One positive aspect of the social media explosion is the ease of staying in touch it affords. Alumni and their parents share their post-Cornerstone experiences more freely, because it’s just a click/send away. For some of our past students, Cornerstone represents twelve years of their life, building a sense of ownership and home that is not forgotten by a mere change of address. In short, we get letters. This came in from a parent, a forwarding of an email they had received from a high school teacher of a Cornerstone graduate: “I just wanted to let you know your son ended the semester with one of the only A+ with Honors I have ever given. On that note, while I know you know how talented he is, I want to throw in my 2 cents that he should take as many AP classes as possible next year. I have tried hard to keep him challenged in my class, but he is so far beyond other students that I don’t think regular classes are the place for him.” Truthfully, this is not uncommon for our graduates, but the parent highlighted the second part of the teacher’s email as being more meaningful: “The other thing I think is great about your son is that even though he finishes his work easily he helps other students. There is one student in particular that sits next to him and she struggles every day. With the patience of a teacher he helps her ALL class. Sometimes I think she is going to wear on his patience but he just gently answers her questions.“
Can kindness, in fact, be taught? As Montessorians, we would answer, “No more than we ‘teach’ geography or arithmetic or science”. Rather, a Montessori school creates an environment, carves a space, maintains a culture that allows a natural process to take place. And while it isn’t quantified on any conference report, the grace and courtesy aspect of our curriculum is an integral component of the fabric of our classrooms. This serves, strongly, as the tapestry on which our lessons are woven. It is so present in fact, that a consistent comment I hear from prospective parents, even after a mere 20-minute observation, is the kindness they witness amongst our students, regardless of class level. Last week, after an especially moving observation, a parent sat with me in the hallway outside of the Junior Class, asking me the how’s and why’s of our school. They enthusiastically embraced the peacefulness and kindness they saw that morning. “Does that happen every day?”, she asked, perhaps a little suspicious. At that moment, Quetzal and Nicholas walked by, hand-in-hand. “Yeh, I said, “Pretty much.”
Walking with great care, the young child brought the stamp game to the table, gently placed it down, and opened the lid. Smiling shyly at me, she carefully began laying out the first and second addends, in horizontal rows, one under the other, carefully aligning the thousands, hundreds, tens, and units by place value. A scene from any lower elementary classroom in the world. In this case, however, the school was Kiara Karitas, and it was located on the other side of the world from me, in Jakarta, Indonesia. The girl, Hee Youn, was a first-year student in their lower primary classroom. A few years back, the larger Montessori community of educators and parents and administrators and children celebrated the centennial anniversary of the first Montessori school. That milestone spoke to the lasting power of a profound pedagogy; one that has truly stood the test of time, allowing children to learn to their potential, to gain an insight to knowledge that is both integrated and internalized, and to develop loving hearts and inquiring minds. In my role as a school board member for my local district in southern Maine, where math curricula, literacy programs, and science textbooks shift with the vagaries of educational reform, I see firsthand the disadvantages of moving from one Newest Thing to the Next Newest Thing, and the expense and learning curve it requires for teachers and children alike. But what struck me in Jakarta (and Seoul, and Nashville, and Sarasota…) was how the span of Montessori not only reaches back 100 years, but also across the world. What does it say of an educational system, that it can speak so forcefully, with such profound results, to parents and children in schools from New Hampshire to California to Asia, Africa, Europe and beyond? What does it say about the Montessori method, that can unite so many schools in a common model, using the same Montessori materials and same prepared environments? What does it say about this worldwide and historical community of teachers, students, and families, who wear different clothes, write in different languages, give and receive lessons spoken in different tongues, but are united all the same? I wouldn’t have bothered to ask Hee Youn. She was too busy, and wouldn’t have cared. And frankly, my Bahasa is lousy.
Many Montessori schools have the great benefit of a full complement of programs. Toddler students, as young as 18 months old, will stay at their school for as many as thirteen years before they graduate as 8th graders. A lesser few Montessori schools may even have programs for infants on one bookend and high school at the other. The familiarity with the building and grounds, the people, and certainly the pedagogy, is of great comfort to both children and their parents. But familiarity is also of great benefit in an educational sense. Piaget, himself the president of the Swedish Montessori Society, did a well-known test called the “mountains study.” He put children in front of a simple plaster mountain range and then asked them to pick from four pictures the view that he, Piaget, from where he was sitting, would see. The test was initially used to show a child’s development in visual-spatial awareness, namely that children younger than age seven were egocentric and unable to see another’s viewpoint. However, a follow-up study using a scene familiar to children, the setting and characters from Sesame Street rather than the completely foreign Swiss Alps showed that the familiarity of setting had a dramatic effect on the child’s learning.
A Montessori environment makes use of this principle in a myriad of ways. A most visible example of this concept would be the “hierarchical colors.” In Early Childhood classrooms, for 3-6 year olds,, an arithmetic material introduces a mnemonic color-coding to place value. Units are green. Tens are blue. Hundreds are red. Thousands? They’re green, because thousands are still units; just units of thousands. These hierarchical colors are then used as a constant device as the child moves towards more complex and abstract math. The Stamp Game, for all four operations, utilizes wooden squares, “stamps”, to solve in all four operations. The bead- frame (for addition, subtraction, and multiplication), the checkerboard (multiplication), and the “racks and tubes” (division) all use the same identifying colors. The colors of the short and long chains, the colors that correspond to each part of speech, all serve as a conceptual grounding for the child, a link to the concrete experiences that preceded it, and a guide to further exploration. The material is new, the concept is more complex, but the familiarity of color (or shape, or timeline, etc..), isolates the difficulty, and frees the mind.
Dad – I’m making a little baby box about Lyra’s birthday with some of her hospital stuff. I wanna include a paragraph from you about remembering when she was born or when you first met her. Skip the trauma part.
There’s a picture of a baby on the kitchen windowsill at my daughter’s house. It’s small and round and cropped in an outline of Lyra’s six-month-old face. Her eyes are scrunched closed. The photograph itself is attached to a popsicle stick, and this allows it to stand in a clay pot, sharing space with an aloe plant. It is just one of a couple dozen shots of Lyra, spanning her first year of life, still to be found, now over two years later, in odd spaces around their house. Lyra looking surprised, Lyra mid-laugh, Lyra in sunglasses. Lyra with a small teddy-bear bandage holding tiny oxygen tubes as small as new stems of grass, onto her blush cheek. There are no photos of the feeding tube, none showing the nickel-sized electrodes dominating her small chest, none taken at the NICU.
All of those photos, all of those Lyra-adorned cupcakes, had once been frosted and laid out on a fold-up table along with the other party food, the pizza and cookies and cake. The birthday cake had a single candle that day, marking a year, a tally of emotions that come with a first child, for Amarinda and Brendan, and a first grandchild for Sandi and me. Gathered that bright and lush September Saturday in Maine, which was bravely holding onto the last of that summer’s warmth, were Lyra’s uncles and aunts, cousins and friends, all there to celebrate an infant turned one. She sat on a dozen different laps, sampled most of the food, and was generally unaware of the fuss and festivities unfurling around her. Every first birthday party is for the adults after all, to see each other, to catch up, the how’s the new job and I can’t believe he’s almost a teenager conversations, and the common question of, “can it already have been a year since Lyra was born?” And if there were thoughts of a year prior, when she arrived eight weeks ahead of expectation, well, those past emotions had healed and changed and grown right along with her.
If we had known, those first fraught hours and days of her life, that in twelve short months we’d be eating Lyra-cupcakes and helping her open the many birthday presents she received from us, her devoted legion, it would’ve been less scary. We could collectively have said, “Oh, THAT’s how this works out.” But life doesn’t work that way, does it. The Buddhists tell us to stay in the moment. To be mindful and dwell in the present. Be now. But in some cases, being able to glimpse just a whisper ahead, would be good for the soul.
Dedicated to Flor, who says I shouldn’t just write about Montessori.
A few summers ago, I was in Charleston, SC teaching Montessori philosophy, methods, and 6-9 (lower elementary) mathematics to a group of aspiring teachers. South Carolina had become a hotbed for Montessori those days, and it was expanding into public schools as rapidly as it was in the independent schools sector. In this particular group, there were both seasoned teachers in regular education, and 3-6 (primary) teachers, and one 9-12 (upper elementary) teacher, all spending a good chunk of their summer learning 6-9 (lower elementary) Montessori pedagogy. Near the end of the week, I was showing some “exponent and powers” lessons, and had a series of squares and cubes arrayed on a mat in front of me, the beauty of the mathematical patterns formed were clearly evident. The 3-6 year old teacher noted, “Those 10 cubes are the same sizes as the pink cube in a primary classroom”. The 9-12 year old teacher remarked, “In upper elementary we use it for cubing and volume work.” There was one of those great thoughtful pauses from the group, and one of the youngest of the cohort, just out of college, said softly, almost to herself, “Wow. I think I’m going to like this…”
Everyone knows that imitation is the sincerest form of flattery. While this may or may not be true, it is certain that imitation is also a powerful learning tool. Studies abound illustrating the human tendency to mimic, both consciously and subconsciously. Participants watching a video featuring rude exchanges between actors are liable to be rude themselves when put in social situations immediately afterwards. It is clear that as a species our behavior profoundly influenced by the people that surround us, and this can impact both our actions and our learning.
How does this manifest itself in a Montessori environment? One clear component is the multi-age classroom itself, an aspect that holds many advantages for students, parents, and teachers. Children enjoy the security and comfort of staying in one room for three years. Parents don’t have to reintroduce their child’s strengths and challenges to new teachers each September; knowing his or her teachers will gain a deeper understanding of a child’s needs given a three-year cycle. For teachers, the variety of ages and developmental stages in the same classroom allows children to move more freely through a scope and sequence of study, as the so-called “shotgun” approach, requisite to a single-aged classroom, is not necessary.
As important as these elements are, Montessorians have known all along that there are also clear pedagogical advantages to a multi-age classroom and the opportunities it affords to use imitation as a tool for learning. Younger students watch older students, hear the language of the lesson given on the next rug over, observe the use of more complex learning materials, and mirror their behavior. This is why we often hear Montessori teachers emphasize to these older students their role as models and peer teachers. And, of course, the teachers
themselves give lessons in such a way, with great care and exaggerated movements, as to stress key elements in any given lesson. For example, the forty-seven steps to washing your hands. We can see how Montessori’s use of the phrase, “the absorbent mind” reflects her understanding of the importance of imitation.
Most adults observing a Cornerstone classroom are quick to notice its strengths. The use of manipulative materials, the small group lessons, the beauty of the prepared environment, the freedom of movement, all form an impressive tableau. A more in-depth observation would also clearly reveal the integration of subject areas, the social interaction, and the element of choice. Within that structure, students move with purpose (most of the time) and ease, seemingly without adult compulsion. Children voluntarily seek out activity, come to lessons willingly and happily, work with peers of their own accord, and, with guidance, take responsibility for their education. The structure for this drive does not come from a draconian adult or some other extrinsic force. Instead, the children appear to have an intrinsic urgency to act upon the environment. Why?
A crucial aspect of any Montessori classroom is perhaps less discernible due to its conspicuousness. The driving force in the child’s interaction and progression through the curriculum is deep interest. It is the tree that can’t be seen for the Montessori forest. This passion is created through creative and impressionistic lessons, the presentation of grand concepts, the use of large numbers, the emphasis on the power of imagination, and the liberty to choose a compelling activity for one’s self. More than a natural incentive, interest further serves as a powerful tool for learning. Studies clearly show that we are much more likely to assimilate information if it holds strong interest. One such study had participants list a series articles in terms of their interest. Not surprisingly, comprehension scores on these readings mirrored the ranking given. Areas of higher interest naturally hold our attention, heighten our focus, and compel us to iteration and practice. Consequently, the learning that takes place is more meaningful, more profoundly held, more deeply understood, more logically connected and synthesized.
And need we mention joy? So, at the end of the day (the metaphorical day, not 3 o’clock dismissal), it is the child’s likely response that speaks volumes in its simplicity. “Why do you like going to school?” “It’s fun.”
A Montessori education provides a rich and integrated curriculum that stresses learning in context. The study of geometry includes a study of its Latin roots, a study of unlike denominators in arithmetic includes the writing of the rule, a study of an ancient civilization coincides with a study of rivers, ph studies evolves into soil testing. Specific Montessori materials can also reflect this sense of context. For example, the Detective Triangle Game, located on the language shelf, consists of a box of triangles of different types (scalene, isosceles, etc…) in different colors and of different size. Labels accompany the work: “Find the large, red, equilateral triangle.”…etc; geometry as a grammar work. Speaking more broadly, the concept of Cosmic Education, unique to this pedagogy, is the overarching theme of a Montessori classroom. It holds the fabric of a Montessori experience together and places everything the child learns in context. Cosmic education states, grandly, that a human developmental process underlies all growth, and further, that education has a role to play in this development. It is a belief that theoretical structures, in all areas of study, should find practical use within our classrooms. Simply put, Cosmic Education presents three concepts; that all things are interdependent; that humans have a role in the universe; and that each of us has a cosmic task.
One aim of Cosmic Education is the development of the whole human being. It would follow then that academic achievement is not the only goal of a Montessori classroom. The child will realize their full natural potential, learning that involves the physical and emotional being, not only the intellect. A second aim is the formation of relationships. By building a sense of marvel and respect for the vast scale of things and appreciating the dignity of all things, we show a relationship between the child and the universe. A third aim is the realization of responsibility, to all life, to the human species, and to the child themselves. And a last aim is one of independent action. In broad terms to take, but to give in return, to share willingly and with compassion, and to appreciate both the conscious and unconscious service of those plants, animals and humans that have come before us. Cosmic Education then, is not a singular area of study, but rather a connective web that unifies the curriculum, providing both respect and responsibility to the child throughout their school years.
With the bead frames and checkerboard, the children were not able to carry out division. The concept of division was given with the decimal system material (golden beads), where they saw the beads distributed into equal parts. Then they saw it with the stamp game as both distributive and group division. The memorization materials worked parallel to these earlier lessons and were an important preparation for this later stage of work. Learning those combinations, as well as their multiplication facts will enable the child to carry out the division quickly.
The Stamp Game, Bead Frame, and Checkerboard, lead the child to abstraction in addition, subtraction, and multiplication, quite elegantly in fact. Division is different. The lessons presented with the decimal system (golden bead) and again with the stamp game used distributive division to find the answer, the quotient in a division problem. To divide abstractly, the child must learn group division. “You get one, you get one, you get one, you get another, you get another you get another…” results in the correct answer, but is a different skill from “How many of these are contained, go into, this?” To lead the child towards this abstraction, we will put special emphasis on the recording of our work.
The Hierarchical Material which consists of seven test tube racks: 3 white (containing green, red, blue beads), 3 gray (with green, red, blue beads), and one black (with green beads). Matching these are seven bowls with matching external colors: 3 white, 3 gray, one black, and internal colors: that are hierarchical, meaning the unit bowls are green, tens are blue, and hundreds are red. In each rack there are ten test tubes 10 beads in each tube, so lots and lots of beads! To perform the distribution, there are four boardsm each with 81 holes/ two green (units and units of thousand), one blue, one red. Finally, there is a box containing many green, and red skittles to represent our divisors
The first division done with the child may be relatively simple, although the material allows children to work with dividends in the millions. Some children will enjoy the challenge of starting with a seven digit dividend while others will find it overwhelming. Our first example is based on a dividend formed of at least two or three digits. The divisor is one digit. There should be no remainder in the initial presentations.
Start with the problem such as: 4)9764
The first time we will only write the quotient. In order to reduce confusion, we will remove from the tray only the necessary holders and bowls.
Form the number by putting the correct hierarchical beads into the bowls to form the dividend and place them closer to the child and to their right. Then put out 4 green skittles on the green board.
Bring the 1000’s bowl up to the bottom of the board and distribute them to the four skittles. We see we can give two to each. But we have one we are unable to distribute. Place it in the 100’s bowl. Write 2 (what each skittle received) above the 9 in the dividend. Pick up the green beads on the board and put them back in their test tubes, referred to as “clearing the board”. Turn the now empty thousands bowl over and slide off to the left of the child.
Look in the 100’s dish. There are 7 reds and 1 green. The green can’t stay so change it for 10 reds (a full tube). Then distribute the reds (100’s). We can give 4 to each skittle. Record a 4 in the hundreds place. We have one red we can’t distribute. Place it in the 10’s bowl. Then pick up and put away all the hundreds beads, turn the bowl over and slide off to the left.
Move the tens bowl up to the board. Look in the bowl; there are 6 blues and 1 red. Change the red for 10 blues. Distribute the tens. This time there are none remaining. Record a 4 in the tens place, pick up the beads and put away the tens bowl.
Move the units bowl up. There are 4 green unit beads to be distributed, and each skittle receives one. Record this final number in the quotient. A good note to keep as a reminder for this presentation is, “beads first, record quotient only”.
Children benefit from lots and lots of repetition at one level of complexity before moving on to the next. I would use the word “facility” meaning there is an ease and confidence in their working through the problem. Perhaps that’s difficult to define quantifiably, but as a parent/teacher, you’ll recognize it when you see it. When it does come time to move ahead, when the child is ready to move from this concrete experience to the next level, we’ll take a small step by simply recording our work, resulting in a problem on paper that will follow the same algorithm we all learned in our childhood classrooms. Here the reminder will be, “beads first, record all work”. To isolate the difficulty, which is a component of Montessori lessons, use the same problem a child has already completed. In repeating the problem above, 4)9764, the work with our hands will be exactly the same. We begin, distributing the nine thousand beads to the four skittles as before. Okay, what did we lay out? Two rows of 4, so 8 green beads. Let’s record that. Write a 2 above the 9, and write an 8 below the 9. We started with 9 and laid down 8 so we subtracted. 9 – 8 = 1. Do we have one green bead left? Yes.
Exchange the green bead for ten red hundreds, adding it to the 7 beads in the hundreds bowl. Repeat the process, dividing, recording the quotient (rows) multiplying the rows by divisor to determine how beads are laid out on the board (or skip count), subtract, bring down, wash, rinse, repeat.
As we move through the next series of presentations, with larger dividends and multi-digit divisors, we’ll move through this same sequence. Eventually, the child will do the writing first, and confirm with the beads. At that point, they will be very close to abstraction, calculating division problems without the material. Once you have crossed the river in a boat, you no longer have to carry it. For more innovative Montessori materials, make sure to regularly visit our website at www.alisonsmontessori.com.Rob Keys
“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 