Elementary Math Night

As requested, here are the notes from the math lecture given last Thursday evening in the elementary room.

Of all that there is to marvel at in the children’s house that Maria Montessori built, it is the math curriculum that so often leaves people gawping in astonishment for her sheer brilliance, her complete understanding of large mathematical concepts and more than anything, in her ability to materialize highly abstract ideas and literally place them in the hands of young children. The more time you spend around the materials, the more you begin to see the connections she has made between them and the power of the notion that it must pass through your hands many, many times on many different days in order for you to achieve real awareness of it.

A child encounters (and absorbs) a sophisticated and myriad array of mathematical preparations during their 3-6 years and they emerge into elementary literally with a “body knowledge” of math. This is acquired over long stretches of time and exposure to the whole casa environment. Very early concepts such as perceiving similarity (by, for example, matching pairs of fabrics) or difference (by serializing units in sensorial materials like the cylinder blocks) with explicit, step-wise instruction indicative of the scientific method Montessori followed, and in the modern context, one of the things Montessori is most often criticized for.

Today we’d like to take a long view from one of the earliest materials children are shown in the casa: the decanomial square.  This is a sensorial presentation given in casa that is prepared for by two of the first presentations a child is given at age two and a half: the pink tower and the brown stair . These serialize, by unit of difference, the three dimensional expression of the decimal system in cubes and rectangular prisms. A child of two and half is shown this material in a sensorial presentation and they are intrinsically drawn to it, often spending great swathes of time with it. The decanomial comes a bit later and flattens the concept into two dimensions while introducing the association between number and colour which will connect later to bead work.

Now we fast forward to a child in their sixth year who arrives in the elementary classroom. If they are well prepared, they arrive literally and figuratively, at a place… ready to manipulate, explore and discover practical applications for mathematics. Geometry and mathematics are given equal weight in the elementary curriculum and this is because of the intense need for children have to hold, manipulate, plan, collaborate and problem solve.

When we thought, however, about which material to show the connection over years of a child’s understanding of math, how they use the work during different planes of their development, one that ties everything together from the foundation of sensorial input in the pink tower to great thinkers like Pythagoras, we decided to walk you through the work of the decanomial square, now called the Table of Pythagoras. It is a beautiful manifestation of the power of Montessori’s math. When they see it again in Elementary, they use it to re-orientate themselves in their understanding of decimal system and multiplication facts before charting a course into new understanding of the commutative law, squaring, binomials, algebraic determination. Eventually, the ground is laid for the cubing material that follows in upper elementary- a simply stunning point of arrival. Interestingly, this material is visually resolved to the pink tower they began with six or seven years prior. As we discovered for ourselves this week, this was not by accident.

It is significant that every step of the way, the material plucks the strings laid deep in the child’s absorbent mind several years ago. And then the great reveal: literally at its centre, the decanomial square contains a magnificent secret: the sum of the squared units of the table are identical to the sum of the cubed material they are prepared for.  Our students spent days disproving various hypotheses (including a² + b² = c² based on the right angle triangle in the square being equal to the hypotenuse of the cubed material running the diagonal followed by a hypothesis which doubled this result) before arriving at the correct arrangement. From here, we will go on to find other mathematical patterns, tie their discovery to the story of numbers, a timeline of Pythagoras’ life and how it fits into the timeline of civilizations. From this seed, we have a cosmic plan to follow in multiple directions through multiple areas of curriculum.