On a recent Saturday, I attended a parent event at a Montessori school called “Journey and Discovery.” During the “discovery” portion, I noticed one parent who was drawn irresistibly to the fraction insets in a Children’s House (3-6-year old) classroom. It happens that she teaches fifth grade in a local public school.

Since you are probably not familiar with the fraction insets, I will describe them for you. They consist of 10 green frames, each with a red circular inset 10 cm in diameter. The first inset is a complete circle with a knob for lifting and removing it from the frame. The other insets are divided into 2, 3, 4, 5, 6, 7, 8, 9, or 10 equal parts each.

The fraction work in the Children’s House classroom begins with sensorial exploration. Children can also use the insets in colorful design work, as they do with the geometric shapes of the metal insets. They can even trace the insets onto colored paper and cut them out for use in collage.

It is only after much exploration of the fractions as shapes, that we move on to defining, naming and writing them. “When we break a unit into pieces of the same size we call those fractions. When we divide the whole unit into two parts, we call each part a half. This is the family name. We write the family name ‘half’ as a 2 under a line. The number under the line, that tells us which family we are talking about, is called the denominator.” In this way we proceed, slowly and with much repetition, to teach the names of the fractions, three at a time.

Once the child can name and write all of the fractions through tenths, we move on to the key exercise with fractions in the Children’s House — substitution. We would remove one of the halves from its frame, and experiment with what combination of other fractions would fit the space. We would find that no combinations of thirds will work, but that two fourths exactly fit the same space as a half, as do three sixths, four eighths and five tenths. The child is then free to experiment with the other fraction insets and discover for himself which combinations of fraction pieces are equivalent and which are not.

If the child wishes, he could write and record his discoveries as 1/3 = 2/6 = 3/9, for instance. Again he could also trace the fraction insets onto paper and cut them out, this time making a poster of the equivalencies he has discovered.

It is only after a great deal of this exploration and discovery, when we judge that the child has internalized a basic understanding of the equivalent relationships of fractions based upon extended experience, that we would take him on to other fraction work. Whenever we move a child on to the mathematical operations of addition, subtraction, multiplication and division with fractions, we can be sure that this new understanding will be built upon a firm foundation of experience, exploration, and discovery with his own two hands.

By this point in my explanation, I noticed that my listener was becoming more and more agitated. Finally, she could contain herself no longer. “I need these tools in my classroom,” she said, “so that my students have an opportunity to understand fractions. Right now, I’m just skipping stones!”

I must have given her a puzzled look, for she went on to explain her metaphor. “No matter how carefully I try to explain fractions to my fifth graders, I can see by their blank looks that it just skips off their brains, with no real comprehension or understanding.”

I was awestruck by her sincerity and frustration, and also by the power of her metaphor. But more than anything I was grateful for the tools at my disposal like the fraction insets, to help children pass from experience to understanding to abstraction. There’s no skipping stones in Montessori!

Peter Davidson was the founding Head at the Montessori School of Beaverton, an AMI school in Portland and currently serves as consultant for Montessori in Redlands, an AMI school in Southern California.