Montessori preschool environments are renowned for introducing big abstract ideas, such as the decimal system, to young learners through precise, beautiful, instructional materials.
In particular, the Montessori math materials, produce more “aha” moments in learners of all ages than any other curricular area. For example, just this last summer, I was witness to a new Montessori guide break down in tears of joy as she laid out bead bars in equivalent multiples.
The effectiveness of early education with high-fidelity to Montessori materials and methodology has been validated by research and I do not presume to question this validity. Yet I do wonder whether specific supplements to the canon of didactic materials is necessary in light of truly following the child’s development. In this case, the development of the mathematical mind in building number sense with dice. In an earlier, related article, I outlined several other precursor activities that cultivate the mathematical mind.
There are NO Dice in Montessori!
I think we could agree that Dr. Montessori was genius in her approach including isolating the difficulty of any given task or step in children’s development. As a scientist, she was likely aware of the research of her time regarding how children acquire number sense, a fundamental precursor to counting and arithmetic. Her deep understanding of how children acquire embodied cognition of concepts is at its zenith in the creation of the brilliant Sensorial materials that concretize mathematical concepts such as Base 10, seriation (the ability to arrange in order of size), and quantity before symbol.
“The senses, being explorers of the world, open the way to knowledge. Our apparatus for educating the senses offers the child a key to guide his explorations of the world, they cast a light upon it which makes visible to him more things in greater detail than he could see in the dark, or uneducated state.” ~Maria Montessori, The Absorbent Mind
What I don’t believe is as accurately personified in the Montessori math materials is the concept of subitizing. Subitizing, as the National Council of Mathematic Teachers emphasizes, “is a fundamental skill in the development of students’ understanding of number” (Baroody 1987, 115) So what is subitizing? It’s what enables you to roll a die and quickly determine the quantity without having to count each pip, the dot on the die. Subitizing is an accurate enumeration of a quantity without internal or external counting. From a Latin adjective “subitus” meaning “sudden”, subitizing is an immediate knowing of a quantity.
Why Would We Bring Dice into Montessori?
The number rods are the first “official” Montessori material that isolates quantity and gives the young learner facility with counting, yet unless it is approached with lots of third period games without explicit counting, is not really isolating the difficulty of subitizing. Even the counters, as pleasingly ordered under each numeral in cards and counters, does not isolate subitizing as the aim of this material is to sequence the numerals and then quantify.
Subitizing is an essential pattern recognition skill that assists in developing “such capabilities as unitizing, counting on, and composing and decomposing numbers, as well as their understanding of arithmetic and place value. Subitizing introduces basic ideas of cardinality – concept of ‘how many’, ideas of ‘more’ or ‘less’, ideas of parts and wholes and their relationships. Furthermore, children who spontaneously focus on number and subitizing number are more advanced in their number skills” (Edens & Potter, 2013)
So what should Montessori guides do to cultivate this developmental milestone and forerunner to number rods?
My suggestion is to introduce the simplest of materials, one that predates recorded history, and present your students with lessons involving dice.
Children are automatically attracted to the idea of rolling the dice, especially if there is a defined space to roll such as a felt-bottomed tray and a game to play! If you’re looking for an excellent resource for beautiful and intriguing dice and is well-known in the Montessori community, check out Great Extensions here.
Here’s a subitizing lesson idea to get you started:
”More or Less” – In a separated tray with two dice, the child rolls one dice in one part of the tray and the other in the other part. Identifying at a glance which one has more, or less. You can extend this lesson with having many dice, maybe of two colors to contrast the two sides, and have the child sort “More” and “Less” into two groups. Further extensions for your older, more experienced mathematicians are to roll the dice and add them together, further developing their facility with quantity.
As you can see, subitizing materials can easily be teacher created; however, there are a set of conditions by which modern research has identified as essential for subitizing:
*the quantity must not be embedded in a picture
*simple geometric forms should be used for the group
*regular arrangements with an eye to symmetry should be used for groups
*clear, neutral backgrounds should be used
With these guidelines, what innovative subitizing materials can you come up with? Share your ideas and success stories and I’ll post them for others to gain inspiration!
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