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Montessori Math

By Katherine von Duyke
Printed in Practical Homeschooling #50, 2003.

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Kathy von Duyke


I'm sure many of you are familiar with this scenario of an early morning school day: Your child begins his math text work, while you are chasing toddlers to the potty and finishing breakfast chores. Suddenly he has a question, perhaps about division with remainders, so you briefly reread the instructions in his text to him, and move on. He still doesn't understand and a slight bit of frustration enters his voice. You go back, sit down, keep one ear out for the child in the bathroom, and draw your student a picture. As you skate off to your next task, you hear your student's heightened confusion and frustration. You do a bodily 180 degree turn while mentally going through your cabinets and teaching aids trying to think of a demonstration on the spot to help him "see" the concept. Your creativity is called upon in the moment you prepare a demonstration that you hope will connect. It does, sort of, but by now he's so frustrated it's hard to get any concepts in, you get frustrated as well. You think, "There really has to be a better way."

There is!

What if, when demonstrating a math concept that you know your child hasn't been taught (because you keep a skill list of what he's mastered) you could begin with materials that are familiar to him and include directions for you? What if you were even given a way to present this lesson so that it honors and delights him as a student? What if the demonstration was so clear, that once you taught one child, he could learn even more by teaching or reviewing with siblings? What if the lesson was just so fun to teach, you couldn't wait to give it to another child?

What if the core of your math method was the demonstrations and skill mastery list and not a textbook? What if you had follow-up work that your child could spend just as short or as long on as he needed to, and you would know how to tell when it's mastered and how to mark it on his record sheet? What if you had several math opportunities available to him and he could self-select the area that most interested him at the time?

If you had all of this, you would have the Montessori approach to math!

Materials That Really Teach... Again and Again

Those of you who are homeschool veterans and math manipulative fans know there is a big gap between Cuisenaire Rods and Mortenson Math or other algebra materials. The other manipulative products I have tried are either too repetitive or piecemeal.

In the Montessori method, there are four or five materials available to teach each concept. This is so the child can approach the concept from several paths and if one doesn't click with him, another surely will. Plus, these materials are familiar to him.

There is, as Harvey Hallenberg notes, "a monogamy in the materials," giving students comfort when new concepts are introduced.

For example, while making a cube chart of numbers 1 to 10 the student finds himself with a bead tower that takes him all the way back to the beginnings of his schooling with the pink tower. The student is filled with awe to find one of his most familiar preschool memories was setting him up to understand higher-order math.

Montessori Training Secret: Albums

I am so enjoying this year, even with a baby due mid-school year and all the inevitable struggles and joys that come with that. I know that when I get there that my children and other students in our cooperative school will be well trained and in charge of their learning. I can't imagine why I ever taught any other way! There is such a satisfaction in giving a child a lesson that connects with his mind and heart and being.

Surprisingly, most of what makes up Montessori learning is not in the materials themselves, but in the training albums - those mysterious albums that have been the "secret handshake" that has kept Maria Montessori's work rather cloaked in obscurity. Even though there are a few sources for upper-level albums, such as Montessori Research and Development (www.montessorird.com), they include only a few pictures. They also assume that the reader is familiar with the materials and has finished Montessori training. Trying to learn with such incomplete materials is like the difference between reading a recipe on how to make bread, or finding a recipe with step-by-step pictures, or better a video demonstration, or best still a friend to show you how and answer your questions while coaching you through the process.

Knowing that direct coaching is the best way to learn is the reason I chose to get official Montessori training. However, the cost and time away from family is huge. I know, because I spent nine weeks this summer getting a super-packed training course towards my AMS (American Montessori Society) elementary certification and earning credits towards my masters degree as well. It was wonderful, but hard to be away from my family. My classmates at school dubbed me "K12," as I learned in the midst of my training that Baby von Duyke number 12 was on the way!

I hope to someday soon be able to offer weekend seminars on album topics to homeschoolers, not for professional accreditation, but so they can enjoy the method at home. In the meantime, you can either choose to hunt and peck your way through the albums that you can purchase, buying materials so you have them to work out the lessons with, or you can take training at Barrie's Institute for Advanced Montessori Studies (www.barrie.org/iams/) in four-week sections.

I have heard that a Canadian company is developing elementary albums with full color photos that are nice but pricey. Also, you can find some text-only elementary albums online for free at www.moteaco.com.

However, whatever way you can obtain training, the method rewards your efforts fully.

I learned that our Montessori cooperative here at my home is in fine American tradition. The Montessori Method was introduced into this country by notables such as Alexander Graham Bell, who turned a room in his house into a classroom so his children and those of several of his friends could learn together. Wow! Just like us.

Infinity Street

So, let me give you some pictures and directions out of my training to help you teach one Montessori lesson at home, and you can make the materials yourself. It would be more fun if we could do this together, but I hope this gets you started.

Most math texts start out with the concepts of "quantity" and "symbol." Students review the words and numerals each year. Usually they are asked to write in words a number such as 123,984,039 in words and then given a number in words that they need to translate into numbers. Not very exciting, but this lesson is!

So here's a verbatim dialogue of this lesson.

MOM: "Who wants to do a lesson on Infinity Street with me?"

KIDS: "What's Infinity Street?"

MOM: "Come to the lesson and find out!"

Invariably, some children will come and others will decide not to, but that's okay because the others always change their mind!

The teacher assembles up to three students around her mats on the floor. NOTE: In the classroom or in the home, if you have this many children, limiting the number to three students allows all the children to answer questions and touch materials as the lesson is being given. I have found it better to give a lesson twice rather than lose this close contact.

MOM: "Thank you for coming to the lesson. What do you see here?"

This is a standard question that allows the students to observe and think and become curious. They note the similarity of the rooms, etc.

Before them are the Montessori numerals and seven houses cut out of art foam paper, each progressively larger than the previous. All have three "rooms" in them - columns of green, blue, and red foam paper. Each house has a mailbox after it except the last one.

Normally, you would lay the houses out in a row with an arrow that shows they can go on forever. I didn't have the room to take a picture this size, but hopefully you can get the idea.

MOM: "This is called 'Infinity Street' [write this out on a slip of paper as you speak and lay it on the top end of the rug] because although we only have seven houses here, we could have houses that go on forever. Please take out your lesson notebooks and write 'Infinity Street' for today's title."

The students now take out their black and white composition books where they record all their lessons. They title a page and add a line to their table of contents. They will later draw everything laid out on the mat and refer to this when they take the materials off the shelf and practice the lesson on their own.

MOM: "Each house on this street has three rooms. Only one numeral can live in each room. Would you like to put a bead bar in the rooms?"

Offer the 1-10 bead bars (or you could use Cuisenaire rods) to the students who put one in each room. Alert students won't use the ten bar, but many will. Students place bead bars in the "rooms." For the purpose of the instructions below, assume the students picked the bars 6, 7, and 2 in the millions house... the bars 5, 4, and 8 in the thousands house... and the bars 1, 2, and 3 in the very last house. (If this doesn't make sense yet, don't worry. Just keep on reading and it will!)

MOM: "Each family that is now in each house has a last name. We have the 8 bar in this room, but this 8 has a last name in this house. The first house is called the 'Simple Family.' They don't have a mailbox because they can't read. We don't use their last name because it would be a little insulting to them, so we just call them by their first names. So who is in this house?"

Have students read 123 as "One hundred twen-TY three," emphasizing the "TY."

MOM: "The 'TY' stands for 'ten,' so we say the name and the room each number is in, except for the units, since we know they are in the last room. Write this on a long slip of paper."

_______________123

[The kids write it.] "But the next house is the thousand family. After we say their first names, we add their last name - 'Thousand.' We can remember that if we say their last name when we see their mailbox.

You write "thousand" and lay it next to the mailbox.

MOM: "When we write this number we put a comma where the mailbox is."

Have students read 548, five hundred four-TY-eight. Add to the paper so it reads

_______548,123

The next house is the million family. [You label a piece of paper 'million' and place it next to the mailbox.] Milli- means "thousand" and -ilion means "thousand." So million means a thousand thousand. Each house on this street is a thousand times bigger than the house before it. This is called the hierarchy of numbers. [You write "hierarchy" on a slip of paper.] Each hierarchy has its own house and a comma to separate it from the other hierarchies. So that's why the Montessori numerals are green for the ones and the thousands, because the thousands live in the one's room of the next hierarchy. The millions are a little snobby because they are very big and rich."

If you have the hierarchy material, which consists of a 1 cube, a 10 bar, and a 100 square, a 1,000 cube, a 10,000 bar, a 100,000 square and a 1,000,000 cube, you could line these up next to the houses. I took Cuisenaire rods and did the same. We imagined the next size and shape of each subsequent room. The children caught on that very quickly that members of subsequent houses could fill our entire house with their size.

MOM: "So these houses represent very large numbers don't they? So now let's name our number 672 in the million house. This means six hundred, seven-ty-two million... "

You read the rest and add it to the strip of paper. At this point your strip of paper should read:

______672,548,123

The lesson continues this way up until the quadrillions.

If you get to a room with a ten bar in it, simply say, "Everyone wants to be a ten friend, everyone wants to be a ten. This guy doesn't want to stay in this room because he's a ten, since each room is ten times bigger than the one before it, he want to move over to that room. But when he does, what does he look like? A one. So we have to exchange."

Continue exchanging and naming large numbers until the students are comfortable with them. Then say, "What do you think about that?" rather than any specific computation questions, as it tends to have them leave the lesson thinking about what they have learned. Say, "Thank you for coming today! This job will be on the shelf so you can lay out the houses, labels, and beads in the correct places."

This lesson is a lot more fun to do than read about. Try it and see if you don't enjoy the wonder of Montessori at home.


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