When one solves an algebraic equation, what authority can one turn to to check the validity of the answer? Unfortunately, there is no easy answer to this question. My most recent attempt at empowering students to be their own mathematical authorities is to use two complementary versions of the same worksheet, with each version reinforcing the other.
This practice works best for problems than can be easily reversed. For example I could ask, "Solve the equation $x^2-2x-3=0$" and receive answers $x=3, -1.$ Conversely, I could ask, "Find a monic quadratic equation with $x=3, -1$ as roots" and receive the answer $x^2-2x-3=0$.
In practice, I create two versions of the same worksheet – one version will ask the questions in the forward direction (given the equation, find the roots) and the other version will ask the questions in the reverse direction (given the roots, find the equation). When I hand out the worksheet, I give each student only one of the two versions. After ten minutes or so, students find a partner with the other version of the worksheet (color-coding helps!) and are able to check their answers. I don't need to get involved at all!
This process works very well, but it does not fully resolve the issue of transferring authority to the students. In the moment, students are the authority on the correct answer to a problem. However, this authority that a certain answer is correct is still coming from me, because I supplied them with both versions of the worksheet.
I started this blog to share my transformation from math nerd to math nerd who loves to share math with young people. I teach high school in Philadelphia, Pennsylvania. Your comments are always welcome.