Design Problem

Montessori was interested in all aspects of child development--physical, emotional, intellectual, social, and spiritual.  Because of my own background and interest in mathematics, I decided to find out about children's needs in mathematics.  I was drawn to explorations in geometry in particular for two reasons:

1. Computer screens offer the opportunity for interaction with geometric objects in a unique way.
2. Montessori developed a number of materials for the child that support explorations in geometry.

From Montessori's point of view, neither of these reasons would be sufficient without evidence that kids need to understand geometry.  For the time being, I'm assuming that geometry is a worthwhile thing to teach and have decided to work on support materials for understanding angles, since this is an area where kids have difficulty.

In my preliminary work, I have relied on observations of others as well as my own, and a short design experiment using Circular Reasoning, software I built based on Montessori and educational software literature and my own field work.  I am now developing a plan to use more extensive ethnography and experiments.  

Why Geometry?

Most of the arguments I see for teaching kids geometry are something like this one from the Improving Measurement and Geometry in Elementary Schools (IMAGES) Team (IMAGES, 2003):

Results from assessments such as TIMSS, NAEP, and PSSA, as well as anecdotal data from team members and the Pennsylvania Intermediate Unit curriculum coordinators, identified geometry and measurement as areas of student performance that continue to need improvement.

In other words, kids aren't performing up to some standard, or American kids are performing at a lower level than kids in other countries or other parts of the US.  This argument isn't terribly compelling from a Montessori point of view, since it is based on imposed standards rather than the needs of the child.

A somewhat better argument is presented by Kynigos (1992, p. 97):

Freudenthal has put forward the case for reconsidering the pedagogical importance of geometry, arguing that it has a characteristic not so common in education, namely, that it is a field within which both inductive and deductive learning can take place (Freudenthal 1973, p. 407)... Furthermore, it has been argued that understanding deduction is at least related to (if not dependent on) experience with inductive thinking (von Glasersfeld 1985, p. 484).

Montessori thought enough of geometry to devote one of seven parts to it in The Advanced Montessori Method (Montessori, 1965, Part IV), a book on her method applied to children aged seven to eleven.  The only other section of the book that deals with mathematics is Part III which deals with arithmetic (Ibid, Part III).  I don't know her reasons for giving such a prominent position to geometry.  I hope to find some argument of hers as I continue my literature review.

Angles

A lot of the trouble that kids have with angles has to do with the representations they have to work with.

Angles are not easy to see or present clearly on paper.  Kids can point to the legs of an angle or its vertex, but the thing that we measure when we measure angles is not easy to point to.  It has something to do with the space between the legs of the angle, but it is different from the area between the legs or length of the drawn legs.

Typically, illustrators try to highlight the angle in a diagram by drawing an arc from one leg to the other with an arrow indicating direction.  But this is not much of an aid to those who haven't yet assimilated the angle concept.  It helps a little, because we can think of the angle as a rotation from one leg to another.  But a rotation is a dynamic motion that can only be suggested on a static piece of paper. 

We might think of the angle as having something to do with the length of the arc that is drawn between the legs in the diagram, but this does not give a completely accurate picture.  We can only use arc lengths as angle measures if the arc drawn on each angle keeps the same distance from the vertex of each angle, and there is nothing in the diagram of an angle (on the perceptual level) that makes this obvious.

Another problem with static diagrams is that they often limit the child's experience of figure to a particular orientation.  For example, Fuys and others have noted that a common misconception among beginning students is that "an angle must have one horizontal ray" (Fuys, et al, 1988, p. 137).

With Logo, we can show rotations dynamically, but the rotations of the turtle don't immediately bring to mind the diagram of an angle.  According to Clements & Meredith (1992):

Logo helps students think about angles. These benefits, however, might not emerge until they have had more than a year of Logo experience. Also, teachers need to help students understand the relationship between 'turtle turn' and 'angle measure.

Manipulatives are also used to help children develop the angle concept.  There are difficulties here as well.  Prescott, Mitchelmore, and White wrote a paper called "Student Difficulties in Abstracting Angle Concepts From Physical Activities with Concrete Materials".  In the study it describes, "it was found that [the difficulties encountered by students exploring angles with concrete materials] could be classified into four categories: matching, measuring, drawing, and describing (2002, p. 586)."

I don't know of specific studies on the use of Montessori materials, so I'll have to find some or collect my own data on this.  My sense at this point is that current materials are limited.  The fraction circles don't help kids distinguish between angle, length of angle legs, and area between legs.  The geometry cards are static.  The geometry sticks permit dynamic illustration of angles, but don't relate readily to the angles in the fraction circles.  I want to address these problems with my design.  I also want my designs to fit in with these materials as extensions and variations of the existing Montessori geometry activities.

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