Class 1: 9th Grade Algebra I
Classroom Environment: The classroom has approximately 29 students; desks are arranged in 7 mixed-ability groups (4-5 students per group), equipped with calculators with graphing and equation-solving tools and whiteboards. The walls display connections between equations, graphs, and real-world scenarios, as well as student projects. Each group is equipped with manipulatives such as algebra tiles and coordinate grids. The teacher for this class is a woman with 6 years of teaching experience, and she says she specializes in foundational mathematics instruction. During group activities, she circulates among the groups, asking guiding and connecting questions, and frequently encourages students to engage in independent learning and to make connections between concepts themselves. The lesson focuses on linear functions (slope-intercept form, graphical representation, and real-world applications such as distance-time relationships).
During a class discussion, a student pointed out, “This is very similar to the proportional relationship we learned last month, but the y-intercept of a proportional relationship is 0—there are no initial costs. So a proportional relationship is actually a special case of a linear function.” Then, each student in that student’s group would search for evidence to prove this point. If the point is correct, and after the teacher confirms it, the teacher will add it as a student-discovered knowledge point to the “keywords” written on the blackboard.
Classroom Two: 10th Grade Geometry
The classroom environment was very traditional, resembling a lecture hall, with approximately 32 students. Desks were arranged in straight rows facing the front of the room, and a projector displayed slides of trigonometric identities and practice problems, without any visual aids to connect concepts. A “Quiet Practice” sign was posted on the classroom wall. The teacher for this class was an older gentleman with 12 years of teaching experience. He preferred teaching through slides, demonstrating problem-solving steps on the blackboard, and then assigning independent practice worksheets that focused on developing problem-solving proficiency. He was very serious in his teaching and rarely joked with the students. The content covered included trigonometric functions of right triangles (sine, cosine, and tangent ratios) and solving for unknown sides and angles.
When students were solving for the unknown sides of right triangles, they simply applied formulas without connecting it to their existing knowledge. One student asked the teacher, “Is this related to the slope we learned in algebra class?” The teacher simply replied, “You don’t need to consider slope in this unit; you just need to focus on the sine ratio.” The discussion on this topic was then abruptly ended. Furthermore, in the classroom, students only solved problems through algebraic calculations, without using diagrams or coordinate graphs to visualize the relationships. When a student drew a right triangle to aid in solving the problem, the teacher told the student, “There’s no need to draw a diagram; you can just directly substitute the values ​​into the formula.”
The classroom culture in this lesson prioritized proficiency in problem-solving and exam scores over conceptual understanding. The teacher would only tell students: “The key to trigonometric functions in algebra is to memorize the ratios and apply them correctly. You don’t need to consider its connection to other mathematical knowledge right now; you just need to learn how to use these things to solve problems.” Independent practice was the main form of classroom activity, and students were prohibited from asking questions that were “unrelated to the topic” or involved making connections to other concepts. This culture led to superficial learning among students; virtually every student could apply the formulas, but they did not understand the meaning and value of the formulas.
Classroom 3: 12th Grade (AP) Calculus
Classroom Environment: The classroom has approximately 26 students; it features group tables and a front demonstration area with a smart whiteboard. The walls are covered with student-made “concept map posters” and “project posters” for each topic, connecting derivatives, integrals, differential equations, and physical applications (such as derivative = velocity/acceleration). Each table has a “Today’s Syllabus” handout for students to record connections between concepts. The teacher for this class is a woman with 8 years of teaching experience and AP Calculus certification; she organizes discussion activities to bridge calculus concepts with real-world applications in other disciplines and encourages students to present problem-solving solutions based on conceptual connections. The lesson focuses on applications of integration (area between curves, volume of solids of revolution, and work and force calculations in physics).
In the classroom, the teacher asks students many guiding questions. When students ask questions, the teacher answers them in detail. After ensuring that the students clearly understand, the teacher uses the question to extend the discussion and ask further questions. The teacher uses a simple question to guide the discussion towards the main topics of the lesson, gradually increasing the difficulty of the questions.
After completing this observation, I found that when teachers design lessons around conceptual connections, students are able to develop a holistic understanding of mathematics, viewing it as a tool for understanding the world and solving problems. However, when conceptual connections are neglected, students only memorize isolated problem-solving steps. This knowledge is not only easily forgotten but also ineffective in real-life situations. I also have several questions: 1. How can teachers balance the teaching of basic problem-solving steps with the cultivation of conceptual connections for students with weak foundational knowledge, without increasing their learning burden? 2. In a teaching environment with significant pressure to cover the curriculum quickly, how can teachers demonstrate the value of conceptual connection activities and cope with the pressure to “keep up with the teaching schedule”? 3. Do different mathematical disciplines require different strategies for establishing conceptual connections?