One
of Maria Montessori’s great contributions to education – aside from her Peace
Education model that honors the child’s rights and dignity – is the
construction of coherent, concrete manipulatives that reinforce place value and
base-10 patterning in mathematics. Her math “games” are a gift to model
learning.
In
years 3-6, Montessori students are introduced to various beautiful materials
which offer visual-spatial representations of number. A giant vanity of beads
hangs in chains of various length, as in the parlor of a queen. The chains
drape vertically as bracelets of squares, and numerical necklaces lay in cubic strands
on wooden terraces. Students are introduced to math operations and the concept
of place value with the golden beads, which are made of either glass or wood.
Maria Montessori observed that children appreciate beauty and care for objects
that are fragile and valuable. She entrusted children with these learning
objects, often giving them trays on which to carry the materials around an
environment, from the place where the object "lives" in the classroom
environment to the mat or rug where a child's learning takes place.
With
ages 6-9, the stamp game is introduced. A box with four vertical sections
(units, tens, hundreds, and thousands), the stamp game contains wooden stamps
painted in green, blue, and red. It also reinforces place value in each numeral
family – simple, thousands, million, billions, etc. This material can be used
to teach all math operations (addition, subtraction, multiplication, and
division) by allowing the child to develop understanding of the nature of
“exchange” (trading ten unit stamps for one ten stamp, for example). Vertical stamp games – “bead frames” in the
form of colored beads movable across a horizontal wire – demonstrate these
concepts in upright form.
In
Montessori classrooms, students use hands-on materials to explore math
operations. Introductory lessons for all
operations are often practiced first with the stamp game. After
the initial lesson, a child may work autonomously either by him/herself or with
a partner. As Montessori said, "Education is a natural process
carried out by the human individual, and is acquired not by listening to words,
but by experiences in the environment."
Two
materials exist that excite students learning multiplication and division
operations: the checkerboard and the test tubes (also called “racks and tubes”).
Very similar to the stamp game, the checkerboard reminds the user of place
value – while also seeming similar to a game. Students practice each level of
sophistication, starting with a single-digit multiplier. They enjoy exchanging
bead bars and eventually moving their products diagonally. The multiplicand
extends to the billions place and the multiplier to the thousands, with
multiple-digit computation manipulated with the same colored unit bars that the
child found at ages 3-6 with the bead cabinet. To a child, it looks like you
are playing checkers, when you're actually multiplying large numbers!
The
test tubes are arranged like the racks in a chemist’s lab, reinforcing place
value yet again with each family of colored beads set in tubes with a dividend
into the millions place, each house in grey, black, or white bases.
“Exchanging” happens with the kinesthetic power of a banker. The idea of
“banking” continues in the multiplication bank game and the money game.
Students enjoy choosing an international name as they play banker with one to
three players. Children love games, and Montessori devised a very crafty way to
inspire interest in math by offering endlessly fun experiences with trading,
exchanging, and playing with numbers.
Still
other materials in the math curriculum demonstrate Maria Montessori’s genius at
connection beauty, intellect, and play. The Decanomial (also known as the Table
of Pythagoras) can be constructed by students on two rugs, then views from an
eagle-eye perspective. The Decanomial uses the same colorful beads found in
other materials (such as the bead cabinet and checkerboard) to construct a
visual multiplication table. Children take great care to make sure that their
rows are parallel and orderly. At the end, they often notice the “backbone” of
the table on the diagonal forms the squares of numbers, while all other
multiplication facts form rectangles. This introduces a concrete lesson about
exponents and helps many students with their multiplication math facts – by
inviting them to count and see the numeric value (rather than simply abstract a
“trick” taught at home by a well-meaning parent). Students follow this lesson
with a plastic Decanomial, which they can also lay out on a rug with a partner.
Maria
Montessori created so many materials that act as hands-on objects for students
to learn math concepts. She was a
scientist and observer first, and she carefully considered what she noticed
about children’s intrinsic motivations to learn: they enjoy beauty and order,
they desire to play through work and to work as a form of "play",
they care for materials that are fragile (in part because children like to be
trusted), and they want to be in charge of their own learning. Maria Montessori
bestowed onto children self-correction, logically-designed materials that they
use every day in their classrooms.
