When you ask a student learning algebra to solve an equation, it is very clear which strategy the student tries to use when solving equations.

The natural way is to figure out an arithmetical solution to the equation when it is not a literal one. Very often, students gifted in arithmetics can guess an approximation of the solution.

Why would they need to understand and learn a new strategy which is more complicated than the previous one that they master and has given excellent results so far.

Solving equations thru an arithmetical strategy can’t be used to solve literal equations. At one point, students need to accept that there is another strategy to solve this kind of puzzle and they need to extend their maps and understandings of the problem.

They need to adopt an object oriented approach.

As a teacher, it is important to know of these understandings tresholds where students need to make a significant effort to accept and manage the cognitive conflicts that arise from a new approach to an old problem or a new way to look at an old problem to generalize it.

Mathematics is full of these jumps in understanding and conflicts of representation.

As a teacher, the first question is how to be aware of these conflicts. Literature? experience? knowledge transfered from other teachers?

The second question is what do you do when you know of these conflicts.

These are very relevant questions for any mathematics teacher. I wonder who is answering them?