Expressing Generality and Roots of Algebra

Abstract

This chapter makes the case that the heart of teaching mathematics is the awakening of pupil sensitivity to the nature of mathematical generalization and, dually, to specialization; that children who can walk and talk have shown plenty of evidence of the requisite thinking; that algebra as it is understood in school is the language for expression and manipulation of generalities; and that the successful teaching of algebra requires attention to the evocation and expression of that natural algebraic thinking. There is no single program for learning algebra through the expression of generality. It is a matter of awakening and sharpening sensitivity to the presence and potential for algebraic thinking. Some examples of tasks which have been exploited in this way are presented with comments about some of the difficulties encountered.