Description and History
Manipulatives are concrete objects that can be viewed and physically handled by students in order to demonstrate or model abstract concepts. They represent a category of mathematical tools that are referenced in mathematics standards such as the Mathematics Process Standards included in Principles and Standards for School Mathematics (NCTM, 2000) or the Standards for Mathematical Practice included in the Common Core State Standards for Mathematics (National Governor’s Association et al, 2010).
John van de Walle and his colleagues (2013) define a mathematical tool as, “any object, picture, or drawing that represents a concept or onto which the relationship for that concept can be imposed. Manipulatives are physical objects that students and teachers can use to illustrate and discover mathematical concepts, whether made specifically for mathematics (e.g., connecting cubes) or for other purposes (e.g., buttons)” (p 24). More recently, virtual manipulative tools are available for use in the classroom as well; these are treated in this document as a tool for teacher modeling and demonstration.
The history of manipulatives for teaching mathematics extends at least two hundred years. More recent important influences have included Maria Montessori (1870–1952), Jean Piaget (1896–1980), Zoltan Dienes (1916–), and Jerome Bruner (1915–). Each of these innovators and researchers has emphasized the importance of authentic learning experiences and the use of concrete tools as an important stage in development of understanding. Piaget (1952) suggests that children begin to understand symbols and abstract concepts only after experiencing the ideas on a concrete level. Dienes (1960) extended this to suggest that children whose mathematical learning is firmly grounded in manipulative experiences will be more likely to bridge the gap between the world in which they live and the abstract world of mathematics. Their pioneering work has led to many studies of the importance of manipulatives for student learning in mathematics.