Perhaps the best exemplification of the layering that occurs in Montessori is the Trinomial Cube. At first glance, a child appears to be putting together a puzzle in a quiet, playful manner. However, the secondary aim of this work is to introduce algebra to the child in the proof of the formula(a+b+c)3.
The Trinomial Cube itself is made up of a lidded box with hinges on its sides, eighteen square-based prisms, six black rectangular prisms, and three cubes in red, blue, and yellow. As children work to first take apart the puzzle, then put it back together just so, they must notice the difference in heights of the prisms.
As seen above, upper left, the square of the trinomial is displayed on the lid of the box for reference for the child as she works.
The hands-on work so consistently seen in Montessori is displayed here so succinctly; this concrete work is building a foundation for an abstract concept (algebra), and works forward as the child develops the math concepts.
“ Whereas most of us had to memorize mathematical formulas, eventually solving the problems we faced, children who grow up with Montessori will understand the problem before they are faced with the formula. ” – Bobby George, Baan Dek
Another thing worth noting in the beauty of this work: the design of this puzzle is reminiscent of Dutch painter Piet Mondrian’s abstract paintings:
“It is through appropriate work and activities that the character of the child is transformed. Work influences his development in the same way that food revives the vigor of a starving man. We observe that a child occupied with matters that awaken his interest seems to blossom, to expand, evincing undreamed of character traits; his abilities give him great satisfaction, and he smiles with a sweet and joyous smile.” (San Remo Lectures, p. 28)