Class 1: 9th Grade Algebra I
This classroom has many stickers of mathematical symbols on the walls, and there are approximately 28 students in the class. The desks are arranged in 7 cooperative groups, with 4 students in each group. Each student’s desk has a whiteboard for drawing, allowing them to write down their ideas during group discussions and share them with their group members. The classroom walls are decorated with diagrams outlining the key steps for solving algebraic problems: understand, plan, solve, and finally check to verify the correctness of the answer. There is a student workstation near the classroom door, containing algebraic puzzles, scratch paper, and the materials needed for today’s lesson. The teacher is a woman with 5 years of teaching experience, and she wants her students to learn in a happy environment. During group work, she walks around between groups, asking students probing questions and recording the strategies they use in solving problems. Teaching content: Solving linear equation problems in real-world scenarios (e.g., “A mobile phone plan costs $30 per month, and each text message costs an extra $0.15. How many text messages can be sent with $50?”).
During the class, I noticed a strong sense of collaboration among the students in each group. They focused on developing strategies to address the problems. Following the four-step process outlined in the key points diagram, the students engaged in frequent dialogue, demonstrating high levels of enthusiasm. Each group and student continuously refined and optimized their problem-solving approaches, striving to find the best solution.
Class Two: 10th Grade Geometry
This classroom is a very traditional one, lacking any particularly prominent mathematical culture elements. There are 31 students in this class; the desks are arranged in a straight line facing the teacher’s podium, and a projector is positioned at the front of the classroom. The teacher displays slides containing theorems and practice problems, and a “Quiet Study” sign is prominently displayed. There is no group work in this class, so the teacher has not set up any group collaboration areas. The teacher is an older gentleman with 12 years of teaching experience. He begins by teaching geometric theorems, then demonstrates the solutions on the blackboard, followed by several independent practice problems. His teaching content includes proving triangle congruence (SSS, SAS, ASA, AAS), and solving congruence-related problems (such as “Given AB=DE, ∠B=∠E, BC=EF, prove â–³ABC≌△DEF”).
In this classroom, the teacher and students employed a very simplistic approach to problem-solving. The entire lesson was teacher-led, with a highly standardized flow and a rigid, one-dimensional teaching method. Both the teacher’s approach and the students’ solutions were extremely simplistic, almost robotic. Every student was required to strictly follow the steps demonstrated on the blackboard. The teacher did not allow students to deviate from the prescribed methods or explore alternative approaches on their own.
During class, a student misidentified the corresponding sides of a triangle, leading to an error in the proof of a problem. The teacher went to the student’s desk, pointed out the incorrect steps, erased them, and wrote down the correct solution, telling the student, “Copy this method.” However, the teacher did not explain the reason for the error, where the mistake occurred, or why it happened, nor did the teacher independently correct the student’s thought process.
Class 3: 12th Grade Advanced Placement (AP) Calculus BC
This classroom is very advanced, with 25 students, each assigned a computer. Group tables are set up with four students per table, and a front-row demonstration area with a smart whiteboard is located at the front. The walls outside the classroom are covered with student-made posters on “Problem-Solving Strategies,” and each table has a “Reflection Sheet” for students to analyze their solutions. The teacher for this class is a young woman with five years of teaching experience and AP Calculus certification. She presents open-ended questions and actively guides students in discussing and developing problem-solving strategies. Teaching content: Applications of differential equations in real-world scenarios (e.g., “A tank contains 100 gallons of water and dissolves 5 pounds of salt. Water containing 0.2 pounds of salt per gallon is poured into the tank at a rate of 4 gallons per minute, and the mixture flows out at the same rate. How many pounds of salt are left in the tank after 20 minutes?”).
In this classroom, problem-solving relied primarily on a combination of student autonomy and group collaboration. Students demonstrated a strong grasp of each defined concept and delved deeply into each problem. During group discussions, they developed a wide range of problem-solving strategies. In group work, each student first explored the problem independently, then considered their own solution approach. Only after developing a clear solution did they share their thoughts and approaches with their group members. Each group then compared different solutions, validated their own, and examined the rationality and correctness of their solutions.
During this classroom observation, I learned a great deal and encountered many different teaching styles. I discovered that these different teaching styles have a significant impact on students. In a classroom environment with good discussion, each student’s ability to work in groups and express their own views is greatly improved. However, in environments without classroom discussion, each student simply does what they are required to do. While this might increase teaching efficiency, it becomes very rigid for students. Students don’t know how to discuss with other classmates; each student works like a machine, using the same methods to solve every problem. However, from the teacher’s perspective, both of these completely different teaching methods are correct; every teacher hopes their students can learn mathematical knowledge and skills in their classroom.