Material Spotlight – the Binomial and Trinomial Cubes

Each material in a Montessori classroom have been carefully curated and designed to serve direct and indirect purposes in a child’s natural development. Dr. Montessori often used the term “Mathematical Mind” when discussing children, referencing French mathematician Blaise Pascal’s theory that “…man’s mind was mathematical by nature and that knowledge and progress came from accurate observation.” (The Absorbent Mind, 17, pg 169)

The binomial and trinomial cubes were beautifully and carefully designed by Dr. Maria Montessori with the Mathematical mind as well as the children’s ages and stages of development in mind.

These materials are introduced to children in Montessori Casa classrooms as sensorial experiences and logical puzzles. The child is first introduced to the binomial cube and later, depending on the child’s readiness as observed by the Casa directress, the trinomial cube.

A child visually discerns and matches each cube and prism based on colour, shape and size. The lid of each box serves as a guide as children begin to explore and experience the material. The binomial and trinomial cube boxes also serve as a control of error – inherent qualities of the material that allows a child to self-correct. In this case, if the cube is built incorrectly, the box will not remain closed. In the photograph below, the binomial cube is on the left and the trinomial cube is on the right.

A child can also demonstrate his or her memory by building the cube outside the box without the aid of the lid to remember the pattern.

Each layer of the binomial and trinomial cubes vary in height. This fact is demonstrated to children as each layer is built individually.

Split vertically, the pattern of each layer remains constant as the widths vary.

The beauty of intricacy of the material lies not only in the various methods in which children can manipulate and experience its concepts, but also in its inherent theorems. These materials are the physical embodiments of some mathematical concepts. Photographed below are the binomial theorem (a + b)2 and the trinomial theorem (a + b + c)3. Although these theorems are not directly explained to children at this age, children experience them sensorially. Children who continue with Montessori are reintroduced to these materials along with the mathematical equations in Elementary. These equations are depicted at the bottom of the post if you are interested.

These complex materials bring forth the child’s inherent desire to build and play while allowing the child to problem-solve and experience mathematics in a very concrete manner.

Binomial Cube Layers

(a + b)^3

Trinomial Cube Layers

(a + b + c)^3  = a^3 + 3a^2b + 3a^2c +b^3+ 3ab^2 + 3b^2c +c^3+ 3ac^2+ 3bc^2+ 6abc

Marie Hui is the Vice Principal of Odyssey Montessori School. She began her tenure with Odyssey as the student teacher in Casa 1. Marie then went on to lead Casa 2 as the Directress for several years. With a love of children and a passion for Montessori, Marie strives to connect parents to the Montessori philosophy - hoping to bridge some gaps between home and school.

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