We need to think as parents and educators about the math we grew up learning. Did we really learn and understand why we do certain operations, or why we manipulate numbers the way we do? The answer if we are truly honest is no. We have an operation or "trick" we can use to get the answer, but we don't conceptually understand the math involved. Case in point recently we've been working with decimals. I have student who came from another district who knows to pull the decimal out when multiplying then just put it back in. But when I asked why we do that, he had no answer. We have got to develop critical thinkers in math just as much as we do in reading or science. They have GOT to understand why, or they will never be truly good at math or problem solving!
We use Investigations in our district. It focuses on the child finding and using efficient strategies to solve a problem. The eventual goal is then to move toward a traditional algorithm when the child fully understands the process involved in solving a problem. We have daily Number Talks where the kids are required to use mental math to solve and explain their thinking. We then record their thoughts and name strategies. Here is an example of a multiplication anchor chart and some questioning we do with students.
As you can see the kids have amazing ways of thinking and working with numbers, often times ways I would never have come up with, but it works for them. As the year progresses we really push our students towards efficiency. Hopefully in years to come you will see your students catch the "Math Curse" and become excited once again about math. We can no longer be a country that is satisfied with mediocre math scores, we have got to push our kids to really grasp mathmatical understanding!