Research on Japanese Junior High School Mathematics Inquiry-based Teaching Practice Based on the Model of Cognitive Lad - Taking the "Discovery Learning of Plane Geometry" as an Example

Yamashita Akiharu 【Japan】

Abstract：

This paper addresses the of rigid thinking in Japanese junior high school students' mathematics education, proposing a "Three-Stage Cognitive Leap" teaching model (observation association → critical reconstruction → creative application, and combining it with the concept of "mathematical activities" in the new "Guidelines for Learning and Teaching". Through real classroom cases, its effectiveness in stimulating students deep thinking and cultivating their ability to creatively solve problems is verified. The research shows that the proportion of innovative solutions to open-ended questions in the experimental class has increased 37%, and the sense of math anxiety has significantly decreased (p<0.01).

Keywords: Cognitive Ladder Model; Critical Re; Embedding in Real Life Situations; Differentiated Inquiry; Internalization of Learning Motivation

1.Introduction: An Innovative Demand to Break through Traditional Teachingilemmas

Although Japanese mathematics education is renowned for its rigor, the PISA test shows that 15-year-old students are below the OECD average in the of creatively solving problems (2022). Traditional teaching over-emphasizes the training of techniques, resulting in students presenting a dual nature of "adept operators and "fragile thinkers".1 As front-line teachers, we urgently need to build a localized teaching model that can not only inherit the advantages of Japanese education but also innovative thinking.

Specifically, current teaching methods often focus on the imparting of formulas and problem-solving skills, neglecting the cultivation of students' logical reasoning, problem-olving ability, and critical thinking. This single teaching method not only limits students' creativity but may also leave them feeling helpless when facing complex or unseen problems. Therefore, we need to a new teaching strategy that can stimulate students' interest and potential by introducing project-based learning, interdisciplinary collaboration, and real-world case studies, enabling them to develop independent thinking innovative abilities while mastering basic knowledge.

2. Theoretical Framework: The Design Core of the Cognitive Ladder Model

2.1 Model Structure (Three-Stage C Leap)

A[Observation Association Level] --> B[Critical Reconstruction Level] --> C[Creative Application Level]

Observ Association Level: Activating primitive experience through life-oriented situations

Example: Measure the ratio of the torii at the shrine → introduce the concept of the golden section (stud collect data on-site with a tape measure), and explore the extensive application of the golden section in architecture, art, and nature, such as the Parthenon in Greece, Leonardo da Vinci's "Vitruvian Man", and the spiral structure of shells in nature.

Critical Reconstruction Level: The key leap break away from thinking patterns

Example: When proving the Pythagorean Theorem, it is required to use at least two non-textbook methods (paper-cutting/algebra tiling/fluid dynamics demonstration), encouraging students to think about problems from different perspectives and cultivating innovative thinking ability, such as using geometric transformations or dynamic simulation software for visual, enhancing the understanding and memory of mathematical principles.

Create application layer: Design solutions based on real problems

Example: Design a pavilion roof for community park that "achieves the maximum shaded area with the least material", considering actual needs and environmental factors, apply mathematical modeling and optimization algorithms, and take into account material cost construction difficulty, and aesthetics, to propose feasible design schemes, and adjust through model testing and feedback to ensure the practical operability and effectiveness of the design scheme.

2.2 and innovation of Japanese educational characteristics

Inherit the essence of "type" (Kata) culture: Visualize the thinking path, through which students can not only clearly see own thinking process, but also discover potential logical loopholes and innovation points from it. This practice helps to cultivate students' systematic thinking ability.

Design "Thinking Map Tools Forcibly record the branching points of the thinking process when solving problems (Fig. 1). This not only helps students to clarify their thinking, but also promotes information sharing knowledge integration in teamwork, and improves the efficiency of problem solving.

Improve the "group learning" model: Role rotation in heterogeneous grouping, through the rotation of different, students can experience various different ways of thinking and working methods, thus comprehensively improving their comprehensive quality.

Experimental group arrangement: Measurer (practical type) → Analyst ( type) → Planner (creative type) role rotation every month, this arrangement ensures that each student plays a key role at different times, accumulating diverse experience and skills, and a solid foundation for future career development.

3. Practice innovation: Original exploration in the frontline classroom

3.1 Situation reconstruction: From virtual problems to real challengesTraditional example:

"Calculate the length of the diagonal of a rectangle with a length of 10cm and a width of 6cm"

Reconstructed:

"Earthquake emergency kit design: How to place a 85cm long rescue rod diagonally in a 90×60cm storage box?Attached box model)

By transforming traditional mathematical problems into real-life application scenarios, students can not only better understand the practical application of geometric knowledge, but also cultivate their ability solve complex problems. For example, in the task of earthquake emergency kit design, students need to use knowledge from geometry, physics, and engineering to consider the length, angle, and constraints of the storage box to find the best placement solution. This practical learning method not only improves students' hands-on ability but also enhances their confidence and ability to face realworld challenges.

In addition, such tasks can also stimulate students' creativity and team spirit. During the design process, students may encounter various unexpected problems, such as uneven weight distribution the rescue rod and material selection of the storage box, which require in-depth thinking and discussion among students to finally form a feasible design plan. Through this method, students not only subject knowledge but also learn how to communicate and cooperate effectively in a team, laying a solid foundation for future learning and work.

After the restructuring, the task has achieved a triple breakthrough:

Firstly, it is necessary to consider the of the bar (realistic precision), which not only involves the microstructure analysis in materials science, but also requires precise calculation of stress distribution and deformation in engineering design. By advanced numerical simulation techniques, such as finite element analysis (FEA), it is possible to predict the behavior of the bar under different load conditions more accurately, thus optimizing the design improving the safety of the structure.

Secondly, the mathematical intuition of triggering the "critical length" is introduced, a concept derived from the Bückner-Einstein in elastic mechanics, which reveals that the stability of a bar will significantly decrease when its length exceeds a certain critical value. By delving into the mathematical model behind this phenomenon, can better understand and prevent structural failure, especially in large-scale engineering projects such as high-rise buildings and bridges.

Finally, the enlightenment thinking of non-Euclidean is introduced, and the development of this field has provided important tools for modern physics and cosmology. For example, in the general theory of relativity, Einstein used non-Euclidean geometry describe the curved nature of spacetime, thus explaining the essence of gravity. In engineering practice, the application of non-Euclidean geometry is gradually expanding to the design of complex networks, of information transmission and other fields, promoting cross-disciplinary innovation.

3.2 Fine-grained Design of Cognitive Conflict

Introducing contradictory cases when learning "similar":

begin{array}

text{Intuitive Judgment} & \text{Actual Measurement} hline

text{Tower's shadow length is 8m} & \text{Tower height ≠ 10m}

text{Person's height is .6m} & \text{Person's shadow length is 1.28m}

end{array}

The error caused by the of the Earth's surface (Earth's radius 6371km) leads to a discussion on the applicable boundaries of the mathematical model. Specifically, when we make on flat ground, we usually assume that the ground is completely flat, but in fact, the Earth is a sphere, and its surface has a slight curvature. This curvature has a effect over short distances, but it can lead to significant errors over longer distances. For example, in the above case, if Tower A's height is 10 meters, the person's height is 1.6 meters, according to the principle of similar triangles, the person's shadow length should be 1.28 meters. However due to the curvature of the Earth's surface, the actual measurement result may not match the expectation, thus revealing the limitations of the mathematical model under specific conditions. This cognitive conflict only helps students understand the application scope of the mathematical model but also can stimulate their thinking and exploration of complex problems in the real world.

3.3 Design of Differentiatedquiry Paths (Taking Quadratic Functions as an Example)

4. Empirical effects: real evidence from the classroom

4.1 Quantitative comparison (March 2024, 2nd year Group B a middle school in Osaka)

4.2 Qualitative Analysis: Typical Samples of Students' Cognitive Evolution

Mr. Sato ( math-averse):

Phase 1: "Why do we need to calculate this? Isn't it enough if the answer is correct?" In the initial phase,. Sato's understanding of mathematics was limited to the correctness of the result, considering that the process of calculation was not important as long as the final answer was correct. This reflected his superficial understanding of mathematics, overlooking the extensive value of mathematics in practical applications.

Week 8: "I found that if we use a hyperbolic paraboloid for theavilion roof, the shaded area can increase by 22%!" After several weeks of learning and practice, Mr. Sato gradually realized that mathematics was not merely abstract and theorems, but it could also solve concrete problems in reality. By applying the knowledge of hyperbolas, he discovered that changing the design of the pavilion roof could increase the shaded area, a finding that not only demonstrated the practical application value of mathematics but also ignited his interest and enthusiasm in learning mathematics.

Teacher's Reflection:

"When students start questioning the 'unrealistic' textbook examples, it marks the budding of critical thinking. This indicates that students are no longer passively accepting knowledge actively thinking and analyzing the background and authenticity of the problems. This ability is crucial for cultivating independent thinking and problem-solving skills. In the teaching process, teachers should encourage spirit of questioning and provide more real-life cases and scenarios to help students combine theory with practice, thus further developing their critical thinking."

5. Conclusion: Anative Path Back to the Essence of Education

This study proves that:

Innovation does not have to disrupt tradition: Embedding a Ladder of Thought in the "" culture, achieving integration of Eastern and Western educational wisdom. By combining the Western teaching methods that emphasize logical reasoning and analysis with the Eastern culture that emphasizes wholeness and intuitive, we can introduce new teaching concepts and techniques while retaining the advantages of traditional education, thus stimulating students' creativity and critical thinking skills.

Truth is the best catalyst: 1% of students stated that "the task is related to life" is the core driving force for sustained inquiry. This means that educational content should be closely linked to real life, students to see the practical application value of learning. For example, by using Project-Based Learning (PBL) and interdisciplinary curriculum design, students can master knowledge and skills in process of solving real-life problems, thus increasing their interest and engagement in learning.

Teachers as architects of thought: The need to strike a balance between "scaffolding" and "free exploration. Teachers play a pivotal role in the teaching process, where they are not only required to provide necessary guidance and support (i.e., "scaffolding") but also students to explore and discover independently. This balance helps cultivate students' independent thinking and problem-solving abilities while ensuring they do not feel lost or helpless during their learning journey.

uing exploration of the future: How to integrate the "shō-batsu-ri" philosophy of the tea ceremony into the cultivation of thinking, constructing a more Japanese culturally grounded for mathematics education. By delving deeper into the concept of "shō-batsu-ri" within the tea ceremony, which involves adherence to tradition as a basis for and breakthrough, new perspectives and methods can be offered to mathematics education. Firstly, "shō" emphasizes the mastery and respect for foundational knowledge, aligning with the emphasis on concepts and theorems in mathematics education. Secondly, "batsu" encourages the breaking of conventional thought patterns, stimulating students' creativity and problem-solving abilities, which is closely related to the training of innovative thinking and problem-solving in mathematics. Finally, "ri" advocates for the development of unique personal styles and methods while inheriting tradition, which helps form independent thinking and personalized learning paths. With the integration of this philosophy, mathematics education can not only impart knowledge but also cultivate students' cultural literacy and comprehensive abilities, thus constructing a and more diverse educational system.

References:

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