Grid paper rules!

If I had only one material with which to teach mathematics from Kindergarten to grade 6, it would be grid paper with 1cm squares. Over the years, I have used grid paper to teach counting, skip counting, multiplication, and area of rectangles and triangles to children in grades 1-6, including children in Special Education in grades 4-6. We have also used it for graphing (see earlier Dice Race post) for children in Kindergarten. If I didn’t have other materials, I would also use it for addition and subtraction.

The paper is simple and appealing to children. It can be used as a full sheet (10 x 10), half sheet (10 x 5), for the dice race (6 x 10), or with 2 rows of 10 squares for early counting exercises. Yesterday, several children in Kindergarten were using the paper for counting. I don’t tell them where they will end up of course, and what they tell me reveals their understanding of numbers. The child writing numbers on the 10 x 5 paper (it was his idea to start this) told me that he knew it would end up at 50, then showed me the rightmost column, saying: “10, 20, 30, then it will be 40, then 50”.

The other child working on the numbers (this time to 20) did not talk about where she would end up. This work was at my initiative, since I wanted to help her with practice on numbers to 20. She enjoyed it and did a second one, quickly, accurately and with confidence.

We also use this paper for practicing writing the child’s name – or any other letters and numbers the child chooses. Often, when the older children are writing numbers to 50 or 100, the younger ones will take out the same paper, sit with the older children, and write 0 or 1, happy to be doing the same work as the “big kids.”

As with many other activities in my posts, this is not from the Montessori curriculum, but rather has evolved through my work with the children. More on this paper and activities to come in later posts…

Teri Courchene

Teri Courchene

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One material, many ways to learn

Within the Montessori classroom, there are endless ways to work with the materials. Some of these ways are set out in the curriculum taught in the teacher training program, whereas others are improvised to match an interest on the part of the child. Several lessons with the colour tablets illustrate this.

Once the child is able to match the colour tablets (there are two of each colour), we try matching at a distance. This involves setting one set of colour tablets up all mixed up at our table, and the other set (the same colours) at a distant table. I set one colour tablet in front of the child and without saying the name of the colour, I ask the child to get the same colour from the other table – without taking the colour tablet along. At times, the child will come back for another look before finding the match. When the child comes back, I ask the child to put the tablet next to the one he/she was matching to and ask if it matches. It is important that the child decides if they match – I do not correct. This work builds the child’s working memory. There is often a distraction (conversation, interesting work to watch, etc.) along the way, which adds to the challenge.

Another lesson – and probably the most fun lesson from the child’s point of view – is matching to the environment (shown below). This activity involves setting one colour tablet in front of the child and asking him to find something in the environment that is the same colour and bring it to the table.  There are usually some very interesting choices of items!

To see if the child knows the names of all 11 colours, I ask the child to hand me the colours that he/she knows and tell me the name as he/she does so. If there are a few yet to learn, we do a 3-period lesson to help the child to learn the names of each colour.

Another way to work with these is labeling. Checking that the child knows the name of each colour, we set up a row of colours in front of the child. Often, I will select only one colour starting with each sound to make this work easier. This time, however, the child was very interested in using all of the colours, so we went ahead and used the entire set. Using the writing box (a narrow box with strips of paper, a pencil, and scissors), I write the name of one colour, cut the label, and place the label in front of the child. This work involves the child sounding out the name and then placing the label it in front of the colour to which it corresponds. This particular time, she was unsure about several labels that started with the same sound (green/grey, blue/black, and pink/purple), so I suggested that she set them aside for now. When all the others were done, she took these out and carefully sounded them out to match them to the appropriate colour. She was happy to bring these labels home to show to her family.

The point of sharing these lessons is to show how the same material can be used to match exactly where each child is in terms of language, working memory, and reading. In a number of cases, one child will watch the other working with a material and then ask to do the same thing, so my challenge is to refine the lesson to meet each child’s interest and ability.

Teri Courchene

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Show, but don’t tell

A twist on the “show and tell” idea, Montessori favours “show, but don’t tell.” For children under 6, her method involves demonstrating an activity, but not giving verbal instructions. Discussion is certainly part of teaching, but not direct instruction. Why does this work for young children?

The answer lies with the mirror neurons. When a child is receiving a lesson, not only must the child inhibit movement (a topic for another blog), the child must also observe the teacher’s actions with the object of enacting them soon afterward. The activity of watching is not passive in the brain, as mirror neurons in the premotor cortex are at hard at work. Brain scanning has shown that when one observes another person performing a movement or series of movements, the observer’s brain goes through many of the motions that it would if the observer were performing the action. The brain is effectively preparing itself to replicate the movements. In research with monkeys, it was found that viewing of a goal-oriented action was required to get the mirror neurons to fire, not the viewing of the objects themselves. This neuron activity may help the monkeys to understand the goal of the observed action (Blakemore & Frith, 2005).

Does this mean that the teacher refrains from talking while presenting? Not necessarily. The focus is on the work, and the role of the Montessori teacher is to entice the child into the work…sometimes by talking, sometimes by altering the lesson to make it more suited to the child. Therein lies the challenge and the art of Montessori.

Teri Courchene

Blakemore, S., & Frith, U. (2005). The learning brain: Lessons for education. Oxford, UK: Blackwell.

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Breaking the rules

The loudest, and in my view the most justified, criticism of Montessori programs is that they are rigid – the teacher shows “the right way” to do the work – and the children must conform. Thankfully, most Montessori schools don’t run this way (certainly not ours!), but there is definitely some rigidity in the program.

There is a fundamental conflict when a Montessori child becomes a Montessori teacher. A true Montessori child learns to question, think for himself or herself, and importantly, look inward instead of to others when making decisions. If such a child becomes a Montessori teacher (as I did), then he or she will be told what to teach, how to structure lessons, and instructed to adhere closely to a method that has been tried and true for more than 100 years. You can double the conflict by making a Montessori teacher out of a Montessori child who is also an economist (me again)! Evaluating choices and trade-offs/opportunity cost is at the core of economic thinking – not following set rules.

What is the solution? Understanding and believing in the Montessori approach means that one can stay with the essence of the lessons and yet break many of the so-called “rules.” A Montessori child takes particular pleasure in breaking rules for a good reason – trust me on that one.

Above is a lesson with the constructive triangles that I showed this morning. At the end of the lesson, the child wanted to structure a lesson for me: I had to recreate the design that he created (shown below). Not in the rule book, but definitely an important lesson nonetheless. The child did reassure me that I had done a good job even though I had difficulty recreating his design. Kindness is a rule that we adhere to happily.

Teri Courchene

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How do children learn to concentrate? To answer this, think of when you have seen a young child completely absorbed in an activity. It might be pouring water back and forth from one container to another or building a sandcastle.  These activities are purposeful and require all of the child’s attention. Providing the child with the freedom to engage in and stay with such activities supports the child’s development of the ability to concentrate.

In a Montessori classroom, there are dozens of purposeful activities that call to the child in such a way. For the youngest children, the button frame or pouring rice may be the activities that hold the attention the longest. As the child grows and becomes more capable of complex activities, it might be polishing, sewing on a button or care of plants.

What is the purpose of these activities? It is not the end result – having a polished dish or a perfectly sewn-on button. Through undertaking these activities, the child develops control and coordination of movement. Working through the many steps of polishing or sewing on a button calls on the child to determine which is the next logical step in the process. The child is the one who decides when the process is complete and returns the activity to the shelf.

In order to entice this wonderful focus in the child, the classroom is filled with beautiful, well-organized materials. The activities on the shelf should call to the child, the materials should be finely crafted and attractively arranged, and the activities should have an end result that the child can work towards (a button attached to a fabric square, or a shiny spoon, strips of cut paper).

Apart from the materials themselves, the other unique aspect of the Montessori approach compared to the pre-school or kindergarten approach is that Montessori allows for endless time – the child is free to repeat as much as he or she wishes. Sometimes I hear, “I want to do it again!” or look up to see that a child has cut 20 serpentine strips in a row.  When I see this, I smile to myself and look forward to all of the interesting work that lies ahead of us.

Teri Courchene

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True or False?

Children have a wonderful sense of humour, especially at the ages of 4 and 5. Since they are so tuned into the language around them (they are picking up grammar, vocabulary, and complex sentence structure at an impressive pace), they find playing with language hilarious.

Early readers love to see their name in print. Sorry, dads, but the children’s next favourite word is often “Mom.” Even before children are reading many words, they learn to recognize the most important ones – the names of the people in their family.

One of the new games I have developed for the students is a True or False Sentence Game. It is both a reading and writing game and builds on the idea that children want to play games with reading and writing before they are fluent readers. Since there is considerable repetition, the children learn to recognize the words we are working with even though the children are (usually) at the early phonetic stage (sounding out individual letters in words).  We make a small pouch for the word labels and the children are excited to take the games home and play with their parents.

I was told these were true:

“Cooper has a toy”

“Carter has a toy”

“Dad has a toy”

But this one was false:

“Mom has a toy”

After I ask the child lots of questions about true and false and move lots of words around, I mix up all the words and ask the child to make some sentences.

This was Cooper’s first true sentence:

“Cooper has a Mom”

In this way, the students can create their own sentences before they can use a pencil and paper to write all the words from memory.  The joy the children get from this game is contagious…I am still smiling about the fun we had yesterday.

Teri Courchene

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The Dice Race

This game has been used with great results at the University of Toronto’s Dr. Eric Jackman Institute of Child Study Laboratory School. It was introduced to me by Lab School Teachers – Julie Comay and Carol Stephenson in my OISE math course in the summer of 2011*. It has been modified a little to work in a Montessori classroom and it is now a standard part of the classroom, ready to take from the shelf when the children wish to use it.

(*The Early Learning in Mathematics course was taught by Dr. Joan Moss and Bev Caswell with support from the Robertson Program for Inquiry-Based Teaching in Mathematics.)

What is the game?

On a piece of grid paper 6 squares wide by 10 squares tall, the numbers 1 – 6 are written below each of the squares at the bottom of the page. When a child rolls the die into the basket, the number rolled is entered in the first square above that number. The first number to be rolled 10 times wins the game.

What can children get from playing the game?

  1. Subitizing – quickly recognizing number, i.e., that 4 dots on the die make the number 4.
  2. Practice writing numbers – when a number is rolled, the child writes that number in the square.
  3. Probability – when the child sees that one number has been rolled more than the others, the child understands that the number in the lead is likely to win. The child tends to be very surprised if the same number is rolled several times in a row.
  4. Graphing – the child sees how to represent which number is winning. The child essentially learns to read a graph by doing this.
  5. Addition and subtraction – the child sees how many squares are left to fill. Sometimes a child will laugh when he/she finds that “4 needs 4 more!” or “3 needs 3 more”.
  6. It is fun! Children start to cheer for certain numbers and often want to play again to see if a different number or their favourite number will the following time.
  7. No one loses! In contrast to many games, it is the numbers and not the children who win and lose.

This is a great addition to a Montessori classroom, since it is a hands-on way to enjoy a broad exposure to mathematical concepts. Children go beyond number sense into another strand of the mathematics curriculum – data management and probability – areas not generally covered within the Montessori mathematics curriculum for this age.

Teri Courchene

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