PURPOSE:This demonstration will present information to acquaint the audience with the mathematics behind an African children's game.

OBJECTIVE: To demonstrate mathematical concepts of binary sorting as used in the bean switching game.

MATERIALS:

magnetic board
artificial beans labeled 1 - 16
switch device (a flowchart model with yes's and no's - in Kpelle: "Oa" and "Gbeh") [a worksheet representation of the device is attached]
two checkers to fall through switch device

OUTLINE:

I. Introduction
  A. Welcome the audience to the Demonstration Stage of the Maryland Science Center
  B. Introduce the topic of the demonstration.- that games African children play have the
       same mathematics as that used by professional mathematicians around the world.
II. The game
  B. Stones/Beans (Kpelle of Liberia)
    1. demonstrate the "mind reading trick"
    2. show the flowchart (related to computer programming)
    3. guide a volunteer through the process to "mind read"
IV. Summary
  A. The children who play games like these can grow up to become
      product distributors, computer programmers, academic
      mathematicians, or problem solvers in their community - no matter
      where their community is.
  B. There will be opportunities to do these and other games at explainers
  C. There are related math activities in the Beyond Numbers exhibit
  D. Any questions? Stick around.
 

DEMO PROCEDURES:
 

1. Check all materials and equipment to ensure that they are there and functioning.
    a. switches are all pointing down
     b. beans are in order

2. Adjust stage lighting and have the general announcement for MATH GAMES OF AFRICA played over the PA system.

We would like to invite our visitors to the Demonstration Stage on the second floor for a free live demonstration on "Math Games of Africa". Learn how the games played by children in Africa contain the same mathematics as that used by adult professional mathematicians around the world. Come to the demonstration stage to experience how play can help us go Beyond Numbers.

3. This game is modeled after a Kpelle game that uses pebbles. We are using cloth beans because they are easier to manipulate on the magnetic board.

This game is just a series of simple choices. With a little practice and a decent memory, you can master it quickly. [I do not have a decent memory, but making mistakes in this part of the demonstration does not hurt the presentation. In my case, it seems to be entertaining. - CB] Think of it as a process of elimination. At each point in the game, half of the remaining choices are eliminated. Since we only begin with 16 choices, it only takes four questions to narrow it down to one possibility.
From 2⁴ - 2 times 2 times 2 times 2 is 16.

Call a volunteer up to the stage to play the game.
Describe how to say "yes" and "no" in Kpelle, since the game is played by Kpelle children. Have the volunteer and the audience rehearse.
.
The 16 beans are separated into two groups: 1-8 and 9-16. Have your volunteer choose a number while you look away. When she has chosen one, request that she point it out to the audience without letting you see. When she is finished ask the audience (who should answer "OA" or "GBEH")if she is ready.

Ask your first question: "Is it in the top row?" (This is the only question you will ask). Suppose the volunteer chooses number 5. She would answer "OA". Once your first question is answered, you have then eliminated half of your 16. In this case you know it cannot be 9, 10, 11,12, 13, 14, 15, or 16. Since you know it to be a choice from 1 through 8, the next moves are designed to cut that group in half. Switch the top and bottom even numbers . This way 2,4,6,and 8 will be at the bottom. Once you make the move it is time to ask the question again.

Ask "Is it in the top row?" Here is where the memory comes in. If the answer is "OA" you know it is 1,3,5, or 7. If "GBEH", you know it is 2,4,6, or 8. Since the number is 5, the volunteer would answer "OA". Now with only four choices from which to choose, make a 1-9 switch and a 5-13 switch. [When youíre proficient, you can switch others back and forth as diversions]

For the third time ask "Is it in the top row?" Since the answer for number 5 is "OA", it must be in 3 or 7. If "Gbeh", it must be 1 or 5. Make the 1-9 switch.

For one last time, ask "Is it in the top row". For our example the answer is "GBEH", so by process of elimination the answer is 5.

Now invite the volunteer (or a new volunteer, if you prefer) to read YOUR mind. Remind him of the question you asked four time: Is it in the top row?

Move to the 'computer'. Then explain that he will ask you the same questions and each time you answer OA, he moves the black pointed part of the appropriate lever on the computer over to OA. When he says GBEH, he moves the black pointed part of the lever to GBEH. (Our levers are the opposite of gas stove levers, so have the volunteer practice to be aware of this). Present the check and say that it will show him which choices to make and which beans to switch.

There is a flowchart available for download that does the same thing as the Maryland Science Center's mechanical "computer"	Because of its size, it comes in four sections that overlap a bit for easy fitting. To download each quarter chart, click on the image.

Place the checker in the hole at the top of the 'computer'. Then instruct the volunteer to look away as you you the number for a bean and show it to the audience.

As the volunteer stands by the ìcomputerî have him ask you "Is it in the top row?". When you answer, help him push the first lever in the proper direction. Now tell your volunteer that the computer is giving instructions on which beans to move in the little yellow square next to the lever you've just turned. Switch the beans indicated or have your volunteer do so. Continue until the end, making sure to help point out which lever is the appropriate one to turn (the one just below the checker), and pointing out which instructions come next.

At the end, explain how this was binary sorting. Every question eliminated half of the remaining possibilities, etc. Mention how it is used everywhere from computer programming to finding a word in the dictionary. Point out the the OA's and GBEH's on the computer are like the on's and off's in a computer's electrical flow.

4. Finish - Thank the audience for coming, make connections with museum exhibit components, invite questioners to stick around, and clean up.

Put all switches back to starting position. Put away ropes and costumes. The beans and flowchart may be placed on a cart and used immediately as an explainer.

Hello there! My name is Paul Roth and I'd like to welcome you to the Maryland Science Center. The demonstration I've got for you today is about the mathematics found in children's games, in particular, games from Africa.

Mathematics is used by everyone. If you can think, you can do mathematics. How many of you have ever played tic-tac-toe? Well, that's a game involving logic and...mathematics. Some people would be surprised to learn that there are many children's games that contain the same mathematics used by professional mathematicians all around the world.

Our game appears to be a mind reading trick but really involves mathematics.. The Kpelle children of Liberia use stones and memory skills to play this game. First, I need a voluteer. [Choose a volunteer].

During this game because it is a game from the Kpelle children of Liberia, you will have to speaking in Kpelle. [!!!] Do you know Kpelle? Well all, you need today is two words and I'll teach them to you. In their language, they say "OA" for "yes". It rhymes with Okay. Can you say OA volunteer? Good. Can you say OA, audience? Great. And they say GBEH, sort of like bah in "Bah! Humbug" for "no". Can you say GBEH? Magnificent! Audience, can you say GBEH? Outstanding!

Please choose to number on the board and point it out to the audience, but don't let me see. [Hide your eyes.] Have you chosen? [Volunteer says "no", you say "what?". Volunteer says GBEH. You say,"Thatís better!"] Audience, don't give away the number... but do you all know it?

Here we go! Is the number in the top row? [Have both the volunteer and the audience respond. Then switch the beans.] Is the number in the top row? [Switch the beans. Do this two more times.] The bean is ____. Let's give our volunteer a big round of applause.

That was NOT mind reading ... it was mathematics. So let me have another volunteer to try to figure out which bean I choose. [pick a volunteer] I'll give you a little help to do this. You can use this machine to help show you what to do. This machine is like a flow chart that computer programmers use. [Pick up a checker] This is the thought process your brain goes through to make the choice. You ask me the four questions I asked you, and if I say "OA", move the lever to the left. If I say "GBEH", move the lever to the right, like this [Show how to use the levers] Now I'll choose a number ... don't look. [You choose a number and be sure the audience and not the volunteer sees it.] The volunteer asks the questions and plays the game. Each time the audience responds "OA" or "GBEH", the volunteer switches a lever]. So what's the number? [Point to the number if necessary.] Let's give our volunteer a hand!

This is called binary sorting. Every time the question is asked, the answer eliminates half of the choices. And if you start with 16 choices and keep dividing in half, you'll get down to one choice. Ant that one choice was my answer. This is used buy computer programmers and probably by you when you're trying to make decisions like finding your place in the dictionary.

The OA's and the GBEH's , yes's or no's are like the on's and off's of the electrical current in a computer.

I hope that you now understand that math is more than just adding and subtracting numbers and stuff like that - that math is a part of everyoneís life and that math can take you on a fantastic journey. A journey BEYOND NUMBERS!
 

Misconception - that the only contributions to modern mathematics from Africa are children's games.
Fact - Countries in Africa have computer programmers, system designers, mathematicians and problem solvers.

Misconception - that the African children play games related to American mathematics.
Fact - Countries in Africa have computer programmers, system designers, mathematicians and problem solvers.

Misconception ? that these games were invented in Africa.
Fact - This is not known. This demonstration stresses the process of mathematics rather than a collection of facts.

Misconception - that these games were played by ancient Africans back in "the old times".
Fact - We don't know. These games are played in present-day Africa.

Misconception - All Africans are alike.
Fact ? There are many many cultures and many many languages in Africa.

Misconception - Africa is a country.
Fact - Africa is a continent.

The following does not need to be said, but the demonstrator should be aware that many African-Americans are looking for information concerning their potential in science and mathematics. Many ethnic and racial groups are under represented in math-related fields.
 

BIBLIOGRAPHY:
Zaslavsky, Claudia; Africa Counts, Prinde, Weber & Schmidt, 1973
Zaslavsky, Claudia; Multicultural Math, Scholastic Professional Books, 1994

Copyright: 1995 Maryland Science Center

Authors:
From the Maryland Science Center - education department

Mary Olenick, while doing research for the Maryland Science Center exhibits department, was able to find out the Kpelle words for "yes" and "no" from the Liberian Ship owners Council.