Your solution doesn’t at all address the problem of students who don’t use the standard algorithm, yet get better results than other students or the teacher. What are they to “show,” then, and why? Of course it would be nice if they showed they’re work and shared it with the teacher and other students. But to be graded on “showing the work” or not, is still, I believe, generally a control issue, and not at all meant for the benefit of the student.

In the case where showing all the students must show the same work, for all the teachers, in all the schools, in the whole state, well, so far I haven’t seen any argument that doesn’t boil down to, “Well, we have to do something, and we don’t know what else to do, so we’ll make the students do the thing that puts the least onus on us.”

I know that’s not what you’re advocating. I think you’re system is best for you, and better than most others, but only when it is practiced by a teacher like you, for the reasons you give. If all other teachers were like that, there would be no problem.

Wait – yes their would. The administrators would never understand it or support it – and they’d make you do something else.

For me, it’s about providing opportunities for metacognition.

For the superior student, work is unnecessary, and the superior student KNOWS it’s unnecessary. The inept student, however, vastly overestimates their own ability, because they lack the metacognition to know where they’re making mistakes. Showing your work puts students gives students something to go back over and check, a foothold for metacognition. It prevents students from forgetting terms and making sign errors in complicated formulas.

Here is a good example. In arithmetic of integers, a struggling student working on -12 – 8 will (some of the time) answer -4, because the answer makes sense with the numbers involved, and feels easy. The student doesn’t realize that there are four easy answers (-20, -4, 4, and 20), and that three of them are wrong.

They need to work the process out on paper in order to arrive at the correct answer, so that they have something they can go back and check and cultivate the metacognition they need to improve. They need to show their work until they demonstrate mastery.

Here is a suggestion: Give an assessment to students who believe they have the method mastered to decide, on the basis of the result, whether or not you should continue to require them to show their work.

I began undergraduate school in engineering coursework, did fine in the engineering practititoner classes, but hated the calculus and chemistry ‘weeder’ courses. Needless to say, I opted out of pursuing a math career. That to me is the failing of our educational structure. Natural talent is not recognized nor encouraged it seems to have turned into a politically correct statist indoctrination program and daycare for absentee parents.