Q&A #522

Teachers' Lounge Discussion: Teaching subtraction

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From: Ray M

To: Teacher2Teacher Public Discussion
Date: 1999020807:55:05
Subject: bits, bytes, bases, and borrowing

The intent of my message is a little deeper.  When a student is having
trouble with borrowing, changing to binary can provide an analogy to
draw on.  The manipulatives are simpler to manipulate too.
4.-1. =>100b - 1b =11b only requires 4 unifix cubes on the desk.

At the other extreme, I can find a base for any given problem that
will not require borrowing.  Notation may be a problem but....     
WHEN I borrow is an artifact of base 10.

And octal and hex provide generalizations of the borrowing.  Hex and
binary are used daily by thousands of EE's so they are "useful".   The
translation from hex to binary and binary to hex is nearly transparent
(FFFh=1111 1111 1111b)   but the borrowing behavior is very different.

One other practical trick, since you mentioned fifth grade.  Some of
the various hand positions in binary finger counting are associated
with certain obscene gestures (645. being particularly rude).   Keep
the back of your hand against a wall or flat on a desk or towards the
ground until you can count very fast.  I generally do this exposure to
binary in kindergarten, so it's not such a problem.

An interesting spin on it is to use the left hand either for larger
numbers OR by crossing the arms a fractional notation:
left thumb = 1/2
left index = 1/4
l. mid =1/8
l. ring =1/16
l. pinky = 1/32
just like on an inch ruler.  And of course now you can demonstrate
carrying from fractions to whole units and from one flavor of fraction
to another.

RE: the dots.  I put the dots on the palm side. 3 and 4 year olds can
count the exposed dots to convert binary to decimal.  The dots are
three dimensional.  Demonstrating that in K-6 is a jaw dropper.  I
have three demos:  a helium lamp with optical flats (very abstract),
an electronic gage (less abstract), and finally least expensive and
most concrete, a hardened steel pin and bushing.  The pin goes in the
hole and spins freely when clean but binds if you draw on it,

Re: semi-concrete.  Much of what Piaget did is bunk.  Sometimes he
didn't understand the physics (billiards), sometimes the methodology
was flawed (water levels), and often his conclusions overreach the
actual experiment.

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