A group of researchers from Italy and the United Kingdom analyzed the development of geometrical concepts, the cognitive processes underlying geometry-related academic achievements, and the educational implications that learning geometry can have.
Irene C. Mammarella, David Giofrè, and Sara Caviola reviewed the literature on learning geometry and evaluated papers from developmental psychology, cognitive psychology, educational psychology, and education.
To thoroughly understand how best to go about this issue, we need to consider not
only the cognitive processes involved in geometry, not only how geometrical
knowledge is developed (are some concepts of geometry innate?), and not only
how geometry is taught, but all of them together.
By first providing a brief review of the principal theories and models relating to learning geometry, the focus was to analyze the secondary role that arithmetic and algebra have in many school curricula, in spite of their role in cognition development. An experiment from Piaget and Inhelder (1967) showed that children can construct perceptual spaces already in infancy, being much later able to develop ideas and concepts about said spaces.
Van Hiele’s model of development for geometrical thinking is also used as an argument in favor of a primary educational role of geometry. The model consists of five levels of understanding:
1/ Precognition (1 to 5 years), when children are starting to form canonical schemes for canonical geometric shapes
2/ The visual level, where shapes are recognized as a whole (by identification and naming) but the brain cannot create mental images of them, being instead described as objects
3/ With informal deduction, children can understand the hierarchies of geometrical figures, are able to distinguish class inclusions (a square is also a rectangle and a parallelogram, etc.), and can also deduce properties of a figure
4/ Deduction, with the understanding of the role of axioms, theorems, and proofs
5/ Learning non-Euclidian geometries
Core intuitive principles of geometry are suggested by making a distinction between those implicit and those associated with schooling. The core knowledge hypothesis provides an argument in this favor, with mathematical abilities seeming to emerge from different representational core systems: one based on intuition and another one relying on the symbolic representation, specific to humans who have received some formal education and are able to create novel abilities (including symbolic mathematics).
The core knowledge that defines intuitive geometry is culture-independent, with people having a preference for geometrical properties and attributes. Studies on Amazonian and North American children and adults (Dahaene et al., 2006), showed that they were able to correctly identify concepts of topology (right angles) and geometrical figures (e.g., squares, triangles, and circles) with no formal geometry training but having the same results in performance. This was interpreted as proof of the existence of core principles of geometry.
Two core systems for this natural geometric knowledge are considered, based on previous studies with children: one for navigating 3d spatial layouts and one for analyzing 2D visual forms), together capturing all the fundamental properties of Euclidian geometry (distance, angle, and directional relationships). There are some limitations, though:
They capture only a subset of the properties covered by symbolic Euclidean
geometry. Children can construct a new, more complete, and general system of
geometrical representation, however, by productively combining the representations afforded by the two systems. This process relies on culturally variable artifacts (such as formal instruction) and children learn to use it as they engage with the symbol systems of their culture.
The researchers studied how Italian children develop the critical skills presented in Hiele’s model. The results showed that while a majority of the children had limited geometric knowledge (e.g., little geometrical education), their spatial skills scored high on identifying, rotating, and distinguishing shapes among distractors. This is consistent with the claim that very young children are already able to solve basic geometrical and spatial problems even if their geometrical education is poor.
The educational evolution of children can benefit in full from understanding these studied core principles. Academic geometry achievement is regarded as one of the most significant areas of mathematical study, particularly at the secondary school level, and is linked to future academic and professional success. Secondary school students must learn basic geometric concepts, definitions, theorems, and other related material, as well as apply their knowledge to solve problems that are often provided in the form of verbal or written language. And without a strong geometrical foundation, students are not prepared for advanced study in the fields of science, technology, engineering, and mathematics.
While teachers report that geometry is a difficult subject to teach properly, they must be able to recognize geometrical problems and theorems, explore the historical and cultural background of geometry, and understand the diverse real-world applications of geometry.
With the help of a proposed curriculum (with steps focused on academic proficiency, geometry, and spatial reasoning and cognition), the educational goal is to develop competence in the two domains consistently identified as fundamental: (1) number concepts (counting, subitizing, or identifying the numerosity of small quantities) and arithmetical operations; and (2) spatial and geometric concepts and processes.
Improving academic achievement in geometry may contribute to growth in other abilities, including higher-order cognition and mathematical reasoning. Academic geometry is a complex domain, demanding a wide range of skills and knowledge of mathematics, problem-solving (which in turn involves higher-order skills related to higher-order cognitive processes, such as intelligence or reasoning), spatial abilities, verbal knowledge of geometrical concepts, and more.
Acquisition of Complex Arithmetic Skills and Higher-Order Mathematics Concepts
Chapter 10 – Learning Geometry: the Development of Geometrical Concepts and the Role of Cognitive Processes, Irene C. Mammarella, David Giofrè, Sara Caviola
Published: September 2017