## Hands on math with the montessori method

### the progression of math materials

Montessori math materials help children approach math with hands-on, visual, and physical learning aids. These materials allow students to attach concrete knowledge to the often abstract concepts in math. As children progress through the Montessori math framework, the materials become more abstract, as students internalize more of the knowledge scaffolded by the materials.

Some Montessori math materials may look familiar to people with no knowledge of Montessori education, like an abacus or a multiplication table, and others may seem new and different. Montessori education uses a wide array of materials to explore the base-ten number system, and the myriad relationships among numbers.

Below, we list some of the most commonly used Montessori math materials, along with brief descriptions of how they may be used in the classroom. This list is not exhaustive, and creative Montessori teachers are developing effective ways to use and combine these materials to create exciting challenges for their students.

### Number Rods

Number rods are similar to the red rods of the sensorial materials–a set of ten wooden rods, the shortest being 10 centimeters long, and increasing by 10 centimeters each time. The number rods make their number of sections apparent by having each 10 centimeter segment alternate between blue and red. These rods introduce students to the concepts of numbers and counting.

### Sandpaper Numerals

The numerals 0-9, created from sandpaper, mounted to smooth wooden tablets. These sandpaper numerals help students to recognize and become comfortable with numbers. Later, students organize the numbers into proper order.

### One Through Ten Objects

Counting from one to ten is an essential part of the early math curriculum. A number of different objects are used to help reinforce this behavior, by counting out specific numbers of different objects, and arranging them under the proper numerical heading. This allows students to practice counting to each number between zero and ten separately.

### Cards and counters

This relatively simple material consists of a set of numbers, and a stock of "dots" used to represent the numeral. Students set out their numbers, organizing them in proper order, and then line up the proper number of dots under each numeral. Students then begin to explore the concepts of odd and even numbers, by pairing up the dots under the numbers.

### Short Bead Stair

A set of cards with the numbers 1-9 written on them, and a corresponding set of bead chains, with the proper number of beads. Students use these beads and cards to practice counting objects and writing numbers.

### BEAD CHAINS

Bead chains are ubiquitous in the Montessori learning environment. Most classrooms are equipped with a bead cabinet that holds a variety of beads and bead chains. These chains can be used in a number of lessons, building from counting and addition to complex algebra. Bead chains are particularly useful in learning the base-ten number system, and the relationships between numbers and their squares and cubes, for more advanced students.

### Bead Trays

The first bead tray is the introduction tray. This tray contains golden beads in orders of ten—a single unit, a chain of ten, a tile of 100, and a block of 1000—with their corresponding numeric labels. This set of materials help students to see the relationship between the numerals and the quantities. The tens tray contains many of the same elements (without the numeric labels). This tray is used to explore the relationships between these quantities; namely, that ten of the smaller unit equals one of the next larger unit. This work lays the foundation for complex work with multiplication, squares, cubes, and factoring.

### Stamp Game

Moving away from concrete manipulatives and into the more abstract representations of numbers, the stamp game utilizes a number of small tiles stamped with the numbers one, ten, 100, or 1000. Students use these "stamps" to replace the beads and chains they had been using, while still using something concrete to scaffold their arithmetic work. Students begin writing out equations, and writing down their answers by transcribing from these stamps.

### Teen and Ten Boards

The teen board and the ten board are technically different materials, but they are used in a similar fashion. The teen board is a wooden frame with ten rows, and a selection of fitting tiles with individual numbers on each. Each row has the number 10 written on the frame, and enough space to insert one of the other number tiles. By inserting a number tile, students can change any of the places where a 10 is written, to a different number in the teens. The tens board works similarly, but rather than 10 being written on each space, they change by 10 units each time. Students combine these boards with beads, bead frames, and writing utensils to explore the relationships between these numbers and quantities, and internalize the next steps in counting.

### BEAD FRAMES

The bead frame is a tool primarily used for exploring the use and power of the decimal place. The bead frame (similar to an abacus) allows students to explore the concepts of place value, decimal place, and numerical powers. Students are thrilled to learn that they can calculate complex equations with the use of the bead frame.

### ARITHMETIC BOARDS

Addition and multiplication boards serve many purposes in the process of learning arithmetic. These chart-like boards show the relationships between numbers and mathematic operations, in a holistic, visible way. Students can follow along on the chart to see what the sum (or product) of any of the two numbers would be by finding the point where those rows intersect. Working with these boards makes the "magic" of math more concrete, and helps students to recall sums and products of simple equations quickly.