Journey from Concrete Reality to Abstract Mathematical Concept
The following two-part discussion describes key areas of the Montessori math curriculum as practiced by our teachers at The Montessori Schools.
We are a community committed to establishing the intellectual, emotional, and physical foundation that will develop the skills for your child to become a self-directed learner, flexible thinker, and creative problem solver. Our math curriculum follows a traditional Montessori education, as developed by world-renowned doctor and educator Maria Montessori.
Our classrooms contain a range of teaching aids for math, but the common factor is their origin: concepts are taught using materials originally designed by Dr. Montessori more than a century ago.
Children begin by counting beads to solve simple addition, subtraction, and multiplication problems. Boards with separate cards demonstrate the base ten system for teen and double-digit numbers. Still more advanced mathematical concepts are made real—comprehensible—through direct manipulation of classroom materials.
This initial series of math works includes some of the most poignant examples of Dr. Montessori’s concept of “materialized abstraction,” as realized in her didactic materials. Works, such as the Number Rods, Sandpaper Numbers, Number Rods with Cards, Spindle Boxes, Cards with Counters, Memory Game, are to be experienced in an established progression.
The math works involving numbers 1-10 have a child physically associate symbols with concrete representations of quantities. The quantity can be fixed—for instance, as per the number rods—but the goal is always to help him or her express abstract mathematical concepts.
A child moves from an understanding of quantity to a meaningful comprehension of symbol.
The Number Rods are one of the first math works to be introduced to a child. Readiness for this work can be measured by the level of interest in numbers as expressed in everyday interactions, as well as previous success with select works in the Sensorial area of the classroom. Ten wooden rods vary in length, with the shortest one at 10 cm and the longest one at one meter.
fig. 1. Child working with the Number Rods. Each succeeding rod increases in length by the length of the first. All other dimensions remain consistent.
In fig. 1., the student is learning her numbers (1-10) in conjunction with the concept of quantity and within the context of an established sequential order. Through working with the Number Rods, the child grows to understand that each number is a separate object in itself.
More advanced works, those involving the use of beads, cards, and “fetching” activities, build on an understanding of how quantity is associated with a symbol. Children are prepped to perform operations using the decimal system. Exercises are repeated over and over again, providing significant exposure and practice and laying the groundwork for an understanding of place value. A child also becomes skilled in reading complex numbers.
The Formation of Numbers, or “fetching,” is a foundation work for a child. With mastery, a child will have success with more complex “banking games,” involving the concepts of static and dynamic addition, multiplication, static and dynamic subtraction, and division.
As a child progresses through a series of steps, he or she experiences and eventually comprehends multiplication, for example, in concrete terms. Similarly, the process of exchanging is introduced using tangible elements. As a child counts quantities in each column, he or she may have to (literally) trade groups of ten, exchanging ten units for one ten bar, 10 ten bars for one hundred, or ten hundreds for one thousand, in order to determine the final addition/multiplication sum.
“Ten units makes one ten,” says the child who has grasped the mathematical concept of exchanging."
fig. 2. Using the Golden Bead materials, children “build” numbers, physically manipulating the materials to arrive at the sums.
Still another series of Montessori math works involve children again working with the concepts of quantity (beads) and symbol (boards) but, this time, in pursuit of developing a sense of sequential order. Part of the genius behind the Montessori Method is establishing a correlation between different concepts.
A child builds on concepts internalized earlier with works that teach the names of numbers, such as the Teens/Tens Beads and Boards (11-19, 10-90). Sensibly, he or she will eventually move on to the next work, the 100 Board, which sequences the numerals from 1 to 100 in their totality.
fig. 3. The Montessori Bead Cabinet teaches linear counting while providing a concrete representation of abstract mathematical concepts.
The Short and Long Chains, or the Bead Cabinet, is also sometimes referred to as the “Cabinet of Powers,” due, in part, to its remarkable flexibility as a teaching tool. For the 4 to 6-year-old child, the work’s primary objective is to practice linear counting and reading numbers, as well as skip counting, from 1 to 100—and 1 to 1,000 with the long chains. However, significantly, the Bead Cabinet also indirectly teaches multiplication, squaring, cubing, and preparation (for the elementary years) for extracting square and cube roots.
Operations of the Decimal System
When a child is ready to work with the operations, he or she has a series of works that progress from a more concrete understanding of adding, multiplying, subtracting, or dividing quantities to a more abstract reinforcement exercise. The Stamp Games represent a shift toward a more theoretical understanding of math.
fig. 4. In the Stamp Game, counter tiles are used to layout the initial quantities. Additional units can be secured by “cashing-in” a tile from the next column to the left.
In the Stamp Games, a child no longer has such tangible, three-dimensional representations of quantities. Counter tiles are identical in size. Differences in color and in labeling are standardized to represent individual units (green “1”), a set of ten units (blue “10”), a set of 100 units (red “100”), or a set of 1,000 units (green “1000”). Consequently, the material’s direct aim, or purpose, is to reinforce the operation with the decimal system at a higher level of abstraction.
At this point in a child’s progress through the math works, he or she is writing work—copying the problem and final sum onto paper. The color-quantity correspondence (place value) is reinforced through using green, blue, and red colored pencils.
Why Understand the “Why”
In a Montessori classroom, children use tangible materials to solve simple and long math problems. Consider how many of us, as children in traditional classrooms, learned how to perform math operations. We likely memorized “tricks,” for example, for long addition: work from right to left, placing the right digit of a multi-digit answer under the line, and then carry the left digit into the next line of numbers to add.
Math tricks may ultimately work, but the process leaves the young child feeling disconnected from the mechanics of the operation. On the road to comprehension, it’s much better to initially connect on more solid terms.
Note on the Next Article in this Series:
Part 2 of 2 delves deeper into how Dr. Montessori’s teaching methods help bridge concrete applications for computation to a more nuanced understanding of mathematics.