Pythagorean and Platonic Bridges between Geometry and Algebra

Stephen Eberhart
Bridges: Mathematical Connections in Art, Music, and Science (1998)
Pages 121–128


Waldorf Schools (best known to U.S. math teachers through work of Hermann von Baravalle[ l]) have continuing class-teachers who grow with their students from 1st to 8th grade, teaching almost everything (except foreign languages) in 1st and undergoing voluntary gradual diminishment from authority figures to friends by 8th grade (with ever more subjects left to specialty teachers). This is intended both to let ideas grow in natural progressions from year to year, with purposeful continuity, and to help ensure that behavioral problems (on part of both students and teachers) are dealt with and not merely passed on, thus helping to keep instruction lively (and life instructive). But subject-specialized high school teachers (9-12) in such schools face the same difficulty that most teachers have at all levels in other schools: how to maintain freshness when teaching basically the same things (the same "material"), year in year out, for many years? One solution I found was to embrace the vagaries of scheduling and make it a challenge to teach things in different ways at different times of year, befitting the changing seasons. An autumn math class on the Pythagorean theorem might apply it to optimizing bird flight paths over warm land and cold water, while the same class in winter might start with crystal forms and then derive proofs of that theorem from tilings. What follows are some things that have occurred to me as possible treatments of classical topics that would tie in either with hand-work lessons or the experiences of students who had been learning, say, about Celtic crosses in 9th grade art history, undergoing the usual year-long Euclidean geometry course in l0th, or being given a survey of astronomy in 11th. The last topic I have also developed further for master's-level courses in advanced Euclidean and non-Euclidean geometry, as part of secondary school teacher preparation at university.