The Diversity of Algorithms

     Algorithms are the short-cuts and tricks we use to complete operations in math quickly, and often without even thinking about it because we have done them so many times. The National Council of Teachers of Mathematics said in 1996 that, "Algorithms are generalizations that embody one of the main reasons for studying mathematics- to find ways of solving classes of problems. When we know an algorithm, we complete not just one task but all tasks of a particular kind... The power of an algorithm derives from the breadth of its applicability." We have been using algorithms from the time we started learning math, and often don't even realize that there are many ways to do a problem because we are so used to taking the same steps. As educators it is very important for us to remember that there are so many different algorithms out there, and they are very difficult to learn and perfect, and that not all of our students were taught the same algorithm the same way.
     In most American classrooms children are taught subtraction using what we like to call the Standard Subtraction Algorithm. This algorithm uses a method of borrowing tens if necessary to always have a larger number as the minuend (top portion of a subtraction problem) and a smaller number as the subtrahend (bottom portion) in each place value. For example lets look at this problem...

In this case we are subtracting 32-15. The Standard Algorithm begins work with with the column that is furthest to the right, which is in this case the ones column. If we were to only look at the ones column as its own problem we would wind up with a negative number because the minuend is smaller than the subtrahend; however, since we are not only working with the ones column, we borrow a ten, literally. We take one ten out of our tens column, turning 3 into 2, and add that ten to the ones column turning 2 into 12. Now we simple subtract down each column. 12-5=7, 2-1=1 to wind up with 17.

     Now if we were to do this same problem using the Austrian Algorithm, the technique that most European students learn when first learning how to subtract,  we will do things a little differently, but still wind up with the same answer. With this technique, when we borrow from the tens column, instead of decreasing the value of the minuend in the tens column by one, we increase the subtrahend in the tens column by one.

With this algorithm, the one that is written in the center of the problem serves as the +1 for the subtrahend of the tens column, and the +10 for the minuend of the ones column. After placing the one in the middle you complete the problem by subtracting the subtrahend of the ones column from the new minuend of the ones column, 12-5=7, then subtracting the new subtrahend of the tens column from the new minuend of the tens column, 3-2=1. Then you put the numbers in their appropriate place values to make your final answer 17.

     Using unfamiliar algorithms can be very frustrating at first, but once you start to feel comfortable with one, you can see the similarities between them. This website, Alternative Subtraction Algorithms, breaks down several different algorithms, including the Standard Subtraction Algorithm, and the Austrian Algorithm and shows how both work using different visualizations of the problems, and is very helpful when comparing algorithms.