Fourth-grade students who participated in a yearlong, whole-class teaching experiment not only reconceptualized natural numbers but also generated flexible solution strategies to perform numerical computations mentally and in writing. Students' reconceptualization of number was mediated by their perceived resemblance between the physical action of splitting an object into parts and the mental action of splitting a number into units. Such a resemblance, evoked by the word "split" as used by the students, can be considered to be a metaphor in the Peircean theory of signs. This spontaneous metaphor mediated students' constructions of numerical strategies. Students communicated such strategies through different signs like numerical diagrams, verbal numerical arguments, and other idiosyncratic arithmetical notations. Students presented the same solution strategy using signs in different semiotic systems and translated signs in one system into signs of another system. Students' arithmetical activity indicated a shift from an instrumental understanding of basic arithmetic to a relational understanding mediated by their "symbolic initiative." The article contains an analysis of the collective construction of this metaphor and its role in mediating the semiotic and arithmetical activity of the students. (Contains 3 footnotes.)