The study of mathematics should instill in students an ever-increasing sense of wonder and awe at the profound way in which the world displays order, pattern and relation. Mathematics is studied not because it is first useful and then beautiful, but because it reveals the beautiful order inherent in the cosmos.
Mathematics stands in a unique position at the intersection of induction and deduction, and as it flowers, it enables the student not only to appreciate more deeply its own subject matter, but also every other discipline since it lends its own intelligibility to their study. This is readily apparent in logic and analytical reasoning, but is no less true for art, music, poetry, history, sports, experimental science, philosophy, and language.
Mathematics can engage all the senses, particularly in the early years, with the direct manipulation of simple objects that illustrate number and counting, similarity and difference, belonging and exclusion, progression, proportion, and representation. Along with this direct experience, students can be coached in observation and taught not only to recognize but to question the relationship of countable to uncountable, unity to plurality, and repetition to progression. They can gradually be introduced to ways in which we quantify the world by applying dimension, magnitude, duration, measure and rank, and also ways in which the world may be analyzed and modeled through mathematical representation, including geometric and algebraic expressions. To the extent possible, students can be encouraged to ‘construct mathematics’ (such as building Platonic solids) as well as work it out on paper, and come to understand that the symbolic writing of mathematics enables us to describe accurately and therefore to predict the outcomes of many real-world events.
The study of mathematics should emphasize its foundational contribution to aesthetics (the study of beauty).
The “mathematics of beauty” can be discerned in every subject.
A love of mathematics naturally leads not only to the development of analytical and critical reasoning skills, but deep creativity.
Most importantly, it fosters a sense of profound reverence for the cosmos and our place within it, and the infinite depth of intelligibility woven into creation. This love is a spontaneous response that arises when a child first discovers math in the world, and must be nourished so that the work of solving math problems does not become tedium. Puzzles, codes, riddles, games, and the direct observation and experience of mathematics in our world are important ways to keep the intrigue and enchantment of mathematics alive while building necessary skills.