I've been working on creating math "labs" that allow students to explore math concepts in a hands-on way. My first lab was in geometry; my goal was to help students understand the relationship between the lengths of three line segments that is necessary for creating a triangle.
First I had students try to make a triangle with three line segments which could NOT create a triangle. They concluded that it was impossible because two of the line segments were not long enough. Then I had students make a triangle with three line segments that were sufficient for creating a triangle.
I then wanted students to record a hypothesis for evaluating any three line segments' ability to form a triangle. This was really tough for students. They were able to describe why three segments could or could not form a triangle, but they had trouble generalizing their findings.
I want to keep working at making my students think about math in a broader sense. They are used to solving problems (as long as the problems are set up in a familiar format), but are not able to apply what they know to scenarios beyond what they are familiar with. My hope is that if I continue to challenge them to think beyond the box, they will begin to learn to see the big picture of what math is really about.