Wrong vs. Not Helpful

In solving a problem, students inevitably make mistakes. This is how we learn. However, there is a huge difference between being wrong and being not helpful.

Wrong

Suppose a student is asked to solve the following for x:

Seeing the 2 next to the x, a student may try to divide both sides by 2. This can be done correctly, but let's assume that the distributive rule is momentarily forgotten:

This leads to the incorrect answer of x = 1, whose falsity can be demonstrated by substituting 1 in for x in the original equation (which would give 6 = 10).

Not Helpful

Now imagine that the student takes the same equation, and subtracts 2x from both sides, giving:

This is not wrong, subtracting 2x is not a violation of any mathematical rule, but neither is it helpful. A student who attacks a problem like this may still be in need of assistance, but a different type of assistance from before.

My Point

This distinction is obvious to educators. It is not, however, necessarily obvious to students. And it needs to be made obvious to students. There is a rampant misconception of math that there is a right way and a wrong way of doing math, and if you're not not doing it correctly, you're wrong. (I previously wrote about that here.) Sometimes you're doing something useless that is not necessarily wrong. And that is ok.

I think a good way to get this distinction across is by way of analogy. Consider chess: there is a notable difference between a wrong move (moving a rook diagonally) and a foolish one (exposing yourself to checkmate).