Casa 3-6 Maths Theory

“The troubles that children ordinarily encounter learning mathematics or grammar, are easily overcome if difficult problems are presented at exactly the right moment…but these difficulties are easily overcome if we use materials that concretely illustrates mathematical abstractions. Such material enables the child to learn according to the laws of mental development.” Montessori, Education and Peace p.81

Mathematics is ‘language’, which enables human beings to understand and explain the laws of the universe. Math is a wonderful cultural heritage and is in our environment all around us even though it might not be obvious. It is particularly involved in nature, art, architecture, music, science and technology.

In the Montessori House we introduce the child to mathematics at a very young age, around four years. We introduce it so early, because the child still has the absorbent mind, human tendencies, and sensitive periods. Simply we offer mathematics activities as they are aligned with the child’s mentality and their development. The child with the absorbent mind will take in everything around them without even trying, therefor introducing the child to math they naturally take it in without it being a forced process. A child’s human tendencies such as abstraction, calculation, and exactness help propel the child to these activities and to further explore then. The child is also in the sensitive period for refinement of senses and language. Through this we can easily introduce the child to the language of mathematics and the material offers the child a sensorial impression of mathematics and therefor meets their needs of refinement of their senses. The mathematics material is also materialized and gives the child a sensorial impression of mathematics. Montessori brings in abstract ideas and materializes them so the child is able to understand and learn. The child is able to work with abstract ideas in a concrete from through the mathematics material which is all tangible. While the materials start off very concrete and tangible, the mathematics area increases abstraction as we continue working our way through. This gradual process of abstraction helps prevent mental blocks. Mathematics is also a part of our culture and it through understanding mathematics where the child will also have a better understanding and adaptation to their environment. Mathematics helps the child understand the world around them.

Montessori uses the term ‘Mathematical Mind’. She was the first to apply this to children. She described that children have the innate capacity to thing mathematically and logically. Therefor teaching them mathematics is possible and the children easily acquire the knowledge.

The child is prepared for the mathematical material through both practical life and sensorial activities. Such as in the practical life area, the child must carefully calculate the speed of pouring as well as aligning the jug with the cup. The child must also calculate the amount of polish they need for polishing something specific. Many of the sensorial activities imply the decimal system with groups of ten as well as all the materials being scientifically constructed. The measurements of the materials are exact and serve a purpose. Similarly with the geometric cabinet the child is learning the language to geometry which is a subheading of mathematics. Both the binomial and trinomial cube give the child sensorial impression of squaring and cubing. Other examples include calculating how much water is needed for washing hands, the direction of sweeping, for flower arranging the child must calculate where to cut the stem of the flower so it is appropriate for the height of the vase. The child must also roll up a table mat appropriately to fit into the napkin ring. Within the environment we also have order, the things in the Children’s House have and order and go in a logical sequence in which we complete our work, such as on the shelves from left to right the activities increase in difficulty. The child also develops concentration in practical life which is important when we come to mathematics. If a child loses their place in counting they must start over again and can lead the child to a wrong answer. Coordinated movement is also developed and is important, as a great deal of the mathematics material is very small and the child must be able to carefully move beads and stamps. Finally, the child gains independence as the materials become less and less reliant on the teacher. Sensorial materials help introduce the child to the decimal system of ten (10 cubes, 10 rods, 10 stairs, 10 cylinders), introduce the child to the concept of dimension ( pink tower, red rods, brown stairs, knobbles cylinders). There is also a direct link from the red rods to the number rods which is the first material in the mathematics area. The Decanomial square introduce the child to algebra as well as the colours match the beads in the bead cabinet. Most of the sensorial materials also meet either pairing where the child goes through the notions of equality, sequencing where the child grades and finds order and then differentiation where the child looks for minor differences. The language of the sensorial materials gives the child the language to qualities that are related to mathematics and it is the games in the sensorial area which help lead the child to abstraction. This is why it is important that parents but also for us teachers recognize and remember that practical life and sensorial activities really are the foundation for mathematics. First the child must learn the basic skills before jumping into some of the “bigger” work.

The basic steps for presenting the mathematics materials are introducing the child to a concrete quantity followed by the introduction to the symbols and then we combine both the quantity and the symbol. Finally we provide the child with further activities to help the child know what they know. The material also allows for manipulation and repetition which helps the child further explore and understand as well as moving from a simple to more complex idea. Doing this allows the child to build their understanding and gives them a solid foundation of mathematics. The process of which we introduce the child to mathematics also follows the natural learning process of the child.

In mathematics we have five groups as well as fractions: the foundation of our decimal system – numbers to 10, Decimal system, counting, memorization of essential combinations and then the passage to abstraction. In the first group, numbers to 10 the child works with the number rods, sandpaper numerals, number rods and cards, spindle boxes, numbers and counters and the memory game. In this group the child gets the foundation to the decimal system, the sequence of our numbers, what our numbers look like and how to write them, odd and even, that a number doesn’t have to be one concrete object but many things can also make up a number, an understanding of zero, as well as an introduction to addition and subtraction. The second group, introduction to the decimal system, the child gains an understanding of the decimal system including the names of each category and their quantities. The child gains knowledge of the relationship to each category. They learn the passage of after 9 units we move to the tens – the passage from one category to the next. They also get a very sensorial impression of addition, subtraction, multiplication and division. The child works both in groups and individually in this group. The child also learns what a number consists of. Here the child works with: introduction to the decimal cards, layout of beads and cards, formation of numbers, formation of large numbers, collective exercises, stamp games and the dot game. The next group is counting which can happen at the same time as group two. Here the child works with short bead stair, teen boards, teen boards and beads, ten boards, linear counting and skip counting. In this group the child learns how to count from 1-1000 and the sequence as well as the symbol and quantities for the numbers. The child also learns more efficient ways of counting and that we don’t have to count one by one. Next is group four which is the memorization of the essential combinations. Here the child works with the addition and subtraction snake, strip boards, charts, bead bars, multiplication board, unit division board. The main aim for this group is to help the child recognize patterns within our numbers. The child gets a significant amount of repetition in this group and is introduced to the commutative law. Finally we have the fifth group which is the passage to abstraction. Here the child works with: short bead frame, hierarchies, long bead frame, division test tubes as well as mental mathematical problems. This group shows the child how to work with little to know materials and really helps the child further develop experience in math. It is in this group where the child ideally learns they can now do math without any materials. Lastly we have fractions which can be introduced after group one. It is very sensorial and just gives the child a basic understanding of fractions, how they work and basic operations with fractions.