Exploration of Singapore Primary Mathematics Classroom Practices Based on Modeling Ideology——Taking "Per and Area Concept Construction" as an Example

Lin Wei  【Singapore】

Abstract

This paper addresses the issue of primary school students confusing the concepts of perimeter and area. on Singapore's CPA teaching approach (Concrete-Pictorial-Abstract), a three-stage teaching model of "Concrete Operation-Image Modeling-Abstract" is designed. By developing life-oriented modeling tools, hierarchical task cards, and a dynamic evaluation system, the concept visualization breakthrough was realized in the fourth grade classroom. The practice shows that the improvement rate of students' modeling ability reaches 83%, and the accuracy of concept discrimination is increased by 42%, which provides a new path for concept teaching.

Keywords: Singapore Mathematics; CPA Teaching Approach; Modeling Thinking; Concept Construction; Life-oriented Teaching

1. Introduction: The Real Dmma of Conceptual Confusion

The abstractness of geometric concepts is a difficult point in primary mathematics teaching. During my teaching in Singapore government primary schools, it was found that about65% of fourth-grade students could not distinguish between perimeter and area (based on the 2024 in-school diagnostic test). Traditional teaching relies on formula (such as the perimeter formula P=2(l w), area formula A=l×w), but students often make mistakes such as "calculating the area using the perimeter unit".

The Singapore Ministry of Education (MOE) "Mathematics Curriculum Framework" emphasizes that "mathematical concepts need to be established relation through real situations".2 This article takes the integration of People's Education Press and Singapore MC textbooks as the basis, combines the CPA modeling ideology, and innovatively designs "Fence Designer" teaching case to break through the bottleneck of concept construction.

2. Textbook Integration Framework

Singapore MC Textbook "Primary 4 Mathematics" Unit3

Chinese People's Education Press Grade 4 Lower Volume "Area"

Cross Point: Life Situation Modeling, Multiple Representation Transformation

There are significant crossfusions in the teaching content of Singapore MC Textbook "Primary 4 Mathematics" Unit 3 and Chinese People's Education Press Grade 4 Lower Volume "Area", are mainly reflected in the two core dimensions of "Life Situation Modeling" and "Multiple Representation Transformation". By deeply analyzing the design ideas and teaching activities of the sets of textbooks, it can be found that they are both committed to closely linking abstract mathematical concepts with students' daily life experience, guiding students to understand the essence of area in concrete, and deepening their mastery of knowledge through a variety of representation methods.

In terms of "life situation modeling", both sets of textbooks select examples close to students' as the starting point of teaching. For example, the Singapore MC textbook may design tasks such as "Calculate the area of the classroom floor to determine the number of tiles needed" "Compare the area of different-shaped flower beds", which allow students to perceive the practical significance of area in the process of solving actual problems; while the People's Education textbooks guide students to measure the size of the desktop, calculate the area of a rectangular handkerchief, and estimate the area of the campus lawn through activities such as "Measuring size of the desktop", "Calculating the area of a rectangular handkerchief", and "Estimating the area of the campus lawn", guiding students to from things around them and experience the extensive application of area in life.

In terms of "multiple representations transformation," both textbooks emphasize presenting and understanding the concept of area through various. The Singapore MC textbooks may guide students to intuitively perceive the size of the area through "counting squares" (such as counting the number of squares covered by a shape on paper), "arranging shapes" (using small squares or rectangles to form a shape with a specified area), and "drawing figures to represent" (drawing shapes but with equal areas). At the same time, it may also introduce the process of "formula deduction," such as by dividing a rectangle into several small squares, guiding to discover the rule that the area of a rectangle is equal to the length times the width. The People's Education textbooks also focus on multiple representations, such as through "operation activities" (such as using 1 cm² square paper to arrange rectangles, recording the relationship between length, width, and area), "transformation of figures" (ing irregular shapes into regular shapes for area calculation), and "symbolic expression" (using the letter formula S=ab to represent the area of a rectangle). These help gradually construct a knowledge system for area calculation from concrete to abstract. This process of transformation of multiple representations not only deepens students' understanding of the area concept but also cultivates their imagination and logical thinking abilities.

In summary, the cross-fertilization of "life situation modeling" and "multiple representations transformation" between Unit 3 of the Singapore textbook "Primary 4 Mathematics" and the "Area" section of the People's Education fourth grade lower volume provides rich resources and ideas for teaching practice. Teachers can draw on advantages of both textbooks to design more effective teaching activities, guide students to explore mathematics in real-life situations, deepen their understanding in diverse representations, and thus comprehensively improve' mathematical literacy.

3. Innovative Class Design: Three-Stage Modeling Teaching Method

3.1 Concrete Stage: From Real-life Objects to Mathematical Per

Innovation Point: Develop a "tear-off sticker film measuring tool" to replace standard rulers. This tool, through the design of low-cost and high-activity materials, allows students to intuitively understand the essential difference between the circumference and area in hands-on operations, breaking through the abstraction of traditional ruler measurements and reducing the difficulty spatial imagination.

Teaching Aids Design: Transparent square grid film (1cm×1cm grid, with slight stickiness on the edges for easy fixing)   tearoff color tape (in red, blue, green three colors, the length matches the different contours of objects, the tape width is 0.5cm to ensure measurement accuracy) When using it, lay the transparent square grid film flat on the surface of the object to be measured, press and fix the four corners with your fingers, and then closely attach the tape along the edge of the object to form a closed figure; then peel the tape off the film to get the entity model of the circumference; At the same time, observe number of grid squares on the film that are not covered by the tape, and perceive the size of the area by counting the squares. Figure 2 shows the assembly diagram and the diagram of the use steps of this teaching aid.

Student Example:

In the activity "Measuring the Perimeter and Area of the Classroom," students first completely cover the surface of the desk with a transparent grid film, ensuring that the four corners of the film align with the four corners of the desk and there are wrinkles; then choose red tape, starting from the upper left corner of the desk, stick along the edge of the desk in turn, passing through the upper right corner, lower right, lower left corner, and finally returning to the starting point to form a closed loop, at which time the total length of the tape is an intuitive representation of the desk's; then gently peel off the red tape, and the red tape traces left on the film outline the contour of the desk, and the areas of the film that are not covered by red tape are composed of complete 1cm×1cm squares, and students can get the desktop area by counting the number of these squares (such as 50 squares) which is about 50 square centimeters (ignoring edge errors). In addition, teachers can also guide students to measure the perimeter of the desk drawer with blue tape and perimeter of the cover of the math book with green tape, through the comparative operation of different objects, to deepen the understanding of the concept of perimeter.

Classroom Recording

During the operation, student A holds the red tape and carefully sticks it along the edge of the desk, pausing slightly to adjust the angle at the corners of the desk to that the tape is completely attached to the edge. When the tape is closed, he excitedly said: "The original perimeter is that piece of ribbon! The empty space left after off the ribbon is the area (area)!" Student B added aside: "I just counted, there are 48 complete squares on the desk, so the area is48 square centimeters." The teacher asked questions along the way: "If you replace the tape with a thinner line, will the measurement result change?" Guide students to about the accuracy of perimeter measurement. Through multi-sensory participation such as tactile operation (tearing and pasting tape), visual observation (counting squares), language expression (describing discoveries), students successfully established the initial concept differentiation of perimeter and area, understood that the perimeter is the length of the outline of the figure for one week and the area is the size of the inside of the figure, laying a solid foundation for subsequent abstract formula learning.

3.2 The Key Leap of Modeling Thinking:

Innovation Point: Dynamic Modeling Task Card (Hierarchical Design)

【Basic Card】

Grid Drawing: Draw a rectangle 3cm long and 2cm

Tasks:

A. Outline the boundary line (perimeter) with a red pen

B. Fill in the internal squares (area) with a blue pen

【allenge Card】

Situation: Xiaoming's room is 2㎡ larger than his sister's, but the perimeter is the same

Tasks: Draw two possible shapes and verify them

Guided Discovery: The core difficulty of "the same area does not necessarily have the same perimeter"

Generative Problem Solving:

When students askHow do I draw a grid for a triangle?", introduce the feature of Singapore textbooks - transformable grid paper (non-equidistant grid), breaking through the limitations regular graphics.

3.3 The Life Return of Mathematical Models:

Innovative Practice: "Community Fence Design" Project

Task Requirements:

- Design an area for pet dogs with a 20m fence

- Goal: Maximize the dog's activity area

- Submission: Design diagram   calculation process   model verification  Excellent Case:

Student B designed a circular fence (π takes 3.14):

"Perimeter 20m → Radius ≈ 3.8m → Area ≈ 31.8㎡

For example, a square only has 5×5=25㎡!"

(Spontaneous use pi, beyond the requirements of the textbook)

4. Innovation-led distillation: Localized practice breakthrough

4.1. Development of diverse representation tools

Tearable film: Dimension reduction of-dimensional concepts to tactile operations

Transformable grid paper: Breaking the bottleneck of unconventional graphic modeling

(Compared to traditional teaching, efficiency increased by 37%, the table below)

Teaching effect comparison（N=45）

4.2. Hierarchical Mechanism for Dynamic Grouping

Referencing Singapore’s streaming education philosophy but its age restrictions:

Rainbow Task Cards: Distribution based on cognitive level (students self-select for upgrading)

Mentor System: Completed taskers guide (achieving 100% class participation)

4.3. Contextualized Assessment Rubric

Abandoning standardized testing and adopting the MOE’s “Gu for Process-Oriented Assessment in Mathematics”2:

【Fence Designer Scoring Rubric】  

◎ Reasonableness of Model (30%)  

◎ of Calculation (20%)  

◎ Innovation of Solution (30%)  

◎Cross-Disciplinary Considerations such as Environmental Sustainability (20%)  

5.Reflection: Generative Resources for Instructional Transformation

Typical mistakes are teaching resources

In mathematics instruction, students’ cognitive biases often contain precious generative resources Taking the unit teaching of “Relationship between Perimeter and Area” as an example, teachers can systematically collect various error models made by students in the process of drawing the “Per-Area Relationship Graph”, such as directly equating the numerical values of the square’s perimeter and area and marking them, confusing the meanings of the horizontal and vertical axes in coordinate system (mistaking the perimeter as the vertical axis and the area as the horizontal axis and not marking the units), or drawing a wrong broken line graph showing a linear in the perimeter and area with the increase in the side length of the figure, etc. These specific and vivid error cases are “windows” that reveal students’ deep-seated misconceptions and can provide precise targeting for subsequent teaching. When carrying out the “Diagnosis Little Doctor” activity, representative errors (as shown in Figure 4, a students incorrect chart of “the perimeter and area values are equal when the side length of the square increases from 1cm to 3cm”) can be selected for students to discuss groups: guide students to first observe the corresponding relationship between data in the chart, and then verify by calculating specific figures (such as the side length of the square is 2cm the perimeter is 8cm, the area is 4cm², and the numerical difference is clear), and finally the “Little Doctor” team marks the error points on original chart with a red pen, and draws the correct chart on the whiteboard normatively, while explaining the core concepts such as “Perimeter is the length of the outline of figure, and area is the size of the inside of the figure, the two have different units and growth rules”. Through this process of “error presentation-independent exploration-operative modification-achievement display”, it can not only enable students to deeply understand the essence of knowledge, but also cultivate their critical thinking and problem-solving ability.

Cultural Context Innovation

To enhance the practical significance and cultural immersion of mathematics learning, teaching content can be deeply integrated with specific cultural contexts. For example, within the typical social scenario of Singapore's public housing estates, a comprehensive practical activity of "Designing a Protective Zone for Stray Cats under the Public Housing Estate" can be designed. First, guide students to clarify the task objectives: it is necessary to delimit a safe and comfortable activity area for stray cats, ensuring that the cats have sufficient space to move around (involving area calculation), while also controlling the cost of using fencing materials (involving the relationship between perimeter and material length). Next, provide real-context parameters: the available space under the public housing estate is rectangular, with a length of 8 meters and a width of 5 meters; the average daily activity area of stray cats should not be less than 10 square meters; the unit price of fencing materials is 20 Singapore dollars per meter. Students are required to work in groups to complete:

① Design different shapes (such as squares, rectangles) of protective zone plans according to the area requirements, and calculate the minimum perimeter required;

② Compare the material cost and practicality of different schemes (such as the side length of a square is about 3.16 meters, the perimeter is about 12.64 meters, the cost is about 252.8 Singapore dollars; the length of a rectangle is 4 meters, the width is 2.5 meters, the perimeter is 13 meters, the cost is 260 Singapore dollars, and analyze which is more economical and reasonable);

③ Incorporate moral education elements to discuss "How to protect stray animals without affecting the use of public space by residents of public housing estates", echoing the empathy and social responsibility advocated in Singapore's "Care for Society" curriculum goals. During the activity, students need to use the perimeter and area formulas to make precise calculations, optimize the plan through group negotiation, and deepen their understanding of mathematical knowledge while cultivating a sense of citizenship that cares for life and serves the community.

6. Conclusion

This paper converts abstract concepts into operable, visualizable, and transferable thinking models through the CPA three-stage modeling, which verifies what Singapore mathematician Lee Bing Yi said: "Mathematics is the science of patterns, and modeling is the bridge for children to approach mathematics." In the future, we will further explore the transfer application of modeling thinking in core concepts such as fractions and ratios.

References

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