![]() Montessori geometric materials support the understanding of patterns, relationships, and functions to represent and explain real-world phenomena by integrating all subjects together to point outward to answer inward questions children ask. Independence in answering those questions through experiment, experience, and personal expertise scaffold the child towards deeper knowledge of math, the world around them, and themselves. The sequential preparation both of the three-year cycle and further planes of development capitalizes on current stages and sensitivities in preparation for the future. Early childhood sensorial materials such as the geometric solids prepare the child sensorially through impressionistic lessons to build vocabulary and synthesis in lower elementary grades. Abstraction, rooted in memory of sensorial experience with their environment, takes flight from a solid foundation. Furthermore, some materials don’t showcase a resurgence until the upper elementary environment, such as the binomial cube, which can now be represented using algebraic functions. Geometry, like all Montessori subjects, requires a careful dance between peaking curiosity in children and overstepping their Zone of Proximal Development. Much like the transition from pre-modernity to post-Cartesian principles, “we may say that the adult works to perfect his environment, whereas the child works to perfect himself using the environment as the means” (Standing, p. 143, 1998). The child, like Aquinas, uses the material world, taken in through sensation, to take on its perfection to fill the previous void and take on the form and perfection of the environment. Descartes, inventor of algebraic geometry and a singular method for math, would later argue for a new world wherein man is master and possessor of nature. If we want children to thrive in a post-modern world, we must first teach them the laws of the universe Descartes pursued through mathematical physics, and then use the standard of certainty found in this singular mathematical method of algebraic geometry as arbiter for certainty in all else. Handing the child the method inspires independence to reason and use the world around them to construct not only their consciousness, but tools and machines as they deem useful and necessary. This mature specimen, raised up according to the perfected environment for education, is set forth into freedom with reason and nature at their disposal. Montessori argued that “his hands under the guidance of his intellect transform his environment and thus enable him to fulfill his mission in the world,” and thus our pedagogy, aimed at the preparation of this, prepares the child for this faculty (Montessori, p. 81, 1936). Our prepared environment molds the consciousness of young children, but is therefore limited to our classroom. However, “to consider the school as the place where instruction is given is one point of view. But to consider the school as a preparation for life is another. In the latter case the school must satisfy all the needs of life” (Montessori, p. 5, 1973). When we emphasize the “real-world” of geometry, we re-center our instruction around children rather than standards, administration, or fleeting fads. Teaching children to gaze out the window during class, Montessori is dedicated to expanding the walls of a child’s classroom as broad as their little feet can carry them. The real world is unshakingly relevant across decades of instruction and future growth of children after they leave our care. The natural world sends a message all around: that math is ubiquitous, useful, and unrelenting in application. Trusting children to take in from the environment real experiences which they are independently capable of developing definitions and drawing conclusions from can feel overwhelming and threatening to both educator and student. Suddenly, struggling students appear as a liability, intelligent students in danger of moving to much further than their peers for you to assist in, and the diversity of experience a threat to standardized anything. Through preparation of the environment and preparation of the inner teacher, however, leaning into the natural perfection of geometry and its availability to the intellect can allow for greater educational acquisition than ever imagined. Through the normalization and extension of geometry into the natural world, educators and families can collaborate to create a community culture of recognizing geometry in all. “Creating an environment in which math is part of everyday life won't transform kids into overnight math sensations,” Ernst argues, “but perhaps it can help kids realize math is a subject for curiosity, discussion and growth” (Ernst, 2018). Other pedagogical masters agree, including Van Hiele’s Levels of Geometric Reasoning. Van Hiele’s levels, which must occur subsequently and over up to a decade of instruction require special activities for facilitating children through each stage through informal geometry activities, which ought to be focused on exploration and investigation, constructing and taking apart, hands-on drawings, and aimed at making “observations about shapes in the world around them” (Van de Walle). Stories and conversations about math ignite curiosity and capitalize on the emotional portions of the brain more likely to enter long-term memory. In addition to serving as an entry point into math, according to Ernst, children tend to “solidify their fear of math” from first to third grade, and they learn it from us; their educators, parents, and role models (Ernst, 2018). It is time we began a new narrative: one where geometry serves the child as a tool to master their environment, leads to creativity, and extends to deeper learning across all subjects. Squealing, “acute! A cute cute cute angle! It is so small and so cute!” may sound alarming to a neighboring classroom, but students will always remember the name of the littlest angle. They remember, not because they necessarily gained something more useful than the other names of angles, but because of how you made them feel, how you made them laugh, how you connected with something within themselves. Geometry connects with a variety of students for a plethora of different motivating factors, but with nature as your leverage, the buy-in power is often greater than other mathematical concepts. Math is the language of reason, but geometry is the language of creation. Patterns found in the universe and natural world remind us that those same patterns make up not only our experiences, but who we are. If the heart is a machine and a sickled cell is a cell which has lost its geometric perfection, suddenly geometry is a matter of life or death. Elliptical eyeballs impede eyesight, mis-calculated angles on bridges crash to the ground, and unevenly sliced pizzas lead to familial brawls. Visual imagery and personal buy-in lend themselves easily to mathematics focused on quality rather than pure quantity. For example, the function of a spider web is to catch and hold the weight of a bug horizontally and vertically to prevent it from falling. In the real-world, the function is to feed the fly to maintain the food chain, however, mathematically, the function of the structure is to provide the necessary strength, through the types of triangles, the distance of the threads between each other, the thickness or thinness of the material. The spider does not have a PhD in engineering, but rather, nature, just like an engineer, works smarter not harder. The real-world patterns that repeat do so intentionally, not necessarily by divine intention, but by the usefulness of the thing. Like Descartes, what is useful is what we will to be, given our present circumstance, with nature as our vessel towards satiating those needs and desires. A new creation narrative centers the child, in awe of the creation of patterns of the universe, creating from nature a new world and a new self: an independent problem solver. The personal interest of children for independence, community, and creation can be satiated by geometry, if taught correctly. Suddenly, children are allowed to self-regulate throughout the day by taking breaks to draw a still life of a geometric solid, take ownership of their research assignments by decorating a border using straight line designs, or finding a piece of themselves in the stories we tell about divergent lines who move to new schools. When we allow students to see that we have made room for all of who they are in every facet of our classroom, we give children our trust in exchange for the most meaningful, engaged work an educator could hope to receive from students. Comprehension skyrockets, quality of work increases, and most importantly, our community practices engage with who children are rather than what they can recite. The implications of Montessori’s geometry method stem beyond mathematics; they create a pattern in our classroom of building relationships of trust that children will function according to their own intellectual perfection, if only given the perfect environment and a teacher willing to sit on her hands. Works Cited Ernst, S. (2018, December 18). “How to make sure your math anxiety doesn't make your kids hate math.” Retrieved from https://www.kqed.org/mindshift/52749/how-to-make-sure-your-math-anxiety-doesnt-make-your-kids-hate-math. Montessori, M., & Carter, B. (1936). The Secret of Childhood. London: Longmans, Green and Co. Montessori, M. (1973). From Childhood to Adolescence. New York: Schocken Books. Standing, E. M. (1998). Maria Montessori, Her Life and Work. New York: Plume. Van de Walle, Elementary and middle school mathematics, pp. 311-349. Van Hiele, P. (1999). "Developing geometric thinking through activities that begin with play," Teaching Children Mathematics 5, no. 6. pp. 310-16.
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Dr. Maria Montessori, philosopher in the footsteps of Aquinas and Descartes, and researcher of children and scientist in the footsteps of Itard and Seguin, serves as revolutionary both in practice and in the significance of her conclusions and method of arriving at them. Where Descartes invented the first singular mathematical method combining algebra and geometry, Maria, in the wake of modernity, pedagogically united arithmetic, algebra, and geometry, and taught it early, trusting the child with The Method (of Descartes) through which they can know themselves directly and immediately independently. Always focused on freeing the child, Maria handed the best to the smallest. However, her emphasis on handing this knowledge to the child remains mediated by the material world prepared initially by someone who already completed this process of personal liberation and actualization. This method, according to a Socialist Congress at Berne, best serves the child, and “to be educated according to Dr. Montessori’s method is one of the social rights of man” (Standing, 1998, p. 57). The philosophical set aside, Maria managed to move the child through the process of habituation while letting them believe they were leading the way, creating a spirited and playful method which best served the child. If your math journey could have been a series of games, stacking towers as high as you could, and moving around the classroom with your friends to create your own bank, I’d imagine you would have loved math, too. The Montessori 6-9 math curriculum supports the child in the second plane of development and fosters a love of learning mathematics by intentionally targeting sensitive periods, skilled application of didactic materials, and storytelling.
Sensitive Periods found in the intermediate stage of childhood, the second plane of development, by targeting their sensitive periods for cosmic order, moral order, analytical thinking, and social peer or herding association (McDermott, 2011). In this stage, the child is mentally “in a state of health, strength, and assured stability” (Montessori, 1995, p. 30). In this stage, direct help impedes growth. Although the child is ready for instruction and remains impressionable to adult intervention, it is best practice to step back and allow the child to interact with the prepared environment. Wherein the stage of infancy contained growth with transformation, ages 6-12 exhibit growth without much transformation and “successive levels of education must correspond to the sensitive personalities of the child” (Montessori, 1973, p. 1). The new growth of the child nods a directress towards informing her instruction by her students’ developmental stages. This second level of education which occurs from 7-12 includes a “veritable metamorphosis” (Montessori, 1973, p. 2). We see this rapid, successive change from our 1st years as they emerge in strength of self-assurance and leadership. In our social community, works like the bank game require students to be patient and communicate with one another to work collaboratively towards a goal while capitalizing on that sensitive period for herding, moral order, and analytical thinking. Careful attention to sensitive periods as the entry-point for calling the child to the content of math is the real strength of Montessori’s pedagogical approach. Activities like this build up executive functioning skills such as perspective taking which serve our students to be that salvation for mankind that our world so desperately needs. Balancing the buzz of socialization with an environment that serves the focusing needs of all children requires careful planning and proactive classroom management, but done well, further amplifies the community’s sense of the needs of the whole community and how to take care of one another, advocate for yourself, and find room in between. The major transitions of children from 3-6 to 6-9 environments may outwardly seem like their noise-level, height, or perhaps even a new-found sassy attitude. However, for Montessori, “while the younger child seeks comforts, the older child is now eager to encounter challenges” (Montessori, 1973 p. 8). Calling on that competitive nature which takes on challenges, Montessori refers to most of her math materials as games. Children in this stage are obsessed with quantity and correlate mass with importance. Concepts such as water, with overwhelming size yet easily accessible experiences and schema, spark the imagination of the child. Thus, materials which involve large impressionistic lessons such as stacking thousand cubes as high as they won’t wobble or moving successively from a tiny cube which delicately sits inside the small hand of a young child all the way to a ginormous box larger than the teacher (hierarchy of numbers), symbol of human completion, can wrap their arms around, ignite the child’s desire for learning. Montessori utilized her knowledge and observations of children to design a method which perfectly served their development. Skilled and intentional creation and use of didactic materials by Dr. Maria Montessori serves as the tool which would free the child and eventually lead them into abstraction. Montessori’s incredible skill in creation, preparation, presentation, and utilization all serve the child in a more fluid and holistic manner than any other math curricula I have observed. Montessori's belief that the hand was the tool to the brain would not go untouched by her math materials. Montessori argued in the early 1900s that “his hands under the guidance of his intellect transform his environment and thus enable him to fulfill his mission in the world,” and research continues to prove her right today (Montessori, p. 81, 1936). Only under the guidance of intellect do the hands function properly. If we hand children meaningless didactics without control (control of error, observation, and assessment), we have defeated the purpose and violated Montessori’s intention. In other words, an untrained or unskilled use of even Montessori's own materials would likely prove fatal. "A second way Montessori instruction allows for children to have extended time with manipulatives is that it uses a limited, but central, set of math materials to represent number concepts and operations, " argues Laski et. al, asserting the importance of both use of the hand as sensorial faculty by which to take in the material world, as well as for intentional, consistent materials (2015). Major critique of other math manipulatives such as plastic bears for counting or unifix cubes with random coloration and non-meaningful shape simply confuse and take away from the intention of the activity. The aim becomes the plastic bear, not the math itself. However, Maria, the scientist and observer she was, dodged pre-emptively this critique. Her materials have consistency and intentionality in size, shape, and color, so that "when the concept of the relationship between unity, tens, hundreds, and thousands has matured spontaneously, he more readily will be able to recognize and use the symbol" (Montessori, 1964, p. 211). That genius of preparation of the materials is the gift which roots the child in the sensorial experience, ignited by imagination to take off into abstraction. "Our children gallop freely along over a smooth road, urged on by the inner energy of their growing psychic organism,” Montessori concludes, “while many other children plod on barefooted and in shackles over stony paths" (Montessori, 2007, p. 261). This essential spark of imagination is drawn out of the child through impressionistic lessons with storytelling as backbone of the cosmic curriculum. The Fifth Great Lesson, adapted from Montessori’s original by Syneva Barrett, published in The Deep Well of Time by Michael Dorer, begins by describing math and numbers as “gifts we have been given, by many people around the world” (Dorer, 2016, p. 129). The true power of both impressionistic lessons and storytelling is that humans rarely remember what we taught, sometimes remember what they did, but they always remember how they felt. It is so uniquely and intrinsically to tell and bear witness to story, that great human heartbeat. All communities tell stories, whether it be with oral or written traditions, sacred or unsacred text, classic symbols, themes, messages communicated by the prepared environment or inward preparation (or lack thereof) of the teacher, but the common thread which transcends all tapestry of the human experience is that great question which 6-9 year-olds love to ask; where do babies come from? Dr. Montessori’s response? Stardust. And where do numbers come from? Human necessities whose solution turned out as vast and diverse as the environments, experiences, and cultures of the person or peoples solving it. It is as if to say, “numbers are tools invented by very human people just like you and me to solve very human problems with very human solutions and this grand cosmic pattern which began long before you can even fully conceive is now yours ripe for the picking. Join me on this mathematical dance filled with laughter, creativity, and joy.” Works Cited Dorer, M. J., & Epstein, P. (2016). The deep well of time: the transformative power of storytelling in the classroom. Santa Rosa, CA: Parent Child Press a division of Montessori Services. Laski, E. V., Jor’dan, J. R., Daoust, C., & Murray, A. K. (2015). What Makes Mathematics Manipulatives Effective? Lessons From Cognitive Science and Montessori Education. SAGE Open. https://doi.org/10.1177/2158244015589588 McDermott, M. (2011). Four Planes of Development. Retrieved from https://vimeo.com/19437369 Montessori, M., & Carter, B. B. (1936). The Secret of Childhood. London: Longmans, Green and Co. Montessori, M. (1964). The Montessori Methods. New York: Schocken Books. Montessori, M. (1973). From Childhood to Adolescence. New York: Schocken Books. Montessori, M. (1995). The Absorbent Mind. New York, NY: Henry Holt and Company, Inc. Montessori, M. (2007). The Advanced Montessori Method II. New York: Montessori-Pierson Publishing Company. Standing, E. M. (1998). Maria Montessori, Her Life and Work. New York: Plume. |
AuthorHi! My name is Brittany Wells, and I am a Montessori 6-9 major. I was born and raised in Cincinnati and attended Xavier University Montessori Lab School, Mercy Montessori, McAuley High School, and now Xavier University! Archives
May 2020
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