The answer is not important. What is important is that one knows a process for solving the general situation. Knowing how to get the answer in a problem with easy numbers is not the point. Demonstrating how to find it no matter what numbers I used is the point.
There is also the fact that I'm not just teaching the student so they can pass this class. I'm teaching them so that they can pass the next class too. In beginning algebra, our lowest level course, we start with basic problems.
For instance, you drive for 3 hours and travel 150 miles. What's your average speed. Almost every person gets this right. But many can't do this problem. You drive a car for 4/3 an hour and travel 100/6 miles. What is your average speed? Now many can't do this problem. They ask, which way does the division occur?
Our goal is that they know and understand a general process. We are setting them up to solve more advanced problems. Problems that can't be done in their head. If you can't write the steps out in the simple case you'll never understand the harder problems. Problems that involve quadratic functions or trig functions.
Mathematics is a human activity and communication is part of it. Knowing how to communicate what is in your brain to another person in a way that they can understand is very valuable. I give the students nice numbers and in exchange they are expected to tell me a general way of solving that type of problem. Instead of feeding them for a day I want to teach them how to fish.
I strongly disagree with your belief that this mathematical abuse.
In this case nice numbers are not necessary. In other types of problems they are. One does not want to burden students with needlessly complicated calculations when the main point is a simple one.
Interestingly, I think many of your points would seem obvious if couched in computer programming terms. For instance, how do you write a program that's correct, but that another human can understand and adapt? What happens when a program gets too big to understand all at once in your head? How do you know that you've found all of the possible outputs?
There is also the fact that I'm not just teaching the student so they can pass this class. I'm teaching them so that they can pass the next class too. In beginning algebra, our lowest level course, we start with basic problems.
For instance, you drive for 3 hours and travel 150 miles. What's your average speed. Almost every person gets this right. But many can't do this problem. You drive a car for 4/3 an hour and travel 100/6 miles. What is your average speed? Now many can't do this problem. They ask, which way does the division occur?
Our goal is that they know and understand a general process. We are setting them up to solve more advanced problems. Problems that can't be done in their head. If you can't write the steps out in the simple case you'll never understand the harder problems. Problems that involve quadratic functions or trig functions.
Mathematics is a human activity and communication is part of it. Knowing how to communicate what is in your brain to another person in a way that they can understand is very valuable. I give the students nice numbers and in exchange they are expected to tell me a general way of solving that type of problem. Instead of feeding them for a day I want to teach them how to fish.
I strongly disagree with your belief that this mathematical abuse.