Geometry 9-12

Geometry 9-12

“The underlying structure and order of the universe must be reflected in the classroom if the child is to internalize it, and thus build his own mental order and intelligence.”[1]

Geometry is basic to both man-made and natural objects, therefore it is easily a natural part of any classroom. To learn and explore 90°angles, for example, opens up avenues of thought previously unknown to the students. They begin to see very familiar objects in a new way and are unstoppable in their efforts to find out even more about those things that they have only recently seen with eyes towards exploration and investigation.

Starting from the very first day, the students learn to classify the shapes of the metal insets they have been working with since pre-primary school. The shapes they know so well, such as the square, triangle, and circle, are approached in a slightly new way at the elementary level by getting more in depth into the types of these objects. The first row in the geometry cabinet are circles of many different sizes, providing the students an opportunity to see and feel the varieties in the known. The next cabinet is made up of many different quadrilaterals—one square and many different sizes of rectangles. This gives the students an opportunity to see and feel why the rectangles are not also called squares—the square is the only one that can fit in the slot in any direction. The succeeding rows also delve into those shapes that the students have been seeing for years, such as the quatrefoil, various triangles, ellipse, oval, and the various polygons (i.e., pentagon, octagon, decagon, etc.).

The next level is to once again observe a shape very familiar to the students and explore it with more depth. Point, line, plane, and solid are shapes these students not only see in their everyday life, they also work with these shapes with their golden bead math work. Now, however, the students are taken on an imaginary journey where objects can be defined and interpreted, but not seen in our three-dimensional world. Next comes the line now defined in multiple ways as opposed to simply “line.” The students can now truly see and understand the difference between a line segment (with a defined beginning and end) and a line (with no beginning or end).

“…the more the children know the more they will see and then the further they will walk.”[2]

Additionally, the triangle, which is any object with only three sides, is now also defined as the most stable object humans can make. This explanation is brought into the language lessons when explaining the reasons for using the triangle as a symbol for the noun. It is also made extremely clear when the students create triangles and other polygons themselves out of linked sticks and realize that the triangle is the only one that cannot be budged or made into any other object. The students are then taken one step further into architecture and realize that the most stable buildings that humans can make are made with triangular supports. History is another off-shoot of the lessons on the triangle. When the students study right-angle triangles, an introductory lesson, discussing Pythagoras and the way he utilized these triangles in ancient Egypt to repartition land after flooding, helps the students see the more practical uses of geometry. Once again, geometry is the basis for a wide variety of researches branching off of a simple concept.

Along with this exploration, the students are using the fraction material to do addition with common, then later uncommon, denominators. This is a very concrete work as the fraction material is already divided up into sections from a whole to ½ to 1/10 and it is left to the student to simply see how many whole circles can be created and how many are left over. It generally takes very little time for the students to be able to perform these basic addition problems with fractions without using the materials. They are then ready for addition with uncommon denominators. Because they have been simultaneously working with similarities, congruencies, and equivalencies, this transition to uncommon denominators occurs naturally and easily.

At the 9-12 level, the exploration of figures becomes increasingly in-depth by learning about areas of simple plane figures, including rectangles, parallelograms, triangles, trapezoids, rhombi, and other polygons. The students are able to take these figures apart, and by using the formulas that are so well known to them, discover how to find the areas of more complex figures. The next step is exploring the area of circles by learning first the nomenclature and then seeing circles as regular polygons and discovering the area and circumference in that light. Then, the volume of solids is explored by using the concrete materials to discover formulas sensorially, and then abstracting some concepts through reasoning. 

“…education is a natural process spontaneously carried out by the human individual, and is acquired not by listening to words but by experiences upon the environment.”[3]

By the end of a Montessori student’s three years in the lower elementary classroom, and another three in the upper elementary classroom, he/she has been introduced to lines, angles, triangles, parallelograms, and polygons, as well as the more complex exploration of all these figures (area, volume, circumference). The students are now able to classify well-known objects in their environment with the additional ability to place that object in its historical, practical, or natural context. They can see beauty in both man-made and natural objects and can see the similarities between the two. Additionally, they are now able to appreciate their environment in a new way and to possibly feel as if they are not only a contributing part of it, but an essential one as well.


[1] Lillard, Paula Polk. Montessori: A Modern Approach. New York: Schocken Books; 1972: p. 56.
[2] Montessori, Maria. The Absorbent Mind. New York: Henry Holt & Col., 1995: p. 163.
[3] Montessori, Maria. Education for a New World. Wheaton, Ill.: Theosophical Press, 1963: pp. 2-3.

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