As you know there is a move in mathematics to have students provide explanations to accompany the steps used to solve various problems.
Sounds good, doesn't it! But have you stopped to think about what they should be explaining? My first impulse is to have them write down the steps such as
1. x + 2 - 2 = 3 - 2. Subtract two from both sides.
This meets the basics of the idea because I have explained what was done but I didn't explain why. Why am I subtracting two from both sides. The why is as important as the step itself. Without the why, you are just following a mechanical process. When I got my degree, the professors never worried about the why's only if we could get the answers.
As I've searched for ways to help my students learn, I've learned more of the why's. I've learned how to provide visualization for many concepts. Visualizations, I didn't need and couldn't have designed by myself because it wasn't important when I got my degree.
To understand why is to understand more. To explain the why is to have the ability to explain more about the concepts which means students know it better. Calculations do not equate to understanding. Unfortunately, its an uphill struggle for me in the high school level because too many of my students had teachers in elementary who knew nothing about math so they just followed the book.
Many of these teachers were scared of math so to them there was only one way to do the problem and only one form of the answer. I include the comment about form because I know a young lady who lost points on a test because she wrote the answer as (x+1)(x+1) rather than (x+1)^2. The teacher neglected to state in simplest form. He just wrote answer the question. She went back and argued the point and got the points back.
Her math teacher had started out as an English Teacher who switched to teaching Math because there was only one way to do the work and only one form of the answer. He did this long before they wanted people to be highly qualified in a subject. He didn't really care about the what or the why. He only cared that they did the problems his way.
So let me know what you think about explaining the process. Do you think the why is as important as the what? I'd love to hear from you. Have a good day.