Geometry and geometry education

Summary: Geometry has been studied since ancient times and continues to develop today.

The word “geometry” is derived from the ancient Greek words geo (Earth) and metron (a measure). In addition to its practical origins, it is also associated with the language and theory of geometric figures, spaces, and forms. Ancient and medieval civilizations from all around the world contributed to the development of geometric concepts, including mathematicians in Babylonia, Egypt, China, India, Mesoamerica, Greece, and the Islamic and Arabic world. At times, the prominence of geometry has declined, such as in Western Europe during the Middle Ages and in certain research areas and undergraduate courses in North America in the twentieth century. Some curricular concepts that were once the focus of investigations have declined in relevance, such as spherical trigonometry, having been replaced in curricula by new fields or notions. Geometry research and education continue to evolve in response to changing emphases. At the beginning of the twenty-first century, students explore the properties of geometric objects and transformations. They learn about deductive geometry, coordinate geometry, and algebraic connections. Visualization and geometric history and applications are also a focus. Some of the curricular topics have been fundamental for millennia, like the Pythagorean Theorem, named for Pythagoras of Samos (c. 569–475 b.c.e.), while others, like vertex-edge graphs, are relatively recent inclusions.

Early Geometry

Some of the first indications of geometry in terms of geometric patterns appeared about 25,000 years ago. These indications have been found in a number of prehistoric sites, such as Stonehenge, spirals in Europe (Ireland and Italy), and various places in Mesoamerica and North America. Geometry also appears in the designs of the pottery, baskets, and mat weaving of many older civilizations and aboriginal peoples in the world. For instance, African tapestry and pottery are filled with symmetric figures. Civilizations around the world, including Egypt, Mesopotamia, China, India, Mesoamerica, and later civilizations, also used geometry to help produce calendars, which, at the zenith of their power, were quite accurate. The Greek historian Herodotus of Halicarnassus (c. fifth century b.c.e.) credited the Egyptians with having originated the subject of geometry, but there is evidence that the Babylonians, the Hindu civilization, and the Chinese knew much of what was passed along to the Egyptians. The earliest extant written records of geometry come from the predynastic Egyptians and Sumerians as early as the fifth millennium b.c.e. Many have connected these papyri and cuneiform stone tablets to art, decoration, and construction rather than to the systematic investigation of figures, patterns, forms, and quantities that has come to be associated with deductive geometry. Some historians and mathematicians caution against interpretations that examine earlier knowledge without considering the contextual language and culture or through the lens of later work in geometry. In many cases, the evidence that survives likely represents an incomplete geometric record.

Egypt

The Egyptians were extremely accurate in construction, making the right angles in the Great Pyramid of Giza precise to what is noted as one part in 27,000. Most of what is known about Egyptian mathematics comes from two Egyptian documents from about 1650 b.c.e., the Rhind Papyrus and the Moscow Papyrus. There are only a limited number of problems from these ancient Egyptian works that concern geometry. The examples therein demonstrate that the ancient Egyptians computed areas of triangles, rectangles, and circles; surface areas of hemispheres; and volumes of cylindrical granaries, rectangular granaries, and pyramids.

Babylon

In the late twentieth and early twenty-first centuries, scholarly work on some of the thousands of extant Babylonian mathematical clay tablets has led to revisions and insight in the understanding of Mesopotamian mathematics and geometry. Some tablets illustrate problems related to lengths and areas of fields, trapezoids, rectangles, right and isosceles triangles, circles, and irregular quadrilaterals. A number of clay tablets provide evidence of knowledge of the Pythagorean theorem long before the Greeks. The Babylonians also computed volumes and used geometric techniques to solve algebraic problems, like completing the square.

China

In the Story of Civilization series, Will and Ariel Durant state that “Chinese mathematicians apparently derived algebra from India, but developed geometry for themselves out of their need for measuring the land.” Geometry was also an integral part of cosmology and astronomy in China. For instance, astronomers from the Confucian time had correctly calculated eclipses and created a basis for the Chinese calendar. One early work is from the Mohists, where one finds a definition of point as the smallest indivisible component, one that cannot be divided into smaller parts. In contemporary terminology, they also explored the congruency of two lines of equal length and provided definitions for the comparison of lengths, parallels, circumference, diameter, radius, and volume. There remain some disagreements about document dating in the history of mathematical development in China. For example, some have dated Zhoubi suanjing (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven) from approximately 1200–1000 b.c.e. during the Han dynasty, but many scholars believed that early versions were written during 300–250 b.c.e. The Zhoubi suanjing has a diagram of the Gougu Theorem (Pythagorean Theorem) that is well-known in twenty-first-century classrooms. The best known of the Chinese mathematical classics may be the Jiuzhang suanshu (Nine Chapters on the Mathematical Art). The book had many applied geometry problems, such as finding areas for squares and circles, the volumes of various solids, and the use of the Pythagorean theorem. Included are mathematical surveying techniques in order to calculate distance measurements of depth, height, width, and surface area. There are also formulas for the areas of planar figures and the volumes of solids that were known by the time of the Han dynasty (202 b.c.e.–9 c.e.). Jesuit missionaries introduced the Chinese to Western mathematics during the Ming dynasty. As part of the Jesuit’s program, part of Euclid’s Elements was translated into Chinese in 1607. The translation of this ancient Greek textbook on deductive geometry led to research and comparison of early geometric knowledge in both cultures, work that continues in the twenty-first century.

India

The Sulbasutras (c. 700–400 b.c.e.), which can be translated as “cord-rules,” are often referenced in the twenty-first century as a source of ritual geometry from India. This and earlier texts gave precise rules for the construction of sacrificial fire altars. The Sulbasutras, specifically the Baudhayana Sulbasutra, may contain the earliest extant expression of the Pythagorean Theorem: “The rope which is stretched across the diagonal of a square produces an area double the size of the original square.” Many historians agree that the statement predates the Pythagoreans in Greece. Some of the material in the Sulbasutras may have originated from the Babylonians or the Chinese or been passed to the Chinese or Greeks. The Sulbasutras also contain other geometric constructions that preserve areas, as well as lists of Pythagorean triples and statements about squaring the circle or “circling the square” needed to construct certain altars for the rituals. There are various theories about the association between the geometric constructions and the religious rituals, such as whether the rituals inspired the geometry or vice versa. Aryabhata’s Aryabhatiya (499 c.e.) includes the computation of numerous areas and volumes. Brahmagupta wrote his astronomical work Brahmasphuta-siddhanta in 628, which included his famous theorem on the diagonals of a cyclic quadrilateral as well as his formula for the area of a cyclic quadrilateral. The Bakhshali manuscript written on birch bark and found in 1881 near the village of Bakhshali in what is now Pakistan is another mathematical manuscript from the Indian subcontinent. The date is uncertain, but many scholars agree that it contains information that is older than the document itself. It includes some geometric items such as the volumes of irregular solids.

Greece

The ancient geometry was passed on to the Greeks, who furthered and transformed the field into an essential component of a liberal arts education. The beginnings of deductive and axiomatic geometry have traditionally been attributed to Thales of Miletus (624–547 b.c.e.). Having studied in Egypt, he was likely familiar with the computations handed down from Egyptian and Babylonian mathematics. The deductive approach was continued over the next two centuries by Pythagoras of Samos (569–475 b.c.e.) and his disciples. Their foundation of plane geometry was brought to a conclusion around 440 b.c.e. in a treatise by the mathematician Hippocrates of Chios (470–410 b.c.e.). Plato (427–347 b.c.e.) founded “The Academy” in 387 b.c.e., which flourished until 529 c.e. and is noted to have included an inscription at the entrance stating the importance of geometry as prerequisite knowledge: “Let no one who is unversed in geometry enter here.” Theætetus of Athens (417–369 b.c.e.) was a student of Plato, who developed solid geometry and the Platonic solids. Menaechmus (380–320 b.c.e.) discovered and developed the conic sections. He was the first to show that ellipses, parabolas, and hyperbolas are obtained by cutting a cone in a plane not parallel to the base.

Euclid of Alexandria (325–265 b.c.e.) collected the theorems of Pythagoras, Hippocrates, Theaetetus, and other predecessors and included discoveries of his own into a logically connected formal axiomatic system, the Elements. So completely did Euclid’s work supersede earlier attempts at presenting geometry that few traces remain of these efforts. His approach to geometry has dominated the teaching of the subject for over 2000 years. Moreover, the axiomatic method used by Euclid is the prototype for what is now called “pure mathematics.” Euclidean geometry was certainly conceived by its creators as an idealization of physical geometry. The entities of the mathematical system are concepts suggested by, or abstracted from, physical experience but differing from physical entities as an idea of an object differs from the object itself. Centuries later, the philosopher Immanuel Kant even took the position that the human mind is essentially Euclidean and can only conceive of space in Euclidean terms.

Some attributed the eventual decline of Greek mathematics to events like the destruction of the library at Alexandria and political and economic factors, and others to a lack of algebraic notation, which was developed by Arabic and Islamic mathematicians and was later to revolutionize geometry theory and applications. However, some Greek mathematicians after Euclid continued to explore concepts that became a fundamental part of school curricula. For instance, Archimedes of Syracuse (287–212 b.c.e.) is regarded as one of the greatest Greek mathematicians. He found the areas and volumes of many objects and explored semiregular polyhedra. Apollonius of Perga (262–190 b.c.e.) was known as “The Great Geometer” for his work on conics and other geometric concepts. Menelaus of Alexandria (70–130) developed spherical geometry in his only surviving work, the Sphaerica. Pappus of Alexandria (290–350) is considered one of the last of the great Greek geometers. His major work in geometry was the Synagoge, or The Collection, a handbook on a wide variety of topics: arithmetic, mean proportionals, geometrical paradoxes, regular polyhedra, the spiral and quadratrix, trisection, honeycombs, semiregular solids, minimal surfaces, astronomy, and mechanics.

Islamic World

Mathematicians in the medieval Islamic and Arabic world preserved and extended classical geometry and astronomy from India, Persia, Syria, and Greece, and developed and applied geometric concepts. Geometric design was found in mosaic tessellations in mosques. Mathematicians extended the astrolabe by adding circles for azimuths on the horizon in order to solve problems in spherical astronomy and trigonometry. Scholars, such as Abu Ja’far Muhammad ibn Musa al-Khwarizmi (c. 780–850) and Omar Khayyam (1048–1131), developed algebraic and trigonometric concepts and applied them to geometric notions. Arabic and Islamic mathematicians explored many geometric topics, including conic sections, constructions, spherical projections, and the parallel postulate. Scholars like Ibrahim Ibn Sinan (908–946) wrote works on geometric analysis and problem solving. Abu Arrayhan Muhammad ibn Ahmad al-Biruni (973–1048) calculated an extremely accurate radius of Earth using the law of sines. There are examples of Greek works that would have been lost if not for copies that were preserved in Islamic libraries or translated as a part of Arabic treatises and commentaries. As Europe emerged from the Dark Ages, these works were translated into Latin and this paved the way for geometry’s return to Europe.

The Changing Nature of Geometry Education and Research Since the Seventeenth Century

While Euclid’s Elements has been standard in mathematics education for thousands of years, geometry curricula were impacted by the development of many new research areas since the seventeenth century. For example, René Descartes (1596–1650) and Pierre de Fermat (1601–1665) explored analytic geometry in the seventeenth century. This exploration allowed for the representation of geometric objects in terms of coordinates and two-variable equations, a topic that begins in primary schools in the twenty-first century and is fundamental in many real-life applications. In addition, analytic geometry is typically paired with calculus courses. However, axiomatic or synthetic perspectives continued, such as through the work of Girard Desargues (1591–1661) and Jean-Victor Poncelet (1788–1867) on projective geometry. While projective geometry declined in some contexts, such as in undergraduate education in the twentieth century, students continue to learn about both coordinate geometry and deductive perspectives. Gaspard Monge (1746–1818) emphasized descriptive geometry at the École Polytechnique, a French technical university, by exploring three-dimensional geometry through two-dimensional images. Descriptive geometry remained important in architecture, engineering, and mathematics classes.

The discovery of non-Euclidean geometry, which can be found in some twenty-first-century high school and college classrooms, was a revolution in geometry. It fell to three different mathematicians independently to show that Euclid’s fifth postulate is not provable from the other axioms and what is derivable from them. These mathematicians were Karl Friedrich Gauss (1777–1855), Nicolai Ivanovich Lobachevsky (1792–1856), and János Bolyai (1802–1860). Gauss’s work appears only in a letter to Franz Taurinus (1794–1874) in 1824, but he seems to have foreseen the results of the other two. Lobachevsky published his work in a Russian journal in 1826, and it was not until 1848 that it came to be published more widely in German. Bolyai’s work received the widest initial distribution, being published in 1831 as an appendix to his father’s algebra textbook. Each of these men, independently, assumed the negation of Euclid’s fifth postulate and developed a consistent geometry: “worlds out of nothing,” as Bolyai described it. Following the pioneering work of these mathematicians, the pieces of geometry began to fall into place. More was learned about non-Euclidean geometries—hyperbolic and elliptic, or doubly elliptic (spherical). For instance, Eugenio Beltrami (1835–1900) helped rigorously establish the subject, and the elliptic geometry studied by Bernhard Riemann (1826–1866) gave rise to Riemannian geometry and manifolds, which gave rise to differential geometry and then to relativity theory. Students may explore Henri Poincaré’s (1854–1912) disk model of hyperbolic geometry. Some undergraduate and graduate students take courses in differential or Riemannian geometry. Mathematicians also started looking at finite geometries (from the standpoint of an algebraic geometry and from an axiomatic process), which led to areas of combinatorics and graph theory that are a curricula part of schools and colleges.

Geometry research continued to revolutionize school geometry. For instance, Felix Klein (1849–1925) greatly influenced geometry through his Erlangen Program, which attempted to unify geometry through symmetries; middle grades students in the twenty-first century explore transformations and symmetries. Kurt Godel’s (1906–1978) work on consistency shook the foundations of axiomatic geometry, and David Hilbert’s (1862–1943) axioms are now explored in some high school classrooms. Donald Coxeter (1907–2003) was noted as preserving the tradition of classical geometry, which then remained a core area in primary school through high school. Likewise, following World War I, the French mathematicians Pierre Fatou (1878–1929) and Gaston Julia (1893–1978) began looking at objects that later came to form the foundation of fractal geometry, introduced in the late twentieth century by Benoît Mandelbrot (1924–2010). Some middle grades students are exposed to fractals, and undergraduate students may take courses focusing on fractal geometries.

Educational theories about geometric learning also had an effect on school geometry. For example, English educator John Perry (1850–1920) advocated an intuitive, inductive approach to teaching geometry, such as graph paper measurements to test Euclid’s propositions. George Bruce Halsted (1853–1922) noted that geometry “always relied upon for training in the logic of science, for teaching what demonstration really is, must be made worthy [of] the world’s faith. There must be a text-book of rational geometry really rigorous.” Halsted’s textbook was based on Hilbert’s axioms rather than on Euclid’s. One well-known geometric learning model was the van Hiele model of geometric thought, which originated in 1957 through the work of Dutch educators Dina van Hiele-Geldof (d. 1959) and Pierre van Hiele (1909–2010). The model encompassed five levels: visualization, analysis, informal deduction, deduction, and rigor. Educational research on the van Hiele model in the Soviet Union during the 1960s and 1970s led to curriculum based on the theory. Some have criticized the structure of the levels and created other geometric learning models.

Recent Developments

In the twentieth century, geometry education was fundamentally transformed because of computers, calculators, and other devices. Geometry for navigation, like spherical trigonometry computations, was built into computer programs or global positioning systems, and so the related topics were eliminated from the curriculum. Other geometric topics were introduced, such as fractal geometry and computational geometry. Mathematicians at places like the Geometry Center for the Computation and Visualization of Geometric Structures produced videos and applets. Teachers and mathematicians discussed topics and shared resources on the Internet. For instance, what began as the Geometry Forum in 1992 was extended in 1996 to the Math Forum. The development of dynamic geometry software programs encouraged mathematical discovery. Students could manipulate geometric constructions while preserving the mathematical relationships that defined the figure. This method enabled students to uncover invariants like the angle sum in a triangle and allowed for inductive educational approaches. Two school programs that originated in the 1980s and remain in use at the beginning of the twenty-first century are Cabri Geometry and the Geometer’s Sketchpad. Jean-Marie Laborde (1945–) headed a team to develop Cabri in order to explore geometric relationships. Nicholas Jackiw (1966–) created Sketchpad as part of a visual geometry project headed by Eugene Klotz and Doris Schattschneider (1939–). Geometry educational software continues to be developed, including open source versions. Graphing calculators and computer algebra programs allowed for easy visualization of sophisticated curves and surfaces.

There has long been debate in geometry education regarding which topics should be taught, including a tension between practical applications and theoretical considerations. For instance, in some locations and time periods around the world, educators concentrated on geometric techniques for construction, surveying, and navigation, while in others Euclidean geometry was the focus in order to train the mind. Teachers point to Euclid’s philosophy, as noted by commentator Proclus Diadochus (411–485): “They say that Ptolemy once asked him if there were a shorter way to study geometry than the Elements, to which he replied that there was no royal road to geometry.” Educators continue to debate how to teach geometry, such as whether two-dimensional perspectives should be taught before three-dimensional perspectives.

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