Editing Transformation geometry (section)

== Use in mathematics teaching ==
An exploration of transformation geometry often begins with a study of [[reflection symmetry]] as found in daily life. The first real transformation is ''[[reflection (mathematics)|reflection]] in a line'' or ''reflection against an axis''. The [[function composition|composition]] of two reflections results in a [[rotation]] when the lines intersect, or a [[Translation (geometry)|translation]] when they are parallel. Thus through transformations students learn about [[Euclidean plane isometry]]. For instance, consider reflection in a vertical line and a line inclined at 45° to the horizontal. One can observe that one composition yields a counter-clockwise quarter-turn (90°) while the reverse composition yields a clockwise quarter-turn. Such results show that transformation geometry includes [[non-commutative]] processes.

An entertaining application of reflection in a line occurs in a proof of the [[one-seventh area triangle]] found in any triangle.

Another transformation introduced to young students is the [[homothetic transformation|dilation]]. However, the [[inversive geometry#reflection in a circle|reflection in a circle]] transformation seems inappropriate for lower grades. Thus [[inversive geometry]], a larger study than grade school transformation geometry, is usually reserved for college students.

Experiments with concrete [[symmetry group]]s make way for abstract [[group theory]]. Other concrete activities use computations with [[complex number]]s, [[hypercomplex number]]s, or [[matrix (mathematics)|matrices]] to express transformation geometry.
Such transformation geometry lessons present an alternate view that contrasts with classical [[synthetic geometry]]. When students then encounter [[analytic geometry]], the ideas of [[coordinate rotations and reflections]] follow easily. All these concepts prepare for [[linear algebra]] where the [[reflection (linear algebra)|reflection concept]] is expanded.

Educators have shown some interest and described projects and experiences with transformation geometry for children from kindergarten to high school. In the case of very young age children, in order to avoid introducing new terminology and to make links with students' everyday experience with concrete objects, it was sometimes recommended to use words they are familiar with, like "flips" for line reflections, "slides" for translations, and "turns" for rotations, although these are not precise mathematical language. In some proposals, students start by performing with concrete objects before they perform the abstract transformations via their definitions of a mapping of each point of the figure.<ref>[https://www.jstor.org/stable/2319962 R.S. Millman – Kleinian transformation geometry, Amer. Math. Monthly 84 (1977)]</ref><ref>[http://unesdoc.unesco.org/images/0013/001365/136586eo.pdf UNESCO - New trends in mathematics teaching, v.3, 1972 / pg. 8]</ref><ref>[http://scholarcommons.usf.edu/cgi/viewcontent.cgi?article=4616&context=etd Barbara Zorin – Geometric Transformations in Middle School Mathematics Textbooks]</ref><ref>[http://unesdoc.unesco.org/images/0012/001248/124809eo.pdf UNESCO - Studies in mathematics education. Teaching of geometry]</ref>

In an attempt to restructure the courses of geometry in Russia, Kolmogorov suggested presenting it under the point of view of transformations, so the geometry courses were structured based on [[set theory]]. This led to the appearance of the term "congruent" in schools, for figures that were before called "equal": since a figure was seen as a set of points, it could only be equal to itself, and two triangles that could be overlapped by isometries were said to be [[Congruence (geometry)|congruent]].<ref name="russianedu">[https://books.google.com/books?id=qwyBPybT4oMC Alexander Karp & Bruce R. Vogeli – Russian Mathematics Education: Programs and Practices, Volume 5], pgs. 100–102</ref>

One author expressed the importance of [[group theory]] to transformation geometry as follows:
:I have gone to some trouble to develop from first principles all the group theory that I need, with the intention that my book can serve as a first introduction to transformation groups, and the notions of abstract group theory if you have never seen these.<ref>[[Miles Reid]] & Balázs Szendröi (2005) ''Geometry and Topology'', pg. xvii, [[Cambridge University Press]], {{isbn|0-521-61325-6}}, {{MathSciNet|id=2194744}}</ref>