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