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.
In history, for example, students can begin to understand the meaning of the architectural design and sacred geometry of classical buildings, in which not only shape, pattern and placement convey meaning, but number also is used to encode philosophical and theological truths.
The mathematical foundations of music can be introduced from the mono-chord to tone relations, and then to the understanding of harmonics and series. In the upper grades, students can be introduced to the mathematics of the fugue and the canon, and taught to hear the voices in their relationship.
In the study of visual art, students can be trained in the geometric and numeric relationships that are at the basis of representational drawing, particularly for creating the illusion of depth through the application of transformation and projection, and can be taught the visually pleasing and dynamic ratios that appear in great art and photography. This visual training can be extended to a broad discussion of dimensionality in the context of iconography and non-representational art.
In the language arts, the mathematics of rhyme and meter can be discussed and practiced, at first through recitation but eventually through imitation. Also, the discovery of the numerological meanings written into great literature can begin with the Bible and advance historically through the various periods studied.
In nature studies, the mathematics of nature can unveil the mysterious occurrences of transcendental constants such as pi and the natural logarithm, the recurrence of biological geometry such as the spiral of Archimedes, and the myriad ways in which relation is communicated in the branches of a tree, the strands of an orb web, or the convergence of streams into a river. Individual plants and animals can be introduced as the basis for understanding growth, and direct observation and measurement can be the basis for understanding numerical and visual representation of change through time. Individuals and populations can be used to illustrate the concepts of rate of change, large numbers, and eventually infinity. Measurement and the mathematical representation of natural systems can become the entry point for a discussion of estimation and precision, order and entropy, probability, and eventually chaos. This can include a discussion of how to represent things numerically, which presupposes an understanding of Aristotle’s four forms of causality, and can culminate in understanding that mathematically representing and quantifying the world depends on philosophical choices.
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.