Why does this student answer $2a + 9$ for the first question, and $x + 49$ for the second questions?

Why does this student express more confidence on the first question than the second question?

This post is part of a series analyzing a bunch of survey results. For previous posts, go here, here and here.

• I wonder what would happen if you gave them (x + 3)(x + 3) and (a + 3)^2? What if you changed up the numbers?

• Zeno

What troubles me about these and many other examples at this website is that the answers given by the students are not simply wrong, they’re OBVIOUSLY wrong. By that I mean there is a simple check which would show that the answer cannot be correct. As in these examples, when the problem is to simplify a given expression, the correct answer must be an expression which is equivalent to the given expression, in the sense that the two expressions evaluate to the same result when any value is assigned to the variable. So, for example, if it is correct that (x+3)(x+3) simplifies to 2x+9, then it must be true that (x+3)(x+3) = 2x+9 for any value of x. If we check this for x = 1, we have (1+3)(1+3) = 2*1+9 then 4*4 = 2+9 and 16 = 11. But that’s not true, so the expressions are not equivalent, hence the answer cannot be correct.

As a general rule, if the problem is to find something that satisfies some constraint, then wrong answers can very often be detected simply by checking whether the proposed answer satisfies the constraint. Do the students not know this? Perhaps it would help if they were consistently required to check their answers.

• Amen, Zeno. The idea of the equivalence for an equation is something that is lost in the thickets for too many students. At the earliest stages, requiring checks of answers should be a powerful and helpful tool to break some bad habits.