Middle School Triad Reflection Method for Deep Teaching Practice in Mathematics: A Case Study on the Teaching of Functions at Mundong Middle School in Seoul, South Korea

Kim Young-Jung   【South Korea】

Abstract：

This study, which addresses the issue of “emph on calculation over understanding” in South Korean middle school mathematics education (Kim & Lee, 2023), developed a “triad reflection method” teaching model Through a controlled experiment at Mundong Middle School in Seoul (N=120), the effectiveness of this model in significantly enhancing conceptual transfer ability in the function unit was (average score improvement of 29.3% in the experimental group, p<0.01). By innovatively integrating three elements: error analysis, situation diagnosis, and situation restructuring, a framework for the development of dynamic learning situation graphs and an Error Value Index (EVI) were proposed, providing a localized solution for mathematics in East Asia. Specifically, error analysis systematically records the types and frequencies of errors that students make during the problem-solving process, helping teachers identify commonly held cognitive misconceptions; situation diagnosis, using methods such as questionnaire surveys and classroom observations, comprehensively assesses students’ current knowledge mastery and learning attitudes; and situation restructuring designs problem situations close to students’ lives, enhancing the practical significance and interest of mathematics learning. The dynamic learning situation graph development framework integrates the above three elements and provides teachers with personalized teaching suggestions through real-time of students’ learning status and progress. The Error Value Index (EVI) helps teachers better utilize error resources by quantifying the educational value of errors and promotes the improvement of students deep learning and reflective abilities.

Keywords: Error Value; Dynamic Learning Situation Graph; Situation Restructuring; Concept Transfer; Function Teaching

1. Introduction

1.1 Research Background

South Korean middle school mathematics education faces a dual challenge:

Cognitive Level: PISA 2022 data show that only 58% of Korean students understand mathematical abstract concepts (OECD average 52%), indicating a significant gap in understanding and applying advanced mathematical concepts. However, 81% of South Korean achieve correct mechanical problem-solving, demonstrating their high skill level and proficiency in solving concrete problems (Ministry of Education, 2023). This phenomenon reflects the on memory and procedural knowledge in the educational system, at the expense of deep understanding and critical thinking skills.

Cultural Level: Excessive tutoring leads to a decrease in participation (weekly extracurricular mathematics learning reaches 14.2 hours, see Kim, 2024). This phenomenon not only increases students’ academic burden also diminishes their interest and engagement in school classroom teaching. Extracurricular tutoring classes usually focus on test-taking skills and quick improvement of grades, rather than cultivating longterm learning abilities and interests. Therefore, although students may achieve better test scores in the short term, their overall mathematical literacy and innovative abilities are not fully developed.

1.2oretical Innovation Points

Breakthrough the traditional “explanation-practice” mode and establish:

A [Cognitive Error] -->B (Error Value Analysis) C [Learning Situation Data] -->D (Dynamic Graph Modeling) E [Life Context] -->F (Conceptual Reconstruction Design) B -->G [Three Element Reflection Model]

2. Literature Review

2.1 Current Status of Mathematics Teaching in Korea

Structural Gaps: The disconnection rate between textbook and real life is as high as 73%, resulting in students' difficulty in applying the knowledge they have learned to solve actual problems (Park, 2024. This disconnection not only affects students' understanding ability but also weakens their interest and confidence in mathematics.

Evaluation Bias: In the midterm and final exams, processoriented questions account for less than 30%, over-emphasizing memory and computational ability, while neglecting the assessment of students' thinking and problem-solving abilities. evaluation method fails to fully reflect students' true learning level and also limits the teacher's attention to process-oriented learning in the teaching process.

2.2 International Experiencelightenment

The transfer efficiency of Singapore's "CPA Model" (Concrete-Pictorial-Abstract) in function teaching is as high as 0.82. This model helps students understand the basic concepts of functions by starting with concrete examples, then transitioning to the representational stage, using graphs and symbols to represent function relationships, and entering the abstract stage to master the mathematical expressions and properties of functions. This process not only improves students' understanding ability but also enhances their ability to apply function knowledge in different situations (an et al., 2023).

Japan's "Museum of Errors" strategy reduces the rate of repeated mistakes by 44% This strategy, by collecting and displaying common error cases that students make during the learning process, enables students to recognize the root causes of these errors and learn from them. This method not helps students avoid repeating the same mistakes but also cultivates their reflective ability and self-correction skills (Sato, 2024).

3. Construction Triadic Reflection Model

3.1 Core Framework

3.2 Innovative Practice Case: Function Concept Unit

Situational Reconstruction Design:

Transform "Directportional Functions" into "Mobile Data Fee Decision Problem". Specifically, a scenario can be set, such as the relationship between the mobile data usage of a user each month and cost they need to pay. Suppose that 1GB of data usage costs 10 yuan, then there is a direct proportional function relationship between the data usage (x) the cost (y) of the user, that is, y = 10x. Through such real-life examples, students can more intuitively understand the concept of direct functions and their applications. In addition, different fee plans provided by different operators can be further explored and compared to deepen the understanding of the rate of change of functions and linear relationships

Error Valuation Path:

Student's typical error: "In the function y=3/x, when x=0, y=0"

In the process learning mathematics, students often ignore the domain and range of functions. For example, in the function y=3/x, when x=0, students may mistakenly think that=0. Actually, when x=0, the function y=3/x is undefined because dividing by zero is meaningless in mathematics.

Conversion Strategy:

a.ablish a "Domain Restriction" Visual Model

To help students understand this concept, a "Domain Restriction" visual model can be used. By graphically presenting and clearly marking "forbidden zone" of x=0, students can intuitively see that the function cannot be calculated at this point. This visual representation helps students better understand and remember the restrictions of functions.

b. Analogy Real-life Situations (Zero Denominator → Zero Fuel Consumption Car Runs Infinitely)

To further deepen the, the mathematical concept can be explained through analogies with real-life situations. For example, the x=0 in the function y=3/x can be analogized with fuel consumption problem of a car. If a car has zero fuel consumption, then theoretically it can run infinitely without consuming any fuel. However, this is obviously impossible, just like dividing zero in mathematics, it is a violation of the basic rules.

c. Generate a Counter-example Database

To consolidate the students' understanding, a counter-example database be generated, including various forms of incorrect examples. These counter-examples can help students identify and avoid similar errors. For example, in addition to y=3/x, rational functions, limit problems, and algebraic expressions involving division can also be included. Through repeated practice and analysis of these counter-examples, students can more firmly grasp the importance of the of functions and avoid common traps in actual problem-solving.

4. Empirical Study

4.1 Experimental Design Optimization

Research Subjects: 120 eighth-grade students Seoul Myeongdong Middle School (ages 13-14) were randomly selected and stratified by their pre-test math scores into an experimental group (n=0) and a control group (n=60) to ensure no significant difference in baseline levels (t=0.87, p>0.05). The group adopted a new teaching method that included interactive learning, group discussions, and online resources, while the control group continued with the traditional classroom teaching model. This allowed for a more assessment of the effectiveness of the new teaching method. During the experiment, regular math tests were conducted to monitor student progress, and a comprehensive math aptitude assessment was performed after the experiment determine significant differences between the two groups. Additionally, student feedback was collected to understand their acceptance and preference for different teaching methods.

Development of Pre-test Tools:

Concept Understanding Diagnostic Scale: Includes three types of questions: function definition discrimination (e.g., "Which of the following is not a function"), representation transformation ( → formula), and real-life modeling (optimization of mobile phone tariff plans). The function definition discrimination section aims to test students' understanding of the basic concept of functions such as distinguishing which relationships meet the definition of a function through multiple-choice or true/false questions. The representation transformation section requires students to transform image information into mathematical formulas or vice, which helps cultivate their abstract thinking and graphical perception abilities. The real-life modeling section guides students to apply mathematical knowledge to solve real-life problems, such as the optimization of phone tariff plans, to enhance their application and problem-solving abilities.

Error Clustering Scale: Referencing Lee (2022)’s taxonomy of errors four core errors were identified:

A[Domain Confusion] --> A1(ignoring the denominator being zero)

B[Variable Relationship Misalignment] -->1(confusing independent and dependent variables)

C[Image Interpretation Bias] --> C1(incorrect association of slope and intercept)

DModeling Detached from Reality] --> D1(not considering constraints)

Experimental Procedure: .

4.2 Deep Data Analysis

Comparison of Function Concept Transfer Ability (Full marks30）

***p<0.01, **p<0.05

Key findings:

Error conversion efficiency: The experimental group achieved a 92% error correction rate the "domain confusion" category (compared to only 65% in the control group), attributed to the concretization intervention of the "zero denominator → zero fuel" analogy strategy. This strategy significantly improved the error correction rate by transforming abstract mathematical concepts into concrete everyday life scenarios, which helped students better understand and remember.

Correlation of learning graph relatedness: The dynamic learning situation dashboard showed a strong correlation between situational reconstruction degree (SRI) and modeling ability (r=0.81), verifying the of the CPA model in function teaching. The CPA model, which includes concrete (Concrete), representational (Pictorial), and abstract (Abstract) components effectively enhanced students' modeling ability and depth of understanding through a gradual transition from concrete examples to abstract concepts.

Gender differences: Females showed more significant improvement in the of counter-example construction (ES=1.53 vs. males ES=0.98), reflecting the role of metacognitive logs in promoting the rigor thinking. Metacognitive logs are a method of recording and reflecting on one's own thinking process, and through regular recording and analysis, females can more effectively identify and correct their thinking loopholes, thus showing higher improvement in counter-example construction.

5. Suggestions for teaching implementation

5.1 Development of a dynamic learning situation graph systemTechnical architecture:

flowchart LR

A[Data collection layer] --> A1(Pre-test error library)

A --> A(Classroom response records)

A --> A3(Reflection log text)

B[Analysis layer] --> B1(Error clustering)

B --> B2(Concept mastery heat map)

B --> B3(Transfer ability prediction model)

C[Output layer --> C1(Individual learning path suggestions)

C --> C2(Group teaching adjustment warnings)

Practical case - teaching monotonicity of:

Pre-diagnosis stage: Through the "comparison of mobile phone package fees" task, students' intuitive understanding of the rate of change is collected. In this stage the teacher can guide students to analyze the change in fees for different mobile phone packages as the usage time increases, helping them to intuitively perceive the concept of functional monotonicity. example, by comparing the fee changes of two packages when the call time changes from 0 minutes to 100 minutes, students can discover that some packages' fees increase as usage time increases, while others may remain unchanged or decrease. This exploration of real-world problems not only stimulates students' interest but also enables them to better understand the basic characteristics application value of functional monotonicity.

Process intervention: When the learning situation dashboard shows that 30% of students confuse "slope" with "total price," automatically the conflicting task: "Package B has a lower cost at x=2GB, why do users generally choose A package?" (Guide to discover the impact of initial cost. Through this task, students need to analyze the difference in initial cost between the two packages and understand how the initial cost affects the total cost. Specifically, although package B has a cost after using up to 2GB, its initial cost may be higher, leading users to be more inclined to choose the A package with a lower initial cost. This task aims help students distinguish between "slope" (i.e., the cost per unit increase) and "total price" (including initial cost and cost after usage increases), deepening their understanding of linear functions and cost structures.

Metacognitive reinforcement: Require students to write "A Brief History of the Development of Function Concepts, outlining key breakthroughs from Al-Khwarizmi to Descartes. In the writing process, students need to describe in detail how Al-Khwarizmi the foundation of algebra by solving linear and quadratic equations, and how he influenced later mathematicians. Next, students should explore the contributions of medieval Arabic mathematicians to the development of algebra especially their innovations in function representation. Subsequently, students need to introduce how the French philosopher and mathematician Descartes in the 17th century introduced the coordinate system, geometry with algebra, thus founding analytic geometry and paving the way for the formation of the modern function concept. Through this process, students will not only be able to understand the evolution of the function concept, but also gain a deep appreciation for the development of mathematical thought and its impact on modern science.

5.2 Quadratic Situation Reconstruction Strategy1. Concrete Anchor Design

A.Direct Proportion Function: Spring Stretch Experiment (Weight vs. Length)

In the spring stretch experiment, we chose a metal spring with elasticity. When different weights of objects are hung at the lower end of the spring in turn, we can clearly observe the change of the spring length. As the weight gradually increases the elongation of the spring also increases proportionally, forming a straight line. In the experiment, we used precise electronic scales to measure the weight of each object and recorded the elong length of the spring with a high-precision ruler. The entire process not only intuitively demonstrates the relationship of the direct proportion function, but also allows students to experience the close connection physical phenomena and mathematical models.

Linear Function: Metro Fare Ladder Calculation (Base Fee   Mileage Fee)

In the scenario of metro fare ladder calculation, we up a reasonable fare system. First, each segment of travel has a fixed base fee, and then an additional fee is charged according to the mileage. For example, the base for the first 5 kilometers is 3 yuan, and an additional 0.5 yuan is charged for each additional kilometer. Through this ladder charging method, students can understand the application of linear functions. In actual operation, we simulated a metro line map, marked the distance of each station, and asked students to calculate the corresponding fares according to starting and ending points. This process not only exercises students' mathematical calculation ability but also allows them to gain a deeper understanding of urban transportation systems.

B. Cognitive Conflict

Trap Case: Showing a comparison between "Uniform Motion Distance-Time Graph" and "Actual Bus Route Graph", questioning the rationality of the linear assumption In the uniform motion distance-time graph, a straight line is presented between distance and time, indicating a constant and unchanging speed. However, the actual bus route graph shows a curve, reflecting the effects of traffic congestion, traffic lights, passenger getting on and off, and other factors. This comparison reveals the nonlinear relationship in the real world, challenging students' assumptions about simple linear models and prompting them to think about more complex real situations.

C. Real-life Feedback Mechanism

Organizing a "Mathematical Modeling Competition" participants apply piecewise functions to optimize the meal plans for school canteens, aiming to reduce waste and lower costs. Specifically, contestants need to analyze the fluctuations in the number of dining at different times and design a reasonable strategy for the distribution of meal quantities. For example, increasing the supply during peak hours and appropriately reducing it during off-peak hours can help food wastage caused by surplus. Furthermore, by precisely calculating the demand for each ingredient, it is possible to effectively control the purchasing cost and improve resource utilization efficiency. Additionally, dynamic adjustment mechanism can be introduced to continuously optimize the model based on real-time data feedback, ensuring the continuous improvement and adaptability of the plan.

6. Conclusions Recommendations

The Tri-Reflection Model significantly enhances learning outcomes by:

Error resource teaching conversion value mining: By deeply analyzing the mistakes made by students during the learning process, educational resources hidden behind these errors can be tapped into and transformed into valuable teaching materials. This not only helps students correct their mistakes but also promotes their in-depth understanding of knowledge.Dynamic adjustment based on real-time learning situation diagnosis: With the help of modern educational technology, such as intelligent learning platforms and data analysis tools, teachers can instantly understand the learning and progress of students. Based on these data, teachers can promptly adjust their teaching strategies and methods to meet the needs of different students and improve teaching effectiveness.

The path of reconstruct life situations mathematically: By introducing real-life problems into the math classroom and processing and analyzing them mathematically, students can establish a connection between mathematics and the real world. This path not only makes mathematics learning more vivid and interesting but also enhances students' application ability and problem-solving skills.

Significantly improve the ability of conceptual transfer (Co's d = 1.27): The study shows that the Tri-Reflection Model can significantly improve students' ability to transfer and apply mathematical concepts in different situations., the Cohen's d value of 1.27 indicates the high effectiveness of this model.

It is recommended to increase the weight of the "Reflection Practice" in the revision of South Korea's curriculum standards: Given the significant role of the Tri-Reflection Model in improving student learning outcomes, it is suggested that in the process ofising South Korea's curriculum standards, the weight of the "Reflection Practice" module should be increased to better support teachers in implementing this effective teaching strategy.

References:

[1] Kim, S. (2024). Shadow Education and Formal Schooling in Korea. SNU Press.

[2] Park, J.2023). "Conceptual Gap in Secondary Math". J Korean Math Ed, 45(2), 112-129.

3] Ministry of Education (2023). National Assessment of Educational Achievement. KEDI.

[4] Lee, J. (2022. Error Analysis in Algebra Learning. Kyoyook Book.

[5] Tan, H. et al. (2023). "CPA Model Effect". Int J STEM Ed, 10(1).

[6] Sato, K. (2024). "Error Museum Approach". Pro ICME-15.

[6] Choi, M. (2025). Dynamic Assessment in Math Class. Journal