# Mathematics

## Overview

Young children are fascinated with manipulating things, with patterns, and with small objects. They enjoys repeating activities, and exploring concrete items with their hands. Our Montessori maths program builds upon this developmental phase, we introduce preschool children to the fascinating world of numbers through enjoyable activities, which are carefully designed to impart mathematical knowledge to our students.

By starting early, and drawing on the child’s natural interests, we enable our students to gain a head start in numeracy and, more importantly, a confidence in their own ability to do math. Instead of the math phobias that many children acquire in primary school, where arithmetic operations are introduced as abstract, mechanistic operations to be memorised even if not understood, our 6-year-old students master the basics of arithmetic using concrete materials, and they therefore acquire a grounded understanding of the meaning of these operations. When they leave Montessori, our students have a double advantage, they have learned many mathematical concepts and math facts typically only taught in Grade 1 & 2, before they even enter primary school and they have learned to enjoy maths.

## Teaching Approach

#### A self paced, sequential progression from concrete manipulative's to abstract operations

Since mathematics is the science of measurement, of bringing quantities outside of our perceptual range into a form we can hold in our minds, we introduce mathematics by having our students experience perceivable quantities and then derive abstract numbers, the essentials of the decimal system, and of arithmetical operations from them.

Throughout our mathematics program, we start with concrete manipulatives. In contrast to many other preschool and primary school programs where the counters are often just that, unconnected counters with no further use, our Montessori math materials are deliberately structured to teach specific mathematical concepts and principles. The bead stairs enable children to progress naturally from counting to abstract operations, the Golden Beads teach the decimal system, the Division Board visualises what division is, sharing a given quantity between people.

The scientifically proven approach behind Montessori also ensures that we teach math in a developmentally appropriate way. For example, the Number Rods enable children to handle “Three” as one unit and thus enable them to understand counting earlier than they otherwise would. They are also structured sequentially, with each material and activity building logically upon the next. For example, once children understands the Decimal System (Place Value), they can progress to adding two four digit numbers, first without carrying and then with carrying. The materials ensure that the children's knowledge remains concrete, that they understands what carrying a number in addition really means (that is, exchanging ten Ten Bars, for example, for one Hundred Square), as opposed to just memorising a series of process steps.

We also understand that especially in mathematics, children progress at different speeds. We therefore equip all of our classrooms with math materials that can take a child all the way through to Grade 2 level, and ensure that our teachers are trained in their use. This, combined with our three-year age grouping, enables children to learn the basics of mathematics at their own pace, to go as fast or as slow as their interests and abilities allow them, fuelled by a genuine desire to understand and master without even a thought of comparing themselves to others.

By offering a developmentally appropriate and progressively challenging series of mathematical activities, we enable our children to advance far in mathematics and to enjoy discovering the capable mathematician within themselves.

## Key Activities

#### Number Rods and related materials

Dr. Montessori recognised that a very young child often has trouble understanding how to count by adding one unit to another. She found that children struggle to realise that the increasing whole must be considered: often, a child would count “1, 1, 1” instead of “1, 2, 3.” Therefore, a child in a Montessori classroom is introduced to numbers with the Number Rods, a series of 10 which increase in units from 1 to 10 each number being represented by one rod, with the units marked in alternating red and blue stripes.

The child can thus handle the Three the rod with three alternating stripes, or the Ten, as one item. The child sees “Three” or “Ten” as a set, a unit of three or ten things, that can be broken down into three or ten individual things, and hence he acquires the basic idea of a number. The teacher helps the child count the alternating red and blue sections of each rod as he arranges them in an attractive stair formation. Later, when he has learned to write the numbers using Sandpaper Numerals, he puts number cards next to the rods to illustrate that quantity. He can also discover many mathematical facts – for instance, many children will place the One rod on top of the Nine rod, to make Ten, the Two rod on top of the Eight rod and so on, thereby demonstrating to themselves the different ways in which two smaller numbers can add up to Ten.

Once the child has mastered counting to ten, he can practice counting using other materials. For instance, he can use the Spindle box to count out 45 units into compartments labeled 0 – 9; he can use the Numerals and Counters, which introduce the concepts of odd and even; he can use the Teen Boards, which represent first the “teen” numbers and then enable the child to construct numbers from 11 to 99.

#### Arithmetical Operations

The Bead Stairs are akin to the Number Rods. Various quantities of beads are strung on wire to represent the whole numbers through 10. The bars are made of different colour beads, according to the numerical value of the bar: The 10-bead bars are orange, the 9-beads are dark blue, the 8-beads are brown, and so on. The bead bars can be arranged by their numeric value to form an attractive bead stair. Children enjoy manipulating these materials, and fill page upon page with math work using them. By working with colour-coded chains, they quickly learn to move from counting to abstract calculations. In Dr. Montessori’s words:

“Little by little the child ceases counting the beads and recognises the numbers by their colour: the dark blue he knows is 9, the yellow 4, etc. Almost without realising it he comes now to count by colours instead of by quantities of beads, and thus performs actual operations in mental arithmetic. As soon as the child becomes conscious of this power, he joyfully announces his transition to the higher plane, exclaiming “I can count in my head and I can do it more quickly!”

#### Place value & large quantities with the Golden Bead Materials

The Golden Bead Materials introduce our student to the concept of place value i.e., the Decimal System. This material includes individual beads or single “units”, strings of ten beads or “ten bars”, ten ten-bars combined into a “hundred square” and ten hundred squares combined into a “thousand cube”. Number cards go along with these beads with Units printed in green, Tens in blue, Hundreds in red, and Thousands in green. With these cards and beads, children build large numbers, the teacher may, for example, make a number with the cards, such as 3,572 and ask the child to bring her the corresponding number of beads, that is, 2 Units, 7 Ten Bars, 5 Hundred Squares and 3 Thousand Cubes.

Children continue to work with these and related materials for a long time learning how to add, subtract, multiply and divide. They also obtain a perceptually grounded awareness of quantity, a related material, the thousand chain, consists of ten hundred-chains strung together and measures twenty feet in length.

#### Further mathematical materials

As our groups are multi-age, and as children progress at different speeds, especially in mathematics, we offer a wide variety of additional mathematical materials in each classroom:

- children learn about fractions with the fraction circles and skittles,
- they learn skip counting and multiplication with coloured bead materials which expand upon the coloured bead stair,
- they learn division with a Division Board, where division is introduced as sharing quantities between people, represented by skittles,
- they practice addition and subtraction with Strip Boards,
- they arrange the numbers to 100 in order using the Hundred Board.

All of our classrooms contain materials appropriate up until Grade 2 (Primary School) and our teachers are trained in their use (in contrast to some other Preschools that may use the Montessori name, where teachers are often unfamiliar with the more advanced math materials and therefore not able to help a mathematically gifted child advance as far as they could.)

## The Results

#### Confidence & natural enjoyment of solving mathematical problems

Confidence in dealing with numbers and a natural enjoyment of solving mathematical problems, this is the most fundamental benefit of our Montessori math program.

Rather than being introduced to bewildering, abstract numbers and being frustrated and confused in math class, our students experience mathematics as a natural, joyful process of discovering how numbers work.

As they work with the materials, as they progress from simple counting, to addition and subtraction with beads, as they first perceive and understand single digits, then teens, then the intricate, amazing structure of the decimal system, our students experience themselves as capable young mathematicians, and come to view mathematics as a fun challenge to tackle.

They acquire, if not always a love of mathematics, then at least an appreciation for its power, and a perception of math problems as rewarding mental exercises that can successfully be completed.

Since we introduce mathematics early, and since our students can work with the materials daily and for as long as they desire, our students are also quite advanced in their understanding of math facts.