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- In the first scene, by far the most common, you will see the students sitting in neatly organized rows, facing the teacher, who stands at the front. On the desk in front of each student you will likely see a textbook, a notebook, a pen or pencil, and perhaps a calculator. At the start of each class, the teacher will spend some time at the whiteboard, explaining some new rule or technique and working through one or two examples. Then the students will open their textbooks and proceed to work through a number of assigned examples whose solutions require the technique they have just been shown. They will for the most part work alone, and in silence. When they run into a problem, they will call on the teacher for help, not each other. When they have completed the task, the cycle begins over again. This teaching method is general known as "the traditional approach." It's an appropriate name, since it has been used since the beginnings of mathematics, some three thousand years ago.
- The other, less common scene appears much more chaotic. Groups of students sit around circular tables discussing how to solve a particular problem, or standing at the whiteboard arguing about the best way to proceed. The teacher moves around the room talking with the different groups in turn, making suggestions as to how to proceed, or pointing out possible errors in a particular line of reasoning the students are following. Occasionally, the teacher will call the entire group to order and ask one group to explain their solution to the rest of the class, or to give a short, mini-lecture about a particular concept or method. This is sometimes called "the progressive approach."
I too find the practice of sub-dividing subjects relentlessly stupid. It gives the impression to students that geometry is not algebra which is not trigonometry, etc, when in actuality all math is just applying operators in predefined ways to determine the necessary but usually unobvious results of a given set of conditions. The first thing that ought to be done in schools is to ban calculators in math class, and probably ban numerical problems in high school math. The more specific a math problem is, the more useless it is. There is no way to expand anyone's mind when detail is king.
I never learned as well as I did when I spent most of my free time in the physics students lounge as an undergrad. We had a huge whiteboard, and worked out problems together. Typically students that weren't in the class would join in for fun. As a result, you could get a number of different approaches, and this would build a genuine context for the specific type of problem. Although we spent well over half of our time goofing off, it was some of the most efficient learning I have ever been involved in.