Gymnasium for the Mind: Geometry in the Montessori Curriculum

By Sarah Fary, Upper Elementary Teacher

Montessori is a curriculum designed to grow with the child.  Often, seeds planted in the primary take many years to reach the fullness of rational understanding in upper elementary.  It was inspiring to witness the interest shown recently by attendees at our parent education night focusing on the Montessori mathematics curriculum, and how it meets the child’s needs at each developmental stage during their time at CFMS.  I would like to take a similar look at the area of geometry, to illustrate how each thread of the Montessori curriculum is masterfully designed to engage the child’s interest and abilities at the appropriate stage of development and to show how the curriculum builds on previously presented concepts to encourage a rich and complex understanding by the time students reach the upper elementary classroom.

Brief overview of geometry in primary

In the primary, geometry work is focused on sensorial, concrete experience and manipulation of geometric figures and forms, and on acquiring the appropriate associated vocabulary.  The exercises in geometric design with the metal insets allow the child to explore relationships between figures while simultaneously strengthening the hand in preparation for writing.  Children also have their first geometric experience of number with the golden bead material:  the unit is a point, the tens bar is a line, the hundred square is a surface or array, and the thousand cube is a solid.  The exercises with the constructive triangle material give the child hands-on experience with the fact that all polygons are composed of triangles.

How the elementary geometry curriculum serves the elementary child

In the elementary the focus shifts, as always, to serve the changing needs of the child in the second plane.  The elementary child is attracted by the story of human progress, and thus is continually reminded of the historical context of geometry as an area of human knowledge and exploration - one of the oldest known.  We explore the roots of the word - “geo” meaning earth, and “metria” meaning measure.  Geometry began as a very practical science focused on measurement of the earth and objects.  In geometry, as in all areas of the montessori elementary classroom, introductory stories told early on in the child’s experience contain the seeds of further lessons, explorations, and concepts not only in geometry but through connections to the entire curriculum.

A few examples are:

·         How geometry got its name - Ancient Egypt, measurement, 3-4-5 right triangles

·         Story of the Babylonians - measurement of degrees of angles, astronomy

·         Euclid of Alexandria - geometric construction and proofs

·         Pythagoras of Croton - the Pythagorean theorem

The study of geometry in elementary also focuses on a more in-depth study of the parts of figures and forms, giving precise nomenclature to aid the child in further exploration of the relationship between these figures later on.  

Congruence, similarity, equivalence

 

One of the most appealing and ingenious Montessori materials is the series of metal inset plates.  These are designed to give the child experience of congruent figures and how their parts relate to each other.  At first the child works with these materials to demonstrate the congruence of a figure to a rectangle in a purely concrete way - then later on these same plates are used to derive the formulae for finding the area of these figures based on the formula for the area of the rectangle.

Maria Montessori described these advanced materials as a “gymnasium for the mind”,  allowing the child to not only make observations, but to make discoveries.  Contrast this with traditional geometry education, where I was taught through memorization and was given no real concrete experience of how formulas for area and volume were derived.

From the early presentations on equivalence and area to the later work with the Pythagorean plates, the Montessori geometry work allows the child to be an active agent in her own discovery of the truths of geometry, rather than a passive observer.  This ownership of one's learning as an engaging, ongoing process of discovery is one of the most precious gifts a child is given in the Montessori classroom, and prepares them to be lifelong, joyous learners.