Teaching Wisdom Facing Classroom Contingencies: A Case Study on Innovative Strategies Based on a "Measurement Class" ——Reflections on Practice in Japanese Primary School Mathematics Classrooms

Kobayashi Keiichi  【Japan】

Abstract

This paper takes an incident of damage to protractors in a fourth-grade "angle measurement" classroom in Japanese elementary school as the research object, and explores how teachers can transform unexpected incidents into teaching resources. research proposes a "three-level response mechanism" (emotional comfort → cognitive transfer → generative learning), and by reconstructing the teaching process, making full use of life- teaching tools, and guiding students to explore independently, it achieves the teaching objectives while cultivating students' problem-solving ability. Practice shows that dealing with classroom contingencies reasonably helps enhance classroom resilience and students' innovation ability, and provides new ideas for individualized teaching under the concept of "relaxed education" in Japan.

Keywords: Class Contingencies; Generative Teaching; Cognitive Transfer; Primary School Mathematics; Japanese Education

Introduction: The Educational Value of Classroom Contingencies

The Japanese "Guidelines the Teaching of Primary School Mathematics" emphasizes "the cultivation of thinking and practical abilities through mathematical activities." In the frontline classroom, unexpected events such as teaching tool malfunctions, student questions, and experimental operation errors occur frequently. Traditional coping methods often focus on "maintaining order" and miss educational opportunities. This paper, based on a fourth-grade " measurement" public class (a municipal elementary school in Tokyo), analyzes how the teacher turned the unexpected damage to the protractor into an exploratory learning resource, and demonstrates unique value of unexpected events in achieving the teaching goal of "applying knowledge flexibly."

1. Case Description: A Measurement Practice "Destroyed"

1.1 Original Teaching Design Plan

Topic: Grade 4 "Angle Size and Measurement" (Tokyo Book Edition Textbook)

Objective:

Use a protractor to measure interior angles of a triangle (skill)

Understand that "the size of an angle is independent of the length of the sides" (concept)

Process:

A[ituational Introduction: Compare the angles of the triangle ruler] --> B[Demonstrate the use of the protractor]  

B --> CGroup measurement of the teaching tool triangle]  

C --> D[Report the conclusion, summarize the nature

Teaching preparation: The teacher prepared a variety of triangular teaching aids in advance, including equilateral triangles, triangles, acute triangles, and obtuse triangles, each with different side lengths but fixed angles; at the same time, standard protractors, triangular paper pieces for student practice and record sheets were also prepared. In the scenario introduction phase, the teacher showed two different-sized set squares to guide students to observe and ask questions: "Are the right angles these two set squares the same size? Why?" This stimulates students' preliminary thinking about whether the size of an angle is related to the length of the sides. When demonstrating use of a protractor, the teacher explained in detail how to identify the center point, zero degree line, inner scale, and outer scale of the protractor, and used a60-degree angle as an example to demonstrate how to align the center point of the protractor with the vertex of the angle, the zero degree line with one side of the, and then read the scale value corresponding to the other side. In the group measurement phase, students were divided into groups of 4, each group was given 3-4 types of triangular teaching aids, and they were required to independently measure the three interior angles of each triangle and record the measurement results in a table, and finally calculate the sum of the angles of each triangle. In the reporting conclusion stage, representatives from each group shared the measurement data in turn, and the teacher guided the students to compare the corresponding angle sizes of triangles (such as equilateral triangles with longer sides and equilateral triangles with shorter sides), and found that the angle values were consistent, thus summarizing the core concept that " size of an angle is determined by the degree to which two sides are spread, and is not related to the length of the sides."

1.2 Unexpected events

During the group operation phase, students from group 3 reported that the protractor was damaged (transparent plastic cracked, causing the scale to be blurred, some scale lines broken and the numerical labels fell off). The classroom fell into a brief chaos, and the orderly group discussions were interrupted by the unexpected event:

Student A: "Teacher we can't see the scale clearly! Our protractor is cracked here, and the scale between 30 degrees and 45 degrees is completely blurred!"

Student: "Can we use a ruler? Isn't there a scale on the ruler? Can we first measure the length of the two sides and then calculate the angle?" ( students began to use a ruler to measure the sides of the triangle, trying to deduce the angle from the measurement of the side length)

Student C: "Is the angle related the length of the sides? For example, if the side is longer, is the angle larger?" (The cognitive conflict appeared, the students proposed intuitive views that contradicted the knowledge, triggering low-volume discussions among the group members)

At this time, the teacher needs to quickly intervene, guide the students to remain calm in the face of unexpected, and turn the unexpected into an opportunity for exploration, such as organizing students to discuss "How to determine the size of an angle without a protractor", thus deepening the of geometric properties.

Key issues:

Lack of teaching aids leading to disruptions in operation

In actual teaching, to limited quantities, monotonous types, or insufficient preparation of teaching aids, it often happens that students cannot operate one set each, resulting in teaching activities being forced to interrupt or can be participated by a few students, and most students can only passively observe, unable to deepen their understanding of knowledge through hands-on operation. For example, in geometry teaching if there is a lack of a sufficient number of angle models, set squares, or modular geometric shape teaching aids, students find it difficult to intuitively perceive the size change of an and its relationship with the sides through actual arrangement and measurement, thereby affecting the teaching progress and effect.

Student's cognitive deviation from the core goal (confusion between angle and)

Some students, when learning the preliminary knowledge of geometry, tend to confuse the concepts of angle and side, mistakenly believing that the angle is determined by the number of, or that the longer the side, the larger the angle. This cognitive bias often stems from a lack of depth in understanding the essential characteristics of an angle—that an angle is of a vertex and two rays, and its size is unrelated to the length of the sides, but related to the degree to which the two sides are opened—when the teaching aids insufficient, students cannot establish the correct conceptual connection through comparing angle models with different side lengths but the same angle, or through dynamic demonstration of the process of angle formation, thus resulting in in judging the size of an angle, drawing an angle, or solving related problems, deviating from the core teaching goal of this lesson, "understanding the definition of an angle the method of comparing its size."

2. Innovative response strategies: a three-level response mechanism

2.1 First-level response: Emotional comfort and freezing (3 minutes)

Strategy:

Reconstruct the nature of the event with a "challenge task":

"Engineers also encounter tool malfunctions! Please pause your for a moment, and think: How did the ancients compare angles without a protractor? They might have utilized the properties of right triangles, methods for determining parallel lines, simulated the size of angles by folding paper. Everyone can try to design a simple angle comparison scheme based on the geometry knowledge they have learned."

Distribute "Emotional Cards" (pre-set emergency teaching aids), requiring students to record their current mood with symbols (☺/☹), and simply write on the card "I feel a anxious now because the tools are broken," to reduce anxiety.

Theoretical basis:

Sudden stress causes prefrontal lobe inhibition, which needs to be released situational conversion to release cognitive resources. When students face sudden situations such as tool malfunctions, the brain will produce anxiety due to stress response, which will further inhibit the prefontal cortex's ability to process information, affecting the efficiency of problem-solving. At this time, by turning negative events into exploratory learning opportunities through "challenge," students' attention can be effectively shifted, and their curiosity and initiative in solving problems can be stimulated; at the same time, the use of "Emotional Record Cards provides an outlet for students to express their emotions, and by symbolizing and textually recording, it helps students to concretize their abstract anxiety, thereby reducing psychological burden and creating for subsequent cognitive recovery and problem-solving.

2.2 Secondary Response: Cognitive Transfer and Tool Reformation (15 minutes)

Strategy

Path of Cognitive Transfer

A[Living Substitution] --> B[Origami Creation of a Simple Protractor]

B --> C[Right Angle Verification Method: Using the Textbook's Right Angle for Comparison]

C --> D[Body Measurement: Simulating Ang with Spreading Fingers]

Origami Method: Fold an A4 paper in half 3 times to generate a 45° reference angle, and compare the angle be measured:

“Is this angle greater or less than 45°? How much greater?” (Guide students to estimate)

Specific Operation Steps: First the teacher demonstrates folding a standard A4 paper along the midline of the long side once to form two equal rectangles; fold again to make the short sides of the two rect overlap, at this time, the paper is divided into four equal parts, and the intersection of the creases is the center of the paper; fold for the third time, fold of the rectangles along the diagonal direction to form an isosceles right triangle, at this time, when the paper is unfolded, you can see two perpendicular creases and diagonal crease, where the diagonal and the horizontal crease (or vertical crease) form the included angle of 45° reference angle. Under the teacher's guidance students independently complete the origami process, and through observing the intersection of the creases, clarify the position and size of the 45° angle. Then, the teacher shows measured angle (such as a 60° angle), and asks the students to overlap the 45° reference angle generated by origami with the measured angle for comparison and visually observe the size relationship between the measured angle and the 45° angle, and then guide the students to think about "How to estimate how much larger the measured angle than 45°". For example, if the measured angle is obviously greater than 45° and close to the right angle, you can guide the students to associate right angle with 90°, so as to estimate that the angle is about 60°-70°. In this process, students not only master the method creating a simple protractor by origami, but also through the application of life tools, they combine the abstract concept of angle with concrete operations, achieving the cognitive transfer from concrete abstract, and at the same time cultivate spatial imagination and preliminary estimation ability.

Body Tool Method:

“What shape is it when the index finger and thumb are spread 9°? What about 60°?” (Connecting with life experience

Generative Discovery:

The second group of students proposed: "Tie two pencils with a rope, and you can make a protractor!" The teacher immediately provided rope and thumbtacks, and the whole class verified the feasibility. In the verification process, the students one end of the thin rope at the top of one pencil as the vertex of the protractor, and the other end was fixed with a thumbtack at the top of the pencil, and the two pencils served as the two sides of the protractor, and by adjusting the included angle between the two pencils, using the length of the thin rope and the effect of the thumbtack, they successfully simulated the basic function of the protractor. The students also tried to use different lengths of thin rope and different thicknesses of pencils for, and found that the length of the thin rope would affect the measurement range of the protractor, and the thickness of the pencil would affect the convenience of the operation. Through this activity, the students not only understood the relationship between the size of the angle and the degree of opening of the two sides, but also cultivated their hands-on ability and innovative thinking and deeply felt the practical application of mathematical knowledge in life.

2.3 Level 3 Response: Conceptual Elaboration and Metacognitive Reflection (10 minutes)

Stries:

Guide students to meticulously compare the results data obtained by each group through the paper-folding method, the body method, and the traditional protractor for measuring same angle, organize group discussions to analyze the specific reasons for the errors generated, such as deviations caused by uneven folds in the paper-folding method, individual differences in the of arm extension in the body method, or line of sight deviations or inaccurate estimation when reading the protractor scale.

Return to the Core Issue:

Propose critical questions:When different measurement methods are used, and the angle value obtained shows differences, is this really due to a change in the size of the measured angle itself, or is it a problem the measurement tools we use and the way we operate them?" Through step-by-step guidance, help students focus and deeply understand the "rigid nature of angles" - the size of an angle is determined by the degree to which its sides are spread, and it will not fundamentally change due to different measurement tools or methods of measurement, thus establishing an understanding of the essential attributes of angles.

Extended Questions:

Further stimulate students' creative thinking, throw out open-ended questions: "If traditional tools like protractors suddenly wiped out from the world, what kind of brand new tools would humans invent based on existing knowledge and needs to precisely measure angles?" Encourage students to combine life experience, principles, and make bold assumptions, such as designing a reflective angle measuring instrument using the principle of optical reflection, developing intelligent angle detection devices with the help of electronic sensors, or new types of measuring tools by imitating the structure of biological bodies (such as the perspective characteristics of certain insects' compound eyes), to cultivate their innovative consciousness and the ability to practical problems.

3. Teaching Effectiveness and Reflection

3.1 Educational Transformation of Unexpected Events

3.2 Core Principles of Innovative Strategies

Guided by Student Questions:

When students exploring properties of triangles mistakenly assumed that "the length of a triangle's sides directly affects the size of its interior angles" and even proposed the hypothesis that "the longer the side the larger the angle," the teacher did not directly point out their mistake but instead sensitively captured this point of cognitive conflict and turned it into a teaching opportunity. The teacher guided to design a comparative experiment: select three sets of similar triangles with different side lengths but completely identical shapes (such as a right-angled triangle with side lengths of 3cm 4cm, 5cm, another with 6cm, 8cm, 10cm, and a third with 9cm, 12cm, 5cm), and use a protractor to precisely measure the three interior angles of each set of triangles. By comparing the experimental data, students clearly observed that the corresponding angle values the three sets of triangles were completely consistent, thus autonomously falsifying the incorrect assumption that "side length affects angle," and deeply understanding the essential attribute of similar triangles "corresponding angles are equal."

Transforming Limitations into Catalysts for Creativity:

During a geometry drawing class, due to a shortage of compasses in the laboratory, some students were unable to use standard tools to complete the task of "drawing a perpendicular bisector of a known line segment with a compass and a straightedge."aced with this limitation of tool absence, the teacher did not simply provide alternative tools but instead encouraged students to break away from the thinking pattern of relying on standardized processes and practice the "まずき (trip-up) teaching method" from Japanese educational philosophy. Students began to think: Without a compass, how can we use the limited tools around us (such rulers, pencils, erasers) to complete the drawing? Some groups tried to draw the end points of the line segment with a ruler and then find the midpoint by folding the multiple times; some groups used the scale on the ruler, found the midpoint position after measuring the length of the line segment; and some groups creatively used the ruler as afixed arm" and slid the pencil tip on the paper to find the symmetrical point. In this process, frustration turned into the starting point of in-depth thinking, and students only mastered multiple methods of drawing perpendicular bisectors but also cultivated their problem-solving ability and innovative thinking under limited resources, deeply experiencing that "limitations" can actually stronger creativity.

3.3 Practical Insights from Frontline Teachers

Emergency Teaching Toolkit: Always keep low-cost, easily obtainable materials such as orig paper, rope, straws, bottle caps, rubber bands, old newspapers, etc., which can be flexibly combined to replace traditional teaching tools. This allows for the rapid of teaching activities when experimental equipment is insufficient or suddenly damaged, such as using straws and rubber bands to make simple bridge models for mechanical principle demonstrations, using origami to explore the of geometric figures, and effectively ensuring the continuity of teaching.

Unexpected Event Record Sheet:

Occurrence Time Event Type Student Reaction Response Strategy Gener Outcomes

Measurement Phase Tool Damage (e.g., caliper breaking) Anxiety/Some students propose innovative methods of combining a ruler and a protractor measurement 

Level 3 Response (Teacher Guidance → Group Discussion → Plan Verification) 

3 Alternative Measurement Methods (Ruler   Protractor Auxiliaryment Method, Rope Marking Method, Object Overlapping Comparison Method)

Teaching research direction: Establish a "Classroom Unexpected Case Bank", systematically collect and organize detailed records of various emergencies (such as experimental accidents, equipment failures, student emergencies, etc.), including event background, handling process, student performance and reflection summary, and regularly organize teachers to carry case discussions and experience sharing meetings, to extract replicable coping strategies and teaching wisdom, to turn accidents into teaching resources, and to promote teacher professional growth and teaching innovation capability improvement.

4. Conclusion: From Accident Control to Generative Learning

Classroom accidents are not teaching accidents, but windows that expose real cognitive conflicts. This paper’s case proves that managing emotions—cognitive transfer—concept sublimation, teachers can turn emergencies into opportunities to cultivate students’ critical thinking (tool critique) and engineering thinking (tool reconstruction), echo Japan’s vision of “problem-centered” mathematics education. In the future, it is necessary to further explore localization strategies, such as incorporating “わざと失敗 (iberate failure)” into teaching design, and activating classroom vitality on a regular basis.

References

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[3] Sakai Tatsuo. Learning from Classroom Incidents [M]. Tokyo: Iwanamistore, 2019. (Classroom Event Research Monograph)

[4] Tanaka Minoru. The Skills of Teachers to Utilize Classroom TroublesJ]. Mathematics Education, 2022(4): 32-35. (Teaching Strategy Journal)

[5] Swan, M. Designingons from Cognitive Load Theory [J]. Mathematics Teaching, 2010(219): 12-15. (Cognitive Load Theory)[6] Tokyo Gakugei University Affiliated Elementary School. Learning Arithmetic through Creation [R]. Teaching Practice Report, 2023. (Localized)