The montessori cosmos is a challenge to question your assumptions about school. About children. about learning.

Is it Montessori? Children’s Abacus Toy

Is it Montessori? Children’s Abacus Toy

In this occasional series, we will look at various materials and methods that are often advertised as Montessori, to see how well they embody Montessori principles.

For this first edition of “Is It Montessori?” I’m going to indulge in a bit of grumbling about a personal pet-peeve of mine, namely those brightly colored abacus “toys” that are often advertised as Montessori math materials. I have even seen photos of them in marketing materials for schools that purport to be Montessori or Montessori-based.

Here’s an example: Children’s Abacus Toy

Now there is a Montessori material that looks similar to this: the Large Bead Frame. It looks like this: Large Bead Frame

So let’s compare these two materials. To the untrained eye, they look quite similar, and it’s not necessarily obvious why one should be considered a “Montessori material” and the other shouldn’t, beyond the fact that one was designed by Maria Montessori for use in classrooms. If that were the only reason, it would be a pretty silly distinction.

First, let’s look at the similarities. Both are wooden frames with horizontal wires or dowels that each contain ten beads. Both can be used for adding, subtracting, and with some cleverness, multiplying whole numbers up to the millions or so, and so both could be considered mathematical tools for calculation. But that is where the similarities end.

Montessori math materials are far more than simply calculating tools (in fact, some of them can’t be used for calculation at all!). They are “materialized abstractions”, that is, they are meant to embody the mathematical concepts under consideration in a clear physical form. The abacus toy does do this to an extent: the beads can represent decimal categories (units, tens, hundreds…) in the base-ten system, hence the ten beads on each wire, but the base-ten system isn’t explicit.

Now consider the Large Bead Frame. On the left-hand side, the decimal categories are explicitly written out in numerals: 1, 10, 100, 1,000, etc, making it clear to the children what quantities each wire represents. Moreover, the left-hand uses three shades of gray, which group together the different families of decimal categories: the simple family (units, tens, hundreds), the thousands family (units of thousands, tens of thousands, hundreds of thousands), and the millions family (units of millions, and we could continue with tens of millions and hundreds of millions).

Furthermore, the coloring of the beads is not random, and is not there merely to be inviting. The colors repeat in groups of three in order to emphasize the repetition of the “units, tens, hundreds” pattern within each decimal family.

So far, not only have we materialized the concept that every decimal category holds ten elements, but we have also materialized the complex pattern of the decimal categories themselves, something that children can have quite a bit of trouble grasping.

In addition, the Large Bead Frame comes with special paper, sensibly named Large Bead Frame Paper, which repeats the color and family pattern, and even contains special lines meant to represent the wires that hold the beads, though the entire setup is turned 90 degrees so the wires go vertically. This allows the children to begin making the transition from the concrete representation of numbers through materials to the abstract representation on the paper. The paper helps children write down the quantities on the wires correctly. It’s also a useful tool for analyzing difficult problems into their component categories, which is essential for multiplying numbers on the Large Bead Frame.

Large Bead Frame Paper

Large Bead Frame Paper

But that just scratches the surface. The decimal category pattern green=units, blue=tens, red=hundreds is repeated throughout the Montessori materials. By the time children begin using the Large Bead Frame, they have already worked with the decimal system in a much more concrete form (the Golden Beads, in which units, tens, hundreds, and thousands are represented by materials that actually contain 1, 10, 100, or 1000 beads respectively), and with the green, blue, red color scheme (the stamp game). In fact, the Large Bead Frame isn’t even the first bead frame the children work with. There is also a Small Bead Frame, which only includes categories up to 1,000.

While the children are learning to use the Large Bead Frame, they are also working with the Wooden Hierarchical Material (or the Wooden Hierarchy, for those of us who are lazy), which physically demonstrates the relative magnitude of the categories from units to millions, beginning with a unit cube 1/2 cm on each side and ending with a 1/2 meter million cube. This material is also colored using the green-blue-red scheme.

This color scheme continues on through material for division (the Racks and Tubes), for studying multiples and factors, for studying decimals (where the fractional categories are represented with pastels), for studying square roots, and for studying binomials and trinomials, which leads naturally into algebra.

As you see, the Large Bead Frame is far more than “just” an abacus. By itself, it is carefully designed to clarify the process of learning operations in the decimal system in as many ways as possible. In the context of a Montessori classroom, it is part of an intricate system of materials that lead the children from performing operations in the most concrete way possible, to performing them abstractly, but with complete understanding.

Verdict: Abacus toys are NOT Montessori materials, though in the hands of a talented teacher, they could certainly be used to teach numeration and operations, but the Large Bead Frame is designed to clarify the concepts far more readily.

Binomials Here, Binomials There, Binomial Theorems Everywhere, Part 1

Binomials Here, Binomials There, Binomial Theorems Everywhere, Part 1

The Joy of Responsibility

The Joy of Responsibility