Historical examples of the systematic use of variation.
Although the studies in this section are not transfer studies,they are relevant to the discussion because they demonstrate the historical roots of the systematic use of variation and invariance in instruction.In China there is a well-known approach to teaching math ematics called teaching with variation (Gu,Huang ＆ Marton,2005).Kangshen,Crossley,and Lun (1999) documented the use of this method dating back nearly 2000 years.Gu (1991) nicely captured its appearance in modern practice by demonstrating that more successful mathematics teachers use the teaching with variation approach more often than do less successful teachers.One example of the systematic use of variation can be found in the use of nonstandard figures in order to investigate aspects of geometric figures that students falsely assume to be critical.Polygons are,for instance,always drawn in textbooks with a horizontal baseline.Rotating them can help students realize that angular orientation is not a critical feature of polygons.Additionally,presenting noninstances of a concept can help students become aware of a critical feature.For instance,by presenting a line perpendicular to a radius and intersecting the circle in two points,the students' attention is drawn to the fact that a tangent of the circle has only one point in common with the circle.
Mason and Watson (2005) analyzed textbooks in mathematics used in England from the 16th century on.They demonstrated that in many cases the authors made systematic use of variation and invariance by building sequences of tasks that bring out critical differences between them.
A final example of the systematic use of variation and invariance can be seen in Maria Montessori's pedagogy.She was inspired by two French physicians,Jean Marc Gaspard Itard (1775-1838) and Eduard Seguin (1812-1880).The former became famous for his treatment of the Wild Boy from Aveyron,an 11—or 12-year-old boy who had grown up in the forest without any contact with human beings.An important principle used by Itard was to try to teach the boy to make distinctions by means of contrasts (e.g.,learning to tell apart a round figure from a flat figure by touch,red paper from blue paper by vision,and bitter from sweet by taste).Itard also tried to link the distinctions between sensory impressions to distinctions between corresponding words by means of simultaneous variation in sensory impressions and words (see,for example,Itard,1801/1962;Lane,1977).I can compare Itard's approach with the more common approach of introducing one concept or word at a time and pointing to different instances of that concept (e.g.,looking at a red rose and saying “red”, followed by pointing to a red apple and saying “red”, and so on).The difference between Itard's approach and the more common approach is the difference between the use of contrasts and generalization.