When we hear the word Algebra, how many of us are taken back to fond memories of secondary school Algebra classes? We learned algebra through memorized formulas and variables plugged into graphic calculators. This helped us calculate various functions, but overall, we were convinced that we would never use the information again! Understanding that children learn through working with their hands, Dr. Montessori and her son, Mario, developed a material that could help make algebra understandable through a concrete and manipulative form.
One of the iconic materials found throughout the Montessori Primary and Elementary classrooms is the Binomial Cube. This material was initially intended for the elementary children for learning algebra. However, Dr. Montessori observed that young children had such a fondness for this material as a puzzle that she included it in the Primary environment.
The Binomial Cube is found in the Sensorial area of a Primary environment, rather than the Math area, because the child works with this material in a purely sensorial way. The child can explore the cubes, prisms, patterns, structure, and qualities of this material by manipulating the pieces with his hands. In addition to using his sense of touch, the child uses his visual sense to discriminate the sizes and colors of the pieces. By matching and assembling the pieces, he comes to know their relationships intimately. He can draw from these experiences with the Binomial Cube in his future work in the elementary Math area with algebraic functions.
The Binomial Cube is a concrete representation of the algebraic formula (a+b)³ and is presented as such in the Montessori Elementary environment. However, in the Primary environment, the Binomial Cube is introduced to the child as nothing more than an intriguing puzzle and the child finds pleasure in repeating the process of constructing and deconstructing it. The series of following exercises allow the child to explore the material and become more familiar with the components and the internal anatomy of a cube. The ultimate point of arrival for the child is when he covers his eyes with a blindfold and builds the cube without looking. This mental stimulation fosters concentration and intellectual growth as the child works intently to solve the puzzle by constructing the cube.
The human mind is mathematical by nature. The mathematical mind that Dr. Montessori discussed does not refer to the ability to count using arithmetic or compute calculations. Having a mathematical mind means that we can use logic, judgment and reason and have the capacity to discern patterns and identify sequences. This exercise supports the development of the child’s mathematical mind. The patterns that are reinforced as the child works with the material support the child’s ability to create and discern patterns as he engages in comparing and organizing the pieces according to their color. This material also supports the child’s reasoning and problem-solving skills through the process of building the puzzle, ensuring the pattern is the same as the picture on the lid and that it fits correctly inside the box.
Let’s briefly revisit algebra in a true Montessori fashion and try to replace our unpleasant algebra memories with something more comprehensible:
The Binomial Cube Deconstructed
Let’s call the length of one side of Red “a”.
Let’s call the length of one side of Blue “b”.
The Binomial Cube continues to be a favorite among most Primary aged children. Children feel an appropriate level of challenge and a sense of accomplishment with each exercise. At almost any given moment in the classroom, a child can be observed working with the Binomial Cube. Once the child has mastered the Binomial Cube, he feels the urge to continue with something a little more challenging – the Trinomial Cube – but we will have to discuss that material another day!
About the Author:
Tiffany Liddell is currently an AMI Primary Trainer in Training residing in Mesa, Arizona. She completed her AMI Primary Training in 2005 and has since been a lead Montessori Primary guide in a charter school and her own private preschool. She has been an active member of the Phoenix area Early Childhood community by facilitating professional development workshops and developing curriculum for a local community college. Tiffany also supports the Southwest Institute of Montessori Studies (SIMS) Primary training courses as a course assistant. Tiffany has been married to her husband for twelve years and has four children who all attended (of which two still attend) Montessori school and, in her spare time, enjoys cycling in the Arizona desert.