So here's a question from the exercises that were assigned during the past week:
You have often wondered how tall the water tower
near Pine Hill Park is, so one day you take your protractor and head for the
park. Looking at the top of the tower,
you measure the angle of elevation to be 30°.
After moving 108 feet closer to the tower, you again measure the angle
of elevation to the top of the tower and find it to be 50°. What is the height of the tower?
A pretty standard exercise, I admit. Here's the thing: the kids came up with at least four different ways to determine the height of the tower based on this information. Some used a system of equations involving tan(30) and tan(50). Some used the law of sines. Some used the fact that the "large" triangle is a 30-60-90 triangle. Some used the Pythagorean Theorem. Many used some combination of the above, but not necessarily the same combination. Most arrived at the correct answer.
Among others, this is one of the things I love best about teaching through discovery and discussion. The kids came into the class knowing basic right-triangle trig, and they had used special right triangles before. Through the course of the exercises, we have emphasized both of these topics, along with discovering the law of sines. That sets us up to work on this exercise. If I had done a similar problem in class as an example to "show them how" the problem is to be done, the next day I would have had classes full of kids who all did this problem exactly the same way I did it. No critical thinking, no real problem solving, no creativity, just mindless imitation. Instead, the kids had to make sense of the exercise themselves, use the "tools" available to them, and solve the exercise. Working through things this way sets them up to handle a wide variety of similar exercises that don't necessarily ask them for the height of the tower, but that could instead ask them for the distance from the base of the tower, the angle of elevation, etc., whereas me giving them as example and having them imitate it sets them up to do an identical exercise with the different numbers and nothing more. To ask them to find the angle of elevation would require a separate example, and asking them to find the distance from the tower would require a third, none of which build or even encourages critical thinking, problem solving, or creativity. Instead, it promotes memorization, and little to nothing else...memorization which, by the way, fades away quickly. And if what the kids memorize is inaccurate, then fixing it is difficult. I have found that it is much easier to correct a reasonable but flawed guess about which a kid isn't completely sure than it is to correct a flawed bit of memorization about which a kid is absolutely certain.
Sadly, I've seen teachers essentially train kids to do exercises only and exactly the way that they are shown in class and then get upset that the kids aren't able to generalize the process on the test. Worse, they don't make the connection that if we don't regularly ask the kids to understand the material well enough that they can explain it and be creative with it, then they won't be able to put the pieces together and generalize the process once every 2-3 weeks on a test. Of course, rock bottom is the teachers whose test questions are only ever the same exercises as those that were specifically shown as examples in class, just with different numbers. This kind of teaching may have cut it in the past, but it won't now, especially with end-of-course exams that require creativity and critical thinking coming soon to our classrooms.
If we want the kids to be creative, problem-solving, critical thinkers, then we need to ask them to be creative, problem-solving, critical thinkers every day. We need to encourage them to make a conjecture as to how a problem can be solved, attempt to solve the problem based on that conjecture, and right or wrong learn from the experience. This requires us to create an atmosphere in our classrooms where it's safe to make a mistake and then try something else. It requires us to see things the way they do and help them find a way that the material makes sense to them. It requires us to accept different but mathematically valid methods of solving an exercise.
It's easier to just ask the kids to do things the way we do them. It's easier to grade a set of papers that all look the same. However, good teaching doesn't focus on the teacher. Good teaching focuses on the learning being done by the students. It isn't easy. But if we don't want the kids to be creative, problem-solving, critical thinkers because it makes things difficult for us, then we need to get out of the classroom.