Dear Jerry,I cannot find en e-mail address to Michael Paul Goldenberg, but I would like to share the following with him in response to his mail via you.
In the 1980s I was approached by a mathematics teacher who taught adults early mathematics. In his class he had a group of gypsies and they had taught him how they multiply numbers between 5 and 9 with the fingers.
The teacher could not figure out why it worked and wanted me to explain it, so I did. Later I gave this task to my secondary students to solve, and I see it as a good modeling task and exercise in use of algebra for them.
I have published the task in a book in 1989 called The Challenge - Problems and Mind-Nuts in Mathematics (in Swedish) and attach here the one page of that book translated into English of the task I call Handy multiplication (fingerfÃ¤rdig matematik in Swedish).
This seems to be old folk knowledge and I have seen old Swedish books where the method is explained and also a method for how to multiply numbers between 11 and 15. We are into ethnomathematics here!
I wrote to Barbro and received permission to reproduce both his message to me above and what he sent from his book:
…to obtain the products for all multiplication facts from 5 X 6 through 10 X 10
How to “read” your fingers and obtain the products can be modeled with children using base 10 manipulatives…each “up finger” is equivalent to a 10-rod, and obtaining the product of the “folded finger(s)” times the “folded finger(s)” can be modeled with unit-cubes. The final product is then obtained by summing these two (2) partial products.
To model with your fingers, begin with a clenched fist, which is worth five (5), and then each raised digit on that same hand is worth one (1) more. For example, the clenched fist is equivalent to five (5), whereas if two fingers are raised, that same hand is now equivalent to seven (7) (…b/c the five (5) for the hand, plus two (2) more for the two (2) raised fingers, 5 + 2 = 7). To more easily “read” your “folded fingers,” it is always recommended to rotate both of your wrists so that your palms are toward your face.
* Each clinched fist is equivalent to five (5).
* Each additionally raised finger on that particular hand is worth “one (1) more,” therefore three (3) raised fingers on the right hand would make the right hand equivalent to eight (8).
* When “reading” your hands, each hand represents a factor from five (5) to ten (10).
* Once a factor is displayed on each hand, remember that each “raised finger” represents a ten (10), therefore skip count by ten (10) for each raised finger and place this final product in your head.
* Now it’s time to “read” the “folded fingers.” Count the number of folded fingers on the left hand, and then count the folded fingers on the right hand…multiply these two (2) smaller factors together.
* To obtain your final product for the two (2) factors, one (1) factor displayed on each hand, sum the two (2) obtained partial products (…one from the 4th step “raised fingers,” and one from the 5th step “folded” times “folded”).
Now here’s the algebra to prove this phenomenon:
Let x = the number of “raised fingers” on the left hand, and…
let y = the number of “raised fingers” on the right hand…remembering that the hand itself is equivalent to five (5), making the left hand now worth 5 + x and the right hand now worth 5 + y.
Therefore, we are finding the product of (5 + x) (5 + y).
(5 + x) (5 + y) =
10 (x + y) = 10 times the sum of the “raised fingers” from both hands and (5 – x) (5 – y) is the product of the “folded fingers” from both hands, and therefore, the final product is the sum of these two (2) partial products.