Computer programs like Sketchpad, Cabri, and Cinderella can be very useful at certain stages of learning. Geometry can be based on coordinates, vectorspaces, transformation groups -- I love them all. But today I wish to put in a good word for old-fashioned, hands-on Euclidean geometry, which has all but disappeared from the curriculum.

One of the most important events of his life "as dazzling as first love" -- that is how Bertrand Russell characterises his encounter with Euclid. No doubt young Bertie's cognitive libido was set ablaze by the seamless perfection and the sheer cleverness of it all. For lesser mortals, the old Greek master's mixture of ontological naiveté and sophistcated problem-solving may form a similar attraction. Moreover, here is a wide panorama of classical mathematics with not a number or formula in sight. As René Thom once remarked: it has no heuristics, every problem starts from zero.

Euclid's geometry has two advantages over more "modern" ones: it is construction oriented, hence gets manual tools involved; it allows (mentally) moving single objects without moving the whole plane (a difficult notion).

The theorems about adding squares (Pythagoras) and about angles on a circular segment (Thales) already suffice for solving a great variety of problems. The concurrence of perpendicular bisectors, altitudes ( = bisectors in the doubled triangle) and medians is unexpected -- and the collinearity of the points of concurrence (Euler) even more so. Proofs are short and elegant: to see Euler's line, one need only reflect the figure through the meeting point of the medians and stretch it by the factor 2. In algebra, mathematical beauty appears much later.

In the course of his strong engagement in curricular reform around 1900, the great Felix Klein, pointed out a double discontinuity: the freshman at university must forget his math habits from school; returning to school as a teacher, the alumnus quickly learns to forget his (useless and alienating) university math.

Universities offer a plethora of geometry courses: linear, algebraic, differential, axiomatic -- each in several flavours -- plus topology, graph theory, etc. Most of them demand a prior, and often advanced, study of algebra and analysis; all of them require a rigour unattainable in school. The student (e.g., the future teacher) has to learn a different way of thinking -- about mental constructs with very scant sensory support.

Mathematics progresses by developping new perspectives. That is why the average student of today can routinely handle problems which Newton found hard. As you climb a mountain your experience remains rich and intense -- but you no longer hear the sounds of the villages below. The prospective teacher, however, must not lose that connection. Teachers can be enthusiastic only about what they know well and love. They cannot be expected to turn group characters into trigonometry or compact manifolds into ellipsoids. Adding water to brandy does not make wine.

There was a time when teachers taught teachers. The example shown on the left (1928) stresses problem-solving, ingenuity, and student initiative (plus ça change ...). Today's math educator, having to publish research articles, can ill afford to write such "expository" books. Math departments must now do their share. The feasibility of a junior/senior level course, thoughtfully and creatively following Euclid and his moderniser Hilbert, is being demonstrated by Berkeley's Robin Hartshorne.

On the first of the pages on the left, the expert teacher (Philipp Maennchen) keeps students interested by asking them to perform constructions around ever increasing obstacles: drawing lines from a given point inside a semi-disc toward its missing corners; drawing a tangent near the edge of a circular arc, etc. On the other page, he uses the surprise effect: every student draws an arbitray triangle, selects any point on each of the three sides, and draws a circle around each of the three resulting corner triangles. They all report that their three circles intersect in one point, and are now curious to know the reason. This is used in preparation for a study of Fermat's Point.

Besides the two already mentioned, Euclidean synthetic geometry has a third advantage: it is well tested, has a large crowd of fans, and a huge body of problems which are neither isolated nor algorithmic. Though scarcely present in recent Western curricula, it is still flourishing in under-developed countries and on the International Mathematical Olympiad. Hartshorne's lecture notes have grown into a stately Springer book. A revival may be in the offing: the drought might soon be over.