Breaking the Math barrier from preschool to adolescence

(Part 5 of a series on Half a century of perpetual re-planning of Philippine education and never catching up)

Four years ago, I wrote a column for parents and teachers entitled "The Montessori Revolution in Math and Geometry – For Parents." I stated that ". . .Even as early as the age of three to six, children can easily fall in love not with Math but also with Geometry. Of all school subjects, Math is the most dreaded since the concepts are very abstract."

It had seven illustrations of materials which a trained Montessori teacher uses to teach numeration in units, tens, hundreds, and thousands as early as preschool – the Number Rods, Counter Chips and the Golden Decimal Beads. Parallel numeration of ten numbers up to 1000 use the Seguin Boards and the Chains of 100 and 1000.

This time, I will show how the abstract high school Math program made up of Algebra, Geometry and Trigonometry is given as early as grade school in an authentic Montessori school to match the strong intelligence of seven to 12 year olds.

Beware of computers making gradeschoolers Math morons

Dr. Maria Montessori was a combination of an engineer, scientist and teacher. She observed that abstract or purely intellectual lessons, especially that of Math, need solid materials to manipulate. A Math computer program is purely visual and flat. It cannot take the place of arithmetic rods, geometry cylinders, and algebra beads.

Since computers correct errors or automatically provide answers on the screen, the child’s mind which is the focus of enrichment deteriorates instead. Thus, the modern child, usually the affluent ones with home or school computer, becomes mentally lazy because he gets addicted to these cyber machines. WHAT IS WORSE IS THAT PARENTS ENCOURAGE THIS. Note that even mentally handicapped children can handle computers.

Traditionally, one is made to memorize the multiplication tables of 2 to 10 in grade school. My public school teacher in Grade 3 at San Andres Elementary School made me stay after class since I could only multiply up to Table 6. Looking back I realize that the act of memorization is similar to the force of a pressure cooker. It does not require comprehension. After a while the child forgets.

I did not realize that the other operations like addition, subtraction and division also had their own sets of tables for memorization until I did the Montessori Elementary School Teacher-Training Course in Bergamo, Italy. Montessori children from Grades I to III concentrate on these memorization tasks aided by special apparata with built in control of error. These save a young sensitive child from many heart-breaking humiliations.

Three painless ways to memorize the multiplication tables

Figure 1. The Pythagoras Multiplication Tables 1 to 10 in bead form

Note how the bead squares along the "vertebra" column can be converted to bead cubes. As early as the second grade when the Montessori arithmetic operations focus on multiplication since first graders master addition and subtraction, what we know as the Pythagoras Multiplication Board (usually printed in the back of old school notebooks) is actually the Decanomial Multiplication Bead Board exercise (see Fig. 1).

This exercise makes use of 55 colored bead bars for each unit of 1’s (red), 2’s (green), 3’s (pink), 4’s (yellow) to 10’s (gold) in a box with 10 sections. The fixed squares and fixed cubes for converting the loose bars, as I will describe later, come from the Board of Powers.

Instead of a child memorizing alone, a group of three students would lay out the multiplication tables in three ways. First, the "vertical presentation" of beads like Table 1– 1x1, 1x2, 1x3, 1x4; Second, the "horizontal presentation" of beads – 1x1, 2x1, 3x1, 4x1, etc. The children recite the multiplication table as they lay them out.

The third way of multiplication is the "angular presentation". Thus, the child multiplies vertically 1x1, 1x2, 1x3, and horizontally by pairs, 1x1, 2x1, 3x1, etc. As this presentation is done, the child will realize that 1x2 is just the same as 2x1 because they have the same products. This is the Commutative Property of Multiplication.

To reinforce the mastery of the multiplication products, the bead bars and the bead squares along the diagonal ‘vertebra" are later converted into numbers to become the Numerical Decanomial Chart (Fig. 2).

Figure 2. "Numerical Decanomial" Multiplication Tables citing products in numerals, replacing the Bead Decanomial Chart

From bead bars to bead squares and bead cubes – Basic Geometry and Algebra

The angular multiplication allows the students to convert adjoining loose bars into fixed squares. Note Figure 1 again, starting with the bead square of 2, the two loose bars of 2 on both sides put together from another square of 2 or 22. With the bars of 3, the bead bars representing 3x1 and 3x2 on both sides, if joined together will form two sets of 3. Thus, 3 squares of 3 will be formed. The same steps are followed with the 4’s, 5’s, 6’s until 10’s. With this presentation, the children will realize that a square can be formed from different product combinations and that the number of squares formed is equal to the number itself.

The second passage involves the "Formation of Cubes" from the squares formed in the first passage. If the squares are put together then a cube will be formed. For example, if the 2 squares of 2 are placed on top of each other then the cube of 2 or 23 is formed. The cube of 3 is done by stacking the 3 squares of 3, the cube of 4 with 4 squares of 4, etc. This passage is capped by putting the cubes on top of each other forming a tower similar to the Pink Tower used in the Montessori preschool.

Extracting the Polynomials from the Algebraic Decanomial

After mastering the Numerical Decanomial, the child is now ready for the next passage, which is an introduction to the Study of Algebra. This is the formation of the Algebraic Decanomial.

Figure 3. Numerical Decanomial converted into Algebraic Decanomial

This passage starts with the substitution of numbers 1 to 10 with the first ten letters of the alphabet – a, b, c, d, e, f, g, h, i, j. The perfect squares along the "vertebra" column of the Numerical Decanomial are changed first – 12-a2, 22 -b2, 32 = c2, 42 = d2 until 102-j2 to form the Algebraic Decanomial. At this point, the terms monomial and binomial should have been introduced like a 2, b2, c2, and ba, ca, da, etc.

The next presentation of this passage is when the child constructs a special notebook, which is called The Algebraic Booklet made up of six pages showing the progression of algebraic lessons on simplifying polynomials. This booklet will now help the child understand the idea of like and unlike terms as well as the application of Commutative Property of Multiplication to Algebra. It also introduces the child to simple addition and grouping of terms.

The study of the square root

Using a rubber band, the bead square of ten is divided into four sections – 2 squares and 2 rectangles (Fig. 4). For example, the division is 3 beads and 7 beads. The child should come up with a figure like figure 4. The child should realize that the small rectangle is equal to 32 and the bigger square is equal to 72 and the two rectangles on each side is equal to 7 x 3 and 3 x 7. So the child will come up with the computation shown below. Other decomposition combinations are done as exercise (decomposition combinations of 2 and 8, 4 and 6 and so on). The child will then proceed to the analysis of the different squares and should come up with the conclusion that the different decomposition of a given square is equal to the square of the number itself.

(Figure 4)

The extension of this presentation directly introduces the Formation of Successive Squares like "from the squares of 6 to squares of 7" and Non-successive Squares like 62 to 82.

Analyzing the Pythagorean Theorem with wooden insets

Figure 5. Pythagorean Theorem. Wooden (or metal) insets are in red, yellow and blue. In black and white illustration, crossed circles, asterisk and plain circles are used instead, while bars and hearts are used for the substitute parallelogram.

Figure 6. Sensorial Proof of Euclid’s Theory

Above Figure 5(A) shows a small parallelogram while (B) shows a bigger parallelogram. Figure 5(C) shows a right-angled triangle with the squares built on its two short sides or the catheti and 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 6, "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. 5) can fit in the empty space, as the triangle is slid downwards to the bottom of the space left by the biggest square. Thus, by sensorial exploration, the First Passage shows that the two rectangles are equivalent to the two common parallelograms.

The Second Passage (a), shows that when the square built on the shortest leg of the triangle is removed and the triangle is slid leftwards, the small parallelogram [from Fig. 5(A)] can occupy the empty space horizontally. This shows that the square built on the short leg of the triangle is equivalent to the smaller parallelogram.

On the other hand, in Second Passage (b), if the bigger square is removed and the triangle is slid rightwards to the upper half of this vacated space, then the lower half of the bigger parallelogram can fit in the empty space plus the space vacated by the triangle. This shows that the square built on the longer leg of the triangle is equivalent to the bigger parallelogram in Figure 5 (B).

The race between education and catastrophe

In 1920, H.G. Wells predicted a race between education and catastrophe. Today, our attention is focused on the race that Wells foresaw, and it is no longer clear that education holds the lead. Catastrophe in two forms – nuclear weapons carried by guided missiles, and governments that blot out that freedom of men – appears to have pulled ahead. It remains to be seen whether the 20th century will go down in history as the century of atomic destruction and the end of human liberties or as the century of universal education in which enlightened men learned to conquer the forces that threatened to destroy them.

(For more information, please e-mail at obmci@mozcom.com)