Brief History of Geometry Education Pythagoras Proclus the

Brief History of Geometry Education

Pythagoras Proclus, the last major Greek philosopher, who lived around 450 AD wrote: • After [Thales, etc. ] Pythagoras transformed the study of geometry into a liberal education, examining the principles of the science from the beginning and probing theorems in an immaterial and intellectual manner: he it was who discovered theory of irrational and the construction of the cosmic figures. • I emulate the Pythagoreans who even had a conventional phrase to express what I mean "a figure and a platform, not a figure and a sixpence", by which they implied that the geometry which is deserving of study is that which, at each new theorem, sets up a platform to ascend by, and lifts the soul on high instead of allowing it to go down among the sensible objects and so become subservient to the common needs of this mortal life.

Plato’s Academy • . . . the exact sciences - arithmetic, plane and solid geometry, astronomy, and harmonics - would first be studied for ten years to familiarise the mind with relations that can only be apprehended by thought. Five years would then be given to the still severer study of 'dialectic'. Dialectic is the art of conversation, of question and answer; and according to Plato, dialectical skill is the ability to pose and answer questions about the essences of things. The dialectician replaces hypotheses with secure knowledge, and his aim is to ground all science, all knowledge, on some 'unhypothetical first principle'. • Plato's Academy flourished until 529 AD when it was closed down by the Christian Emperor Justinian who claimed it was a pagan establishment. Having survived for 900 years it is the longest surviving university known.

Seven Liberal Arts • • Based on classical studies Grammar Rhetoric Logic Arithmetic -- Number in itself･ Geometry -- Number in space･ Music, Harmonics, or Tuning Theory -Number in time･ • Astronomy or Cosmology -- Number in space and time

Alva Walker Stamper • Educational Aims in the Teaching of Elementary Geometry, Historically Considered (1909) • Practical versus logical location dependant Ex: Euclid (England) versus Spherical Trigonometry for Navigation (Italy) • United States 1844 first required geometry for entrance into universities

Dr. Jennifer Bergner • 1865: Geometry made part of college entrance requirements (Yale 1865), hence entering secondary curriculum as “formal, demonstrative” subject. “Informal” geometry advocated for lower grades but makes minimal appearance. • 1894: NEA’s Committee of Ten convened (wanted geometry to replace some arithmetic in lower grades)

Dr. Jennifer Bergner 1899: Committee on College Entrance Requirements-encourages geometry to be included in grades 7, 8 1910: Junior high movement, informal geometry gains a foothold in the curriculum of grades 79 1911: National Comm. Of 15 on Geometry syllabus: “algebra in the 9 th, plane geometry in the 10 th, …. ” 1963: Cambridge report- integrate geometry throughout curriculum starting in grade K

David W. Stinson • “Geometry is divided into the speculative and practical. The former is a science that teaches the mind how to form ideas, and demonstrate the truth of geometrical propositions. The latter, or practical Geometry, conducts the hand in working. ” – Sebastian Le Clerc, Practical Geometry – 1690 • “Geometry has often been the area of mathematics where many of the most important developments in mathematics first appeared. It is not an accident that geometric approaches to problems have often been followed by later ‘algebraization’ of the same ideas. ” – Joseph Malkevitch, Geometry for a New Century, Focus: The Newsletter of the Mathematical Association of America - December 2000

David W. Stinson NCTM Yearbooks • The Teaching of Geometry – 1930 • Geometry in the Mathematics Curriculum – 1973 • Learning and Teaching Geometry, K– 12 – 1987

David W. Stinson (1987 Yearbook) How is geometry best taught? How does a student learn geometry? Van Hiele Model • Level 0: Visualization – students are aware of space, but only as something that exists around them. • Level 1: Analysis – through observation and experimentation students begin to discern the characteristics of figures. • Level 2: Informal Deduction – students can establish the interrelationships of properties both within figures and among figures. • Level 3: Deduction – the significance of deduction as a way of establishing geometric theory with an axiomatic system is understood. • Level 4: Rigor – non-Euclidean geometries can be studied and different systems can be compared.

David W. Stinson • “At the secondary level dynamic geometry environments can (and should) completely transform the teaching and learning of mathematics. Dynamic geometry turns mathematics into a laboratory science rather than the game of mental gymnastic, dominated by computation and symbolic manipulation, that it has become in many of our secondary school. As a laboratory science, mathematics becomes an investigation of interesting phenomena, and the role of the mathematics student becomes that of the scientist: observing , recording, manipulating, predicting, conjecturing and testing, and developing theory as explanations for the phenomena. ”

NCTM Standards • Geometry and Measurement Standards • Role of dynamic geometry software and non-Euclidean geometry • Best practices in learning and teaching geometry?

Example: Why 2 -D before 3 -D?

Example: Why 2 -D before 3 -D? • 2 -D is easier than 3 -D if one thinks of geometry as analysis using equations • Piaget and recent studies confirm that children enter school more adept at 3 -D and that 2 -D is in fact both a mismatch for our cognition and very artificial in terms of children's experience. • Inuit do not even have words for 2 -D objects. A square is one image of a cube. Known for spatial abilities.

Dr. Walter Whiteley • The Decline and Rise of Geometry in 20 th Century North America • Many teach college geometry as an important past accomplishment (axiomatic study and exercise in logical proofs) instead of a continuing source of new mathematics. • Geometry relegated to a service course for future high school teachers by the sixties and many majors never take a geometry course in college.

Dr. Walter Whiteley • Argues that geometry is on the rise, although often centered outside of mathematics • Applications: Computer Aided Design and Geometric Modeling, Robotics, Medical Imaging, Computer Animation and Visual Presentations, Linear Programming, Science and Engineering • “Geometry is out there and is essential for application. Geometry will be practiced, with or without mathematicians, and with or without an education in ‘geometry. ’ I believe this geometry would be done better if the future practitioners of geometry receive an appropriate preparation in geometry. The ‘geometry gap’ will haunt North America.

Dr. Walter Whiteley • Human abilities - visualization. Geometry is central to a basic human ability - visualization and reasoning with visual and spatial forms. For a variety of reasons, often associated with computers, this ability is playing an increasing role in learning, in memory, in communication, in problem solving and the practice of many professions.

Dr. Walter Whiteley • Suitable resources for learning. The development of dynamic geometry programs for teaching and for research is dramatically changing what researchers, students, and therefore teachers, do when they solve problems in geometry. Companion resources for teaching geometry in a rich way are accelerating the impact within the undergraduate and secondary classrooms of the rise in geometry at the level of research and applications