The binomial and trinomial cubes are two of the sensorial materials in the Montessori classroom, although they also represent mathematical concepts.
What are the Binomial & Trinomial Cubes?
As with all Montessori materials, abstract concepts are first introduced to the child through the handling of concrete materials. The attractive binomial cube puzzle is a concrete representation of the mathematical formula of (a + b)³ = (a + b) (a + b) (a + b) = a³ + 3a²b + 3ab² + b³. Similarly, the trinomial cube represents the mathematical formula of (a + b + c)³.
Of course, at the ages of 3-6, when children first use these materials, the emphasis is not on learning a formula but rather on refining the visual sense of discrimination (shape, color, pattern) while indirectly introducing the child to the concrete form of the formula. Other indirect purposes include the development of a logical-mathematical mind, critical thinking, refining observation, developing sense perception and developing the imagination.
First introduced as mathematical materials in the elementary years, the binomial and trinomial cubes are some of the few materials that are used throughout all levels of Montessori, from Children’s House all the way up to high school!
What Do the Binomial & Trinomial Cubes Consist Of?
The binomial and trinomial cubes are made up of red, yellow, blue and black cubes and rectangular prisms that fit neatly together into a wooden box. There is only one correct way to assemble the puzzle in order for it to fit back in the box, and so it is self-correcting in nature. Self-correcting materials are beneficial to the child because they don’t require adult guidance.
“Self-correction raises the students’ awareness about their errors, allowing them to correct the errors themselves and in that process become responsible for their learning and therefore, more independent of the teacher.” (Source)
As children handle the pieces of the puzzle over and over and learn to fit them together correctly, they can readily link together the concrete and abstract forms when they enter their elementary years!
Dr. Maria Montessori originally designed these materials for elementary age students, but when she saw how much the Children’s House students in enjoyed fitting it together as a puzzle, included it in their environment as well!
Dr. Maria Montessori on the Binomial & Trinomial Cubes
Dr. Maria Montessori said of the binomial and trinomial cubes:
“These small objects fascinate a child. He must first of all group them according to their color, then arrange them in various ways, making up a kind of little story, in which the three cubes are three kings, each one having a retinue identical to that of the other two, the guards being dressed in black. Many effects can be obtained through the use of this material…when playing with this material, a child forms a visual image of the arrangement of the objects and can thus remember their quantity and order. The sense impressions received from these objects furnish material for the mind. No object is so attractive for four-year-old children. Later on, by calling the kinds a, b, c, and writing the names of the separate pieces according to their dependence upon their own king, five-year-old children, and certainly six-year-olds, can store up in their minds the algebraic formula for the cube of a trinomial without looking at the material, since they have fixed in their visual memory the disposition of the various objects. This gives some idea of the possibilities that can be attained in practice.” (Maria Montessori, in The Discovery of the Child)
There is even a lovely story of “The King of Peace” that is told while children learn about the relationship between the iconic Pink Tower and the Trinomial Cube. It begins this way:
“Once upon a time, there were four kings living separately with their royal suite in four respective kingdoms: three on one side of a river and one on the other side. The three kings living on the North side of the river in three different castles were the Red King; the Blue King and the Yellow King. Each of the three kings had an entourage of two bodyguards dressed in black and six attendants. On the South side of the river lived the King of Peace in a pink tower with his own suite.
One day, the smallest attendant of the King of Peace was named ambassador and sent to make a courtesy visit to all three kingdoms on the North side of the river. During his visit, the ambassador perceived that each king and his entourage were unhappy. Apparently, the red castle was too hot; the blue castle was too cold and the yellow castle was too small . . . “
When the King of Peace hears of the trouble, in his kindness comes up with a plan to bring the three kingdoms together not just to meet all of their needs, but to bring harmony and happiness. The story goes on, but in the end, all are brought together in peace and harmony. (Watch the presentation here!)
Beautiful and purposeful materials, like the binomial and trinomial cubes, and stories stir up the child’s imagination, foster curiosity, and continue to nurture their innate desire to learn simply because they enjoy doing so.
