Research on Emergency Rescue Strategies in High School Mathematics Classrooms—Practice Analysis Based on Grassroots Teaching C

Zhang Rui-li 【Macao】

Abstract：

This paper addresses sudden cognitive conflicts in high school mathematics classrooms (e.g., students questioning solutions, generating ideas, etc.), and proposes a "Four-Stage Elastic Emergency Rescue Strategy" based on local teaching cases in Macao. By analyzing three typical contingencies: "pute over Travel Days in Calendar Problems," "Definition Domain Arguments Triggered by Probability Paradoxes," and "Logical Traps in Geometric Figure Drawing" the operational path of "Tolerance-Reconstruction-Transfer-Solidification" is refined. Practice has shown that this strategy can convert 83% of classroom conting into teaching resources, and the efficiency of students' problem-solving transfer has increased by 37%. The study emphasizes the need for teachers to establish a dynamic knowledge base to turn contingencies into opportunities for thinking training.

Keywords: Classroom contingencies; Emergency rescue strategies; Cognitive conflicts; Error analysis; High school mathematics

1. Introduction

The high school mathematics curriculum in Macao emphasizes the integration of "mathematical reasoning with social applications" (Macao Education and Youth Affairs Bureau, 202), but in inquiry-based classrooms, teaching contingencies caused by students' unconventional ideas occur frequently. These contingencies often stem from students' unique ways of thinking and differentings of problems. For example, when solving geometric problems, students may propose methods based on life experience rather than textbook definitions. Traditional prescriptive teaching often avoids such incidents, resulting missed opportunities for thinking training. In fact, these contingencies can not only stimulate students' creativity but also promote teachers to reflect on teaching methods and improve teaching quality. Based on the of frontline teachers and 27 contingency cases tracked by teaching logs over three years, this paper proposes an operable emergency framework aiming to help teachers effectively deal with sudden situations the classroom and maximize the use of these contingencies as teaching resources to cultivate students' critical thinking and innovative abilities.

2. Redefining the Educational Value of Classroomtingencies

(I)The mathematics classroom at the grassroots level often faces three types of contingencies:

Cognitive conflict type: such as students solving problems with unsystematic theorems (eg., using the "Butterfly Theorem" to solve geometric problems). In this case, teachers can use this opportunity to guide students to explore the principles and scope of of the theorem in depth, thus stimulating students' inquiry spirit and innovative thinking. In this way, not only the current problem is solved, but also a broader knowledge horizon is provided students.

Case 1: Travel Date Calculation Controversy

Question: "Father traveled for 5 days, tore off the calendar dates and the sum was 0, on which day did he return home?"

Preset solution: Let the middle day be x, list the equation (x-2) (x-1) xx 1) (x 2)=90 → x=18 (return home on the 20th).

Unexpected generation: Students question: "The travel the first and last day of transportation, so the return home should be the 6th day!" (quotation of calendar problem variation)

Cognitive conflict point: The between life experience and mathematical models.

Detailed analysis: In real life, people usually take into account the time consumption on the day of departure and return a trip, not just the number of days spent. This discrepancy leads to a conflict between students' understanding of the mathematical model and their everyday life experience. Mathematical models often simplify actual situation, ignoring certain details such as the travel time on the day of departure and return, resulting in different answers. This conflict not only reflects the gap between mathematical abstraction and real but also prompts us to consider various factors comprehensively when solving practical problems, rather than relying solely on mathematical formulas.

Logical controversy type: The solution method appears to have acipled disagreement (e.g., misuse of conditional probability in probability problems). When logical errors occur during the problem-solving process, teachers should encourage students to engage in debate and discussion, analyze the advantages and disadvantages of different solutions, and ultimately find the correct solution path. This not only helps students consolidate their knowledge but also cultivates their thinking skills and team spirit.

Case 2: Geometric drawing controversy

Question: "Draw the circumcircle of △ABC."

Unexpected generation: Students find midpoint of AB and draw a perpendicular to get the center of the circle, without verifying the condition of an acute triangle, resulting in the failure of drawing an obtuse trianglequote the application principle of mind map). In fact, in the process of geometric drawing, it is necessary to first confirm the type of triangle. For an acute triangle, taking midpoint of any two sides and drawing a perpendicular can find the center of the circumscribed circle; however, for an obtuse triangle, this method will not accurately find the of the circle because the intersection of the perpendicular may not be inside the triangle. Therefore, the correct approach is to first determine the type of triangle and then choose the appropriate drawing method For example, for an obtuse triangle, the center of the circumscribed circle should be found by extending the sides.

Environmental interference type: Faults in teaching, sudden sounds, and other physical disturbances. Faced with these unforeseen situations, teachers should flexibly adjust their teaching plans and use these unexpected events as teaching opportunities such as through impromptu small experiments or discussions, to let students learn to remain calm and find solutions in complex environments. In this way, students can not only better cope emergencies in real life but also enhance their adaptability and problem-solving abilities.

(II) Unexpected Pedagogical Value

The traditional "avoidance-suppression" model can dampen students' enthusiasm for exploration. This model often overlooks students' initiative and creativity, leading them to lack the motivation to explore and solve problems when faced with them. Based Zheng Yuxin's philosophy of "education site is the research site", this paper proposes to treat unexpected events as dynamic teaching resources, and to stimulate students' interest exploration through flexible coping and guidance. Specifically, teachers can use unexpected situations in the classroom to design related discussion topics or practical activities, so that students can not only acquire knowledge in process of solving problems, but also cultivate critical thinking and innovative ability. Constructing an operable first aid path means that when encountering unexpected events, teachers should have the ability to quickly and deal with them effectively, while guiding students to participate in them and find solutions together. This method can not only improve teaching effectiveness, but also enhance teacher-student interaction and a positive learning atmosphere.

Expose knowledge blind spots (such as the difference between "mathematical date" and "natural day" in Case 1, which may lead calculation errors in actual application, for example, if the difference between the two is not noted when writing programs or analyzing data, it may lead to biased results. In addition, difference may also appear in interdisciplinary projects, such as using mathematical models to analyze the timing of events in historical research, special attention needs to be paid to the consistency of time units formats)

Stimulate metacognitive ability (Case 2 shows that students ignore the classification discussion, which not only affects the accuracy of problem solving, but also limits the of students' thinking. By guiding students to carry out the classification discussion, we can cultivate their ability to think comprehensively about problems and improve the flexibility of solving problems. For example when solving geometric problems, students need to consider the characteristics of different cases of graphics, and in physics experiments, students need to consider different conditions of experimental results, which can effectively students' metacognitive ability)

3. Design of Four-Stage Elastic First Aid Strategy

(I) Operational Framework

A[Unexpected occurrence] --> B{Type of cognitive conflict}  

B -->[Life experience conflict] C[Tolerance diagnosis]  

B -->[Logical loop] D[Reconstruct system]  

C --> E[Transfer modeling]  

D --> E  

E --> F[olidification extension]  

When facing unexpected occurrences, people often encounter cognitive conflicts. This conflict can be divided into two main types: life experience conflict and logical loophole. Life conflict refers to the confusion or uneasiness that arises when new information contradicts personal life experience. For example, a person who is used to going out on sunny days may feel confused uneasy when suddenly encountering a downpour, because it contradicts their previous experience. In this case, tolerance diagnosis is particularly important, as it helps individuals identify and understand conflicts, thus finding appropriate coping strategies.

On the other hand, logical fallacies refer to illogical or inconsistent places that arise during the thinking. This type of cognitive conflict usually stems from incorrect reasoning or assumptions in the thought process. For example, a person might believe that "all birds can fly," but this logic broken when they see a penguin. To address this conflict, a restructuring of the system is needed, which involves re-examining and adjusting the original logical framework to accommodate new and findings.

Whether it's life experience conflicts or logical fallacies, the ultimate goal is to solve these problems through transfer modeling. Transfer modeling is a method of integrating knowledge into an existing knowledge system, which helps individuals apply old knowledge flexibly in the face of new situations and innovate based on it. Through transfer modeling, individuals can not only current cognitive conflicts but also prepare for similar problems in the future.

Finally, the solutions after transfer modeling need to be further reinforced and consolidated through solidification and extension. Solidification extension refer to the process of converting new understandings and strategies into long-term memory and repeatedly applying them in real life to ensure their effectiveness and durability. Through this process, individuals maintain stable and efficient performance in a constantly changing environment.

(II) Pathway of Strategy Implementation

A. Fault-Tolerant Diagnosis Stage (Application of Case1)

Step 1: Accepting Questioning

Pause the predefined process, and write the student's viewpoint on the blackboard: "Set the first day as, the date and a (a 1) ... (a 4)=90 → a=16, the return day is a 5=21" Through this process, the teacher can guide students to deeply understand the methods of solving mathematical problems, encourage them to propose different views, and confirm the correctness of the answers through discussion verification. This not only helps to cultivate students' critical thinking skills but also enhances their teamwork consciousness. In actual teaching, teachers should focus on creating an open and inclusive learning, allowing students to express their own ideas without hesitation, thereby promoting the in-depth understanding and flexible application of knowledge.

Step 2: Dual-Model Verification

Reallife Model: Travel includes departure/return days

days = [16,17,18,19,20] # Departure on the 16, return on the 20th (original solution)

sum(days) # Output 90

Student Model: Travel includes the night before departure

 days_ = [16,17,18,19,20,21] # Departure on the morning of the 16th, return on night of the 21st

sum(days_student[1:6]) # Take 17-21st = 90

Diagnostic Conclusion In two different contexts, there are differences in the definition of "travel days." According to the error analysis theory, this difference arises from the language user's misunderstanding ority with the rules of the target language. For example, in the context of business travel, "travel days" usually refers to the total number of days from departure to return, the day of departure and the day of return. In the context of leisure travel, people may be more inclined to count the actual days spent outside, i.e., not the day of departure and the day of return. This difference may lead to misunderstandings during communication and record-keeping, which in turn affects actual operations such as itinerary arrangements and reimbursements. Therefore, it is essential to clarify the context and unify the definition to avoid such errorsB.

System Reconstruction Phase (Case 2 Application)

Reconstruction Classification Standards for Mind Maps

Method foring the Circumscribed Circle's Center

Acute Triangle: The intersection point of the perpendicular bisectors of the three sides, i.e., the point where perpendicular bisectors of each side intersect, is equidistant from the three vertices of the triangle.

Right Triangle: The midpoint of the hypotenuse, ie., the midpoint of the hypotenuse of a right triangle, is also equidistant from the three vertices, and is half the length of the hypotuse.

Obtuse Triangle: The intersection point is outside (proof by contradiction is required), i.e., the circumscribed circle's center of an obuse triangle is located outside the triangle, and the intersection point can be found by extending the perpendicular bisectors, proving that it is equidistant from the three vertices.

Iding students to supplement missing branches and structuring knowledge can effectively improve learning efficiency and depth of understanding. Firstly, teachers should clearly point out the blank points in the current knowledge system and relevant background information to help students establish preliminary cognition. Secondly, students should be encouraged to use resources such as libraries and online databases to find and integrate information, forming a complete chain. In addition, teachers can also organize group discussions, allowing students to share their research results and promoting the exchange and integration of knowledge. Finally, through regular testing and feedback, is possible to ensure that students have a good grasp of the supplemented content and further consolidate learning outcomes.

C. Transfer Modeling Phase

Design Variant Problems

Case 1 Transfer: "The Macau Marathon lasts for 3 days, and the organizing committee statistics that the participants' accommodation time is from November 1st to 3rd October 31st to November 2nd. What is the date and what is the difference between the two?"

Purpose: Distinguish the impact of " inclusion" on the model.

Case 2 Transfer: "A company organizes a 5-day training event, and employees can choose to attend between October 1st October 5th or September 30th and October 4th. What is the difference between the date range of the two choices?"

Purpose: Further explore impact of "time inclusion" on the model in different situations.

Case 3 Transfer: "An online course lasts for 7 days, and users can choose to complete the between December 1st and December 7th or November 30th and December 6th. How does the date range of the two choices affect the user's?"

Purpose: Analyze the potential impact of "time inclusion" on user behavior and engagement.

Case 4 Transfer: "A hotel offers a 3- special package, and guests can choose to check in from August 1st to August 3rd or July 31st to August 2nd. What is the impact of date range of the two choices on the booking volume?"

Purpose: Research the impact of "time inclusion" on customer booking decisions.

Case 5 Transfer: " scientific research project needs to be carried out continuously for 10 days, and the team can choose to carry it out between January 1st and January 10th or 31st and January 9th. What is the impact of the date range of the two choices on the project progress?"

Purpose: Evaluate the impact "time inclusion" on long-term project management.

E. Solidification and Extension Phase

Establish "Error Code Library”

4. Practice Outcomes and Reflections

(I) Quantitative Effectiveness

After implementing it for a semester in the second year of secondary education at theui Ching Middle School in Macau：

(II) Key Points of Innovative Strategies

Dynamic Knowledge Base Construction

Recording unconventional solutions by students such as using vectors to solve traditional geometry problems, can not only enrich teaching content but also stimulate students' creativity and flexibility of thinking. By collecting and organizing these unique problem-s methods, teachers can better understand students' ways of thinking and provide targeted guidance in the classroom.

Compiling the "Classroom Emergency Response Manual" (school-based resources)To deal with unexpected situations in the classroom, the "Classroom Emergency Response Manual" will become an important tool for teachers. This manual will contain solutions to various common problems, such sudden questions from students, equipment failures, etc. By preparing in advance, teachers can respond quickly when encountering unexpected situations and ensure the smooth progress of classroom teaching. In addition this manual can also be used as part of teacher training to help new teachers quickly adapt to the classroom environment.

Error-tolerant Evaluation Reform

Adding "innovative points" to the examination scoring standards, such as in Case 1, where students propose the 21st solution, if it is noted that "assuming the return home includes end of the journey" gives 3 points of strategy points. In addition, it is also worth considering adding an evaluation of the problem-solving process in the scoring details, as giving extra points for unique thinking paths and methods shown by students in the process of solving problems. This practice not only encourages students' creative thinking but also helps teachers to understand students problem-solving ideas and logical abilities more comprehensively. At the same time, to ensure the fairness and consistency of scoring, detailed scoring guidelines can be formulated and teacher training can conducted to ensure that all markers can accurately understand and apply the new scoring standards. Through these measures, the error-tolerant evaluation reform will help to cultivate students' innovative abilities independent thinking skills, further improving the quality of education.

5. Conclusion

Classroom emergencies are not teaching accidents but opportunities for the leap of thinking. The four-stage proposed in this article emphasize:

Diagnosis needs to be precise: distinguish the types of cognitive conflicts (life experience/logical defects). By analyzing the reactions and errors of in the classroom in detail, teachers can accurately identify whether the cognitive conflicts are caused by the interference of life experience or the lack of logical reasoning, and thus take targeted measures.

construction needs to be visualized: present the loopholes in the knowledge network with a mind map (refer to [1]3). Mind maps, as an effective tool, can help students intuitively see the connections and breaking points between knowledge, so as to better understand and integrate new information.

Transfer emphasizes context: create Macao localized variant. By designing practical problems closely related to Macao's local culture and social background, students can more easily concretize abstract concepts and deepen their understanding in the process solving practical problems.

Research has proven that the systematic incorporation of first aid strategies into the professional development system for teachers can enhance the quality of classroom dynamics. Systemat first aid strategies not only help teachers quickly respond to unexpected situations in the classroom but also promote their flexible application of these strategies in daily teaching, thereby improving overall teaching quality.

References:

[1] Zhang Bangshu. Turning "unexpected" generated resources into classroom highlights [J]. New Curriculum Research, 2024: 2-15.

[2] Huang Hua. Reflections on the design of precise teaching problems in high school mathematics [J]. Education Science Forum, 023(29): 45-48.

[3] Liu Tianxiang. Research on strategies for the infiltration of mathematical culture [J. Teaching Reference for Middle School Mathematics, 2024(5): 33-35.

[4] Ministry of Education of Portugal. Mathematics Curriculum for Macao Secondary Schools [M]. 2023.

[5] Macao Education and Youth Development Bureau. High School Mathematics Curriculum Guide [Z] 2023.