Concrete Math Materials for Assessing & Learning

For the past three weeks, I’ve been away from my Montessori learning environment (and my students) due to the COVID-19 pandemic. Since I am feeling nostalgic for the hands-on didactic manipulatives that help children learn without abstract formulas, I want to describe why these materials are so wonderful. (I last wrote about Montessori math materials on this blog over seven years ago, in case you want to search under the Math tab on the right hand side to learn more!) The following are Montessori math materials most relevant to the youngest learners in a lower elementary classroom, the 6-year-old first graders.

100 board

This material is often introduced in primary (age 3-6) Montessori learning environments, and I also find it a helpful assessment of number sense for first grade students and children new to the Montessori lower elementary classroom. The hundred wooden tiles sit randomly in a two-column rectangular box and fit neatly on a board with a hundred blue grid boxes. Children often enjoy doing this work in partners, although it can also be done independently. One way of giving this lesson is to ask the child to find all of the numerals that end in zero, then all the numerals that end in five – and lay them out on the board first, followed by other numbers per teen.

I have observed some very linear children like to lay them out in order, some children group them by teen, and some children who enjoy just taking their time with it choose tiles at random and find where they go. There is no need for an adult to intervene – by showing or asking the child where the tiles go. This is an intuitive experience that is only disturbed by interruptions and interventions. All Montessori materials are naturally self-correcting.

Golden Beads numeral composition

Another Montessori math material used frequently in primary classes that is also very useful as an assessment for new lower elementary students is the set known as the Golden Beads. One of each – a unit bead, a golden ten bar, a golden (or wooden) hundred square, and a golden (or wooden) thousand cube – sit in a rectangular presentation tray. Montessori guides (teachers) lay out a Golden Mat (actually usually colored green with place value columns) and write, on a board or piece of paper (or using wooden numeral tiles), a four-digit numeral. Children take an empty tray to the shelf where the Golden Beads are stored and retrieve the correct amount, sometimes all at once, and sometimes very slowly, one place value at a time. There is even a little golden dish for the unit beads to sit in.

The child places the Golden Beads in the correct place value column and reads the numeral aloud when they are ready. This lesson assesses number sense and place value sequencing, as well as demonstrates the child’s spatial balance and math-language abilities.

Infinity Street

The first official Math lesson I do with first graders and children new to the Montessori lower elementary environment also reinforces place value understanding. Infinity Street is not an original lesson created by Maria Montessori in 1907, however it was introduced in my Montessori training. I have used it with children for 15 years with great success. It takes awhile to do, mainly because it has a few steps, a lot of coloring, and requires some fine motor skills.

Infinity Street is a basket containing twelve yellow houses that are in ascending order of size, much like a two-dimensional set of nesting dolls. Each house comes with a label (Simple, Thousand, Million, etc. up to Decillion, the largest house in the basket – not the largest possible number family!) and a yellow “mailbox” comma to put between each house. There are three “doors” on the front of each house – green for units, blue for tens, and red for hundreds – as each house contains these three place values. In the first lesson, the Montessori guide tells the story of Infinity Street and invites children to help read or lay out the labels and commas.

In the first extension (maybe a day or two later), the children work in partners to lay out the houses, labels, and commas in order. Size is the control of error on the houses, and the labels and commas have number keys on the reverse if children need to check their work. In the second extension, children lay out their house and trace it onto a long swath of butcher paper. They write the label name for the number family (such as the “surname” Trillion), and color the house yellow and each of the doors green, blue, and red. They even draw the comma “mailbox” between their house and the next child’s. At the end, we display Infinity Street on a wall so all children can access it for reading large place value numerals!

Curriculum: Math Operations and Montessori Materials

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.