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The previous blog post (found here) gave an introduction to the Montessori Math curriculum. This month’s post will focus on how the child experiences and progresses through the four operations – addition, subtraction, multiplication, and division. The math curriculum, as with most Montessori introduced concepts, progresses from the concrete to the abstract.
The golden bead materials introduced the child to the decimal system through concrete quantities.
The number cards introduced the child to the decimal system’s numerical symbols.
Once the child is able to associate these symbols to the appropriate quantities, the child is now open to a new world of possibilities – collective exercises.
A smaller version of the number cards is introduced for the following activities.
For collective addition, two or more sets of small number cards represent the quantities to be gathered by each child. These separate quantities are brought to a mat and combined or collected together. The resulting quantity is counted and the matching large number cards are brought. As seen below, the children end up with a full picture of the concept of addition – smaller numbers combined or added together to result in a larger number. This is seen through the smaller number cards resulting in a large number card and the smaller quantities resulting in a larger quantity after addition.
The opposite can be seen when the child experiences the process of subtraction. A child first gathers a large number and the second child takes away or subtracts his or her smaller number. The remaining quantity is counted with the matching number card brought to the mat. The tray pictured below holds the quantity that has been taken away. Again, the golden beads and the number cards work together to provide the child with the clear impression of the number decreasing in value versus the increase seen in addition.
Multiplication is introduced in much the same way as addition with the difference being that the number being added together remains constant. The number of times each number is brought is emphasized to give the impression of the multiplier. Division is set up in a similar way to subtraction; the difference being that the number is shared out equally and fairly between the number of children or the number of trays brought to the activity to denote the divisor.
It is also with these materials wherein a child is provided with many opportunities to experience exchanging (carrying over) from one category to the next. Once a child counts ten units for example, the child exchanges these ten units for one ten bar. If a child must take away or share out five units from only one on the mat, the child takes a ten bar and exchanges that for ten units instead.
Each quantity is now represented by a uniformly sized stamp – differentiated by colour and symbol. The child experiences the concepts of the four operations just as he or she has with the golden bead materials. Note that the colours for each category are consistent with those colours found on the number cards when working with the golden beads. Pictured below is a stamp game division question in progress.
Many of us have learned our addition, subtraction, multiplication and division facts through rote memorization. The Montessori classroom is equipped with a large array of materials to help the child learn these number facts in a variety of interesting and engaging ways. The child has numerous opportunities to not only remember these facts, but to naturally internalize and understand these numerical relationships.
The addition strip board shows the child, in a more concrete manner, how each answer is obtained. 8 + 2 = 10 is pictured below.
The child also begins to explore the various combinations that result in the same sum using the addition strip board.
The child’s exploration continues with the beautiful bead materials introduced before; this provides concrete quantities for each number fact. The multiplication bead bar layout, for example, is pictured below. The child counts and builds the equivalent quantities underneath each multiple. The resulting quantities are similar to those the child experienced through the teens and tens curriculum area shown in last month’s blog post (seen here).
A variety of charts are then introduced to the child to further provide opportunities to internalize the various number facts associated with the four operations. The second addition chart is pictured below.
The child is given full independence with control charts available on the math shelves for each operation by which he or she can check each fact.
This blog post merely aims to provide an overview of the process by which children are introduced to and experience addition, subtraction, multiplication, and division within the casa classrooms. As such, only a handful of materials and presentations were showcased. As always, parents are more than welcome to ask their casa teachers for more in depth information or for an opportunity to experience the materials first hand. As Dr. Montessori said in The Absorbent Mind,
He does it with his hands, by experience, first in play and then in work. The hands are the instruments of man’s intelligence.