From the Concrete to the Abstract, Part Two

In this way, in these years, the child develops the mental facility and imagination to see a digit as the quantity it represents. This growth continues as children move into the Upper Elementary environment. In Upper Elementary classrooms, we can see this evolve both in the use of the materials and in the curriculum itself. For instance, many of the math materials, e.g., the Checkerboard for multiplication and the Racks and Tubes for division, are designed to allow children to move from calculation using beads and bead bars to calculation done abstractly, using only pencil and paper. The repetition of process leads to memorization of algorithm. Other math materials allow the child to see difficult concepts, such as operations with fractions, percentages, geometry proofs, and the squaring of a binomial, as physical manifestations, making the abstract more understandable. That is, Montessori children can quite literally grasp the meaning of the math while wrapping their heads around the abstract concept being presented. And while math is a convenient and straightforward example of this process, it is by no means the only area of the curriculum that uses this structure. The study of grammar and sentence analysis at the Upper Elementary level, by its nature an abstract topic, is tethered to earlier concrete experiences, making the step away from the manipulative materials just that, a step. And a small one at that.

  In a larger sense, the Upper Elementary child is also developing the ability to think more abstractly in terms of time. Whereas for a younger child each moment is either “now” or “not now,” the older elementary child can think beyond the here and now to other lands and cultures existing across time. Further, Upper Elementary students can use this nascent imagination to see how their own lives perhaps would have unfolded across time and space. The curriculum responds in kind. History, geography, and science, known collectively as the cultural subjects (in the Montessori vernacular), have the study of ancient civilizations as a concentration. The 9- to 12-year-old child “takes these seeds of culture and germinates them under the heated flame of imagination.” Leaving Upper Elementary marks the end of the second plane of development (6 to 12 years old). The child moves into the Junior Class, i.e., the third plane or early adolescence, and the concomitant development from concrete to abstract continues in new ways.

The entire pedagogy, encompassing children of 2 1/2 to 14 years of age, reflects the development in each child. A toddler enters a Montessori community, touching, stacking, and mouthing his or her way through the environment. Some dozen years later, the student graduates, but now postulating, considering, and coming to conclusions about the universe. Toddler to Junior: the former manipulates the physical, real, concrete environment, the latter the mental, psychic, abstract environment.

  This transformative process also takes place within each plane of development. Smaller-scale evolutions from the concrete to the abstract in Toddler, Primary, and Elementary classrooms provide key manipulative materials at each juncture, matching the child’s development with each step taken forward. The materials themselves do not teach. Rather, they provide a vehicle for children to make their own discoveries, create their own “a-ha” moments, and learn for themselves. The third plane of development, from 12 to 18 years of age (the first plane was birth to 6 years, and the second  was 6 to 12 years), marks the start of adolescence. But even in these years of young adulthood we see the same change from concrete to abstract thought; it can be seen in several areas of the curriculum, but notably in the subjects of the humanities. The study of Shakespeare is often one of the stronger components of a Junior Class curriculum. As new readers to these classic plays, 13-year-olds are challenged by the vocabulary, often struggle with the meter, and take the histories, comedies, and tragedies at face value. The play is the plot, and the characters represent themselves. Under the guidance of a gifted teacher, these young adults begin to see the action as metaphorical, the plays as allegorical, the plots as representative of larger, more abstract concepts. Macbeth is more than a Scottish king, he is any power-hungry politician. Prospero is any puppet master adult. Iago is evil.

  This harmony of purpose, providing the developmentally appropriate concrete materials to allow the child to internalize a great body of knowledge as the mode of thinking becomes more abstract, is a profound characteristic of a Montessori education. 

From the Concrete to the Abstract, Part One

An overarching theme of movement from the concrete to the abstract in a Montessori pedagogy can be discovered at each level. At the Primary level (3- to 6-year-olds), this can be seen reflected in the environment itself. Primary rooms contain three distinct areas: the sensorial, practical life, and academic areas, the latter of which is itself composed of arithmetic, geometry, language, and cultural areas. While these learning opportunities are available to all the children, they tend to have a particular developmental appeal and are selected by the students accordingly. For instance, a 3-year-old, who is likely a more concrete thinker, will be drawn more intently towards the sensorial materials. The Pink Tower, the Red Rods, the Broad Stair, and the Sound Cylinders are just a few of the many materials and activities that allow children to learn for themselves the distinctions between dimension and form, i.e., size and shape. Likewise, the practical life area, with its pouring works, polishing activities, and buttoning and zippering frames, also calls to a developmentally younger student. As the child matures, his or her attention is drawn to more challenging, more abstract, and more conceptual learning. In the Primary environment, this translates to the areas of language, arithmetic, geometry, and the cultural subjects (biology, geography, and history). 

  The advantages of a multiage classroom are manifold. An environment that encompasses learning materials ranging from the concrete to the abstract, and available to children over a 3-year age span, allows them to be successful where they are developmentally, regardless of their chronological age. This is not to say that a 5-year-old never uses the Lacing Frame, but one can speculate that for a 3-year-old it is a work of great concentration, while the 5-year-old uses it to take a mental break. And while the different areas of a Primary classroom are separated spatially, it is important that they are well integrated with each other. For example, the sensorial area serves as an indirect preparation for math, while the practical life area guides the child’s fine motor skills towards the development of writing skills.

At the Lower Elementary level, this development can be understood most clearly if we discuss the scope and sequence of the arithmetic curriculum. The Lower Elementary arithmetic curriculum includes, among many other concepts, computation in whole numbers, using all four operations. As they enter the environment, children are, by and large, in the nascent stages of “formal” computation. By the end of this 3-year cycle, many, though certainly not all, will gain a facility with math facts and will be able to add, subtract, multiply, and divide abstractly, to some extent, using just pencil and paper and no materials. This example represents a remarkable academic, psychological, and spiritual journey taken within each student. 

Through the use of manipulative materials, the child constructs an understanding of concept and a memorization of computation that eventually lead to abstraction. When we explore the materials and activities, it becomes obvious how they lead the child from stage to stage. Let’s look only at addition and allow this specific study to serve as a microcosm of the whole. The Golden Bead material consists of individual golden beads, beads strung into ten-bars, bars wired into hundred-squares, and squares wired into thousand-cubes. Thus, a child adding two four-digit numbers is adding actual quantities of beads. As the child gains understanding, he or she will be presented the Stamp Game, which represents a large step towards abstraction. In this material, individual tiles are stamped with various place values (1, 10, 100, and 1,000) in the hierarchical colors for units, tens, and hundreds. The child, however, still lays out each quantity, making exchanges if a column’s total exceeds nine. Next in the sequence of materials is the Bead Frame, an abacus that uses the same hierarchical colors as those in the Stamp Game and that represents another step towards abstraction. A bead on the Bead Frame gains its value by its placement on the wire, and there are only nine of each type; thus, this work is a triumph of representational thought!

The Great Lessons

Separate from the vast number of presentations given to children over their tenure from the Toddler to the Junior classroom, Montessori identified six Great Lessons, given in the second plane of development (between the ages of 6 and 12). In a sense, they are stories. At a time when a child’s development is making full use of a burgeoning imagination, beyond the here and now and into the vastness of space and time, these lessons are meant to capture the child with an impressionistic presentation. They are an opening of a curtain to the drama of the universe.

  The first Great Lesson is a Creation Story or its alternative, Life and Its Beginnings, which is the Montessori version of the Big Bang. This leads the child into discussions and work in geology, astronomy, and history. The second Great Lesson is the Story of Life, which uses the Timeline of Life as the chief material to present the evolution of life on Earth, from the early Paleozoic to the present day. This is followed by the Story of Humans, a lesson which employs two timelines, the first examining humans from the early australopithecines to Homo sapiens sapiens and the second inviting further study into just the last 25,000 years. The fourth Great Lesson is the Story of Communication, sometimes referred to as the History of Writing. A series of pictures and descriptions accompany this study, but it is also expected to be woven into the study of all subjects, including language, certainly, but into other curricular areas as well. Similarly, the fifth Great Lesson is the Story or History of Math. Much of this lesson is present in the myriad math and geometry presentations that explore the history of a particular concept at the same time that its presentation is given. These two lessons open up the study of ancient civilizations. And lastly, the sixth Great Lesson is The Great River, a metaphor for the human body system. The Great Lessons integrate and spark an interest in all areas of our classrooms and are part of a larger framework that Montessori referred to as “cosmic education.

CJMTE

Brocha and I never get tired of telling the story. Of how we met, how we first collaborated, and how it all took off from there. It was August of 2017 and I got a call from a dear friend, Khurram, the owner of Alisons Montessori. He had received a call from Brocha Baum, the owner and head of school for Darchei Noam Montessori near Baltimore. Could I help her with launching a new elementary program? Work with her new teachers? Support a new Montessori classroom? “She’s good people. Help her if you possibly can”. I thought ahead to what the Fall had in store for me by way of obligations and felt that I could probably manage a trip down from Maine sometime in October, and said, “Sure, give her my contact information and we’ll go from there”. In the ensuing conversation, I quickly learned two things: one, Brocha wanted someone to come down over Labor Day weekend (!) , and two, it saves a lot of time if you just agree with what she needs to get done in the first place. Brocha must mean “relentless energy towards a goal” in Hebrew. Consulting turned into workshops, turned into in-house teacher-training, turned into a full-blown course with 12+ adult learners coming from five different Montessori programs three years later. Mazel Tov!