PLACE VALUE MAtEriALs - Center for Innovation in Education

Chapter 11: Place ValuePlace Value ManiPULA tivesPlace value manipulatives fall within two dis tinct de velopmentallevels: concrete and rep resentational. Con crete place value experiencesmust come first. Representa tional activities are appropriate later in theelementary grades — after children have engaged in a variety of experienceswith concrete materials.Concrete MaterialsConcrete materials — beans and cups and Unifix cubes — allow thechildren to construct groupings. Experiences with these materialsstress one-to-one correspondence be tween the number and the materialit represents.Kathy Richardson addresses the issue of one-to-one corre spondenceat the place value level in her book, De veloping Number Concepts Us ingUnifix Cubes (Chapter 5, p. 133):Counting groups [of objects] requires a dif ferent kind of thinkingfrom counting sin gle objects. Children’s first counting experiencesrequire an understanding of one-to-one correspondence.They learn that one num ber word goes with one object. But,when dealing with numbers above ten, children are requiredto count groups as though they were individual objects. Thequestion “How many tens in thirty-four?” as sumes the child canconceive of ten objects as one en tity. The question “How manyhundreds in 346?” as sumes the child can conceive of one hundredob jects as one group (all the while remem bering that eachhundred is also ten groups of ten).Representational MaterialsBean sticks and multi-base blocks are “fixed” materials. The materialsimpose a ready-made structure on the stu dent. Although childrenmanipulate materials — e.g., chips, multi-base blocks and bean sticks— they are re quired to think abstractly. A child with insuffi cient experiencesusing concrete materials may be confused when a numberis represented by multi-base blocks. They must already under standthat one block in base ten represents ei ther ten flats, a hundred longs,or a thousand units. Rather than understanding that a num ber like362 is rep resented by three flats, six longs, and two units, a child mayonly see eleven separate blocks of varying sizes.Chip trading is also representational but at a more ab stract level.Children must be able to understand the “abstraction” that one chipcould represent a group of ten or a hundred chips (chips can alsorepresent groupings in other bases). The necessity for children to besurrounded with an abundance of concrete ex periences before theymove on to representa tional materials cannot be empha sized stronglyenough.Page 11.2© 2011 Center for Innovation in Education, Saratoga, California