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.
So, don't use easy numbers?
> I give the students nice numbers and in exchange they are expected to tell me a general way of solving that type of problem.
If this quid-pro-quo is explicit, that seems fair. But I still don't see why "nice numbers" are necessary.
In general, if the step that I took to solve a problem was "I looked up a memorized fact in my brain", then that is the truth, and writing down anything else to "show my work" is a lie.
As other people have said, a proper way to ask that question is "Show a derivation of the answer to [problem]." Bonus points if you specify what assumptions they're allowed (e.g., in the above example, "addition").
