“The child gives us a beautiful lesson – that in order to form and maintain our intelligence, we must use our hands.” – Maria Montessori
The beauty and efficacy of Montessori pedagogical materials is well-established. Designed specifically for the developmental stage of the children in a given classroom, they meet a child where they are in the moment. There are a handful of Montessori materials, however, that follow the child from one classroom level to the next. Their presentation to children, as they age, grows in complexity and deeper in understanding. One such material that finds a home both in Primary and Elementary classrooms is the Bead Cabinet, that big and beautiful collection of squaring (short) chains and cubing (long) chains from one to ten. Many fused squares and a single fused cube for each number is also displayed. An emphasis on counting was present in those younger environments, but this would surely be “baby stuff” to a six-year old. Instead, our work in the Elementary years, the Second Plane of Development, will range from seeing patterns, skip counting, multiplication tables, and a nascent understanding of squaring and cubing.
The colorful array of linked chains, squaring chains arranged on horizontal shelves that gain in length from one to ten, and cubing chains hanging vertically, are all presented similarly. Each chain, each number, has a corresponding box of arrow labels, in that chain’s color, that always include the cardinal numbers up to that number and then its multiples. The box that accompanies the squaring four-chain for example, contains the labels 1, 2, 3, 4, 8, 12, and 16. The last label arrow in a series is wider, to denote that it is the square of that number. The cubing chain has a matching color box with cardinal numbers, wider arrows for the squares it contains, and the widest of them all, the mother of all labels if you will, is the cube. The arrows contained in the light blue cubing box to match the five-chain for example, will have 1, 2, 3, 4, 5, 10, 15, 20, 25 (wider arrow), 30, 35, etc… ending with the widest arrow for 125. The arrows for 50, 75, and 100 are wide, and children will often place a fused square above each of these arrows, placing the cube at the very end.
The work is somewhat intuitive. The child lays out a given chain (while there is no set sequence, squaring chains precede cubing chains; the one and two chains are actually a bit trickier!). Once the chain is completed, the child recites the multiples, skip-counting to the end of the chain. Some classrooms have children write these as multiplication tables, while others have pre-printed recording sheets. Once laid out, there are a myriad of possibilities that two or three children can investigate, all in service of memorization. Children can recite the multiples progressively and then regressively. The first child flips an arrow over and another child has to name the missing multiple. Every other arrow can be flipped over and the children recite the missing multiples. Eventually all the arrows are flipped or removed, and the child(ren) skip count the multiples.
As is usually the case, the ten chain, which has one hundred beads, gets special treatment. Lay the squaring chain of 100 on the rug. Ask the child if they can fold it into a square. Superimpose the hundred square on top to show equality. Ten taken ten times is one hundred. Unfold the chain. Lay out the green unit arrows and have the child place them under the first nine beads. What comes next? The child places the 10 – 90 arrows appropriately. And finally? The large red 100 arrow. Have the children close their eyes while you remove an arrow. What’s missing? Everyone closes their eyes while a child removes an arrow. What’s missing? Later, Where is 37? Where is 84? Where is 61? Or, someone (teacher or child) points to any one bead and everyone else names it. Note if a child can count backwards from 60 to get 59 as opposed to starting with 50. Skip count by tens backwards.
The cubing chain for ten often represents the final chain in the sequence, though this should not be misconstrued. The squaring and cubing chains should be done repeatedly, using the full complement of the work over and over. Carefully give a lesson on how to transport the thousand chain (it’s heavy!). Lay the long chain on a rug in ten parallel rows of ten ten-bars each, as it is when hanging. Cover with the 100 squares to show equality. Stack the 100 squares into a cube. Ten tens make one hundred, and ten hundreds make a thousand. Place the thousand cube next to the stack of hundred squares. How many beads does this chain have? One thousand! Let’s see what that looks like. Let’s carefully stretch this chain out. Note: Some classrooms have super long and narrow rugs just for this purpose. Unfold the chain of 100 to compare. Count together from 1 – 9, then by tens to one hundred, laying arrows down as you go. When you get to one hundred, place the red 100 arrow, but also a hundred square next to it. Continue counting with the children by 10’s to 200 One hundred ten, one hundred twenty, one hundred thirty… Place the 200 arrow and a second hundred square. When you reach 1000, have the child place the largest arrow (green), a hundred square, and then the thousand cube. All the activities we employed with the chain of 100 can be used here.
As a teacher educator, if I have a group of thirty adult learners in an Upper Elementary course, and I’m covering squaring and cubing, the notation, the superscript 2 or 3, there is almost always two or three students who will come up to me during a break, and somewhat abashedly tell me that up until that day, they never understood why we called a number times itself “squared” and when multiplied, “cubed”. There is no underestimating the lasting impact of using your hands to manipulate materials such as the bead cabinet, with engagement, over many years.
Follow the Child 