Scaling Equivalencies in the Table of Pythagorus

A Math Story

The Table of Pythagoras (Decanomial Square) is a beautiful manifestation of the power of Montessori’s math. It introduces the decanomial: the concept that all numbers are organized by decades (tens). Children are usually given the sensorial presentation of this material around the age of four or five in casa. When it re-emerges in Elementary, it is used to situate and reinforce multiplication facts before charting a course into new understandings of the commutative law, squaring, and binomials. It lays the foundation for algebraic determination and eventually, for the cubing material that follows in upper elementary. Interestingly, this material is visually resolved to the pink tower they are first introduced to sensorially at the age of two and a half. We discovered that none of this is by accident.

Every step of the way, the materials pluck strings laid deep in the child’s absorbent mind years prior. Materials are auto-didactic, not just to correct errors made as they work with it but also to lead them to the next place of discovery. Midway through this year, some of our Upper Elementary children encountered the great reveal within the decanomial square. At its diagonal centre, it contains a magnificent secret: the sum of the units of the bissected square divided as a right angled triangle are identical to the sum of the cubed material they are prepared for. This is a cool math fact to be sure but the days the children spent solving it were breathtaking. They will remember what they “learned” because they experienced and discovered it, not because they memorized it for a test. They worked together to test different theories and approaches. They got a little frustrated and in the end, they came together and found a solution.

The Math Curriculum at the Lyceum

The children are privately tutored in math by an Engineering graduate from UofT. Currently, they are working on studies in Functions and Relations from the grade 10 curriculum which is augmented by real-world problems and case studies. They are a currently working as a group but there the flexibility and resources are here to work to individualized plans tailored to each student’s needs. They continue to study geometry with Holly.