As I go through my days teaching and preparing I often hear the questions of "What is math" It is no easier to answer than "What is culture?" or "What is art?" but because math is based in some definition of logic people feel like it should be easily explained. The answer varies far and wide. It is most surely not limited to what we teach in schools, particularly in middle and secondary education where it is often distilled into a laundry list of rules. Primary school studies are closer to the point when they encourage kids to explore numbers and shapes to identify patterns and relationships. What happens when you change the numbers? Can you derive certain rules from what you observe?
In reality, education has failed in some way when we start hearing, "Why do I need this?" and "What is this?" The former question moreso for it means someone has lost his curiosity or interest, that a course of study has become a chore with only an end result. We must strive to find ways to retain that curiousity and enable learning. This happens when students really learn and see the horizon of possibility expand with each step. Something as minute as adding fractions may seem small but it can have a real and positive impact.