The Montessori materials all share certain qualities. They are beautiful. They are precise. They are didactic. They offer an isolated concept to master. They are self-correcting. Because the materials share these qualities all at once, they often address concepts that may be more complicated than you'd expect for a preschool classroom.
A perfect example is the Trinomial Cube, a cube constructed of twenty-seven different wooden prisms, each representing a different combination of variables from the expanded trinomial cube equation: (a+b+c)(a+b+c)(a+b+c).
When the child is first presented the Trinomial Cube, the teacher unpacks it from its box and demonstrates it as a simple color matching puzzle. By matching the colors of the faces of the prisms and the heights, the child will be able to construct the Trinomial Cube completely. Each layer of the cube corresponds to the height of one of the variables a b or c, and is grounded by a cube of that same dimension. By starting with the largest cube, the red "a cubed," the child can build all three layers in an orderly and precise way. If the box lid closes, the child knows that he or she has completed the work accurately.
A child who has mastered constructing the larger cube inside the box can attempt to build it outside of the box, relying more on his or her visual discrimination of size and color. In the primary classroom, the Trinomial Cube often is presented with the Sensorial materials, emphasizing the visual and tactile discrimination through which it will be mastered. As the child advances, however, the cube can be presented with names for each of the prisms: a-cubed, b-cubed, c-cubed, a-squared-b, a-squared-c, b-squared-a, b-squared-c, and so on. Eventually, the child will be offered a specific chart that represents how the quantity (a+b+c)3 is expanded, with each of the prisms placed in the same relationship as in the abstract equation. At each new level, the child works off his or her previous mastery to understand more deeply the algebraic relationships here. Algebra, then, becomes as concrete as the weight of the largest pink cube in the Pink Tower.
When concepts are presented concretely and precisely, there is no limit to what children can learn. Likewise, when we learn new concepts without a sense of how they work in the "real world," it's that much harder for us (at any age!) to truly understand the concept. If you've ever struggled with algebra, come spend some time with our Montessori materials... you may be surprised at how simple some of these ideas are when they're just presented differently.