Date: Apr 10, 1995 4:55 PM
Author: Joan Reinthaler
Subject: Manipulatives

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