Nienhuis USA_Geometry 2024

Unlocking the Child’s Mind: Montessori’s Vision of Geometry

Maria Montessori’s groundbreaking insights, as presented in “Psychogeometry,” Chapter: “Introduction to the Elementary Period” (p. 55), challenge traditional geometry education. Her approach generally starts with an inspiring story and prioritizes curiosity and self-discovery. Montessori’s method departs from conventional systematic geometry, acting as a mental gymnasium. Advanced geometry materials engage young minds, enabling them to unearth relationships and kindle authentic interest. Instead of clear demonstrations, Montessori introduces tangible, interconnected geometric shapes, bridging the gap between innate cognition and abstract concepts. This approach creates an environment conducive to spontaneous mental exercise, fostering independent thinking aligned with each child’s abilities. Montessori’s method prepares young minds for active learning, cultivating curiosity and profound geometry comprehension. It shapes adept thinkers and future science innovators.

How Geometry Got Its Name A story by Maria Montessori

This story begins many, many years ago in the land of Egypt (help the children to locate Egypt on the globe.) This land was and still is a vast dry sandy desert. The only place where life was possible was by the banks of a very great river. Without this river, it would have been impossible to live in Egypt. The river was so important for the people who lived there that they simply called it “The River” (Can you hear the capital letters?). The waters of the Great River Nile – this is its name – run from south to north. The river has its source in some mountains quite far away. Every year, at springtime, it rains and rains in these far away mountains. All the water from the rains cannot be absorbed by the land and much of it flows into the river, which rises high within its banks. By the time the waters reach Egypt, the river overflows its banks and floods nearby fields. You might think that these floods were a disaster for the Egyptians. No! This flooding was a real blessing for the people in Egypt in ancient times. When the river water came over the banks, it carried debris and mud down from the distant mountains. The waters spread the rich black mud, this humus, over the lands it flooded. This was a natural way to fertilize the farmlands. The mud from the Nile River made the land very fertile. This is why the river was called “The Gift of Egypt”. But the flooding caused some problems as well. Every year the boundary lines that separated one farmer’s fields from his neighbor’s fields were washed away. When the water withdrew and it was time to cultivate the land, the farmers argued about which land was whose. So every year a certain group of people had to solve this problem all over again. These people were called the “harpenodapta”. It is a Greek word and it means “the rope-stretchers.” This is what they did: they stretched ropes according to a pattern. The ropes they used were knotted at regular intervals, just like this rope (showing the rope with the knots). The harpenodapta had slaves help them stretch the rope out in a triangular shape. Who will be the slaves? We need three slaves to stretch out the rope. (Have three children make a triangle with the rope; hold the larger knots. The triangle is placed on the floor and held in position by the weights). Look at what you have made with the strings! Yes, it is a righthanded scalene triangle. Now the ancient Egyptians did not know, or did not seem to know that the shape they made was a right-angled scalene triangle. Maybe it had a different name in their language. However, the harpenodapta could make a rectangle by reversing the original triangular shape after the vertices of the triangle were marked on the land. To be a rope stretcher in Egypt was a very honorable profession. This is how geometry began: by measuring the earth. Indeed geometry means in Greek: geo=earth and metron=measurement. Well, geometry can be said to have its beginnings with this experience of measuring the land after the yearly flooding of the Nile River. Geometry grew from the discoveries the people made while working. Whenever a problem arose, they had to find a practical solution. So, when people needed things, for example how to build houses and temples, the palaces of their kings and pyramids, I am sure they used the same critical thinking. We all know that there is more to geometry than a right-angled scalene triangle (pointing to the rope triangle). But this is another story for another day. Rope stretching was a very important profession in ancient times and these skilled people were held in high regard. Measuring the earth thus started with the rope stretchers. The Greek word for measuring earth is Geometry. Today we are going to try to measure earth like the Egyptians used to do.

The Greek word for measuring earth is Geometry

Three ways to use the story in your class

2. General lesson

3. Activity

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Material: • A piece of rope (2 m or more) • 12 beads

We need 4 children and 4 pawns. The children are holding the rope on the spot of the marked knots. In for example a rope of 12 meters, after every meter there is a knot, but knot 1, 4, 9 and 12 are marked by another color or size. It can be a tied rope but it is better to use an untied rope. Three children form a triangle first (rectangle, not equilateral triangle).

The pice of rope needs to be divided by knots or beads of 12 equal distances.

Possibility A The piece of rope is divided by knots.

Child 1 holds the beginning, child 2 holds knot 4, child 3 holds know 9 and child 4 holds knot 12. After knot 12 a piece of rope is left which should be connected to knot 1.

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Possibilty B The distances are being marked by beads: • Start with a knot; • Add a bead; • Make an knot directly after the bead so that is stays on its place; • Choose a distance of 10 cm or more; • Make another knot;

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After this the knots 1, 4 and 9 will be marked by pawns. Then child 4 takes the rope at the point of knot 12 and mirrors the entire shape by walking to the opposite edge. If the rope is tied, the child is supposed to let go the rope but the pawn stays. If it is an untied, child 4 puts knot 12 exactly above knot 9 and connects the last part to knot 4. After this a pawn has to be placed on the right top corner at knot 12.

1. Doing research by asking questions

Possibilities are: • Can you think of a way how the Egyptians can use a rope like this to measure the land with 4 people? • Do you have an idea what the red knots would be meant for? Can you find a solution for this? • What shape did you make? • Are you able to make another shape?

• Another bead; • Another knot; • And measure a similar distance again.

A perfect rectangle has been created, marked with pawns or children. When you do this exercise outside you could cover a larger surface.

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Learning is in the details

Download poster here

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