I first observed an Algebra 1 (9th grade) class where the teacher used a group work format. There were 4 to 5 groups, each with 5 to 6 students. Because there were many students in the class, each group had a large number of students. In the group work, the teacher gave students 10 to 15 minutes to solve a system of two linear equations in two variables, requiring students in each group to discuss and try different methods. I saw some students use the substitution method, while others used the elimination method; the whole process was very friendly. Students also compared which method was more efficient. After completing the group work, the teacher organized a class discussion, with each group presenting their solution, and then the groups compared their solutions. When groups shared their solutions, if there were different solutions, the teacher would ask, “Are there any other methods?” This encouraged responses from other groups and made the class discussion more active. Furthermore, the teacher will ask students to write their answers on the blackboard individually. If a student makes a mistake, the teacher will gradually correct the error through targeted dialogue, allowing the student to find the correct answer on their own. I think this is a very good approach, and I especially like it. Because the teacher doesn’t directly give the student the correct answer, but guides them to find the correct answer through dialogue.
I also observed a preparatory calculus class (grade 12). In this class, the teacher divided the students into three groups of two to three students each. While the students were divided into groups, there was no formal group work. The teacher allowed students to work and think independently, and when students had questions, they were encouraged to ask the teacher directly. If a student encountered a problem, they simply raised their hand, and the teacher would approach them and help them solve the problem. During the class discussion, students only needed to answer the teacher’s questions, and the teacher would tell them whether their answers were correct or incorrect. There was no dialogue between students, and when students shared their solutions, no peers followed up or responded; everyone solved problems alone. In one-on-one teacher-student interactions, the teacher did not provide guidance. When a student gave an incorrect answer, the teacher would only give the correct answer and explain how to arrive at the answer or the formula. The teacher did not engage in guided dialogue or allow students to find the correct answer independently; instead, the teacher directly provided the correct answer. I think that this approach will not allow some students to learn real mathematical knowledge.
I also observed a geometry class (Grade 10). In this lesson, the teacher also used a group learning method. When learning how to prove a parallelogram, students in the groups came up with many different approaches, then discussed them with each other, considering whether each other’s methods were correct. During the whole-class discussion, they also agreed on and considered other students’ viewpoints. All students were very active in the discussion. The teacher also analyzed the students’ answers and told them whether the answers were correct. If a student’s answer was incorrect, or if there was a mistake in a part, the teacher would analyze it for the student, guiding them to find their error and explaining the correct solution steps and final solution method. This allowed students to find the correct answer through their own guidance and taught them how to solve similar problems in the future. I think this is one of the teaching methods I should learn.
I observed the teaching methods of seven different teachers. I found that some teachers would provide several solutions and processes for the same problem, including correct and incorrect solutions. The teachers would then have students observe these different solutions, divide them into groups, and identify which solutions were correct and which were incorrect. If a solution was incorrect, the teacher would ask students to find the reason for the error or the steps that led to it. Students would then analyze the error, find the correct steps, and correct the incorrect steps. I think this is a very good teaching method because it exposes students to more problem-solving methods and teaches them various ways of thinking. The process of identifying incorrect solutions helps students remember them, effectively preventing them from making the same mistakes on similar problems in the future. It also encourages active discussion among students, making the mathematical concepts in the lesson more solid.
Some teachers provide students with a syllabus for the upcoming math lesson before class. This syllabus includes the key points of the lesson, definitions and examples of what will be learned, expectations for students, and what each student should learn. Students can also write down their perceived difficulty level and what they have learned in class. The teacher then uses this syllabus to guide the lesson. This allows students to know what the lesson is about and what they need to understand and what mathematical knowledge they need beforehand. When explaining mathematical definitions, the teacher provides key words to help students remember and comprehend the meaning. In such a classroom, everything becomes very orderly; the teacher knows what they are teaching, and the students understand what they need to learn. This approach makes the classroom relaxed and simple, making teaching and learning much easier for both teachers and students.
However, I also noticed that in class, many students knew how to solve the problem, but when it came to writing down the answer, they didn’t know how to write out the correct solution steps. Furthermore, they often became very anxious when faced with difficulties with the solution steps. They knew the correct answer, but their ability to write out the correct solution steps was particularly poor. In most cases, they could give the correct answer, but their solution steps contained numerous errors. Moreover, their expression was also very flawed when the teacher asked questions. I think this problem is very serious because if this kind of problem exists when analyzing a problem, it means that the analysis of the problem will often be very flawed.
During this observation, I learned many new teaching methods and also gained insights into the different perspectives of various teachers on teaching.