Class 1: 9th Grade Algebra I
This is a cheerful classroom with about 27 students. The desks are arranged in seven groups, each group’s desks clustered together. Each group has four to five students, and each desk is equipped with a traditional whiteboard and markers, allowing students to write their opinions directly on the desks during group discussions. Posters about methods and techniques for reasoning mathematical formulas or equations are displayed on the classroom walls, along with reminders of classroom discipline and rules. Algebraic blocks, number lines, and other teaching aids are available near the classroom door for students to borrow during group work, making the process simpler and more enjoyable. The teacher for this class is a young gentleman with four years of teaching experience, who mentioned receiving training in inquiry-based mathematics teaching at university. During group work, he would circulate among the groups, observing each group’s problem-solving approach and progress. He was very patient, listening to each group’s discussion and asking follow-up questions. He also recorded the students’ reasoning processes in his notes, making it easy to share each group’s problem-solving approach during class discussions. This lesson focused on proving the rationality of solutions to linear inequalities (e.g., “Solve the inequality 2x + 3 < 11 and explain how each step maintains the inequality”) and analyzing real-world problems (e.g., “A store is offering a $5 discount on shirts. If you have $25, what is the maximum price of shirts you can buy?”).
In this class, the reasoning groups exhibited strong collaboration among members. Group reasoning was based on evidence presented in the problem and was relevant to the context. Students explained each step of the solution in groups, guided by prompts from the teacher. For example, group members corrected each other’s errors through reasoning. If one member made a mistake when dividing by 2 by flipping the inequality sign, their groupmate would question, “Why flip the sign? Did you divide by a negative number?” The group member would realize their error: “Oh right, you only need to flip it when dividing by a negative number. Our reasoning was wrong, so the answer is wrong.” They would then correct their solution and explain their correct reasoning process to the rest of the group.
Class Two: 10th Grade Geometry
This class was held in a very traditional lecture hall with about 30 students. Each desk was arranged in a straight line facing the teacher’s podium. The projector was in the center of the classroom, displaying slides with pre-prepared proofs and theorem definitions. A “Quiet Study Area” sign was posted on the wall. There were no group projects or discussions in this class; each student worked independently and was required to follow the teacher’s pace and rhythm. The teacher was a woman with 15 years of teaching experience. When explaining problems, she would demonstrate the proof step-by-step on the blackboard, explaining why each step was necessary. She would also randomly call on a student to recite or explain mathematical theorems. After the lesson, she would assign an independent worksheet, informing students of the due date and important points to note. This lesson focuses on completing and memorizing the proof of the triangle inequality theorem (such as “proving that the sum of any two sides of a triangle is greater than the third side”) and applying the theorem to solve common problems and some questions that may appear in the Regency Examination.
In this lesson, the reasoning was entirely teacher-led, and the entire process was highly formulaic and lacked mathematical proof. Each student acted like a robot, simply copying the proofs from the blackboard and reciting theorems without any independent thought. The teacher didn’t explain the validity of the steps or establish a connection between the evidence and the conclusion. In one instance, when faced with the question, “Does a triangle with sides of lengths 3, 4, and 8 exist?”, a student simply used the relevant theorems to answer “No,” without explaining why the theorems held true or their reasoning. One student even told the teacher, “3 + 4 equals 7, which is less than 8, so it’s impossible, but I don’t know why this rule holds—I just memorized it.” I felt this kind of classroom lacked flexibility; each student seemed rigid, only knowing how to answer the teacher’s questions and memorize mathematical definitions. None of them understood the reasons behind their actions or the justifications for doing so. I think this kind of classroom is too traditional and lacks the skills to cultivate students’ self-thinking abilities.
Class 3: 11th Grade Advanced Placement (AP) Statistics
This class was held in a flexible classroom with approximately 24 students. There were group round tables with four students per group, each with a whiteboard. The walls were covered with posters of individual or group math projects, such as “Proving Problems,” “Using Data,” and “Drawing Statistical Tables.” Each table also had peer feedback forms for evaluating arguments, allowing students to provide simple feedback to their group members. The instructor, a woman with seven years of teaching experience, stated that she was certified in AP Statistics, which allowed her to teach the subject. She presented real-world datasets, explained the basic information and themes, and guided students to debate the conclusions. She also helped each group refine their reasoning process. This lesson focuses on using basic data to make inferences (such as “A survey of 200 students showed that 65% support the new lunch menu. Does this prove that the majority of the 1500 students in the school support the menu?”) and on demonstrating the validity of statistical claims using margins of error and confidence intervals. I found it very interesting.
In this lesson, the reasoning was largely student-led, based on fundamental data and iterative. Students conducted simple analyses of the dataset, then presented their arguments to the teacher and classmates, refining their reasoning based on peer feedback and learned statistical principles. For example, the teacher had two groups debate test scores: Group A argued that “new learning methods improved scores,” citing an increase in the average score from 75 to 80; Group B countered, “We need to verify whether the difference is statistically significant—it might just be random fluctuation.” The class discussed how to verify this (e.g., using a t-test) and concluded that without measuring variability, Group A’s reasoning was incomplete. I found this class very engaging, with every student participating seriously.
This classroom observation allowed me to see how different teaching methods can affect students. This will be of great help and influence in determining my own teaching methods in the future.