I love math manipulatives. I really do. Manipulatives allow students to physically model mathematics concepts. But manipulatives are no panacea. Manipulatives have limitations.
The first limitation is that math is far more powerful than the physical manipulatives. Manipulatives should be a bridge to that power allowing students to imagine math beyond the three dimensions. Some Montessori schools have a manipulative that physically models a quadratic equation, you know the one that goes Ax^2 + Bx + C. That "x squared" is called "squared" because it forms a 2-dimension geometric object. If the factors of the quadratic equation are equal to each other, the quadratic equation models a square. If the factors are unequal, the quadratic models a rectangle.
I first saw the intriguing quadratic equation model in a Montessori school in Japan where preschoolers were enthusiastically absorbing the geometrics of the quadratic equation with any resort to pencil and paper. FOIL? Who needs it? The factors were perfectly obvious to them. Add height to the model and now it models three dimensions and volume in the sense of multiple layers of the base (base x height). If the height is "x," now we have a model of an equation in the third degree. We have an "x-cubed." Cubed! How cool is that? Can we build a model in of an equation in the fourth degree? Well, now we have bumped up against a limitation. Children can discover the twin powers of math and imagination to overcome the limitation.
A second limitation of manipulatives is that sometimes it requires substantial training in the symbolism and design of the manipulative before the child can use the manipulative. For some children, imagining that one thing stands for another can create an obstacle to the mathematics itself. Constance Kamii, a researcher of the ways children learn math, found that when young children were given a problem for which they had received no instruction and free access to a variety of manipulatives, writing instruments and paper, children preferred their own constructions over those imposed by others. Children preferred to express their problem solving with pictures they draw themselves to modeling with the manipulatives.
It is an adult myth that children have superior imaginations. Children represent, pretend, or re-enact what they already know. They have trouble with pretending something they do not already know. Adults can learn math with the incomplete sets of manipulatives often found in classrooms. Children may be stymied. Children especially have trouble with strings of representations. Dr. Kamii says manipulatives can end up being "abstractions of abstractions" rather than the concrete representations usually intended. For example. "We do not have enough hundred-flats for every group to make their number. You can use a teddy bear to stand for a hundred-flat if you need to." Young children need the real thing to be quite closely inferred from the representation.
Math manipulatives are analogies. Every analogy breaks down at some point. Math manipulatives are no exception.
Perhaps the most dangerous limitation of manipulatives is the fun. Student teachers have often reported to me that their math methods courses were little more than a term's worth of "playing" with manipulatives. They loved their methods course, but when they got into the classroom they found to their chagrin that they were woefully ill-prepared to actually facilitate the acquisition of mathematics concepts in their students. I have often observed teachers use manipulatives as a fun diversion without ever getting to the point of the mathematics involved. I have seen educators demonstrate the use of manipulatives without ever building the bridge to the concept.
Manipulatives cannot substitute for the teacher's own profound understanding of the fundamentals of mathematics(PUFM). Sadly, nearly every college of education has a version of the course "Principles of Mathematics for Elementary Teachers" because so many elementary education students lack PUFM.
Crossposted st School Crossing.