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""The teachers are eager and able to learn. I vividly remember one summer class when I taught why the multiplication algorithm works for two-digit numbers using base ten blocks. I have no difficulty doing this with third graders, but this particular class was all elementary school teachers. At the end of the half hour, one third-grade teacher raised her hand. 'Why wasn’t I told this secret before?' she demanded. It was one of those rare speechless moments for Pat Kenschaft. In the quiet that ensued, the teacher stood up." The "Secret": http://www.tech4mathed.com/MAT156/topics%20test%202/twodigit...

Since there is a strong, strong chance that many of the readers who were taught in a U.S. classroom never saw this, I have the above link included.

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for each digit in the bottom number, starting with the ones digit: begin a new line beneath the current state. write a number of trailing zeroes equal to the number of digits of the bottom number you've already processed. for each digit in the top number, starting with the ones digit: write (carry + top digit * bottom digit) % 10 to the left of the leftmost digit on the bottom line of the state. the carry for the next digit of the top number will be (carry + top digit * bottom digit) / 10 use the addition algorithm to compute the sum of all the numbers you wrote this way.