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Math in the Montessori Classroom
The following presentations on mathematics were given by staff members representing each level of the school at the Parent Education Evening in October of 2008. The Montessori math materials used in the classrooms are beautifully designed and lead a child from concrete to abstract concepts. Join us on the journey.
by Elisabeth Raasch
To fully understand Dr. Montessori’s approach to offering math concepts to young children, it is vital to recognize that her method and her materials are rooted in a psychological, rather than a pedagogical, foundation. I remember so clearly the day that my nephew, Peter, turned two. His entire extended family gathered at his home to celebrate his birthday. Now, Peter called both of his grandfathers ‘Ba Ba’. Although he knew both quite well, he had never seen them both in the same location. He looked at one, then the other, then back at the other. He squealed with delight, and shouted, “TWO BA BA’s!” We all marveled at his brilliance and first understanding of quantity.
Dr. Maria Montessori’s son, Mario, who lectured internationally promoting the Montessori method, spoke of her contribution and cultivation of the mathematical mind in an address given in Germany in the early 1960’s. He states, “…the absorption of mathematical knowledge can be natural, easy, and a source of joy: the joy of one who discovers in himself powers that he had not even suspected.”
Maria Montessori’s observations of children’s capabilities, and the existence of the concept she called ‘sensitive periods’, led her to the research and implementation of the math materials in our primary environments. The math materials were some of Montessori’s first materials, but she did not set out to teach mathematics to young children. Through her keen observation and tireless inquiry, she was intent on finding out what the natural process was in children’s development. She discovered that young children, during certain sensitive periods, were quite eager and capable of understanding mathematic concepts through repetition and exploration. She received criticism at the time from educators and psychologists who thought these children were being forced to do tasks that older children found difficult or disliked.
A significant discovery that Dr. Montessori made was the importance of offering indirect preparation for the math materials while children were in the sensitive periods for movement and the refinement of the senses. It is through children’s work with the Exercises of Practical Life and Sensorial materials that they first encounter and experience the concepts of measurement, sequence, exactness, and calculation.
All children have human tendencies that are related directly to the mathematical mind. Montessorians recognize the necessity for order and exactness. We place materials quite intentionally on trays, we color code activities, materials are displayed in a logical sequence, and we break down movements during presentations into series of sequential steps. Children practice calculation skills when determining how much water to pour or precisely how many drops of polish to squeeze out of a dropper.
Concrete, scientifically prepared math materials are presented to children with the understanding that it is the individual experience that leads a child to abstraction. The child’s interest and attraction to a material is not necessarily determined by the efforts of the adult but guided by her sensitive periods.
There are fundamental abilities necessary for children that will lead to success later in the area of mathematics. Children need to be able to discriminate differences. In the Montessori classroom, they experience this concept when they encounter size and dimension with the sensorial materials. These materials also help a child recognize similarities and differences, find relationships, understand terminology, make judgments, and come to conclusions as they construct and compare a series.
Exploration of the math materials begins with presenting concrete impressions of abstract concepts, paired with movement and the use of the child’s muscular memory. We begin with quantity. Ten square prisms are shown to the child, sub-divided in 10 centimeter segments up to one meter, with alternating colors; the ‘Number Rods’.
Montessori recognized the difficulty children may have in understanding quantity. After making the association between the symbols and quantities of numbers 1 through 10, rather than following with 11, 12, 13, etc., we offer next the gift of the decimal system, “The Golden Beads”-- loose glass unit beads, bead bars of tens, squares made of tens bound together, and a thousand cube made of ten squares. Dr. Montessori discovered that children who knew the quantities and symbols 1 through 9 could count in the hundreds and thousands. This exploration in the decimal system eventually gives the child the vision of the whole and that everything is related to a unit.
There are certain math materials that need to be presented in a sequential manner while others are presented parallel to one another. Children learn linear counting, skip counting, essential combinations, and an introduction to fractions. They perform arithmetic operations with materials that assist in the passage to abstraction.
Math comes alive in our environments, and children find joy in the materials that meet their needs, interests, and mathematic potential. Dr. Montessori recognized the power of the mathematical mind in human development and created tools to help children explore the keys to mathematical thinking. This thread is carried through Elementary One and Two, the Junior High, and beyond.
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After more than 30 years in the classroom, I continue to be inspired by the principles that guide and inform my work with my students. Revisiting these principles in my training albums has already yielded fruits within my classroom community. The richness of our Montessori pedagogy can take a lifetime to plumb.
The ongoing discovery and perhaps invention of mathematics is considered one of the crowning achievements of human history. Evidence of this extraordinary discipline can be seen all around us. From the moment of our birth, we bring to our exploration of the world around us a tendency towards precision and exactitude. It is part of our human nature and compels us to measure, compare, quantify, create, invent. Montessori called this tool of exploration our mathematical mind. And each one of us possesses it.
There are some of you who may be having a little internal conversation. You might be thinking that you are not so sure you are drawn to mathematics. In fact, you are actually pretty well repelled by mathematics. Were you passed over when the universe was busy dishing out mathematical minds?
Some of us for a variety of reasons have lost touch with our natural urge to explore mathematically. Perhaps we were not given opportunities to explore when we were most sensitive to the environment or perhaps we were the recipients of unsound pedagogy. We encountered obstacles in the path of our mathematical development. Not many of us were lucky enough to have our life aided by a pedagogy whose most basic features included: first, a recognition of sensitive periods; second, the belief that discovery is the “engine of learning;” and third, profoundly honored and protected children by recognizing that the adult’s role was to address the periphery of the child, to provide him experiences and allow his center being, or nature, to direct him. To re-establish this flow, step inside a Montessori classroom with me and observe the full flowering of the mathematical mind of the child at the elementary level.
We know that the child under six possesses an absorbent mind. At the elementary level, this absorbent mind is replaced by a most extraordinary and powerful reasoning mind, which takes the rich fund of facts build up in the Children’s House years and looks for relationships between them, patterns that may emerge, often thought of as laws. The elementary years are the age of law and order in every respect, including mathematics.
My wonderful new six-year-olds came to me after three years of counting chains in the Children’s House, knowing many facts. For example, if you take that yellow colored bead bar of four, six times, it will make 24. That is a fact.
I might give a child a key lesson that focuses on a related fact. Six could be taken four times, and the product of these two numbers is also 24. Hmmm! Isn’t that interesting… ? I wonder, is that a coincidence, or could it be true of other numbers? Hmmm! Let’s investigate. Let’s see…3 taken 7 times is 21, and 7 taken 3 times is 21. Look, it happened again!
Now, the elementary-aged child would think to himself, “ I know! I can get all my friends to investigate and see if we can find other numbers that fit this pattern. I wonder if someone can find two numbers that don’t fit the pattern?” Off they go.
Oh, oh! Sara in the far corner of the room found a rule breaker and she is excited. Yes, she has found numbers that are not behaving according to the law! “Look,” she calls across the room (unlike she would have done in Children’s House just a few months earlier), “6 taken 8 times is 48, AND 8 taken 6 times is 42!!! Ha, ha, ha!”
Most likely there would be a moment of silence before another, perhaps mathematically more experienced, child might respond, ”No, 6 x 8 is 48, not 42!”
This, of course, would be followed by a flurry of activity to prove a point. “Oh!!! Those numbers do follow the pattern. Oh, wait. If they all follow the pattern, it looks like there is a law governing them. And now….what is it that Patricia is writing on the board? Com-mu-ta-tive Law. What??? And she now says it comes from that Latin word, Commutare. Who were the Romans that lived so long ago and did they know about this law? Way cool!!!”
And it is cool because the child’s reasoning mind found what it seeks…order, pattern, laws.
Another gift is the imagination. This powerful tool of exploration is particular to elementary-aged children, just as the senses were the tool of exploration for the very young children. Our imagination allows us to envision something that is not there in the concrete.
Using the Wooden Hierarchical Materials, for instance, a child can imagine beyond what we can show in concrete form. A simple unit is a point. The units stand together and form a line called the ten. The tens line themselves up across a plane and form a square of a hundred.
Next we have the thousands family, a unit of thousand, a tens of thousand (again a line), and a hundreds of thousand, which forms a square.
Now look at a million, a unit of million. It is like a point. If this is a unit of million, can anyone imagine what a ten million would look like, a hundred million?
What about a billion???? This hierarchical material is a springboard allowing the child to imagine the unlimited possibilities of our decimal system. It gives them the nudge to imagine what is not clearly and immediately visible to them. Appealing to the imagination of children inspires them.
Another important characteristic of children at the elementary level is that they are first and foremost collaborative workers. They move in small hordes and are always trying to figure out who they are within the small community of their classroom.
Recently a group of three boys made a decanomial square out of graph paper. The very first week of school I had reviewed the layout of the decanomial square using our bead cabinet with a few second and third year students. I did this because I wanted to find something that would engage them for awhile, that would resurrect those rusty math facts that they were working on last year. I also wanted to create an opportunity for them to work with the new boy in the classroom, to help him settle in. After a short lesson, the group remembered the activity and were off and running.
After they completed the decanomial with beads, I showed them that they could, if they wanted, recreate it with graph paper. Off they went in a flurry of activity. They had to organize and they did, “You do the tens; I’ll do the sevens and twos. Rory, do the…” There were a number of kerfuffles. So and so wasn’t doing his share. Another child misunderstood and did the wrong ones. But over the course of a couple of days things got sorted out and they produced a chart, which they brought to me, beaming.
“Grand,” was my response. Together we admired how well it all fit together. One of them noted that the coloring job was kind of messy. Just as they were about to leave I asked. “And, oh, what is the value of this decanomial cube?”
“What??? What do you mean?” they asked.
“I mean, how many squares are there in the entire decanomial?”
“Well, uh, a lot but we are not sure.”
“Hmm. I wonder, could you find out?”
Again, another flurry of activity.
“Well, we could count them. It would be faster to add. Who is going to add the fives? I’ll do the tens.” Yada yada yada.
Another few days of investigation ensued. A couple days later they showed up at the lesson table again, Cheshire grins on their face. They told me the big number.
“Congratulations,” I responded. “What a lot of work went into this! Now... are you sure?”
Of course, off they went in another scurry of activity investigating various methods of proving their work. Several breakdowns happened and some tension mounted. At a certain point they again reappeared at my lesson table in a huff. As they began to tell me their troubles and mounting frustrations, I could see they had fallen into a significant “twit.” I noticed the new boy attempting to get a word into the fray.
And he said, “Couldn’t we have just counted the squares on the side of the decanomial and on the base and then multiply the two numbers. Wouldn’t that give us the answer?” The other boys stood there with blank looks on their faces.
The new boy further explained, ”Didn’t we do that with all the squares and rectangles inside the square already?”
Then, slowly but surely, understanding dawned.
“Oh ya, he’s right, that could work. We can solve this.”
(Smiles, smiles, smiles!)
This was a great aha! moment mathematically for the group AND at that exact moment the “new boy” found his niche, became known to his peers for the contribution he made to the group, and perhaps most importantly, recognized his contribution himself. He became in that moment a bona fide member of the Class C community.
Lastly our elementary child needs a field of exploration much bigger than he met in the Children’s House. He is not limited by the physical world; he needs to expand his exploration to the Universe itself. And that exploration needs a context- the Great Lessons. These stories can never be given in written form. They must be told by the classroom teacher, who plays a crucial role within the classroom as a storyteller of truth.
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“The true spirit of delight, the exaltation, the sense of being more than Human, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.” - Bertrand Russell
By the time students reach the Junior High, they have discovered a wealth of mathematical knowledge. The foundations laid in Children’s House and Lower and Upper Elementary foster a curiosity which we in the Junior High strive to maintain.
Mathematics at the Junior High level is an exciting experience for the adolescent. Students have math classes and the classes have familiar, yet mysterious, names to them: Algebra. . . Geometry (what do they even mean?). They use calculators and take notes. They work with variables and equations, expressions and inequalities. It is a transitional time for students, one of exciting new opportunities.
Of course, math at the Junior High level exists outside of Pre-Algebra, Algebra and Geometry. On the Odyssey, students measure cups of water in proportion to the oatmeal for breakfast crew. In Science, they calculate the force and momentum of tennis balls. At the Land School, they compare the expenses and income of their produce for micro-economy. During Interim, they take measurements to construct the set for the play. And for Marketplace, they calculate the rent on their spaces and the tax on their products.
Mathematics is truly a language students spend years working to master, and as with any new language, it must be practiced.
Junior High students encounter a more traditional atmosphere in their math education. We take note of Minnesota state standards and make a conscious effort to best prepare them for high school. Students have daily lessons and assignments, lectures and activities. They actively work together from the concrete to the abstract. They identify patterns and relationships and describe them symbolically. Pre-algebra students examine breakfast menus and baseball lineups to investigate concepts of probability. Algebra students recognize the distinguishing characteristics of linear, exponential and quadratic relationships through interest rates and the flight of projectiles (such as a baseball). Geometry students master compass and straightedge techniques and refine communication skills through the logic of proof writing.
As at any level of the school, errors are essential to the learning process. Students’ ability to recognize their own mistakes is as vital a skill as recognizing the correctness of their answers. And over time, students come to appreciate the power and beauty of solving a complex equation and knowing the solution must be correct, regardless of what the teacher says. There exists an elegance in mathematics, separate from other disciplines, in which truth exists a priori. These mathematical truths are timeless. It is as if they have existed in some upper stratosphere, invisible to the casual observer, yet waiting for us to rise to the level where we can see them. Together, student and teacher must climb this mountain of previous discoveries to interact with and appreciate these timeless truths. And there’s more than just mathematics up there! Fundamentals of the human conditionbeauty, order, logiccornerstones of philosophy as well as math! From the top of this mountain, we feel this “highest excellence,” this “sense of being more than Human,” to which Russell refers.
Back in my college days when I decided to pursue the teaching of mathematics, I had to write a paper as to why I was choosing this particular profession. I wrote something about having lousy math teachers when I was in school, wanting to make the beauty of mathematics more available to students today, seeing math as (forgive the pun) a great equalizer for students from disadvantaged backgrounds. Today I summarize my answer in two words: math anxiety. Math anxiety among students has permeated today’s schools. As a society, we have married students’ abilities to their test results. Students are hypersensitive to their scores and their “ranking” relative to their peers. It is unfortunate these scores have become such gatekeepers in our educational system. And although our students are prepared to succeed in testing, this focus on scores runs counter to our aims as a school, which is to foster their intrinsic motivation and love of learning. Rather than allow our scope of what is important to narrow, we seek to maintain their interest through continued exploration centered on their curiosities. At Lake Country, we serve to nurture students’ innate problem-solving abilities, so when they come across a new problem, they find it exciting and challenging (and manageable) rather than uninviting or impossible because we “haven’t covered it yet.” This way we may soothe the math anxieties that rise in our students.
Throughout the school, students have rich and varied opportunities to revel in the wonder and awe inherent in mathematics. One of the strengths of the Lake Country math program is that we have problem solvers, not just problem doers. And isn’t that what we want?
The above are transcripts of presentations given at the October 2008 parent enrichment evening and were published in the Fall 2008 issue of the Lake Country School Courier
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