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Teachers' Lounge Discussion: Multiplication with your ten fingers

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From: Andy Harris <aharris@cs.iupui.edu>
To: Teacher2Teacher Public Discussion
Date: 2004040512:58:42
Subject: Re: Chisenbop Tutorial

Hi all, I'm Andy Harris (of the 'Chisenbop Tutorial' web site)

I was doing a 'random' web search in class this morning, and came
across this group's discussion of Chisenbop.  I'm absolutely amazed at
how many people have linked to my chisenbop page.  Thanks for your
references to it.  

Since I seem to have become an unintentional (and unqualified)
authority on the subject, I've been thinking about the problems of
multiplication and division on fingers.  I have two thoughts, one very
traditional, and the other much more profound.  I'd love your comments
on them.

I mentioned a hybrid multiplication system on the web site (breaking a
multiplication problem into repeated additions and using pencil to
store intermediate steps)  I've since heard of another inventive
solution.

Create a paper sheet with the numbers 1 through 10 printed on it in
large text (or of course braille.)  For successive additions, simply
touch the appropriate square with whatever finger is down.  For
example, to multiply two by three, you can add two three times or
three two times.  Let's go with adding two three times.  Move your
zeroed hands over the zero square.  Add two to your hands, and touch
the one, because you've done one addition.  Add two again and move to
the two.  keep going until you reach the three, and your hands will
contain your product.  This technique has the advantage of retaining a
sensible relationship with the mathematical process.  
Of course, division will be very similar.  Start with the dividend
stored on the fingers.and place a finger on zero.  Subtract the
divisor and move to another square.  When you can no longer subtract
the divisor with a positive result, you will be pointing to the
quotient and you will hold the remainder on your fingers.

It seems the basic problem with multiplication (and division) in
finger math is the way chisenbop insists on the base 10 representation
system.  Most of the finger 'tricks' for math rely on switching to
other bases temporarily (whether the user knows it or not)  This is a
baffling prospect for a person struggling with basic multiplication.

However, the base 2 binary system has some unique advantages when it
comes to mathematics (part of the reason it was chosen for most
computers)  In binary math, you only use zero and one.  It's trivial
to perform arithmetic with these digits.  The hard part is converting
them to and from the binary notation.  

In binary, every digit is a power of two, so 

101 is equal to 

    1 * 2^2 (4)
  + 0 * 2^1 (2)
  + 1 * 2^0 (1)

which is 4 + 0 + 1.  

You could represent this value by placing all ones on the table and
raising all zeroes, so 5 would be 101, or the middle finger and pinky
of the right hand.  

I agree it would be difficult to teach students this way of
representing numbers, but it has some profound advantages.  First,
take a look at some basic multiplications

In base 10, 5 * 2 = 10

Look at the same problem in binary:

 5  *  2 =  10
101 * 10 = 1010 

To multiply by two, simply shift all values to the left one digit and
place a zero in the right pinky.  

To divide by two, shift to the right.  (if your pinky is down when you
shift to the right, you have a remainder of 1).

To multiply by four, shift two digits.  8 is a three digit shift, and
so on.

Let's say you want to multiply 3 * 4.  Simply place 3 in binary, then
shift to the left twice.  This works great when you want to multiply
or divide by factors of two (2, 4, 8, 16, ...)  To do other
multiplications, you'd need to work to the closes factor of two then
add or subtract a different factor.

For example, multiply 3 * 5.  This can be thought of in a couple of
ways that work well for binary:

   5 * 2 = 10 (5 shift left once)
 + 5 * 1 = 5  (5 no shifting)

Write down these values then switch to traditional chisenbop to add
them: 10 + 5 = 15 (or of course add them mentally) 

-- 
  You can add directly in binary, but binary addition involves 
  LOTS of carrying, so it would be frustrating for struggling 
  students. 
--

The same problem could be thought of in these additional ways

3 * 5 = 

    5 * 4  = 20 (5 shift left twice)
 -  5 * 1  =  5 (5 no shifting)
            ---
             15

or

3 * 5 = 

    3 * 4 = 12 (3 shift left twice)
  + 3 * 1 =  3 (3 no shifting)
           ---
            15


All these problems require the student to think of multiplication as
factored additions, and force the learner to see factors, a key part 
of efficient mental math.  The most important part of any mental
prosthetic (as calculators and all forms of finger math are) is that
they support good mental models of mathematics rather than replacing
them.  
  
A side effect of the binary technique is its incredible range of
values.  With the traditional chisenbop techniques, the largest value
you can hold on two hands is 99 (10^2 -1)  This limitation is due to
working in base ten, because each hand stores one digit of base ten. 
When you move to a binary system, each FINGER holds one digit of base
two, so the two hands can store an incredible maximum value of 1023
(2^10 -1)  

Let me know what you think of these musings.  I may add them to my
chisenbop site, or (eventually) turn them into a book.

-Andy


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