# Background

I taught for three years in Montessori classrooms for children aged 3-12.

I spent six years as an educational software developer for public school kids at Northwestern University, where I now teach gifted kids in grades K-12.

In creating the software and activities I will discuss, I drew on ideas from Montessori, constructionism, Dynamic Geometry, and Turtle Geometry.

A specific connection can be seen between the Montessori fraction circles and the fractures and fraction circles in Circular Reasoning.

As with traditional Montessori exercises in geometry, Circular Reasoning activities begin with perceptual level work with whole objects (sectors and fraction circles) and then move to analysis. Analysis is supported by supporting the child's ability to color (and so highlight) various parts of the sectors (radii, arc, area), to select different representations of angles (sectors, segments, two lines joined at a point, arc, orbit of satellite, inclination of a line), and to change spatial attributes (lengths of radii, central angle, and heading) with sliders and text boxes.

## Constructionism and Montessori

What distinguishes constructionists from other constructivists is the idea that this construction of knowledge, which "takes place 'in the head' often happens especially felicitously when it is supported by construction of a more public sort 'in the world'-a sand castle or a cake, a Lego house or a corporation, a computer program, a poem, or a theory of the universe (Papert, 1993a)."

On the other hand, for Montessorians, "children construct their own knowledge through exploration and discovery assisted by the introduction of efficient experiences planned by the teacher (Loeffler, 1992)." Because Montessorians are constructivists, the term "efficient experiences" is not used to refer to efficiency in transferring knowledge or "training" the child in any way. Rather, the term "efficient experiences" as applied to the child may be compared to experiences that would efficiently support scientific discovery.

For example, when conducting an experiment, the scientists generally try to reduce the number of variables as much as possible. A corresponding behavior in the Montessori classroom is the isolation of difficulty, in which Montessorians try to make aspects of the world apparent by providing opportunities to compare, contrast, serialize objects and otherwise manipulate objects that are identical except for the attributes to be studied. It must also be noted that the most efficient route to a scientific discovery is never a straight line. Scientists must spend considerable time repeating experiments, examining results, and "mulling things over" in a process that is largely subconscious. When Montessorians speak of "efficient experiences", they refer to materials and exercises that support these processes in an efficient way. Montessorians do not look for "shortcuts" around these processes which are critical for the construction of knowledge.

Neither the Montessori nor the constructionist community sees "construction in the world" and "efficient experiences" as mutually exclusive paths to the construction of knowledge. Constructionists generally acknowledge that scaffolding provided by the teacher and the environment are important for successful construction and reflection on construction. On the other hand, Montessori children are typically free to explore materials in ways that were not presented by their teacher (Chattin-McNichols, 1992) and Montessorians recognize the importance of supporting open-ended, creative tasks, especially for children older than five.

In devising her approach to geometry, Montessori took note of the fact that children are aware of objects as a whole before they grasp components like sides and vertices or measurements such as length, area, and angle. She then proceeded to develop materials to give them "efficient experiences" with shapes and gradually guide them toward analyses of different geometric objects in preparation for work in formal Euclidean geometry.

Seymour Papert's work on Turtle Geometry is based on the child's awareness of her own body and other mobile bodies such as dogs, cats, and boats. Living things and motorized vehicles can change their positions on a surface by changing their heading (turning right or left some number of degrees) and then moving forward or back by some distance.

## Logo Problems

Reports of children spontaneously performing mathematical analysis with Logo are rare (Hoyles & Noss, 1992). Children tend to work with Logo at the perceptual level, tweaking their programs by trial and error until the desired effect is produced. In a literature review on the effects and efficacy of the Logo learning environment, it was found that "without guidance, misconceptions [in geometry] can persist (Clements & Meredith, 1992)." Further, "some studies show limited transfer to activities outside of Logo (Clements & Meredith, 1992)."

Clements and Meredith further reported:

"Exposure [to Logo] alone is not completely adequate. A more satisfactory approach features teacher mediation and a sound theoretical foundation (e.g., for geometry: Piaget and van Hiele). Mediation implies clarification of the mathematics in Logo work and the extension of the ideas encountered; construction of links between Logo and non-Logo work; and provision of some structure for Logo tasks and explorations. Structure does not imply authoritarianism. For example, it is often useful to allow hesitant students to accept or reject suggestions until they build confidence."

"Construction of links between Logo and other mathematics activities [is] challenging-- research shows that teachers find it extremely difficult to create a learning environment that fosters creativity within [traditional] school and curricular structures." Further, children must typically be exposed to Logo for more than one year before any noticeable effects can be observed (Clements & Meredith, 1992). This is less likely to happen in traditional schools where children spend only one year with a particular teacher and there are seldom school-wide policies about the kind of software to be used or how its use should be integrated into other classroom activities.

As I read about (and experienced) this problem reported by Clements and Meredith, it occurred to me that a Montessori approach could provide the structure that is so often lacking in Logo activities. In the next section, I will describe a set of Montessorian "efficient experiences" that I developed to support children's work in Turtle Geometry. These activities provide "links between Logo and non-Logo work [as well as] provision of. structure for Logo tasks and explorations" (Clements & Meredith, 1992) using a Montessori approach.