A New Look at Psychogeometry: The Construction of Geometric Knowledge from Absorption to Abstraction

 A New Look at Psychogeometry: The Construction of Geometric Knowledge from Absorption to Abstraction

Montessori's Psychogeometry, originally translated into Spanish and published in Barcelona in 1934, sat in relative obscurity for nearly eighty years before its English translation and subsequent publication in 2011. While the “series of unfortunate events” behind its delayed distribution to a wider audience is well documented in the foreword and preface, a closer look at the title and what it represents in 2023 is warranted:

  • What, exactly, is psychogeometry?
  • What does psychogeometry look like in the first and second planes of development?
  • How does psychogeometry relate to other developmental theories?
  • Are the ideas and lessons in Psychogeometry still relevant today?

What is Psychogeometry?

David Kahn and Michael Waski, in Deep Dive on Psycho-disciplines, define any psycho-discipline as:

… the study of a discipline (subject) based on the psychology of the child. It connects the psychology of the developing human with the qualities and attributes of each discipline … [including] which aspects of the discipline we share at each stage of development, how we approach the discipline to support the developmental goals of each stage, and an awareness of the human tendencies and how they can be supported by our approach to teaching the discipline (2019).

In this respect, a word that might come off as esoteric, or even insular, really only reflects what can be described as a sensible approach to teaching—that of taking into account the psychology of the developing child and the characteristics of the corresponding developmental stage. Since such an approach reflects turn-of-the century and contemporary educational psychology, psycho-disciplines should not be regarded as exclusive to Montessori education. It's just good teaching with an unconventional name.

Additionally, when taken in context of the educational psychology movement at the time of its writing, the title Psychogeometry might also be understood as a reflection of a renewed interest in the field, with new theories about learning and child development being put forth by the likes of John Dewey, Edward Thorndike, and Jean Piaget—ideas which were front and center in the world of education during this time. A highly educated and eclectic thinker, it is certain that Montessori was aware of these trends; she writes in Psychogeometry, for example (1934/2011):

These new ideas regarding the interested child have caused the old psychological concerns to change, opening up a more dynamic field of educational methods …The old ideas were not wrong, but they corresponded to a preconception developed by the adult. If we consider the child as the cornerstone of education, and if guidance lies in the choice made by the child, rather than the teacher's logic, brand new principles are necessarily brought to education [original italics].

Regardless of the degree to which Montessori may have taken into account any of the then-prevailing theories of other developmentalists, her book title reflects the idea that the teaching of geometry should be informed by the psychology of the child. This should not surprise us.

Geometry and Development

Like every subject in the Montessori classroom, geometry is presented in keeping with the sensitive periods, or natural dispositions that children have toward certain activities during specific times in their development. It is noteworthy that Montessori addresses the sensitive periods in Psychogeometry with specific reference to the subject of geometry itself. In so doing, she stresses that introducing geometry to younger children is more in keeping with natural development than waiting until they are older, when the subject is often more difficult to learn.

Like Aristotle and John Locke before her, Montessori believed that nothing exists in the intellect that does not first exist in the senses. The study of geometry begins, therefore, with sensorial exploration. The child in the Early Childhood (EC) classroom is in “a stage of marked sensory and motor achievement” and “proceeds by ordering images, continuously making ever finer distinctions between things, and moving ahead with surprising potential for improving his coordination of fine, delicate movements [original italics]” (ibid.).

It is significant that the materials used for preparing the child for future studies in geometry are on the sensorial shelf in the EC classroom. The following examples demonstrate how specific sensorial materials, presented at around age 4, are precisely targeted at skills for ordering, distinguishing differences, and the development of motor coordination, while simultaneously and indirectly preparing the child for both writing and geometry studies.

Geometry in the First Plane of Development

The earliest materials presented are the wooden geometric plane insets—first the square, the triangle, the circle, and later the other polygons—each with a corresponding frame containing a recessed space for the inset. The child carefully removes the inset from its frame using a tripod grasp on the small peg at its center. The figures are manipulated such that the child distinguishes between properties such as straight and curved edges, obtuse and acute angles. Visual discrimination and motor coordination are practiced as the child returns the inset to its corresponding frame, puzzle-like. The child is concurrently given the name for each figure, thereby beginning to absorb the vocabulary of geometry. Later, the same figures, plus additional curved figures, will be explored in metal inset form, where they will be carefully traced and used to create a variety of designs, all of which promote coordination and fine motor control as indirect preparation for writing.

This is only a small sample of the range of sensorial materials targeting sensory and motor skills at the EC level. The graded figures, for example, promote the discrimination of size in two dimensions while the Pink Tower does so in three dimensions; both advance seriation skills and fine motor coordination. Work with the constructive triangles allows the child to discover that two triangles, depending on which kind, will make any number of different quadrilaterals, and six equilateral triangles will form a hexagon.

Discovery may, in fact, be the cornerstone of Psychogeometry. In Montessori's own reflection on her “magnum opus on geometry” (Lockhart 2019, 2), she underscores that it is not

… a systematic study of geometry. We only offer the means to prepare the mind for systematic study … We simply offer geometric shapes, in the form of material objects, which have a relationship to each other. These shapes can be moved and handled, lending themselves to demonstrating or revealing evident correspondences when they are brought together and compared … The discovery of relationships is certainly most likely to arouse real interest (ibid., 56).

It is important to recognize that these relationships between figures are not demonstrated by the teacher, but rather discovered by the child through her work, thus stimulating interest. It is this dynamic of discovery driving interest that becomes the foundation of the geometry work of the Elementary-age child.

Geometry in the Second Plane of Development

Having worked sensorially with the geometric figures in the EC classroom, the Elementary-age child explores their properties on an abstract level, consistent with his emerging power of reason—one of the sensitivities, or distinguishing characteristics of the second plane child. Montessori reminds us of this transition in Psychogeometry, where she “talks about how the child’s sensorial experiences lead to his ability to 'logic out' relationships, theorems, and formulas” (Lockhart 2019, 81). This transition is congruent with the observations of Jean Piaget, who said:

… the essential thing is that in order for a child to understand something, he must construct it himself, he must re-invent it. Every time we teach a child something, we keep him from inventing it himself. On the other hand, that which we allow him to discover by himself will remain with him visibly… (1972).

Reg Britz brought Piaget's words to light in his presentation at the annual AMS conference in Washington D.C. (March 2019), wherein he shared lessons from Psychogeometry on congruence, similarity, and equivalence. Using the red and green square metal insets to demonstrate a variety of exploratory exercises for the second plane child, Britz's lessons illustrate that “the geometry materials are not designed to teach; they are designed to enable children to make discoveries” (Britz, 2019).

In one example presented by Britz, the child proceeds through a series of exercises in which she constructs a square exactly half the size of another square, eventually leading her to the construction of an inscribed square. She will be given the vocabulary—inscribed and circumscribed—if this is a new concept, and will be able to explain, based on what she sees, the definition of an inscribed square: one in which each of the vertices touches one side of the larger (or circumscribing) square. From here, the child is able to formulate the following theorem: Given two squares, if one is inscribed within the other, the inscribed square is equal to exactly half the size of the circumscribing square.

These exercises exemplify what Montessori meant when she wrote that the advanced geometry materials “could almost be described as a gymnasium for the mind” (1934/2011, 55). When a child discovers a relationship on his own, “formulating a theorem and possessing the words to describe it correctly, is truly something able to fire the imagination” (ibid.) and surely a key to the psychology of learning.

Relevance to Today's Geometry Instruction

Like much in Montessori education, her genius bears out over the years as more contemporary approaches “catch up to” (and independently test, validate, confirm—falsify) her ideas. In 2009, the National Council of Teachers of Mathematics (NCTM) put geometry front and center in its seventy-first yearbook, Understanding Geometry for a Changing World. In the yearbook's preface, the authors trace the transition of geometry instruction in the United States from a high-school based program based on axiomatic Euclidean geometry, highly content driven, “to a consideration of issues related to students’ learning” (Craine & Rubenstein 2009, xii). This strikes a familiar chord.

Many of the successive changes endorsed by the NCTM since this shift are based on the van Heile model of geometric thought, first articulated for the NCTM 1987 yearbook by Mary Crowley. The model is based on five levels of geometric thinking—visualization, analysis, abstraction, formal deduction, rigor—and is highly compatible with psychogeometry in its demonstration of the successive phases through which students pass as they gain knowledge in geometry. These ideas, akin to those introduced by Montessori half a century earlier, eventually came to influence the American NCTM Standards and Common Core State Standards for geometry.

While the word psychogeometry is obscure, and perhaps even obsolete, its ideas are anything but outdated, as evidenced by one entire section of the NCTM 1987 yearbook devoted to geometry activities appropriate at the Elementary level. This too does not strike the Montessori practitioner as novel or groundbreaking. Montessori education has always been about “connect[ing] the psychology of the developing human with the qualities and attributes of each discipline.” Psychogeometry is yet one more indicator that Montessori's ideas were well ahead of the times in which she lived and wrote, and while it might be an unconventional name for a common discipline, in today's classrooms, as in those of the past, it simply amounts to good teaching.

References

Britz, R. (2019). “Geometry in the second plane of development.” Lecture. American Montessori Annual Event, Washington, D.C., March 2019.

Craine, T. & Rubenstein, R. (2009). Understanding geometry for a changing world: seventy-first yearbook. Reston, VA: National Council of Teachers of Mathematics.

Montessori, M. (1934/2011). Psychogeometry (Montessori Series Volume 16). Netherlands: Montessori-Pierson Publishing Company.

Piaget, J. (1972). “Some Aspects of Operations” in Play and development, ed. Maria W. Piers. New York: W. W. Norton & Company.

About the Author


Cynthia Brunhold-Conesa

Cynthia Brunold-Conesa, MEd, is an educator of adult learners at two AMS teacher education programs. She has 23 years experience as a lead guide at the Elementary and middle school levels. Cynthia also publishes on a variety of Montessori topics. She is AMS credentialed (Elementary I – II). Contact her at cynthia.conesa@meipn.org .

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