Math and Geometry in the cosmic world for K-12 BEC education plan

(Part IV of a series on P-Noy’s K-12 Education Plan)

The life of mankind comes from the Cosmic world. The Cosmos refers to the whole planetary system with Earth within it. With her scientific orientation Dr. Maria Montessori appropriately used Cosmic Education as the theme for the basic education curriculum matching the nature of 3 to 6-year-old preschoolers, the 6 to 12-year-old grade school students, the 12 to 18-year-old teenagers and the 18 to 24-year-old adults.

The origin of Cosmic Math

Math and Geometry are taught in highly abstract level in conventional schools. Galileo, Michelangelo, Marconi are Italian historical giants who have mastered the art of Mathematics. Their contributions have become the basis for modern engineering and technologies.

While they have influenced the scientific thinking of people, another Italian doctor, Dr. Maria Montessori, sought to unravel the learning difficulties of conventional education which set back the progress of young children. She transformed the approach in the teaching of all subjects integrating them into a unified and continuum curriculum from preschool to adolescence. She made Mathematics easy to comprehend using a complete set of colorful apparata for numeration, the four arithmetic operations, Geometry, Algebra, Trigonometry and Calculus.    

How the abstract concepts of Math were materialized

A philosopher and scientist, Dr. Montessori enthusiastically envisioned children using hands-on materials as a sensorial ladder, to intellectualize Mathematics. Aristotle himself stated the principle “Nothing is in the intellect that does not pass through the senses.” For the preschool child who learns best through movement and the senses, Montessori had the wooden blue Geometric Solids made for the 3 to 4-year-olds to classify and name them as the “rolling family” of spheroid, ovoid and ellipsoid; the “sliding family” of cube, brick, triangular and rectangular based pyramids and the “rolling and sliding family” of cylinder and cone.

Another preschool Math lesson, introduced after the child masters numeration 1-10 using rods, spindle box and counters, is learning how to count from units to thousands. Whole numbers are presented through the Golden Decimal beads and cards: a single unit gold bead, the golden bead bar of 10, the golden square of 100 and the golden bead cube of 1000. Conventional education presents the Decimal fractions right away, which confuses children for they are not aware that this system divides unit 1 into a tenth .1, a hundredth .01, a thousandth .001, etc.

Gradating Math difficulties from gradeschool to high school

The Cosmic Math curriculum finds its richest throve of apparata in the elementary school. In Grades I to III students go through an intensive arithmetic memorization using several charts of Addition, Subtraction, Multiplication and Division tables, the final “blank table” is used as in the Bingo Jackpot game.

Multiplication is intensified using the Decanomial Bead Box, a wooden box where 10 sections contain the color-coded square bead bars of numbers 1 to 10 (red, green, pink, yellow, light blue, brown, light aquamarine, indigo blue and gold). Complementing this is the Board of Powers where the chains of 2, 3, 4 to 10 squares are laid out on ten narrow shelves with corresponding fixed squares. Grades II to III use the chains for skip counting. The longer chains hang like rosary beads under the top shelf where the fixed bead cubes from 2 to 10 are laid out. 

Building squares and cubes with the decanomial bead box

Using the Pythagoras bead bars, squares and cubes (Fig. 1), Grade III to IV children work on three multiplication processes: vertically 2x1, 2x2, 2x3, 2x4 ... 2x10; horizontally 1x1, 2x1, 3x1, 4x1... 10x1 and angular combining both processes. Meanwhile, their classmates keep busy with various Geometry materials: the Fractions Insets (i.e. Equivalence and Congruence, Similarities, etc.), Classified Nomenclature Cards (i.e. Point to Solid, Study of Lines, Study of Angles, Triangles, Quadrilaterals, etc.). The Pythagoras bead layout is converted into Numerical Decanomial Cards (Fig.2). Thus lessons on Commutative and Associative Properties of Multiplication, Properties of Real Numbers, Exponential Notations, Laws of Exponents are comprehended well.

The Analysis of Squares and Cubes for Grades IV to V introduces the concepts of Binomial, Trinomial and Quadrinomial Squares and formulas. Example, a rubber band maybe stretched crosswise on an 8-bead square to form 5 square and 3 square or 6 square and 2 square. Using the former, sample equation of 52+(3x5)+(5x3)+32 = 52+2(3x5)+32 would be the result. The final conversion of the Numerical to Algebraic Decanomial is done substituting 1 to 10 with letters a, b, c, d...j. It introduces the idea of combining like terms (Polynomials) and the Number Theory which is linked to Factoring (GCF and LCM). This serves well for Algebra I for freshmen high school which includes definition of Algebra, Algebraic Expressions, Linear Equations/Inequalities, Polynomials, Rational Expressions/Equations, Rational Roots and Radical Expressions/Equations.

Finally the ascent to Advanced Algebra of sophomore high school students is achieved. Systems of Equations/Inequalities, Complex Numbers, Quadratic Equations/Functions, Mathematical Induction and Exponential/Logarithmic Functions are taken up which prepares the students for Differential Calculus.

In third year, students learn the Logic of Geometry. This covers proving of the similarities and congruencies of triangles/quadrilaterals and parallelism of lines. Solid Geometry is likewise studied.

Trigonometry is taken up in fourth year. It is introduced by the Pythagorean Insets.

Trigonometry made easy with the Pythagorean insets

Figure 3 (A), shows a small parallelogram, while (B) shows a bigger parallelogram. Figure 3 (C) shows a right angled triangle with the squares built on its two short sides or the catheti and on its longest side or the hypotenuse. Notice that the biggest square is divided into two rectangles by the extension of the altitude of the right-angled triangle.

With Figure 4, “Sensorial Proof of Euclid’s Theory,” if the two rectangles of the biggest square were removed, the small parallelogram and the bigger parallelogram (from Fig. 3) can fit in the empty space. Thus by sensorial exploration, the First Passage shows that the two rectangles are equivalent to the two common parallelograms. Figure 3 (A) shows that the square built on the short leg of the triangle is equivalent to the smaller parallelogram. The square built on the longer leg of the triangle is equivalent to the bigger parallelogram in Figure 3 (B).

The urgent need for cosmic math

This exposure to universally and scientifically tested materials, are not incidental but psychologically appropriate for the 6 to 12 grade school children. This is the “years of plenty” that can only be satisfied with a well-integrated curriculum that will fulfill their potential for excellence.

From 12 to 18 years, the stage of adolescence replaces the graders’ strong reasoning power with creative and work-oriented energies. Our experiences with Cosmic Science and Cosmic Math have enabled the publications of two teachers’ manuals for preschool to high school.

WARNING: The inadequate curriculum programming in grade school has been handicapping high school students causing the prevalence of drop-outs in public high schools.

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