The Mysterious Trinomial Cube

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Today, we’re going to highlight on one of our students’ favorite works; the trinomial cube. There’s real beauty behind the intricate details of this mathematical work, details that are hard to notice at first glance.

The direct aim of this work is rather obvious; to build a puzzle in a box. The indirect aim, an introduction to algebra and preparation for the formula a+b+c³. DSC_0349 (1)

The child approaches the box with curiosity. It’s elegant details can spark the interest of anyone. Children are inherently captivated by mathematical materials. The colors, the difference is sizes and heights…”Why these specific colors?”, “Why do they fit together like that?”…DSC_0343

While there are many hidden lessons in this puzzle, the child is actively working towards building a concrete foundation for the abstract nature of the formulation, which they will later expand upon. The puzzle helps the child grasp the concept of a+b+c³ at such a young age, rather than simply memorizing the formula.

Through the child’s discovery, they come to unlock the intricate layers of the trinomial cube.DSC_0356 (1)

To begin, the guide carefully opens the box, and lays out all of the pieces, organizing them in different manners. With little, to no words, she places the cubes upon the lid, pairing the colors to one another. Each layer is placed in the box upon completion. The child is then invited to do the work on their own. There are many variations of the trinomial cube that call upon the child’s creativity, imagination and critical thinking.

“These small objects fascinate a child. He must first of all group them according to their color, then arrange them in various ways, making up a kind of little story, in which the three cubes are three kings, each one having a retinue identical to that of the other two, the guards being dressed in black. Many effects can be obtained through the use of this material…when playing with this material, a child forms a visual image of the arrangement of the objects and can thus remember their quantity and order. The sense impressions received from these objects furnish material for the mind. No object is so attractive for four-year-old children. Later on, by calling the kinds a, b, c, and writing the names of the separate pieces according to their dependence upon their own king, five-year-old children, and certainly six-year-olds, can store up in their minds the algebraic formula for the cube of a trinomial without looking at the material, since they have fixed in their visual memory the disposition of the various objects. This gives some idea of the possibilities that can be attained in practice.” (Montessori, The Discovery of the Child)

To see the lesson in full, click on the link below:

https://www.youtube.com/watch?v=ZEZnNFxsnS8#t=13

Why we Take Photographs in the Classroom

When we take photos of our children in the classroom, it’s a very meticulous and careful process. Each photo is taken and shared with a specific goal in mind; to help our parents understand the natural learning process experienced by each precious child that comes to our school. It’s hard to fully grasp and understand all of the magnificent things your young child is learning each day, by a simple, brief conversation with them at the end of the day. Most of the time, they’ll repeat the last thing they did just before you picked them up, rather than highlight on a special memory from the day. It’s hard, as parents, to entrust our children in the care of others for 8-11(+) hours per day, and not know exactly what they’re doing at any given point. That’s why we love to share these memories in photos! Whenever given the opportunity to do so, and depending upon the level of disruption, admin. will try to sneak into the classroom and photograph these special moments; two friends working together, an older child helping a younger child, a classroom celebration, or a child so engrossed in concentration that they don’t even notice we’re there. There is so much that we look for as “photographers”, rather than simply snapping a quick photo of the environment. Our photos are meant to educate, inspire and motivate, all the while helping you feel the natural emotions embodied within every portrait.

We observe and respect the child’s concentration…the child’s concentration is spontaneous and precious. It’s not necessarily “rare”, but beautiful whenever caught on camera.

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We try to photograph periods of “normalization”, when the child, or group of children are under intense concentration, working on something that engages their interest. Through this process, children gain self discipline and peace. This is the work of the materials in the environment; truly something to be captured on film.

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Some children have been with us 5+ years, so we’re able to capture photos from infancy through young childhood. We have the opportunity to photographically document their growth over several years.
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Lessons and mastery. Many of the works in our environment require weeks, even months to master. We’re able to photograph the entire process, from the preliminary lesson all the way to the child’s mastery and application. We focus more on the process of learning, rather than the “final product”. We want to capture the child’s expression when they’ve reached that level of self accomplishment. These moments are extremely precious to us.DSC_0305DSC_0310

On special occasions, we’ll even have the opportunity to capture a child’s first steps, illustrating the child’s new-found independence.

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Dr. Maria Montessori spent a lifetime observing and documenting the developing child. I always tell our families, the best way to truly understand what your child is doing in the environment is to observe. It is so difficult for us as Montessorians, to document their growth in a weekly progress report, because there is so much more depth to it than simply stating what they’re currently working on. Pictures truly speak louder than words! We take photographs to help our parents understand the amazing things their child is doing in the classroom, and to further educate them on the power of the pedagogy.

We thank you for sharing your little ones with us, and giving us the amazing opportunity to witness them grow independently, spiritually, and academically. Our gift to you is these precious memories!

Parent Resource: the Trinomial Cube

There is so much to discover using the trinomial cube; a favorite work in our primary community. This blog article, courtesy of the Baan Dek Blog, unfolds all of the intricate details involved in the trinomial cube, and the true beauty behind this awesome mathematical work. Definitely a “must read”!

THE TRINOMIAL CUBE

Baan Dek

“There are so many layers to Montessori.

No where 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.

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.”

…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.

…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…”