This symposium explores the role of placeholder structures—systems of words, non-linguistic symbols, or procedures—in the construction and manipulation of numerical concepts. The structure supplied by a placeholder system – like the count list in English – critically constrains the potential for creating and manipulating conceptual content. A series of four talks will describe (1) the role of placeholders in creating exact number meanings using pragmatic inference, (2) the effect of semi-productive count structures on number word use in Momu, a language spoken in Papua New Guinea, (3) the creation of novel placeholder systems in deaf Nicaraguan homesigners, and (4) the use of visual imagery and gesture as structures for doing mental abacus. Each talk will discuss how placeholder structures guide and constrain learning, whether by facilitating the association of symbols with quantities, guiding inductive inferences, or facilitating operations that are unique to a particular structure.