I did not understand that math formulas had a visual or concrete form as a high school student taking classes in algebra and trigonometry. I did have an ah-ha moment when I was taking a Montessori Math class during training to become an Early Childhood teacher.
The Montessori Math materials evolve out of concepts taught with the Montessori Sensorial materials. Children learn to discriminate by size and form. The red rods are followed by the red and blue rods which teach the hierarchical property of quantity. One is part of two, one and two are part of three, etc. These perfect materials allow the smallest hands to feel the difference in quantity – the lessons incorporate movement. The ten rod is carried to the work mat, held end to end in small hands and is a stretch for the young body.
To me, the most exquisite of all the Montessori Math material is the Trinomial Cube. In high school, I could memorize the formula of (a+b)³. But I never had an image in my mind that the result would be a larger cube! Our Early Childhood children ages 3-6 years work with a Trinomial Cube that is made up of cubes, rectangular prisms, each of which represent the components of the formulas’ answer. This material helps build a foundation for later mathematical instruction. When a Montessori educated child takes algebra and is introduced to the Trinomial Cube, I hope that s/he is reminded of the puzzle box of the same name that s/he worked with in preschool. The visual image will come to mind and the process of solving the equation will be understood.