### Geometry One of the most enjoyable pursuits in a Montessori classroom would be geometry. Beginning atage three, students can explore geometric shapes sensorially, gaining both visual and muscular knowledge of the various shapes. This leads to awareness of similar shapes in the environment.

The first deliberate introduction comes with the Geometric Cabinet, a six-drawer compartment that classifies circles, rectangles, triangles, polygons, and more. The drawers serve as puzzles (with knobs, of course!), but they also offer
material that the teacher uses to isolate vocabulary concepts: “This is a triangle. This is a circle.” The child moves from concrete to abstract with accompanying cards that show just the outline of each particular shape. This is indirect preparation for reading, which is to say the recognition that an outlined shape can stand for something else. It is also indirect preparation for writing, because every line and curve in the cabinet may be found in a letter of the alphabet. In a couple of years the children will be using the same drawers to learn to identify and classify types of shapes, such as a right-angled scalene triangle, an acute-angled scalene triangle, an obtuse-angled scalene triangle, a right-angled isosceles triangle, an acute-angled isosceles triangle, an obtuse-angled isosceles triangle, and an equilateral triangle! Do you know the difference? Our kindergartners would! There are also baskets of solid blue polyhedra, prisms, pyramids, and curved surface solids. These provide additional opportunities for expanding vocabulary. Students explore with these, identifying various plane surfaces, counting edges, vertices and more. Second-graders might be discoveringEuler’s formula (V-E+F=2)for them.selves. The Constructive Triangle boxes soon follow. With these, the children explore to discover proportions, dimensions, and math patterns to be found in various combinations of the triangles. Older children use the same material to learn about similar and congruent shapes. Upper elementary students combine three of the sets of constructive triangles to prove the Pythagorean Theorem. The Geometry Sticks lesson offers opportunities to explore angles that are acute, right, or obtuse, as well as how to use a protractor to measure angles. For adults, the most “mysterious” of lessons are the binomial cube and the trinomial cube. For preschoolers, these are three-dimensional puzzles of matching colors and congruent surfaces. They are unaware that they are actually absorbing the algebraic formula for cubing a binomial!

Here we see Ember constructing the cube of the binomial as a puzzle: Upper elementary students are able to use the trinomial cube puzzle to concretely represent the cube of a trinomial, as Ruby is doing here: a3+3a2b+3a2c +3ab2+6abc +3ac2+b3 +3b2c +3bc2 +c3