The Cosmic Math book for ‘EFA’ Teacher Quality Education

My daughter Sara Francesca Soliven de Guzman, Executive Vice President of the OB Montessori Center, Inc., and I are co-authoring the UNESCO-sponsored "Cosmic Math book for EFA Teacher Quality Education" to help ease the training of Math teachers for elementary and high schools. In general, both Math and Science are dreaded subjects in Basic Education. (The Philippines has been in the bottom rank in Math and Science of student competency in the world.)

Dr. Ester Ogena, director for Science Education Institute of DOST (Department of Science and Technology) and concurrently a new UNESCO Commissioner, organized the Forum on "Market Study and Monitoring System on Teacher Education in Science and Mathematics" to analyze why there are few Science and Math teachers, as well as how teachers can be attracted to major in Math and Science. Dr. Merle Tan, Executive Director of NISMED (National Institute for Science and Math Education) has been organizing similar seminars and workshops for the past decades.

Why high school students fail?

The annual United Nations country report noted that both substandard education and illiteracy have incapacitated underdeveloped countries. Two-thirds of their population, mostly farmers and fishermen residing in rural areas, are generally functionally illiterate. Majority of their children drop out of school as early as Grade I and eventually add to the illiterate adult population who fail to get employed. The few who get to high school remain weak students.

Since the turn of the century, the UNESCO Regional Office of Bangkok, which specializes in Education, has focused on secondary school teacher training because the 12- to 18-year old students have difficulty learning. What compounds the problem is the young adults psychologically tend to argue, disobey and even rebel against their teachers, particularly in Math and Science.

Notice that if all of us look back to our childhood, we easily recall that seldom are there failing students in Grade School, particularly in private schools which consistently keep good standard curriculum. However, in both public and private high schools, failures are more common.

Psychological and educational researches on pupil competence reveal three major facts. First, Early Childhood Education definitely spurs the competence of grade school children. Second, the six- to 12-year old grade school children are by nature very intelligent. With ease they get promoted from Grade 1 to Grade 6, although by fourth grade when they reach puberty level, intelligence tends to weaken. Third, the intelligence weakens during the high school years of 12 to 18.

Why American and European high school students are employable

Teenagers’ personality is more inclined to have more technical work and livelihood training. All over Europe, America and Canada, technical high schools enable the young adults to work part time even as students. Their government allows high school schedule to end early in the afternoon so that students can go straight to work.

From the ’60s to the ’80s, as a recipient of travel grants to visit schools, I have visited several high schools in Europe whose curriculum is loaded with more skills training than intellectual matters.

British students work right after their "O" level exams of high school. They work as assistants to professional bakers, cooks and butchers. In the Netherlands, one of two KLM airline buildings employs high school graduates. They also work in the famous Tivoli garden cafes and restaurants. In Germany, they assist professional masons, carpenters and plumbers. In Spain, they are often store attendants. In Italy, they usually work in trattorias or are store attendants. In Denmark, they work either in cafes or are employed as farm apprentices. Besides working in bistros, bars, and cafes, Parisienne students assist in suburban farms, such as St. Germaine en la Haye outside Paris.

The Math and Geometry Continuum: Cosmic Math in preschool

In a Montessori classroom, each child is guided to work on his own. One will see students as young as three to five years old working with the Montessori Golden Decimal beads and happily identifying them – "This is a unit bead. This is a bar of 10. This is a square of 100. This is a cube of 1,000!" Simultaneously, he lays the beads on the mat with the corresponding decimal number cards. Without prodding, he works with them repeatedly in the following days - never getting tired.

In contrast, a primary school child using the conventional method would need three years to do this lesson. He is made to count 1 to 10 with the teacher or mark a whole workbook illustrated with cats, flowers, etc. Then, he memorizes by class recitation 1 to 100. Later in Grades II, III and IV, he is made to memorize the thousands, without any materials to make him understand logical ascent to units of 1000, tens of 1000, hundreds of 1000, and thousands of 1000 (or a million).

The Cosmic Math program for three- to six-year old preschoolers is divided into four lessons. Group I - numeration 1 to 10 using the Number Rods and sandpaper numbers, the Spindle Box which introduces 0 before 1 to 9 and loose counters to teach even and odd numbers within ten.

Group II - Teens and Tens using the Seguin Board for numbers over 10, Skip Counting color bead chain establishes counting by twos, threes, fours… up to tens. The Decimal numeration is linked to the basic arithmetic operations: addition, subtraction, multiplication and division.

Group III - Numeration, the Decimal Whole Number using the Golden Beads. The Golden Decimal unit represents a dot; the ten bead bar, a line; while the bead square of 100, the figure of a square; and the bead cube of 1000, the figure of a cube. Group IV - introduces the Family of Fractions (wooden fraction insets), Addition and Subtraction with three digits, Multiplication and Division using static problems without remainders as well as the Jumbo Decimal Box (an expanded version of the Decimal Introduction Box).

The Cosmic Math in grade school to high school

With the use of the Jumbo Decimal Box, the four basic Arithmetic operations (addition, subtraction, multiplication and division) continue but focus on "dynamic" (sums which involve remainders). The Game of Change prepares the "banker" child to change a 15 into a 10 bead bar plus 5 unit beads; 46 to four ten bead bars and six units or 213 to two 100 bead square, one 10 bar and with remainder 3 unit beads.

There are memorization games using charts to master not only multiplication tables, but addition, subtraction and division tables as well between Grades I, II and III. The Short and Long Bead frames, similar to the Chinese abacus, are used to master addition, subtraction and multiplication from thousands to millions. Multiplication exercises is linked using the Pythagoras or Decanomial beads. The Snake Game makes use of colored bead bars to check how many beads from 1 to 9 can make a 10 to gauge "the age of the snake".

As early as Grade IV, Geometry and Algebra of high school are introduced. They have been illustrated in my past articles (Montessori Revolution in Math and Geometry - 11 Sept 1997; The Missing Geometry in the Philippine Math Arts - 26 Nov 1998; Breaking the Math Barrier - 4 Oct 2001). The Pythagoras bead layout of the multiplication tables 1 to 10 is converted into the Numerical Decanomial and later into Algebraic Decanomial. The students substitute the numerical products with letters (1=a, 2=b, 3=c… 10=j) until they come up with the Algebraic Chart.

The Pythagoras Decanomial beads are complemented by the Board of Powers. This four-foot high by three-foot wide board has colored bead chains resembling rosaries, but are actually short bead chains (squares) and long bead chains (cubes) of numbers 2 to 10 with shelves that carry their corresponding fixed bead squares and fixed bead cubes.

The Cosmic Geometry from preschool to high school

Since preschool, Montessori students are already exposed to the idea and concept of Geometry. The major apparata is the six-shelf Geometry Cabinet (each tray has geometry wooden insets within a frame): the first tray introduces the circle, squares and triangles; second tray - circles gradating from large to small; third tray - six quadrilaterals; fourth tray - six triangles according to angles and lines; fifth tray - six polygons, pentagons and decagons; and the sixth tray - irregular shapes of rhombus, trapezoid, ellipsoid, ovoid, triflora or curved triangles.

The Geometric Solids is a box with three-dimensional blue wooden solids: ellipsoid and ovoid (the rolling family); the square-based pyramid, the triangle-based pyramid, the rectangular brick and the cube (the sliding family); and, the cone and cylinder (both rolling and sliding). Note the activities that go with this exercise of preschoolers.

In Grade School, the Geometry Classified Nomenclature Cards help students to comprehend the concepts of lines, angles, triangles, quadrilaterals, etc. Concepts of Similarity, Equivalency and Congruency are introduced and reinforced in the Equivalency Cabinet where triangular, square and rectangular insets are laid out. (One inset in this cabinet introduces the Transitivity Property of Equivalency.) The same materials are used to link higher concepts in Geometry like proving of equalities and characteristics of geometric figures. This is also used to prove the three hypothesis of the Pythagoras Theorem.

The formulas for getting the measurements of Areas and Capacity (volume) of geometric figures are derived by the students using the wooden Yellow Bricks for Areas and the Blue Hollow Metal for Volumes.

Note that all these grade school materials introduce the Math programs of high school. The proof of the Pythagorean Theorem is sensorially and mathematically presented using the Pythagorean Theorem Insets in Grade School, while the Euclidian Proof using the concept of projection is presented in High School. These materials further clarify and illustrate the concepts of triangles in terms of sizes and measurements that extend to Trigonometry in High School that allows a detailed exploration of triangles other than a right-angled triangle.

The experience of children in deriving formulas and proving of theorems is a complete sensorial introduction of the abstract concepts leading to polynomial functions and eventually derivatives of Calculus in High School and College.

Truly cosmic

The cosmos that embraces our planet moves in clockwork precision. Life is mathematical. Everyday the sun rises in the morning, then sets in the afternoon. Many things are constant. The birds sing never barks, the trees stand upright not upside down, water will always be wet, colorless and tasteless.

All our activities at home, in the office or in our community must go on schedule. I look forward to using the Cosmic Math book for teacher training and workshops. All its mental and active exercises will contribute to the mathematical awareness of quality life.

(For more information or reaction, please e-mail at [email protected] or [email protected])