What is mathematics?
Defining what mathematics is has apparently not been easy, but understanding the purpose and ways of math is necessary, if we are going to believe in what we are teaching and how we are teaching it.

Through history, wise persons have spoken, and books have been written about this question (i.e. Courant, R. et. al., 1996; Hersh, R., 1999).  Here are some interesting definitions given:
Mathematics is the alphabet with which God has written the universe.  Nature's great book is written in mathematical symbols.
[Galileo Galilei (1564-1642)]
Mathematics is the queen of the sciences and arithmetic is the queen of mathematics.
[Gauss, Carl Friedrich (1777-1855)]
Mathematics is the rock upon which the arts and sciences of the world rest.
[D. E. Smith]
[Mathematics:] the logical study of shape, arrangement, and quantity.
[Common dictionary]
Let us grant that the pursuit of mathematics is a diverse madness of the human spirit, a refuge from the goading urgency of contingent happening.
[Whitehead, Alfred North (1861-1947)]
Mathematics is a magnificent example of the beauty, richness, complexity, diversity, and importance of human ideas - a marvelous testament to what the ordinary embodied human mind is capable of - when multiplied by the creative efforts of millions over millenia.
[George Lakoff, Rafael Nunez (Where Mathematics Comes From, 2000)]
Mathematics are all the things we do to understand, predict, and transform the environment efficiently and reliably, using accuracy (exactness) and precision (reliable procedures) to help us use resources with the best balance of safety, speed, and economy.
[an engineering instructor]
A PRACTICAL DEFINITION OF MATHEMATICS