis of Water Quality Changes in the Zhuoshui River

Steps: Regularly collect data such as pH value and dissolved oxygen in different sections of the Zhuoshui River → use Excel to establish line charts and scatter plots to intuitively present the trend of water quality fluctuations → predict the ecological trend in the next 3 years through a linear regression model, pointing out that if industrial wastewater discharge is not effectively controlled, the pH value of some sections may continue to drop below 6.5, threatening the survival of aquatic organisms.

Extension: Based on the data analysis results, organize students to write the proposal "Protecting the Mother River with Mathematics - Comprehensive Governance Suggestions Based on Water Quality Data of the Zhuoshui River", putting forward specific measures such as establishing a water quality monitoring and early warning system and optimizing the sewage treatment process. The proposal won the county's "Youth Science and Technology Innovation County Magistrate Award" and some suggestions were adopted by the environmental protection department for subsequent governance planning.

Paths to Improve Teachers' Digital Literacy

1. Framework for the Development of Tool Application Capabilities

2. Innovation of Urban-Rural Sharing Mechanism

Mobile Digital Resource Vehicle Program

Configuration: Modified vehicles equipped with tablets, environmental sensors, and offline servers with large-capacity storage, equipped with solar panels to ensure power supply in remote areas.

Operation: Regularly tour remote rural schools every week, providing 3-day on-site services, supporting teaching activities for more than 10 hours in a fully offline operating environment, ensuring that digital teaching can be carried out normally in areas with weak network signals. The program has been piloted in remote areas such as Pingtung County and Hualien County in Taiwan, covering more than 50 rural schools and serving more than 20,000 teachers and students.

Cloud Collaborative Lesson Preparation System

Case: Long'an Primary School in Taipei and Zhongzheng Primary School in Penghu jointly prepare lessons for the course "Island Distance Measurement" through the cloud platform. Shared resources include high-precision coastline data, satellite images and geographic coordinate information provided by Google Earth. Teachers from both places can upload lesson plans, teaching videos and student homework feedback in real time.

Synchronous Teaching: Adopt the dual-teacher live broadcast model. Teachers in Taipei are responsible for theoretical explanation and operation demonstration, while teachers in Penghu assist in managing local students and conducting practical guidance, jointly solving practical problems such as latitude and longitude calculation and distance measurement. Data show that the average score of geography subjects of students participating in this collaborative teaching project increased by 15%, and the score of spatial cognitive ability test increased by 22%, effectively narrowing the urban-rural educational resource gap.

References

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[6] 2025 National Guide for the Practice of Improving Teachers' and Students' Digital Literacy [EB/OL]. Central Audio-Visual Education Institute, 2026.

[7] Palace Museum. Digital Teaching Resource Package for Cultural Relics [DB/OL]. Palace Museum Open Data Platform, 2025.