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Re: Manipulatives
Posted:
Apr 11, 1995 8:23 AM


If I may be a little philosophic,...
One possible problem of the misuse of manipulatives is because some teachers believe that manipulative "represent" and "embody" mathematical concepts. If we take the constructivism epistemology seriously, knowledge/concepts do not exist in things out there. They are constructed by individuals. So, we as adults can "see" a red Cuisenaire rod representing/embodying the notion of a half when it is compared to a purple rod, there is no guarantee that children also "see" that relationship because that relationship we "see" is our own construction. It is not in those blocks.
To me the most important criterion for a good instructional material is whether or not students can "think with" it. I think Seymour Papert discussed this idea in his book, Mindstorm. Cuisenaire rods, pattern blocks, base10 blocks, etc. can be wonderful materials if they are available for students to use in the way that makes sense to them. It's just that we can't assume that children will make sense of those materials in the same way we do. When we make that assumption and structure the activities, then, IMHO, manipulatives are being misused.
************************************************************************ * Tad Watanabe email * * Dept. of Mathematics watanabet@toe.towson.edu * * Towson State University watanabet@towsonvx.bitnet * * Towson, MD 21204 tad@midget.towson.edu * * (410) 830  3585 (410) 830  4149 FAX * ************************************************************************
On Mon, 10 Apr 1995, Joan Reinthaler wrote:
> I agree with Linda Coutts that working with manipulatives can become as > rote as other excercises, and certainly one would not want to become > "dependent" on them in the sense that you had to carry them around with > you. I do have a couple of other thoughts about manipulatives, however. > > 1. Too often they are contrived and become an end in themselves  their > connections to the mathematics they are to represent just never really > get made. > > 2. Where well used, however, they can enable students to visualize the > mathematics they are doing in ways that pushing symbols around does not > do. I think that a student who is dependent on *visualizing* pattern > blocks when working with fractions probably has a far better sense of > what is going on then the student who has just learned some procedures. > > For older kids  > When I teach trig, I give my students large unit circles on graph paper > with a radius of ten graph units of .1 each. They approximate the > coordinates of the points on the circle at the intersections of the radii > at pi/6, pi/4, ... etc and write these on the circle. From then on they > can visualize the important symmetries by referring to the circle and > they use the circle on tests and in doing work involving trig for several > years afterward. For these kids the unit circle is a manipulative  they > depend on it, not for values (they have calculators for these if > necessary) but for visualizing and understanding the nature of circular > functions. A kid who doesn't happen to have her circle available will > often sketch it quickly when working on a problem  which is exactly how > I would hope she would think > > Joan Reinthaler > Sidwell Friends School >



