Abstract
In my Algebra 1 class, solving for the unknown in a linear equation occurred when they had to deal with reversal tasks in patterning situations such as item 4 in Fig. 4.1. Figure 4.2 shows the written work of Dung (eighth grader, Cohort 1), who understood the process of solving for the unknown in the context of finding a particular stage number p whose total number of objects is known. As shown in the figure, he initially took 1 away from 73 and then divided the result by 3. For Dung and his classmates, this particular process of undoing has been drawn from their everyday experiences in which doing and undoing form a natural and intuitive action pair. When my Cohort 1 participated in a teaching experiment on patterning and generalization in sixth grade, the first time I saw them use the undoing strategy occurred in the context of the patterning activity shown in Fig. 4.3. When they were confronted with the situation in item 21, the first thought that came to them was to take away the height of the original cup hold and then divide the result by 3.
A symbol is symbolic if it describes or expresses or stands for an idea but has not yet become an enactive element. It can only become an enactive element if it has meaning, in other words, if there is associated with it at least one icon, an image, metaphor, picture or sense with itself [that] captures a pattern or relationship. Thus the state of being symbolic is highly relative.
(Mason, 1980, p. 11)
It is one of the essential advantages of the sign … that it serves not only to represent, but above all to discover certain logical relations – that it not only offers a symbolic abbreviation for what is already known, but opens up new roads into the unknown.
(Cassirer quoted in Perkins, 1997, p. 50)
The differences between sign-types are matters of use, habit, and convention. The boundary line between texts and images, pictures and paragraphs, is drawn by a history of practical differences in the use of different sorts of symbolic marks, not by a metaphysical divide. And the differences that give rise to meaning within a symbol system are similarly dictated by use; we need to ask of a medium, not what “message” it dictates by virtue of its essential character, but what sort of functional features it employs in a particular context.
(Mitchell, 1986, p. 69)
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Rivera, F.D. (2011). Visual Roots of Mathematical Symbols. In: Toward a Visually-Oriented School Mathematics Curriculum. Mathematics Education Library, vol 49. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0014-7_4
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