 # Mathematics

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.

## Lower Elementary (Lower Grammar)

-Acquire basic numeracy

-Understand equivalent forms of the same number using diagrams, objects, and numbers

-Recognize basic geometrical shapes and parts of shapes

-Solve word problems

-Count, read, write, and compare numbers up to 1,000, both symbolically and through

physical construction

-Acquire facility with basics of place value

-Perform basic addition and subtraction functions of one-, two-, and three-digit numbers

-Understand basic fraction concepts

-Count by 2s, 3s, 4s, 5s, 10s

-Identify and construct circles, squares, rectangles, triangles, ovals, cubes, tetrahedral

pyramids, cylinders, cones, spheres, and rectangular prisms

-Recognize and describe the appearance of basic patterns in nature

-Recognize equivalency in number, shape, pattern, and other physical characteristics

-Construct basic sets and groupings of objects in the environment and nature and be able to articulate the criteria for inclusion and exclusion

-Recognize and solve simple replacement codes

-Solve simple geometric puzzles

-Recognize the relationship of tone to the size, length, shape, and material of the object

being sounded (e.g., bells of different size being rung or the length or thickness of a

string being plucked)

-Take linear measurement and be able to articulate changes in measurement over time;

introduce basic means of recording measurement

-Tell and record time and changes in time in seconds, minutes, hours, days, weeks,

months, and years

-Recognize and calculate basic currency; introduce coin-tossing scenarios as an

introduction to probability

-Recognize the ubiquity of number and shape in the world around us

-Develop a sense of wonder at recognizing how the world can be expressed mathematically

-Develop a love for constructing math, numerically and geometrically

## Upper Elementary (Upper Grammar)

-Deploy numeracy/counting: whole numbers into the millions; decimal place value

-Recognize geometric shapes and calculation of perimeter and area

-Have facility in addition, subtraction, multiplication, division whole number operations

-Add, subtract, multiply, and divide decimals up to the thousandths place

-Use fractions (reducing, adding, subtracting, multiplying, dividing)

-Measure accurately using both customary and metric systems

-Estimate measurement when measurement tools are not available by comparison of

surrounding or similar objects

-Solve word problems

-Count money and basic decimals

-Acquire basic algebra skills (looking for unknowns)

-Begin to understand proportions

-Comprehend basic averages and ranks (median and mode; mean by grade 5)

-Introduce classical geometric and architectural design (choose a building from historical

time period being studied and analyze its geometric and proportional properties)

-Analyze rounds and simple canons to identify simple progressions

-Recognize and construct fundamental shapes in plane geometry: points, lines, rays,

angles, parallels, perpendiculars, quadrilaterals and regular and irregular polygons

-Analyze composition and use of light in art in relation to geometry

-Analyze perspective in art in relation to angle measurement

-Construct Platonic and Archimedean solids

-Use Euler’s formula for the number of vertices, faces, and edges of polyhedral

-Solve more complex codes such as a single replacement and translation code

-Apply numeric methods in describing natural phenomenon—for example, estimate the

number of leaves on a tree by modeling the splits in a branch

-Memorize and master addition/subtraction tables (0-10)

-Memorize and master of multiplication tables and division (0s-12s)

-Use mental arithmetic

-Multiply single- and multi-digit numbers

-Divide multi-digit numbers by one-digit numbers

-Tell time to the quarter- and half-hour and to five minutes and one minute

-Add and subtract decimals, and compare decimals and fractions

-Multiply multi-digit numbers by two-digit numbers

-Divide larger multi-digit numbers by one-digit numbers

-Find the area of two-dimensional shapes

-Reason mathematically both orally and in writing through word problems

-Use problem-solving strategies to solve real-world math problems

-Add and subtract fractions and decimals

-Identify and describe three-dimensional shapes, and find their volumes and surface areas

-Use long division to divide large numbers by multi-digit numbers

-Recognize numerical patterns in music and nature and geometrical patterns in nature and art

-Solve simple probabilities, including independent and dependent events and simple truth tables for conjunctions, disjunction, negation, and implication

-Read and use bar, line, and circle graphs

-Measure shape and position over time, such as tracking the phases of the moon and

simple astronomy, including solar measurements (measuring shadows and angles at

different times of the year)

-Count back change up to \$100

-Recognize basic Biblical numerology

-Acquire a foundation for logical reasoning through math

-Be attuned to the relevance and significance of number and shape

-Begin to appreciate the 'aesthetics' of number through recognition of patterns

## Middle School (Logic Stage)

-Master arithmetic necessary for algebra: order of operations; fraction, decimal, and

integer operations

-Develop more advanced number sense (integers, irrational numbers, percentage,

scientific notation, absolute value, exponents, roots and radicals)

-Understand factors and multiples; find greatest common factor and least common

multiple/denominator

-Understand measurement concepts

-Master developmentally appropriate algebra and geometry

-Read and use a coordinate plane

-Recognize mathematical and geometrical patterns in nature and art

-Begin to understand the philosophical and theological history of mathematical symbolism

-Think algebraically and geometrically

-Use logic and hands-on experience to solve problems

-Convert fractions, decimals, and percents

-Rewrite fractions using factors and multiples

-Solve problems using rate, proportion, common formulas, and percentage applications

-Use estimation techniques

-Use mental arithmetic

-Use and convert customary and metric measurements

-Solve developmentally appropriate functions, equations and inequalities and graph them on a coordinate plane

-Calculate slope

-Write and use formulas to solve problems

-Combine like terms

-Add, subtract, multiply, divide, and factor polynomials

-Identify properties of and congruency between angles, parallel lines, triangles,

quadrilaterals, other polygons, and common three-dimensional figures

-Calculate area and perimeter or circumference of two-dimensional figures

-Calculate surface area and volume of three-dimensional figures

-Use the Pythagorean Theorem to solve problems

-Use a coordinate plane to translate, rotate, and reflect a given image

-Calculate simple probability

-Read and create bar graphs, line graphs, circle graphs, and stem-and-leaf plots

representing data; make predictions from statistical data

-Analyze musical compositions for mathematical properties, particularly Baroque music

(Bach, Vivaldi, Pachelbel, Albinoni, etc.)

-Understand Christian iconography in relation to dimension

-Analyze poetic meter

-Recognize sacred number in writing and art

-Appreciate mathematics as one way humans give an account of reality

-Appreciate relevance of math to music, art, science, and architecture

-Enhance logical reasoning

-Acquire a foundation for logical reasoning through math

-Be attuned to the relevance and significance of number and shape

-Begin to appreciate the 'aesthetics' of number through recognition of patterns