Research on the Application of Multiple Representation Theory in Elementary Mathematics Teaching in Korea

Luo Shuyan  【Korea】

Abstract：

This paper addresses the issues of fragmented concept understanding and one-sided thinking training in elementary mathematics classrooms in Korea, proposing a localized practice for "Multiple Representation Theory." The study, based on a sample of third-graders (n=45) from Seoul's S Elementary School, develops a "ased Problem Chain" teaching strategy through the design of a four-dimensional representation conversion path of "Physical Object-Diagram-Symbol-Language." The empirical evidence shows the experimental group's accuracy rate in the concept transfer test increased by 27.8%, and the indicator of thinking flexibility was significantly better than the control group (plt;0.01). This research provides an operational solution for deepening the "process-oriented" teaching advocated by the "2022 Korean Mathematics Curriculum."

Keywords: Multiple representation; Process thinking; Hierarchical problem chain; Concept transfer; Elementary mathematics

1. Introduction

With the Korean Ministry of Education' implementation of the "Mathematics Literacy Promotion Plan" (2021-2025), traditional teaching models that emphasized computational proficiency are facing the challenge of. In my five years of frontline teaching, I have observed:

Shallow concept understanding: 62% of students can correctly calculate fraction multiplication, but only 2% can explain "why the numerators and denominators are multiplied separately" (based on the school's 2024 diagnostic test). This phenomenon indicates that although can master basic calculation skills through mechanical memory and repetitive practice, there is a clear lack of deep understanding and logical reasoning.

Rigid thinking phenomenon: The solution of word problems on fixed routines, such as the "keyword triggers formula" strategy, leading to an error rate as high as 41%. For example, when solving problems involving proportions and, many students habitually look for specific keywords, such as "each" and "total," and directly apply formulas, ignoring the actual context and internal connections of the problem, in a large number of errors.

International comparative research reveals: Singapore's success in mathematics education is closely related to its "CPA (Concrete-PictorialAbstract) progressive model," while Japan's "linguistic representation of mathematical ideas" training significantly enhances students' reasoning ability. Specifically, Singapore's CPA model helps build a solid mathematical foundation through a gradual transition from concrete physical operation to diagrammatic representation and then to abstract symbols; Japan, on the other hand, emphasizes the development of students' thinking and expressive ability by describing mathematical concepts and processes through language.

This study draws on Lesh's Multiple Representation Theory to construct a teaching framework that adapts to the context. Lesh's Multiple Representation Theory advocates presenting mathematical problems in various ways (such as text, graphics, symbols, etc.) to promote students' comprehensive understanding flexible application of mathematical concepts. This theory will provide new perspectives and methods for mathematics education in Korea, aiming to improve students' mathematical literacy and comprehensive abilities.

Localized Multiple Rep Model:

Physical operation → Dynamic diagram → Mathematical symbol → Language explanation → Real-life application

(Circular interaction rather than linear progression)

2. Practice Strategy Innovation

2.1 Four-dimensional Representation Transformation Design

Case: Reconstructing the "raction Multiplication" Unit for Grade 3

Physical Layer: Distribute the colorful strip model, require covering 2/5 blue strips with 1/3 yellow strips and observe the overlapping area. Students can touch and move the colorful strips to sense the ratio relationship between different color strips, deepening their understanding of fraction multiplication through visual and tact means.

Diagram Layer: Transform the operation process into a rectangular area diagram, marking "the shadow part = the desired result". In the diagram, the length and width the rectangle represent the numerator and denominator of the two fractions, respectively, and the shadow part intuitively shows the result of the product, helping students establish a connection between the graph the calculation.

Symbol Layer: Derive the formula (a/b)×(c/d)=(ac)/(bd), emphasizing the principle of calculation rather than memorization Through specific examples, such as 1/3 multiplied by 2/5, guide students to understand why the result is 2/15, rather than simply memorizing formula, and cultivate their logical thinking ability.

Linguistic Layer: Use "fractional unit repeated accumulation" to explain 3×(1/4)=3/, such as "3 parts of 1/4 apple make up 3/4 of an apple". Through actual examples in life, let students feel the practical application of multiplication, and enhance the interest and practicality of learning.

2.2 Hierarchical Problem Chain Development

To address the issues of "high repetition of question types and insufficient gradient" in Korean classroom exercises, a three-level problem chain is constructed. The first level problem chain aims to stimulate students' initial interest and basic understanding, guiding them into the through simple and intuitive questions; the second level problem chain gradually increases the difficulty, requiring students to think and analyze more deeply and cultivate their logical reasoning ability; the third level problem chain students' advanced thinking ability, involving complex situations and multi-dimensional analysis, prompting students to apply the knowledge they have learned to solve problems. Through this hierarchical design, not only diversity of classroom exercises is improved, but also the thinking gradient and deep learning experience of students are effectively enhanced.

Table 1: Example Design of Fraction Multiplication Problem

The core value of this design lies in:

Avoiding the mechanical stage theory of Singapore’sPA model, emphasizing the flexible transformation of representations. Specifically, this design encourages students to freely switch and apply various mathematical representation methods, such as graphics, symbols, and physical operations, different situations through diverse teaching tools and interactive activities, thereby cultivating their profound understanding and flexible application of mathematical concepts.

In line with the South Korean “process evaluation” reform requirements it provides observable evidence of thinking. This design not only focuses on students’ final answers but also values the thinking paths and strategies they demonstrate in the problem-solving process. detailed observation records and reflection reports, teachers can fully understand each student’s learning progress and thinking patterns, and then tailor personalized guidance and support for them to promote deep learning and sustained.

3. Empirical Effect Analysis

Conduct a 16-week controlled teaching at S Elementary School (Experimental group n=45, Control group n=4):

Concept transfer test: The experimental group’s “non-routine problem” solving rate reached 82.6%, which was 27.8 percentage points than the control group. Students showed stronger adaptability and innovative thinking when facing complex and novel problems.

Representation of flexible thinking: Using Kim (2023) coding to analyze the problem-solving process, the proportion of multi-path solution in the experimental group reached 64%. Students were not only able to find one solution but explore a variety of different methods, showing flexible ways of thinking.

Changes in affective attitude: The math anxiety index decreased by 31%, and the proportion of students took the initiative to draw mind maps increased from 12% to 53%. Students showed less tension and pressure during the learning process, participated more in classroom activities, their autonomy significantly improved as they actively used mind maps to organize and present their thinking process.

In the typical lesson case “Constructing the Meaning of Fraction Division,” students used the pizza distribution model to explain the principle of “reciprocal multiplication” (Figure 4). They cut a complete pizza into several equal parts, each representing a, and through actual operation, they showed how these fractions are multiplied by another fraction, thus revealing the mystery of reciprocal multiplication. This intuitive visual and hands-on experience not only helped better understand abstract mathematical concepts but also promoted their deep understanding of fraction division through multiple representations.

4. Discussion and Suggestions

4.1 Key Points for Local Implementation

C Adaptation Strategy:

Utilizing Korean food culture (such as cutting up rice cakes, kimchi ratio) to create representation transformation situations, and by showing how to evenly up rice cakes and adjust the ratio of different ingredients in the kimchi-making process, students can understand the concept of proportion and partition in mathematics. At the same time, can be organized to practice hands-on, increasing the fun and sense of participation in learning.

Adapt traditional mathematical game "윷놀이" (Yut Nori) to develop probabilistic thinking, allowing to calculate the probabilities of different outcomes during the game by incorporating more dice or changing the numbers on the dice, thereby deepening their understanding of probability. You can also design different rules, so that students can predict and analyze possible outcomes based on rule variations, cultivating their logical reasoning abilities.

Reform evaluation interface:

Process evaluation record representation example

Student name]: Park Ji-young

[Unit]: Area of polygons

◎ Physical operation: Derive the trapezoid area formula using tangram ✓

◎ explanation: Explain the relationship between "cut-and-paste method" and "double-piece method" △ (needs to be strengthened)

◎ Innovative application Design an optimized plan for the classroom storage area ✓

4.2 Reflection on existing problems

Class time allocation contradiction: Representational activities take 35% more time traditional teaching, which may lead students to feel time pressure when understanding complex concepts, affecting the overall learning effect. Teachers need to be more flexible in adjusting the teaching pace to ensure that link has sufficient time.

Bottleneck of teacher guidance capability: It is necessary to master decision-making wisdom such as "when to intervene in graphical transformation," which teachers to have keen observation and rich teaching experience, so as to provide effective guidance and support at critical moments, helping students better understand and master knowledge.

It is recommended to the "micro-circulation" model: Focus on 1-2 types of representational conversion per class, such as "Monday: Physical → Graphical", through specific display, guide students to transform actual items into graphical representation, enhance intuitive understanding; "Wednesday: Language → Symbol", through explanation and practice, help students transform abstract language descriptions into symbols or formulas, and improve logical thinking ability. This model helps students gradually master different types of representational conversion and enhance comprehensive quality.

5. Conclusion

This study confirms the theory of multiple representations effectively breaks the dilemma of "algorithms are skilled but thinking is weak" in Korean classrooms by strengthening the process-oriented construction of mathematical concepts. Its not only lies in the innovation of tools (such as hierarchical problem chains) but also in reshaping teachers' understanding of "the essence of mathematical understanding" - understanding arises from the transformation of different forms of representation rather than a single transmission. In the future, we will continue to explore the design of differentiated representation paths to respond to the educational vision of " mathematical literacy" in Korea. Specifically, by introducing a variety of representations such as visual, symbolic, and linguistic, students can flexibly switch when solving complex problems, thereby enhancing thinking ability and creativity. In addition, this teaching method also encourages teacher-student interaction and in-depth discussion, making the classroom atmosphere more active and productive.

References：

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