Why is it important to understand the foundations of mathematics?

Math is about sense-making. "This the most fundamental idea that a...teacher of mathematics needs to believe and act on. It is through the teacher's actions that every [learner] in his or her own way can come to believe this simple truth and, more importantly, believe that he or she is capable of making sense of mathematics. Helping students come to this belief should be the goal of every teacher. " (Van de Walle, Teaching Student-Centred Mathematics, 2006, p. ix)