# Fraction Circles

Here is Montessori's description of the Fraction Circles:

"There are ten metal plates, each of which has a circular opening ten centimeters in diameter.  One inset is a complete circle; the other circular insets are divided respectively into 2, 3, 4, 5, 6, 7, 8, 9, and 10 equal parts" (Montessori, 1916).

Below is an illustration of the first four fraction circles from Shu-Chen Jenny Yen's web site:

Shu-Chen Jenny Yen presents a lesson that encourages the child to compare the different fractions.

Older children are given instruments to measure the fractions.  These include a 360-degree protractor and a centesimal circle that is used to reduce fractions to decimals.  Students are also shown how to use the fraction circles to perform operations involving fractions.

### Design Implications

In developing an understanding of geometric objects, we proceeded from three-dimensional objects to two-dimensional objects.  Within Montessori's web of activity there is also a sequence of three- two- and one-dimensional objects (exercises with pink cubes, brown prisms, and red rods) to draw the child's attention from three dimensions (size), to two dimensions (thickness), to one dimension (length).  In general, we proceed from consideration of the whole to consideration of components.

After the child learns to recognize and name an object, we introduce tools for measuring the object.

The sequence of activities above suggests a linear path from one activity to the next.  In general, lessons have prerequisites, but there are a number of alternative paths through the web of activities.  The teacher observes the child and introduces new activities according to the child's interests and performance level.

In general, puzzle activities precede game or design activities.  However, this is not due to some primacy of puzzle activities.  Rather, it is due to the fact that Montessori found puzzle activities more suited (for the most part) to children from three to six years of age.

In addition to the methods above, we can further reduce the cost of mistakes by

• Giving feedback that is frequent enough so learners don't invest too much in a fruitless path
• Working in small and frequent iterations
• Make it easy to reverse missteps

For the six to twelve-year-old child, we can increase the rewards of participation by relating activity to the outside world, or bringing some part of the outside world into the classroom.