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Wednesday, November 24, 2010

I Love Math Manipulatives...But

I love math manipulatives. I really do. Manipulatives allow students to physically model mathematics concepts. But manipulatives are no panacea. Manipulatives have significant, often overlooked, limitations.

Mistaken Modeling

Many teachers view math instruction as teaching standard algorithms, that is, teaching students the conventional step-by step recipe for computing an answer. Thus teachers use manipulatives to model algorithms. However, teaching algorithms is not the same as teaching math. For example, the most common explanation for dividing fractions is to multiply by the reciprocal. Multiplying by the reciprocal works because something mathematical is going on. However, we usually teach the superficial procedure and ignore the mathematics. The purpose of manipulatives is to model the mathematics, not the algorithm. The difference is subtle, but crucial.

Manipulatives Cannot Model Everything

Math is far more powerful than physical manipulatives. Manipulatives are merely a bridge to that power. Manipulatives cannot model beyond three dimensions, but manipulatives can lead students to math beyond the three dimensions. Some Montessori schools have a manipulative that physically models a quadratic equation, Ax^2 + Bx + C. 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 geometry of the quadratic equation without resorting to pencil and paper. FOIL? Who needs it? The factors were perfectly obvious to them. Add a “height” factor to model three dimensions. If the height is “x,” we have a model of a third-degree equation. 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. Mathematical representations can express math much more powerfully than physical models.

The Training Curve

It can sometimes require 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. 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 manage 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 models usually intended. For example, a teacher might say “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.” Such instructions only make things more perplexing for the kids.

Impractical for Problem Solving

If manipulatives are used as algorithm aids, students may not be able to solve problems when they have no manipulatives, like during a test. Constance Kamii, who researches 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 think their way through problems with pictures they draw themselves rather than with manipulatives.

Broken analogies

Math manipulatives are analogies. Every analogy breaks down at some point. Math manipulatives are no exception. Manipulatives have lots of features which may or may not be salient to the math. Children may have difficulty understanding which features to pay attention to and which to ignore. For example, Cuisenaire rods are different lengths. Each length is a different color, but the color is arbitrary and has nothing to do with the math. However, the colors are sure convenient because kids can use them to express math without numerals.

Too Much Fun

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 a real classroom with real kids, they found to their chagrin that they were woefully ill-prepared to actually facilitate the acquisition of mathematics concepts. 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.

The over exuberant adoption of manipulatives is yet one more instance of educational pendulum swinging. Good ideas get over-used and misapplied all the time, often turning what could have been promising strategies into just another education fad.


  1. You mentioned preescholers in a montessori school working with algebra blocks...which is actually earlier than when I was exposed to them at a montessori school in america.

    I'll never forget what the teacher said to me: "this is x. nobody knows what it is". I didn't quite get the abstraction right away-I rebelled and said "except God, of course"-but aside from the teacher sheepishly agreeing with me on that point, that was that-I wasn't going to worry or care about the value of x.
    I now believe that by not caring about the value of x, and focusing on the way x interacted with normal numbers, I avoided all sorts of conceptual traps about x. After all, believing "x is a gray bar and nothing else" is considerably easier to self-rectify than believing that x has a definite value and you'll find it by the end of a problem, and it's easier to teach that x sometimes does have a definite value than to teach that it sometimes does not.

    At any rate, after two weeks I transitioned myself to factoring without the algebra blocks, simply because I had the problem solved in my head before I could even touch the blocks, let alone rearrange them into a rectangle.
    I am wondering if you have any idea on when the typical kid will be capable of going from factoring with blocks to without. typical, I mean 5-year old's who don't scream when their parents turn off a feyman lecture so they can go get their thanksgiving dinner. No, I didn't understand what I was looking at, I just knew I liked it.

  2. I don't know when 5-year-olds will be capable of going from factoring with blocks to without. That would be a great study---with a lot of caveats. It might be hard to find a sufficiently large control and treatment sample. Then there is the problem of replication in the wider school system where many elementary teachers are weak with math. Another difficulty is American society's lack of commitment to academic achievement regardless of all the kip service to the contrary.