Written and Oral Mathematics.

TLDR: For instance, this paper found that children use oral calculation procedures not taught in school, such as decomposition and repeated grouping, to demonstrate their solid understanding of the decimal system, and that children were far more successful in using oral mathematics than written mathematics.

Abstract: Previous studies have demonstrated that children use oral calculation procedures not taught in school. The present study provided evidence for situational variables that strongly influence the tendency to use such procedures. It also provided a qualitative analysis of the oral mathematics used by Brazilian third graders. Concrete problem situations were powerful elicitors of oral computation procedures, whereas computation exercises tended to elicit school-learned computation algorithms. Oral computation procedures involved the use of two reliably identifiable routines, decomposition and repeated grouping, that revealed the children's solid understanding of the decimal system. In general, the children were far more successful in using oral mathematics than written mathematics. An understanding of children's oral procedures may be useful in developing more successful programs for elementary mathematics instruction.