By Burt Kaufman (auth.), Hans-Georg Steiner (eds.)

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Kriewall, Perspectives in Elementary School Mathematics, Chas. , 1969, pp. 50-55. THE PROBLEM OF RELATING MATHEMATICS 49 just what objectives are supposed to be achieved by the instructional program- what, exactly, do you want the learner to do, once he has learned whatever it is that you're teaching. Like faculty psychology, the idea isn't entirely without merit, but - again like faculty psychology- it can easily lead to undesirable classroom practice. There are recorded opinions of some educational theorists to the effect that visiting Westminster Abbey, or listening to Beethoven's last five quartets, cannot be defended as "educationally valuable" unless one states behavioral objectives for them.

OBP = <}::. APB = 90•, showing that the inverse of y is the circle with diameter A'B'. This proof is easily modified to cover the cases when y is a circle passing through 0, or a line not passing through 0. The inverse of such a line y is a circle y' through 0 whose tangent at 0 is parallel to y. 4. 39 INVERSIVE GEOMETRY a point P are equal to the angles formed by their inverse circles, which intersect at 0 and again at P', as in Figure 4. In other words, inversion is an angle-preserving (or "conformal") transformation.

6. INVERSIVE GEOMETRY 5. 41 MID-CIRCLES AND ORTHOCYCLIC POINT PAIRS We recall that two intersecting, tangent, or non-intersecting circles can be inverted into two intersecting lines, two parallel lines, or two concentric circles, respectively. When a circle is inverted into a line, inversion in the circle is transformed into reflection in the line. Given any two distinct circles a and /3, consider the locus of a point P such that two circles, tangent to both a and /3, are tangent to each other at P.