THE TRINOMIAL CUBE
There are so many layers to Montessori.
Nowhere is this more apparent than in the simplicity and elegance of the trinomial cube. While the primary aim may seem rather obvious, to build a little puzzle in a box, the secondary aim, an introduction to algebra and preparation for the proof of the formula (a+b+c)3, just blew us away.
We are increasingly, and at the risk of sounding cliche, exponentially, interested in the layered nature of the activity. Neat, orderly, and succinct, at first blush the trinomial cube appears almost obvious. Of course, it feels like there’s a certain level of mathematical rigor to it, but it remains largely unpronounced.
The mystery of the material summons our attention. We approach the box with curiosity. We are enchanted by what it promises. We are allured by what might be inside. “What is inside?”, we ask ourselves. With our fingertips, we start to explore.
As we unfold the wooden box and engage, an entirely new world slowly reveals itself. The colors seem to call out to us. Black, red, blue and then yellow. “Why these colors?”, we wonder. Little by little, or more literally, piece by piece, we begin to see more.
“There seems to be an order here.” “Why these shapes?” “What’s going on here?” Then, as if naturally, we begin to form a rough idea of the concept behind the work. “I think I understand what’s happening.” “I can figure this out.”
We thought it might be helpful if we laid out some of the basics of the trinomial cube:
- A box with hinged sides and a lid
- Three cubes: red, blue, and yellow
- Eighteen square-based prisms
- Six black rectangular prisms
At heart, the initial task of the activity is to take out the pieces of the box and find a way to steadfastly, tidily, systematically, put them back together again – a feat Humpty Dumpty might enviously rejoice in! Once assembled, all of the individual pieces form a cube of a trinomial.
“ The trinomial, as many adults might recognize, is represented by the mathematical formula, (a+b+c)3. ”
While children are inherently captivated by the mathematical precision of this seemingly innocuous puzzle, noting the different heights of the prisms, they are actively working to build a concrete foundation for the abstract nature of the formulation, which they will later expand upon.
It’s also important to point out that the square of the trinomial is painted on the lid of the box, which serves as both a reference and a control of error for the child, as they assiduously work to complete the challenging activity, glancing as needed. As you can see, there are many different layers to the trinomial cube. In a way, this material accurately illustrates the multi-faceted nature of the Montessori approach to education.
How does this concrete Montessori activity known as the trinomial cube, meant as an introduction to algebra, stack up to the abstract world of mathematics?
As adults, we’re always working backward with Montessori, trying to connect the dots from Z to A. We take for granted that we understand the “concept” and when we “teach” children, we expect them to just “get Z”.
With Montessori, children are constantly working to create the foundations they will need to explore abstractions on their own terms. While many of us encountered this auspicious formula at some point, (a+b+c)3, very few of us had the opportunity to understand and grasp the concept, in such a visceral, hands-on type of way. What’s so key in this process is that it is by discovery alone that children come to unlock the intricate layers.
“ 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. ”
In many respects, they’ll be able to see what it is they are solving…