In simplistic terms, an algorithm is a series of mathematical steps for completing a computation. Each math operation (addition, subtraction, division and multiplication) has such steps to solve a math problem. The series of basic steps are known as algorithms. Over the years, different kinds of algorithms have been developed and renamed. In the Everyday Mathematics Program, the breakdowns are called focused algorithms because they look at specific steps and focus on one basic concept.
To complete a problem using the traditional subtraction algorithm, students need to understand the concept of borrowing. However, in a 1984 study by Dr. James Hiebert, it was revealed that only 60 percent of 10-year-old children in the United States achieved mastery of the math concept of borrowing from a higher number. Similar results were obtained in studies completed in Japan, and better methods of teaching subtraction were developed.
In the partial differences method, students are taught to operate on each column of a math problem individually. In a two-step procedure, students are taught to keep track of the sign if borrowing is needed and combine the results using mental arithmetic. The example here shows the mental arithmetic. Keep in mind that students learning this subtraction method would already have been trained to do partial sums addition. If you have a problem of 325 -- 58, in traditional subtraction, you would need to borrow to get the correct answer. In the partial differences method, you are trained to think of the problem in a series of whole numbers. The number 300 would be put off by itself. Then you mentally add together +20 and -50. Next, mentally combine +5 and -8. Finally, combine all the numbers from each value and you get your answer. The sum of 300 plus -30 plus -3 gives you 267. Although this seems more complicated to those who have already mastered the traditional borrowing method, for children new to math and the concept of borrowing from tens and hundreds, this type of focused math is working.
Partial sum methods are used for teaching subtraction, addition and multiplication. Studies indicate that children who are taught using these focused algorithms seem to understand the concepts of ones, tens and hundreds more quickly. Since partial sum methodologies tend to help with student understanding, the concepts will be used more extensively in teaching everyday math and focusing student learning.