If you spend any time whatsoever on social media, and you happen to be in the online world of Montessori, you might be faced with the kinds of questions like I field from parents, newly trained guides, and homeschoolers.
“I’ve introduced all the number rods and numerals, why isn’t my student remembering them?” “If my student is able to count to ten, are they ready for the Golden Beads?” “My child’s teacher is holding my child back in math. Should I insist they introduce the next lesson to them?” and possibly my favorite, “I thought Montessori meant my kid would be doing multiplication by the time they’re four!”
You might have even asked the question yourself, “There’s a math lesson that comes before number rods?” If you’re a classroom guide, you are most definitely in the position of sharing information with your student’s parents regarding their child’s whole development, and although parents may not be expressing their concerns or asking questions like these, you have an opportunity to share how the mathematical mind develops and is nurtured by Montessori pedagogy.
Sensitivity to Numbers and Math Peaks in the First Plane
Dr. Montessori was a keen scientific observer of children and of how people learn best. She identified windows of time in children’s development when they are more sensitive to a particular stimuli or type of interaction than at other times. One of the most significant periods, and relevant to this topic, is the sensitive period of order which begins at birth, peaks in early toddlerdom, and begins to wane by the child is about five years of age. The sensitive period for numbers is between 4 and 5.5.
Stanford University Neurobiologist Eric Knudsen reiterates Dr. Montessori’s findings, “When the effect of experience on the brain is particularly strong during a limited period in development, this period is referred to as a sensitive period. Such periods allow experience to instruct neural circuits to process or represent information in a way that is adaptive for the individual.” (Knudsen, 1412).
Montessori pedagogy leverages this sensitivity by enticing young child to engage with materials that promote profound number sense. So what is the foundation, or basic concepts that come before learning numbers and what do they entail?
Building Basic Concepts Before Number
Although our Montessori math curriculum for early childhood often begins with an introduction to number rods, the emblematic blue and red striped individual rods that isolate quantity from the shortest “1” rod to the longest “10′ rod, a compelling case can be made that the entire Sensorial Curriculum is truly where the mathematical mind is intentionally being nurtured. And for that matter, Practical Life as well! There are also a few “teacher-made” materials that I’d suggest adding to your repertoire of early math materials that broaden the range of materials that come before number rods.
The basic concepts that come before number rods include conservation, seriation, classification, one-to-one correspondence, and spatial and positional ideas. Let’s break those down a little further.
Conservation is the idea that although you might have one large pot of soup, it can be split into multiple bowls and that the quantity doesn’t increase. It was just distributed differently. Another example would be two rows of objects, each with the same amount, yet one row has bigger spaces between the objects hence making the row longer. The child who has not yet acquired conservation would say the longer row has more. Montessori students receive lots of practice conserving as they pour (and stop pouring) from one pitcher to multiple smaller cups in the Practical Life preliminary exercises. Conservation cannot be taught, it’s developmental.
Seriation is simply the ability to create an order of objects or pictures that increase or decrease in size. There are tons of examples of this throughout the Montessori environment, the most obvious is in the Sensorial area where students grade knobbed and knobless cylinders, arrange cubes and prisms and rods in order by size, and even work with concentric figures organized by size. Facility with seriation increases in difficulty as the child develops, beginning with ordering objects, then quantities, and eventually numbers.
Classification is closely related to seriation and these two concepts are often taught in conjunction with another. Classification is grouping sets based on a specific attribute or attributes. Montessori Sensorial materials are all about classification and students are intrigued with the many perfectly calibrated objects that isolate an attribute to sort including shape, size, weights, length, width, height, texture, and volume!
What Number Concepts Do We Need to Create in Montessori?
Noticing similarities and difference between objects and images (more abstract) through plenty of matching and sorting exercises builds critical thinking skills and the visual discrimination necessary for greater mathematical computation. I suggest that Montessori guides think about how these materials can be integrated into every area of the environment and changed out frequently enough to cultivate interest and engagement.
There are a few basic math ideas that are helpful for Montessori guides to introduce to the environment and come before number rods. I introduced several in previous posts, the Sharing Game and Equivalent Sets and Subitizing, and two more, those that involve spatial and positional concepts, and one-to-one correspondence I’m introducing to you today.
I think of spatial and positional concepts as less of a material I’m creating for the environment and more about how I embed these concepts both into the everyday life of the classroom and into my lessons. Think of introducing these concepts in three levels that are steps to mastery:
- Concepts of top, bottom, around, middle, center, corner, line, straight, curved, next to, and beside
- Concepts of parallel, diagonal, perpendicular, intersecting, angles and rotation
- First, second, third, next, last, before, and after
The first concepts are easy to introduce into your lessons such as when introducing the idea of line in art. Use and model directional language when creating an order of materials on the work space before beginning the lesson, and use this language to increase awareness and exposure. The second and third steps here are formally introduced at the elementary level in Montessori, generally through the geometry materials (2.) and after the student has more mastery over numbers (3.).
Is One-to-One Correspondence About Number?
One-to-One Correspondence happens all around us, and is a concept the child engages in daily whether they know it as such or not. Every day, the child puts one sock on one foot and then puts that one foot into one shoe. One thing to one thing. One-to-One Correspondence is a necessary foundation to understand numeration.
Early childhood teachers often create counting materials that have a “spot” reserved for the object to be placed. While this doesn’t actually isolate the difficulty of quantifying and transferring to the idea of counting, it does provide excellent practice with one-to-one correspondence!
Consider creating this easy one-to-one correspondence material adapted by a lesson suggested by Bee Pape in her article Before Number found in the Summer 1994 Montessori LIFE magazine:
Cover It All Up Create a card with 15 spaces or sections and a basket with just as many counters as spaces on the card. A die with each side marked with a set of 1, 2, or 3 dots. The die is rolled, and the child covers with counters as many sections as indicated by the top of the die. Increase the difficulty with a die having 1-5 dots. When all the counters are covered up, the child’s roll of the die indicates how many to take off.
Once your student has shown mastery with classification, seriation, classification of two or more attributes, one-to-one correspondence, and the Sharing Game, take a few minutes to assess their mastery with equivalent sets. If they are consistently making equivalent sets, then it’s time to introduce the number rods!
I hope that you are better equipped to both leverage what we know about the building blocks of the mathematical mind to amplify your ability to engage your students thoughtfully before they’re ready for number rods, and of course, better able to answer all those questions coming your way!
-by Tammy Oesting
Tammy Oesting has spent the last 25 years delivering professional development workshops, consulting schools, and educating new Montessori teachers. Her passions include issues of social justice, educating support staff, art education, neuroscience as applied to educational practices, and exploring the magnificence of the world. She is location independent and serves Montessori globally through her company ClassrooMechanics. AMS certified 3-6, 6-12