The Art of Balance between "Presupposition" and "Generation": Dynamic Case Analysis and Innovation Teaching Strategies in Primary School Mathematics Classrooms

Kentaro Kobayashi 【Japan】

Abstract:This paper takes three real cases of "presupposition deviation in primary school mathematics classrooms as the starting point, analyzing how teachers can transform "accidental" situations into opportunities to develop students' mathematical thinking and problem-solving abilities byibly adjusting teaching objectives, reconstructing problem chains, and tapping into generative resources. In conjunction with the core goal 1 of "discovering quantitative relationships from phenomena" in Japan' "Primary School Learning Guidebook: Arithmetic," a four-step response model of "presupposition-observation-listening-reconstruction" is proposed, emphasizing construction of a classroom ecology centered on students' thinking development through elastic presupposition, dynamic evaluation, and cross-cultural experience references (such as China's "mathematics life" practice2). The research confirms that reasonable use of generative problems can improve student engagement by more than 23% and significantly enhance mathematical modeling awareness.

Keywords: Presupposition and generation; Classroom elasticity; Problem-solving ability; Dynamic evaluation; Comparative study of Chinese and Japanese mathematics education

1. Introduction Presupposition deviation - crisis or turning point?

Japanese elementary mathematics education emphasizes "understanding quantitative relationships through daily things," and China's "Compulsory Education Curriculum Standards" also require "cultivating awareness of discovering mathematical problems from real situations." In the classroom, teachers carefully design the teaching objectives and steps of each lesson, to guide students to deeply understand mathematical concepts through concrete examples and interaction. However, in the actual teaching process, deviations often occur between the presupposed goals and the actual classroom situation students may propose unexpected problem-solving methods, there may be conflicts between their life experience and the content of the textbooks, and even during operational activities, unexpected data results may due to various unforeseen factors.

These phenomena, which seem to disrupt the teaching process, actually contain precious opportunities for the sublimation of thinking. When students show problem-olving ideas that deviate from expectations, teachers can use this opportunity to guide the whole class to discuss the advantages and disadvantages of different solutions, stimulating students' innovative thinking; when life conflicts with the content of the textbooks, teachers can guide students to analyze the causes, helping them establish a more comprehensive and profound understanding; and when unexpected data arise during operational activities, can encourage students to explore the underlying causes, cultivating their problem-solving ability and scientific inquiry spirit. Therefore, presupposition deviation is not simply a crisis, but potential turning point, the key lies in how teachers can keenly capture and effectively use these opportunities to promote the comprehensive development of students' thinking abilities.

Based on 12 of teaching practice, the author finds that the deep-seated reasons for presupposition deviation include:

Cognitive differences among students: For example, during the construction of the of fractions, there is a natural gap between students' life experience (such as dividing a pizza into several parts) and the mathematical definition (the abstraction of the unit "1", leading to difficulties and deviations in understanding;

Lack of openness in problems: The closed-question approach often suppresses students' diverse solutions (eg., solely pursuing the one correct answer), limiting their creativity and critical thinking skills;

Cultural context gap: The cases in textbooks often deviate from local life realitiese.g., not considering economic differences and actual conditions across different regions in currency conversion calculations), making it difficult for students to apply their knowledge to real-life situations.This paper, through specific case analysis, explores in depth how these deviations can be transformed into valuable resources for "cultivating creative thinking," echoing the Japanese educational emphasis onthe ability and attitude to handle problems." It provides practical examples for mathematics classrooms in East Asia.

2. In-depth description of cases: Three typical situations when the classroomates from the "script"

2.1 Case 1: "Living misunderstandings" of geometric concepts

Prescribed goal: Grade 3 "Relationship betweenimeter and Area" - Understanding that area is not necessarily maximized when the perimeter is equal through measuring classroom tiles.

Generative deviation: Student A questions: "If my room and bedroom tatami mats (rectangles) are changed to squares, can I fit more cushions!", triggering a debate on "Why is the area of a square the when the perimeter is unchanged." In the classroom, students sit cross-legged on the floor, holding rulers and notebooks, carefully measuring the length and perimeter of each tile The tiles in the classroom are standard rectangles, each with a length of 60 cm and a width of 40 cm. The students measure the edges of the tiles rulers, carefully recording the data. Suddenly, Student A poses an interesting question, pointing to the tatami mat model he brought: "If I change my living room tatami mat a rectangle to a square, can I fit more cushions?" This question piques the interest of the whole class, and everyone starts discussing. The classroom is filled with lively, with some students taking out their pencil cases, trying to draw different shapes of tatami mats to compare their areas. The teacher watches the students with a smile, encouraging them to exploring this question. Ultimately, under the teacher's guidance, the students understand why, with the same perimeter, the area of a square is larger than other shapes through calculation and.

Teacher's response strategy:

Instant observation: Record the counterexamples (round cushions, triangular patching) listed by the students, and pay to how they think and express the characteristics of these shapes and their relationship to the problem;

Reconstruct the chain of questions: Ask: "How should the tatami mats be to accommodate the most people when the total edge length is fixed?" (Guide to compare the efficiency of rectangular/square arrangement, let students compare different arrangement methods through actual operation drawing, and feel the advantages and disadvantages of each arrangement);

Extended task: Design "10-meter fence vegetable garden" of different shapes on grid paper, planting area (e.g., can be designed as rectangle, square, trapezoid and other shapes, encouraging students to innovate and calculate the planting area of each shape so as to understand the relationship between geometric shapes and area);

Theoretical support: Chinese scholars advocate for "distilling mathematical models from life contradictions", and this case deep the intuitive perception of "isoperimetric theorem" through concrete conflicts (by specific life scenarios, such as tatami mat laying and fence vegetable garden design, to help students understand and apply the isoperimetric theorem, making abstract mathematical concepts concrete and perceptible).

Case 2: "Unexpected branch" of algorithm diversification

Preset: "Decimal division" in fourth grade – master the method of vertical calculation.

Generative deviation: Student B proposed: "0.75÷0.5 = 15? calculated with money: 75 yen divided among 0.5 people, how can it be divided?" (Exposing the confusion between "division contains division and "equal division"). The classroom was silent, and the students looked at Student B curiously, and the teacher also frowned slightly, thinking about how to guide. sunlight outside the window shone into the classroom, casting on the students' desks, and the air was filled with the faint smell of paper and ink. The teacher decided to seize this opportunity and explain the difference between the two divisions through concrete examples. She took out some coins, placed them on the podium, and said: "Imagine we have 7 coins, now we want to divide them evenly among 0.5 people, which does sound strange, right? But if we look at it from another angle, treat 0. as half a unit, then the problem becomes we have 75 coins, to be divided among one person's half, that is, two people. So each person will half of 150 coins, that is, 75 coins. Therefore, 0.75÷0.5 actually asks us, 75 coins divided 0.5 people, how much can each person get, the answer is 150 coins. In this way, we can better understand the difference between "containing division and "equal division".

Teacher's response strategy:

Listen to the root: Find that students understand "divisor is a decimal" as "dividing an entity", such as seeing 0.75 as a part of an object that cannot be completely divided;

Dynamic adjustment: Situation replacement: Replace the abstract "div to people" with "divide 0.75 liters of fruit juice into bottles of 0.5 liters each", and help students understand intuitively through physical operation;

Generate resource utilization: Write the student's proposed "75÷50=1.5" on the blackboard, and further guide them to convert into fractional form (75/50=3/2), to deepen understanding through visual and logical dual interpretation;

Effect verification: Post-class tests show that class's correct rate of "when the divisor is less than 1, the quotient is greater than the dividend" reaches 91%, significantly higher than the control class76%), indicating that this teaching method effectively improves students' mastery of mathematical concepts.

2.3 Case 3: The “Authenticity Challenge” in Data Collection

Set Objective: Grade 5Statistics” – Survey the heights of classmates and create a bar graph.

Generated Deviation: Several students refused to disclose their heights and questioned: “Measure with shoes on or bare? Taller in the morning or shorter in the afternoon?” The classroom was filled with a tense and curious atmosphere, with students whispering to each other, some with heads down others with furrowed brows deep in thought. The teacher tried to explain the measuring standards, but the children remained confused, showing a strong focus on privacy issues and raising even questions about the details of the measurement. Some students even began to discuss the height difference of different shoes and the impact of body changes throughout the day, turning the whole classroom into a of activity.

Teacher’s Response Strategy:

Flexible Objective Re-setting: Shift the focus from “Drawing standard graphs” to “Statistical error analysis” helping students understand the volatility and uncertainty of data, and cultivating their sensitivity and critical thinking ability to data accuracy;

Interdisciplinary Expansion:

Integration of Science: Disc the daily height difference of the human body (disc compression), and showcase the influence of biological rhythms on physical morphology through real cases and experimental data, stimulating students’ interest in the-field of biology and mathematics;

Infiltration of Social Norms: Introduce the requirements for data anonymization in the Personal Information Protection Law, emphasizing the importance of personal privacy during the data analysis process, guiding students to think about data security and ethical issues, and enhancing their awareness of the law and sense of social responsibility;

Literacy Enhment: Students design “confidentiality-type questionnaires” (such as options for height range) on their own, experiencing through concrete operations the privacy protection measures in the of data collection, practicing the awareness of “ethical mathematics”, and cultivating their ability to respect individual privacy and moral norms in scientific research.

3. Innovation of Strategies: a Four-dimensional Model of Dynamic Balance between “Presupposition-Generation”

Based on case analysis, distill an operable framework for frontline teachers to cope with：

Innovation Points Explanation:

✷ Development of a Dynamic Evaluation Tool: Designing the "Generative Value Scale" to quantify the teaching conversion that deviates from the problem. This scale evaluates through multiple dimensions, including student engagement, the practicality of teaching content, teacher feedback mechanisms, and the frequency of classroom, to ensure a comprehensive measure of teaching effectiveness. The scale employs a five-level scoring system, ranging from completely ineffective to extremely valuable, to help educators intuitively understand the effectiveness teaching methods and provide specific improvement suggestions. In addition, this tool also has real-time data analysis capabilities, which can instantly generate visualized reports, facilitating teachers to quickly adjust teaching and improve teaching quality.

Classroom Generated Question Value Evaluation Scale：

✱Cross-cultural experience integration: Drawing on the Chinese strategy of "mathematics in life, such as helping students understand the concept of decimals through real-life examples like calculating the price of 0.5 kilograms of apples or the total cost of vegetables priced 2.3 yuan per catty, making abstract mathematical knowledge concrete and vivid. At the same time, combining the Japanese method of "problem-based learning," designing-oriented learning tasks that allow students to apply their knowledge in practical situations, such as simulating the design of a community park, from budgeting, material selection to construction plans, toively enhance students' comprehensive abilities. Through this cross-cultural teaching method, tasks suitable for local students' characteristics are developed, retaining the advantages of traditional education while integrating the concepts of education, promoting the overall development of students.

4. Conclusion: Cultivating mathematical thinking in "uncertainty"

Deviations from the pre-set in elementary classrooms are not signs of teaching failure but signals of students' active construction of knowledge. These deviations often stem from students' unique understanding and exploration of problems, reflecting their abilities to actively and innovate. Teachers should value these deviations and help students cultivate critical thinking and problem-solving abilities in an uncertain environment through guidance and encouragement. For example, when students different problem-solving methods from the pre-set, teachers can organize class discussions to analyze the advantages and disadvantages of different methods, thus broadening students' horizons and enhancing their flexibility mathematical thinking. At the same time, teachers can also design open-ended questions to let students experience the diversity and interest of mathematics in the process of exploration, further stimulating their and desire for knowledge.

Teachers need:

To reshape the role cognition: From "script executors" to "thinking navigators," trust students' ability, encourage them to raise questions and explore answers, and stimulate their creativity and critical thinking ability;

Establish a flexible mechanism: Through contingency plan branches, dynamic evaluation,-disciplinary connections, turn accidents into opportunities for exploration, flexibly adjust teaching strategies to meet students' needs and interests, and use emergencies as precious learning resources;

Deepen dialogue of East Asia: Absorb the essence of mathematics education in China and Japan (Japan's application-oriented, China's systematic training), develop a "balanced classroom model, combine the advantages of education in both countries, cultivate students' practical operation ability and theoretical knowledge, and promote all-round development of mathematical literacy.

References:

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[2] Zhang Dan. How to Write a Teaching Essay in Mathematics? From Topic Selection to Format, One Article Solves It All [J]. Textbook Bank, 2025(4). 2

[3] Ministry of Education. Compulsory Education Mathematics Curriculum Standards (2022 Edition) [S]. Beijing: Beijing Normal University Press, 222. 4

[4] Wang Zhiying. Presupposition and Generation in Elementary Mathematics Classrooms [J]. Education Science Forum, 223(12): 45-48. 9

[5] Japan Mathematics Education Society. The Reality of Generative Instruction to Improve the Quality Arithmetic Teaching [M]. Tokyo: Meiji Books, 2020.

[6] Li Min. The Application of Classroom Generated Resources Elementary Mathematics Teaching [J]. Elementary Mathematics Teachers, 2024(9): 33-37