1. Bibliographic Information

1.1. Title

Teaching Design Exploration Based on the History of Chinese Mathematics (Original Chinese Title: 以中国数学史为主线的教学设计探索)

1.2. Authors

• Hao Lianming: Associate Professor, Master's Supervisor, School of Mathematics and Computer Science, Jilin Normal University.
• Wang Dan: Associate Professor, Master's Supervisor, School of Educational Science, Jilin Normal University.
• Chen Yanjie: Vice Principal, Master's Supervisor, Affiliated Experimental School of Jilin Normal University.
• Ding Simin: Teacher, Affiliated Experimental School of Jilin Normal University.

1.3. Journal/Conference

China Professional Degree Case Center (Published/Included on February 15, 2024). Note: This is a peer-reviewed teaching case study intended for professional degree education (e.g., Master of Education), focusing on pedagogical practice rather than a traditional theoretical journal article.

1.4. Publication Year

2024

1.5. Abstract

This paper presents a case study of a young mathematics teacher, "Teacher D," who designs a lesson for the Pythagorean Theorem (known as the Gougu Theorem in China) aimed at junior high school students. Guided by the Mathematics Curriculum Standards for Compulsory Education (2022 Edition), the study details the iterative process of lesson planning—from preparation and mentoring to group discussion and final demonstration. The core objective is to shift from a traditional teaching approach to one that deeply integrates Chinese Mathematics History. The study demonstrates how integrating historical materials (specifically from The Nine Chapters on the Mathematical Art) can enhance students' understanding of mathematical concepts while fostering patriotism and cultural confidence.

1.6. Original Source Link

/files/papers/69228dc1d8097f0bc1d01430/paper.pdf (Status: Case Study Publication)

2. Executive Summary

2.1. Background & Motivation

• Core Problem: In the context of China's basic education reform, there is a significant gap between the policy requirement to "integrate history and culture" and the actual classroom practice. Teachers often view the history of mathematics (HPM) as mere "decoration" or "seasoning," lacking the methods to integrate it effectively into the core logic of mathematical learning.
• Importance: The Mathematics Curriculum Standards (2022 Edition) explicitly mandate the inclusion of ancient Chinese mathematical achievements to foster "Cultural Confidence" and national pride. However, frontline teachers face challenges: lack of historical resources, time constraints, and insufficient pedagogical strategies to blend history with mathematical reasoning.
• Innovation: This paper documents a collaborative teaching research process. It moves beyond a static lesson plan to show the evolution of a design, highlighting how a teacher transforms a standard lesson into a history-centric inquiry-based class through mentorship and peer critique.

2.2. Main Contributions / Findings

• Pedagogical Pathway: The paper outlines a specific pathway for integrating history into math teaching: Goal Setting $\rightarrow$ Material Curation $\rightarrow$ Task Design. It validates the approach of "Education through History."
• Concrete Design: It provides a detailed, validated lesson plan for the "Gougu Theorem" that replaces the traditional Western introduction (Pythagoras) with Chinese historical narratives (The Nine Chapters on the Mathematical Art, Zhao Shuang's String Diagram), proving that indigenous history can support rigorous mathematical inquiry (Core Competencies) just as well as, or better than, Western examples for Chinese students.
• Professional Development: The case illustrates that novice teachers require structured guidance (mentorship, collective teaching research) to transition from "knowledge-based" teaching to "literacy-based" teaching.

3. Prerequisite Knowledge & Related Work

3.1. Foundational Concepts

• Gougu Theorem (勾股定理): This is the Chinese name for the Pythagorean Theorem ($a^2 + b^2 = c^2$). In a right-angled triangle, the shorter leg is called Gou (勾), the longer leg is Gu (股), and the hypotenuse is Xian (弦). The theorem states: $Gou^2 + Gu^2 = Xian^2$.
• HPM (History and Pedagogy of Mathematics): An academic field focusing on the relationship between the history of mathematics and mathematics education. It advocates using historical evolution to help students understand mathematical concepts, overcome cognitive obstacles, and appreciate the cultural value of math.
• Core Competencies (2022 Standards): The current Chinese math curriculum emphasizes nine key competencies. This paper focuses on:
  • Geometric Intuition: Perceiving shapes and spatial relations.
  • Reasoning Ability: Logical deduction (both inductive and deductive).
  • Cultural Confidence: A patriotic appreciation for one's national heritage/history.
• Saying Lesson (Shuoke - 说课): A specific Chinese teaching research activity where a teacher explains their lesson design, pedagogical philosophy, and expected student outcomes to peers and experts, rather than teaching actual students. It is a meta-analysis of one's own teaching.

3.2. Previous Works

The paper discusses the evolution of curriculum standards regarding math history:

• Experimental Standard: Required preliminary understanding of history (e.g., introducing Zhao Shuang's proof).
• 2011 Standard: Encouraged introducing background knowledge (e.g., The Nine Chapters, Golden Ratio) to stimulate interest.
• 2022 Standard (Current): Explicitly requires reflecting the achievements of ancient Chinese mathematicians to enhance patriotism.

3.3. Differentiation Analysis

• Traditional Approach: Often uses history as a "slide 1 intro" (e.g., "Pythagoras lived in Greece... now let's solve equations"). It treats history as disconnected trivia.
• This Paper's Approach: Adopts a Reconstruction/Adaptation Model. The historical problems (from The Nine Chapters) serve as the driver for the entire lesson. Students solve the actual historical problems to discover the theorem, making history the vehicle for learning, not just the scenery.

4. Methodology

This paper utilizes a Case Study Methodology, tracking the professional development of "Teacher D" through five distinct phases. The "Methodology" here is the process of instructional design iteration.

4.1. Phase 1: Preparation (The Traditional Baseline)

Teacher D initially designed a standard lesson based on old materials.

• Core Logic: Introduce the topic $\rightarrow$ Show Western history (Pythagoras floor tile pattern) $\rightarrow$ Present Theorem $\rightarrow$ Drill textbook exercises.
• Critique: This design was deemed "outdated" by the mentor (Teacher C) because it failed to address the Core Competencies of the new 2022 Standards and the historical integration was generic.

4.2. Phase 2: Guidance (Aligning with Standards)

Teacher D analyzed the 2022 Curriculum Standards to identify specific competencies.

• Targeted Competencies: Abstraction, Operation, Geometric Intuition, Reasoning, Application.
• Innovation Strategy: The mentor suggested using Chinese Mathematics History not just for interest, but as the primary innovation point to fulfill the "Patriotism" and "Cultural Confidence" requirements of the new standards.

4.3. Phase 3: Discussion (Deepening the History)

The teaching group (Teachers L1, L2, Y, S) reviewed the revised design.

• Feedback: Using history is good, but simply mentioning it is "decorative." The group questioned: Does the history actually help learn the math?
• Refinement: The consensus was to move beyond simply "knowing" about ancient mathematicians to "experiencing" their reasoning. The lesson needed to use original historical problems as the engine for inquiry.

4.4. Phase 4: Re-exploration (Historical Research)

Teacher D conducted literature research into HPM and primary sources.

• Source Material: The Nine Chapters on the Mathematical Art (Jiuzhang Suanshu).
• Historical Fact Check: Discovered that the Chinese mathematician Shang Gao described the theorem relation ($3^2 + 4^2 = 5^2$) significantly earlier than Pythagoras.
• Design Shift: The "driver task" changed from the Pythagorean floor tiles to a specific wood-measuring problem from The Nine Chapters.

4.5. Phase 5: Demonstration (The Final Methodology)

The final lesson design was structured into a narrative flow. The following details the logical flow of the final proposed lesson.

4.5.1. Step 1: Historical Intro

• Action: Display The Nine Chapters on the Mathematical Art.
• Formula/Task: Present the "Wall and Wood" problem (described in Section 5).
• Purpose: Contextualize the learning in a real historical Chinese puzzle.

4.5.2. Step 2: Analysis & Conjecture

• Action: Present specific numerical cases from the ancient text.
• Data Points:
  • $Gou=3, Gu=4 \rightarrow Xian=5$
  • $Gou=5, Gu=12 \rightarrow Xian=13$ (Implied generic examples)
• Method: Inductive reasoning. Students observe the relationship $a^2+b^2=c^2$ from these ancient examples.

4.5.3. Step 3: Proof (The Core Algorithm)

• Algorithm: Out-In Complementary Principle (出入相补法) / Cut-and-Paste Method.
• Visual Aid: Zhao Shuang String Diagram (赵爽弦图).
• Logic:
  1. Construct a large square using four congruent right triangles around a small central square.
  2. Calculate area in two ways:
     • Area = $c^2$ (Area of the large tilted square).
     • Area = $4 \times (\frac{1}{2}ab) + (b-a)^2$ (Sum of parts).
  3. Equate and Simplify: $c^2 = 2ab + (b^2 - 2ab + a^2)$ $c^2 = a^2 + b^2$
• Significance: This provides a rigorous geometric proof rooted in Chinese tradition, contrasting with the algebraic approach often emphasized in Western texts.

4.5.4. Step 4: Application

• Task: Return to the "Wall and Wood" problem and solve it using the now-proven theorem.
• Method: Modeling real-world objects (wood, wall, ground) as a right triangle.

5. Experimental Setup

In the context of this pedagogical research, the "Experiment" is the Saying Lesson (Shuoke) competition/demonstration where the design was tested against expert peer review.

5.1. Datasets (Historical Materials)

The "data" used to drive the lesson are specific problems from The Nine Chapters on the Mathematical Art.

• Problem 1 (The Driver):

  "今有高垣一丈。倚木于垣，上与垣齐。引木却行一尺，其木至地。问木长几何？"

  Translation: "Suppose there is a wall 1 zhang (10 chi) high. A wooden pole leans against the wall, its top level with the top of the wall. If the pole is pulled back 1 chi at the bottom, the pole falls to the ground. What is the length of the pole?"

• Problem 2 (The Rule/Theorem Description):

  "勾股各自乘，并而开方除之，即弦"

  Translation/Formula: "Multiply Gou by itself ($a^2$), multiply Gu by itself ($b^2$), add them together, and take the square root to get Xian ($c$)." $c = \sqrt{a^2 + b^2}$

• Problem 3 (The Door Problem - Homework):

  "今有开门去阔一尺，不合二寸。问门广几何？"

  Translation: Deals with a double door opening and the geometry formed by the door panels.

5.2. Evaluation Metrics (Qualitative)

The design was evaluated by peers and experts based on:

1. Alignment with Standards: Does it meet the 2022 requirements for core competencies?
2. Historical Authenticity: Is the history accurate and meaningful (not just decorative)?
3. Feasibility: Can students actually perform the inquiry?
4. Cultural Value: Does it effectively foster cultural confidence?

5.3. Baselines

The baseline was the Initial Design (Phase 1), which represents the standard teaching approach:

• Intro: Pythagoras floor tiles (Western context).
• Content: Standard textbook examples.
• Focus: Calculation skills ("Two Basics") rather than cultural/historical literacy.

6. Results & Analysis

6.1. Core Results Analysis

The primary result is the validated, finalized instructional design. The evolution from the baseline to the final design demonstrates a successful integration of HPM.

• Shift in Focus: The focus shifted from "How to calculate side lengths" to "How ancient Chinese wisdom solved geometric problems," achieving the dual goal of Math Skills + Moral Education (Patriotism).
• Expert Validation: Expert H concluded the design was "unique," "culturally thick," and "highly feasible." The use of The Nine Chapters was praised for being challenging yet accessible.
• Critique/Refinement: A critical finding during the evaluation was the need for historical rigor. The expert noted that common images of "Zhao Shuang" online are often fake or misattributed (e.g., actually Liu Hui or generic figures). Teachers must verify visual sources to avoid spreading historical misinformation.

6.2. Data Presentation (The Final Lesson Plan)

The following is the Finalized Teaching Design (Table 3 from the original text), transcribed via HTML to preserve structure. This represents the optimized "algorithm" for the lesson.

            1. Understand the content of <em>The Nine Chapters on the Mathematical Art</em>; conjecture the Gougu Theorem using grid paper; develop <strong>Geometric Intuition</strong>.<br>
            2. Experience the verification process of the Gougu Theorem; appreciate the cultural value and the thought of "Combination of Number and Shape"; develop <strong>Reasoning Ability</strong>.<br>
            3. Through examples, apply the Gougu Theorem for simple calculations; develop <strong>Application Awareness</strong>; be able to express the real world in mathematical language.<br>
            4. Understand the math history embedded in the Gougu Theorem; appreciate Zhao Shuang's proof process; enhance <strong>Patriotic Feeling</strong> and <strong>National Cultural Confidence</strong>.
               </td>
    </tr>
    <tr>
        <td><strong>Teaching Focus</strong></td>
        <td colspan="3">1. Proof process of the Gougu Theorem.<br>2. Simple application/calculation.<br>3. Understanding ancient Chinese math history.</td>
    </tr>
    <tr>
        <td><strong>Teaching Difficulty</strong></td>
        <td colspan="3">Understanding the relationship between the square of side lengths and area; conjecturing the Gougu Theorem.</td>
    </tr>
    <tr>
        <td><strong>Teaching Methods</strong></td>
        <td colspan="3">Lecture, Inquiry-based learning, Group cooperation.</td>
    </tr>
    <tr>
        <td colspan="4" style="text-align:center;"><strong>Teaching Process</strong></td>
    </tr>
    <tr>
        <td colspan="2"><strong>Main Segments</strong></td>
        <td><strong>Teacher-Student Activities</strong></td>
        <td><strong>Design Intent</strong></td>
    </tr>
    <tr>
        <td colspan="2">(1) Intro History, Create Context</td>
        <td>
            1. Show image of <em>The Nine Chapters</em>. Ask students to guess the book and content.<br>
            2. <strong>Problem 1:</strong> "A wall 1 zhang high. A wood leans... pulls back 1 chi... falls to ground. How long is the wood?"
               </td>
        <td>Introduce history to spark interest. Lead with the question of how ancient Chinese people solved math problems.</td>
    </tr>
    <tr>
        <td colspan="2">(2) Analyze Materials, Induce Conjecture</td>
        <td>
            1. Show Problem 2 (The Rule): "Gou 3, Gu 4, Xian is 5... Formula: Gou squared plus Gu squared..."<br>
            2. Ask: Is this relation true for other triangles? (Not just 3-4-5).<br>
            3. Guide students to use grid paper to explore.
               </td>
        <td>Use real problems from <em>The Nine Chapters</em> to let students recognize the theorem through induction. Grid paper fosters "Number-Shape Combination."</td>
    </tr>
    <tr>
        <td rowspan="2">(3) Understand History, Prove Theorem<br><br>(4) Solve Problems, Feel History</td>
        <td rowspan="2"></td>
        <td>
            1. <strong>Proof:</strong> Guide students to use cut-and-complement methods.<br>
            2. Introduce <strong>Zhao Shuang</strong> (Ancient Mathematician) and his "Out-In Complementary Principle."<br>
            3. Explain historical timeline: Shang Gao (<em>Zhoubi Suanjing</em>), Liu Hui, Hua Hengfang. Note this is 500+ years earlier than Pythagoras (questioning Western priority).
               </td>
        <td>Through Zhao Shuang's String Diagram, appreciate ancient wisdom. Diversity of proofs. Deepen cultural confidence by contrasting timelines with Western math.</td>
    </tr>
    <tr>
        <td>
            2. Solve the "Door Frame" problem (Modern application).<br>
            3. Solve the "Ladder Sliding" problem (Modern application).
               </td>
        <td>Echo the start of the class. Enhance enthusiasm for solving historical problems. Convert real problems into Right Triangle models.</td>
    </tr>
    <tr>
        <td colspan="2">(5) Summary, Extend History</td>
        <td>
            1. Summarize Gougu Theorem.<br>
            2. Homework: Textbook P28.<br>
            3. <strong>Extension:</strong> "The Door Problem" from <em>The Nine Chapters</em> (See Section 5.1).
               </td>
        <td>Supplement with ancient problems to further strengthen understanding of ancient culture and wisdom.</td>
    </tr>
</tbody>

6.3. Parameter Analysis (Comparative Impact)

While not a quantitative ablation study, the comparison between the initial and final designs acts as a qualitative parameter analysis:

• Parameter: Type of History Integration.
• Setting A (Western/Decorative): Resulted in "bland" engagement and missed requirements for cultural literacy.
• Setting B (Chinese/Integrated): Resulted in strong alignment with the 2022 Standards, high feasibility, and clear "Moral Education" value.

7. Conclusion & Reflections

7.1. Conclusion Summary

The paper successfully demonstrates that integrating the history of Chinese mathematics into classroom teaching is a viable and powerful strategy for meeting the 2022 Curriculum Standards. By transitioning from a superficial mention of Pythagoras to a deep, inquiry-based exploration of The Nine Chapters on the Mathematical Art and Zhao Shuang's proofs, teachers can effectively cultivate students' Cultural Confidence and Patriotism without sacrificing mathematical rigor. The case confirms that history can serve as the cognitive vehicle for learning, rather than just background noise.

7.2. Limitations & Future Work

• Implementation Verification: The paper describes a "Saying Lesson" (a presentation of design). The authors note that the design has not yet been fully field-tested in a live classroom with large sample sizes to quantitatively measure student performance improvements.
• Resource Constraints: The study highlights the difficulty teachers face in verifying historical accuracy (e.g., fake portraits).
• Future Work: The authors suggest that how to effectively integrate history into daily teaching (beyond just special demonstration lessons) remains a key focus for future research.

7.3. Personal Insights & Critique

• Transferability: The "Reconstruction" method used here (using historical problems as the main task driver) is highly transferable. It could be applied to teaching $\pi$ (using Zu Chongzhi's method) or solving equations (using the Nine Chapters "Fangcheng" method).
• Critical Reflection: The paper strongly emphasizes "Cultural Confidence." While valuable, there is a risk of over-correction. A balanced view (acknowledging both Shang Gao and Pythagoras) is ideal scientifically. However, in the context of the stated educational goals (Patriotism), the focus on Chinese priority is understandable.
• Pedagogical Value: The use of the Out-In Complementary Principle (Cut-and-Paste geometry) is actually superior pedagogically for young students compared to algebraic proofs, as it relies on visual intuition. This proves that historical methods are often didactically sound, not just historically interesting.