New Squeak Objects Based on Montessori Design Patterns to Help Children Explore Angles

Maria Montessori used iterative design experiments to make environments, materials, and activities that help children to create the adults they will become. As a starting point for the development of some of her materials, she took existing artifacts used for learning or assessment, made them available to children, observed children's work with the artifacts, and refined them accordingly. Using this approach, together with interaction patterns I have begun to capture in Montessori’s work, I developed software that I used successfully with elementary students at the Center for Talent Development at Northwestern University and in private tutoring sessions.

I started with learning materials used in Montessori classrooms to present the angle concept. From learning sciences literature and my own classroom observations, I have found two important difficulties that children have in constructing this concept. First, for most representations of angles, children have trouble distinguishing angle measurement from measurements of length or area. For example, if an angle is represented by two rays emanating from a single point, it was found that third grade children confuse measurement of angles with measurement of the length of the representations of the rays or the area between them.

Second, children have trouble relating various representations of angles, as illustrated, for example, by the hands of a clock, the turning of a doorknob, the rotation of the moon around the earth, a slice of pizza, or the internal angles of a polygon. There is an added difficulty in relating these angles to implied angles, such as the slope of a hill or the external angles of a polygon.

Starting from existing Montessori materials and using Montessori design patterns, I wrote a Java application called Circular Reasoning, which allows children to dynamically change representations of the same angle and directly (and separately) manipulate the angle, its orientation, and its size. Children can also manipulate whole fraction circles to create a camera shutter effect that relates the 360 degree circle to the external angles of a regular polygon. Children have reported that they enjoy working with the software, they persist in using it, and they demonstrated tremendous gains in performance on assessments of understanding of angles.

Circular Reasoning activities can be used in Montessori classrooms as extensions and variations of existing geometry materials to specifically address the problems mentioned above. They can also be used outside of Montessori environments as a self-contained set of activities.

I am now in the process of porting the software from Java to Squeak, and plan to demo the Squeak version. This promises a tremendous increase in flexibility and extensibility of my original virtual manipulatives. With Morphic representations of angles, children and teachers and others can use the eToys interface or “under the hood” programming to work in various ways with different representations of angles, and better relate angles to turns.