Learn about Dr. Montessori’s genius materials and curriculum progression of mathematical learning in the 3-6 year old classroom (Children’s House).
Dr. Maria Montessori believed that observation and experimentation are tools that children naturally use to take in information about their environments. The child’s brain aspires to make sense of the world, and identify order and patterns. Children explore their environment to learn, and while there is much for children to learn from time spent in nature and from their daily environments, we can enhance and support the development of the child’s observation skills by providing an environment of precise materials for the child to explore.
Dr. Montessori spoke of the development of the mathematical mind, and to an extent, all work in the Montessori primary classroom is intended to develop this aptitude. When Dr. Montessori spoke of the Mathematical Mind, she referred to the development of, and appreciation for precision, order, abstraction, intelligence, and perhaps even imagination. She believed that the human “ …mind was mathematical by nature and that knowledge and progress came from accurate observation” (The Absorbent Mind, 17, pg. 169).
Developing an understanding of mathematics provides a common language for quantifying, measuring, and discussing the world around us. The Montessori classroom environment is full of tools that help the child, through manipulation with the hands, to develop a mental representation of the world that provides the keys to practice observation skills.
While every Montessori classroom contains an area that is considered the “Mathematics” area, preparation of the mathematical mind happens throughout the classroom, but especially in the Practical Life and Sensorial areas. Practical life and Sensorial works reinforce mathematical concepts such as ordering, sorting, patterns, getting a sense of shapes, logical sequence, and even some basic one-to-one correspondence (for example, tonging one pom-pom into each opening of a container). The Sensorial materials are designed to emphasize sequential relationships with standard increments. Manipulating these precise objects helps to form the way that the child observes and analyzes their environment. Also in the Sensorial area are the Geometric Cabinet and Geometric Solids. These materials provide the child with early impressions of shape and form, and give the child language for these observations. Throughout the Sensorial area are numerous opportunities for the child to compare and contrast, using multiple senses to do so.
“Articles of mathematical precision do not occur in the little child’s ordinary environment. Nature provides him with trees, flowers and animals, but not with these. Hence the child’s mathematical tendencies may suffer from lack of opportunity, with detriment to his later progress. Therefore, we think of our sensorial material as a system of materialized abstractions, or of basic mathematics.”Dr. Maria Montessori, The Absorbent Mind – p.186
Once children have had many experiences successfully manipulating materials within the Practical Life and Sensorial areas of the classroom, they may be ready to delve into the math materials. Children are, of course, ready for these materials at different times, but it is important that those foundational skills of maintaining the order of the task, and the ability to compare and grade materials is established so that the child can be successful with the early mathematical materials.
When introducing children to working within numeracy, we move from concrete to abstract, and as such, we introduce the manipulative quantity first, and then the numerals separately. Many children come to us having prior experience with rote counting and may even recognize many of their number symbols. The quantity is the more concrete and therefore more meaningful aspect of learning about numbers because it is through the quantity that children can develop that internal representation Dr. Montessori called materialized abstraction that is part of the mathematical mind. Therefore, we want to give children ample opportunity to practice counting with only the material before tying the quantity and material together.
The first material that children encounter in the math sequence is the number rods. The number rods are an extension of previous work that the child has done with the red rods, and so the child begins the exploration of the math materials with a familiar and successful experience. The red and blue number rods represent quantity from 1-10 in equal increments (developing both an overall sense of quantity and one to one correspondence through the changes in color on the rods as the child counts). Only after the child has successfully worked with the number rods will we introduce number symbols.
This 0-10 numeracy work is foundational to all the other work in the mathematics curriculum. Further materials that help children to successfully master understanding of 0-10 number concepts include the Spindle Box, Numbers and Counters, and Colored beads. These materials are useful in helping the child determine one to one correspondence, and to establish the relationship between quantity and number. Once the child has a firm grasp of 1:1 number concepts and is working successfully with the colored beads, they are ready for a multitude of other mathematics works.
Around this same time, the child is introduced to the decimal system work, first with an introduction to the beads that represent units, tens, hundreds, and thousands, and then in retrieving various quantities of each of these beads. After the child is familiar with the quantity we introduce the number symbols that correspond to the quantities and the child repeats the process with the symbols. Only after they are familiar with both quantity and symbol do we join the two in the formation of complex numbers. One of the milestone works in the math curriculum progression is the 45 layout, where the child demonstrates their understanding of the connection of quantity and symbol by retrieving every quantity and symbol (1-9 thousands, hundreds, tens, and units) and matching the two. Also important in this initial work with the decimal system is developing an understanding of exchanging quantities. The nature of our decimal system is that when you reach the quantity of ten in any place value, you exchange for one of the next greater place, and there are beautiful lessons and extensions that help children to develop this important number concept.
It is typical to introduce the decimal system work in the Montessori curriculum before the comparable linear counting work. This is because the decimal work gives the child the big picture of our number system, and really allows them to develop a mental representation of place value in ways that the linear counting cannot reinforce until much later in the sequence. However, it is typical for children to be working on their teens around the same time that they are exploring the decimal work in the Montessori environment.
The teens board is another example of how we first introduce quantity, and then symbol with children, allowing them to make those mental representations of the quantity first, then work with the symbols, and last putting the two together. There are wonderful materials available to children in linear counting, including the tens boards, the short bead chains (squaring chains), the long bead chains (cubing chains), and the hundreds board. These materials give children many experiences with finding patterns in counting and plant the seed for later mathematical concepts such as square roots and prime numbers.
Both the decimal system works and the linear counting works lead children to explore operations. In the Montessori curriculum, the order of introduction of operations is first addition, then multiplication, followed by subtraction and division. Additionally, all operations are first introduced statically, which means that initially we do not give children equations that require regrouping, and then add regrouping later, which we call dynamic operations. In the decimal system work, the bank game is the basis for operations work.
Other works that reinforce operations concepts include the addition strip board, the multiplication bead board, the division bead board, the subtraction strip board, the positive snake game, and negative snake game. There are also lessons with the colored beads for all of the operations, and these can be done both statically and dynamically. Once the child has spent a lot of time with the concrete materials and is very comfortable with them, it may be an appropriate time to move the child’s work in math further into abstract mathematical work. Materials that support this movement toward abstraction include the stamp game, the charts for each operation, and the bead frame. Not all children will reach this point in their three year cycle in the Montessori classroom.
Other areas of mathematics in the Montessori classroom include the study of fractions. The initial material for this work are the Fraction Skittles, which introduce the concepts of halves, thirds, and quarters. These and the Fraction Circles are the main materials in the curriculum, and can be used to learn about both parts of a whole and fraction equivalents. Other work in the classroom can contribute to an understanding of fractions as well. Both food preparation and sewing have a lot of potential for introducing fraction concepts. Also, literature is a great way to reinforce fraction concepts.
Measurement, time, and money are other branches of mathematics that should be included in the Montessori mathematics curriculum, although there are not classical Montessori works that were designed for these purposes. Measurement and time can be integrated into the Cosmic curriculum in many ways. For example, when studying Ocean creatures, the class might learn that a certain shark is up to 9 feet long, and measure that out, creating a yarn representation of 9 feet and attaching it to a representation of a shark. This creates a concrete representation for 9 feet in the child’s mind as they unwind and wind the yarn around the shark figure.
Money and time are much the same, in that the two concepts can be integrated into other areas of the classroom. Time ties in to Cosmic education, helping children understand cycles. Calendar work helps children understand years, seasons, months, weeks, day/night. The use of clocks and timers helps children to understand hours, minutes. For example, some classes set a stop watch when the begin making silence, to see how long the class were all making silence together. Money and monetary equivalencies can be studied using the hundreds board, with pennies = units, dimes = ten bar, dollars = hundred square, add in the relationship between quarters/fractions, use hundreds board, classroom fundraiser and making change games.
Some educators argue that three and a half or four is young to be introduced to formal number concepts, but in the Montessori curriculum, the majority of the materials are very concrete, and working with these materials supports the child’s development by appealing to their sensitive periods for order, movement, small objects, spatial relationships, and of course the sensitive period for mathematics, allowing the child to absorb these mathematical representations and concepts easily, and to greatly enjoy doing so.
Montessori, M. (1967). The absorbent mind. New York: Holt, Rinehart and Winston.