When we teach any science, we recognize the importance of lab work, yet often math is taught only in the abstract. However, in real life, math is nearly always hands-on: we count the money, double the recipe, measure the wood, or estimate how to make the treats last until the final kid has come to the door on Hallowe’en.
As I was talking to instructors in preparation for writing this manual, they told me that they found many barriers to using manipulatives. Some said their programs didn’t have any, and there were no funds to buy any; some instructors themselves weren’t comfortable using them, so they weren’t comfortable using them with students; some were afraid that the students using manipulatives might get into places that the instructors had no explanation for; many said that their students resisted using them. These are major barriers, and the results of using manipulatives had better be worth the time and mental energy it takes to overcome them.
I think most people would agree that manipulatives such as base ten blocks or fraction pieces provide a model of mathematical operations to supplement verbal explanations (themselves abstract) of abstract processes. For example, the two cubes that show 1 and 1 000 are clearly not the same size, and illustrates physically the difference in the value of the digit 1 in the numbers 1 and 1 000.