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!

The First Great Lesson: the Coming of the Universe

“Let us give the child a vision of the whole universe… for all things are part of the universe and are connected with each other to form one whole unity.” – Dr. Maria Montessori

One of the hallmarks of the Lower Elementary Montessori classroom is the sequence of Great Lessons about the Coming of the Universe, Earth, and Humankind as central to the scientific studies in the Cosmic Curriculum. This is where concepts from physics, chemistry, and astronomy are introduced -- often via experiments or easily replicable demonstrations. This is where myths from cultures across place and time are shared; this is where the child begins to wonder about the grandest question: how did the universe originate?

“It is not enough for the teacher to love the child. She must first love and understand the universe.” – Dr. Maria Montessori

While respectful of global belief systems, Montessori education in the First Great Lesson – the Coming of the Universe – is based solidly in science. In fact, the juxtaposition of cultural myths and evolving hypotheses helps children ages 6-9 delineate what is factual and what is imagined, what is provable and what is improbable.

“The things he sees are not just remembered. They form part of his soul.” – Dr. Maria Montessori

These experiments and materials help the child see the inter-relationship between the unfathomably large multiverses and the smallest known spark of life, the atom, made up of even smaller particles – protons, neutrons, and electrons. Children are excited to see the order in this infinitesimal set of concentric circles and to use materials to recreate this brilliant structure of the building blocks of matter, the things that constitute stars and bananas and mountains and babies.

“Regard the child’s intelligence as a fertile field in which seeds may be sown, to grow under the heat of flaming intelligence.” – Dr. Maria Montessori

Freedom of movement in Montessori education is not only the ability of children to work on mats, as well as tables. Freedom of movement is not only their choice within limits of sequential shelf materials to grasp math, language, and science concepts. Freedom of movement is not only their right to work alone or with a partner; it also means that children can freely move between concepts. The Parts of an Atom connects to experiments in osmosis to pin-poking constellations or drawing them using oil pastels. This freedom of movement in thought stimulates the imagination and makes space for the connections that exist amongst all things over time and through space.

What the Hand Does, the Mind Remembers

“What the hand does, the mind remembers.” – Dr. Maria Montessori

Children remind me that they learn with their hands, something Dr. Montessori understood through quiet observation over a century ago. This continues to be true in the current time of digital technology (literally: tools used by the hands) and an adult focus on abstraction as a metric of success. I have grappled with the inclusion of technology and testing in the Montessori classroom, in both private and public charter environments, and I continue to believe that they are incongruous with the beautiful and patient process of a child learning by holding actual objects – clay, fossils, fern fronds – with their hands.

“Do not tell them how to do it. Show them how to do it and do not say a word. If you tell them, they will watch your lips move. If you show them, they will want to do it themselves.” – Dr. Maria Montessori

A few years ago, I asked a mother of one of my students to come in and demonstrate how she makes her own kombucha. There were simple ingredients and a live bacteria, which we stored in our room for weeks and watched grow, documenting its bubbling surface and the layers that it formed in a controlled environment. It is much easier to just buy a bottle of kombucha at a store, however the children looked forward to being a part of the process – using gloves to touch the SCOBY, selecting the flavors to add. Convenience is not always possible nor preferable. It often stalls understanding.

"The senses, being explorers of the world, open the way to knowledge." – Dr. Maria Montessori

In the Montessori learning environment, especially the expansive Cosmic Curriculum, there are many varieties of concrete materials – from the wooden Bohr diagram modeling the inside of an atom to the Timeline of Life with era boxes full of fossils to the Land and Water forms (used with colored water poured from a jug). Many of these initial materials have extensions which allow interested children to go deeper. For example, many children choose to make their own Land and Water booklet with brown and blue paper, or (in this case, shown above) models formed by painting dried clay models.

When I think about the ways in which public education (and, often, adult thinking in general) prioritizes large quantities of superficial, one-time-only lessons – some of which are never reviewed nor returned to – I feel thankful that Montessori education functions in the exact opposite manner. Children are encouraged to go deeper, to be reiterative, to be creative with a concept, making the trench of the neural pathways surrounding it that much further ingrained.

“Education is a natural process carried out by the child and is not acquired by listening to words but by experiences in the environment.” – Dr. Maria Montessori

Children learn about Parts and Kinds in Montessori education, from math to language to science:

Parts of the Atom and Kinds of Atoms

Parts of a Line and Kinds of Lines

Parts of Speech and Kinds of Words

Children are attracted to the largest things and the smallest things. When we study the Coming of the Earth, we find an inter-relatedness with the Inner Earth and the Plate Tectonic activity that caused and causes land forms to exist, that allowed animal and eventually human migration to occur. Children are fascinated by superlatives: the highest, the furthest, the smallest, the coldest, the hottest. Again, it would be easier to simply purchase a model of a volcano from a craft store. However, witness these children forming volcanoes with their own hands!

“The hands are the instruments of man’s intelligence.” – Dr. Maria Montessori

It is amazing to be alive, and children in Lower Elementary also come to a knowledge that death is something that belongs to all things. We study life cycles of living things, such as a tree or a jellyfish, and we also know where our Universe is in its own lifespan. (Like me, it’s middle-aged!) This knowledge of the ephemeral quality of nature is deepened when children see the Timeline of Life, specifically how old the world is, how old other organisms are (like the jellyfish, one of the longest lived creatures on the planet), and how young we humans are as a species. When we study the life cycle, we see the same phases and know that each organism is unique and special.

“The human hand allows the mind to reveal itself.” – Dr. Maria Montessori

One of the most inspiring things to see is what children create from their own imaginations. When they use geometric building blocks as a Practical Life activity, they are resting their reading mind and engaging their body – their kinesthetic and tactile intelligences. They are using the concept of gravity when they balance an arch on a cone. They are using the concept of symmetry when they build a structure out of rectangular prisms, pyramids, and cylinders. Mainly, though, they are free to experiment, make mistakes, try again, and eventually hopefully innovate while creating an architecture of their own happiness.