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.