Place Value

The way your child is doing mathematics in school likely looks somewhat different from what you remember from your own elementary-school days. Most of us learned to add and subtract using a particular algorithm (a rule or procedure for solving a problem). To add, we were taught to "carry," and to subtract we learned to "borrow." We did pages and pages of computation problems that were unrelated to any particular mathematical context. These assignments were designed primarily to help us remember the steps of the procedure that we had seen in class.

Because these methods are familiar to us, we tend to think of them as a standard for judging computational competency. Unfortunately, students frequently learn these algorithms without connecting them to the meaning of the numbers in a problem. And many adults who learned math this way are unable to figure out simple real-life problems. algorithms were invented to streamline the process by which we compute. They are useful tools, but because they allow us to bypass understandings about place value, they are a place to end, not the place to begin.

The shortcoming inherent in our standard carrying and borrowing procedures is that they focus attention on the individual digits in the numbers rather than on the quantities that the numbers represent. Students who forget the steps of the procedure find themselves making fairly outlandish errors without even realizing they've made a mistake. And even when they do follow the procedures accurately, they often don't understand why they got a correct answer. Here are some examples of mistakes that students commonly make:

58 53

+25 - 16

713 43

The good news is that there are many efficient ways to solve computation problems. In fact, second graders are very capable of constructing their own procedures. Suppose a problem calls for adding 58 and 25. Second graders often solve this type of problem as follows:

Add 50 and 20 to get 70.

Add 8 and 5 to get 13.

Add 70 and 13 to arrive at the correct answer of 83.

This method is as efficient as the "carrying" algorithm, is easy to keep track of, results in numbers that are easy to work with, and takes seconds to carry out. It is superior to the standard algorithm from a mathematical standpoint because the problem solver never loses sight of what the digits represent. And it can be generalized to any problem.

Most of us don't know that other cultures have historically used algorithms that are different from those currently taught in U.S. schools. The following examples, from an article by Randolph A. Phillip titled "Multicultural Mathematics and Alternative Algorithms," published in the November 1996 issue of Teaching Children Mathematics, shows that some adults from other countries were taught the same procedure in their schools that many of our second graders devise:

An older man educated in Switzerland and a man schooled in Canada in the early 1970s both demonstrated that they had learned to add by starting from the left-most column. The man from Switzerland worked the following two problems:

59 481

+16 +926

60 1300

+ 15 100

75 + 7

1407

This algorithm is one that many elementary-school children in the U.S. invent when encouraged to do their own thinking. That is, when asked to add multidigit numbers, most children naturally begin with adding the digits with the largest place value. This is quite natural for adults as well. For example, if two friends emptied their wallets to pool their money, would they first count the $20 bills or the $1 bills?

Of course, solving the problem with the approach shown above requires knowledge of how a two-digit number is composed of a multiple of 10 and 1s, and how numbers can be taken apart and recombined. Children learn these concepts in class through games, opportunities to build mathematical models using manipulative materials, classroom discussions, and the chance to solve many problems. When faced with the task of adding two double-digit numbers together, the children use what they've learned about our number system to come up with a procedure that they understand in order to arrive at an accurate answer. Students have a profound understanding of an approach that they've constructed themselves, and they make fewer errors by using it.

In the case of both addition and subtraction, it is not possible simply to tell children a procedure for doing a problem. Truly understanding what it means to combine two quantities to get a new quantity is a mental relationship that children have to forge themselves. The logical-mathematical knowledge needed to solve both addition and subtraction problems develops over time, arising out of many experiences. We need to respect and encourage children as they move through the natural stages of learning. The process can be uneven and is likely to include periods of confusion as well as learning. Children need a chance to form and reform their thinking as they develop understanding.

Students typically go through several stages when learning to add and subtract. For example, some might solve the problem above by starting at 58 and counting on 25 ones (59, 60, 61, 62,...83). These students have developed an understanding of the meaning of 58+25. If carried out accurately, this method will give a correct solution. However, counting by 1s becomes unmanageable and is prone to errors as numbers get larger. Our goal is to help children find more efficient methods of adding and subtracting. Over time, they will learn to chunk numbers in an addition or subtraction problem so that the numbers are easier to work with. We find that students frequently develop efficiency with subtraction.

The procedures that students develop in the primary grades can be applied to larger problems. When faced with a problem like 1462+1745+278, students have no need for the old carrying algorithm. Instead, they might approach the problem like this:

- 1000+1000=2000 and 400+700+200=1300. That brings the total so far to 3300. (Jot that figure down to keep track of it.)

-Next, calculate 60+40+70=170. Now the total is up to 3470. (Jot that number down.)

-Now it's a simple matter of adding 2+5+8=15 to bring the total up to 3485.

Note that it hasn't even been particularly important to line the numbers up vertically. The child has jotted down only three figures to keep track along the way. And most important, the problem solver can feel confident about the answer because he or she has remained focused on the quantities represented by the numbers, not the individual digits. Approaches like this one are efficient and accurate for solving virtually any problem we might reasonably encounter in life. You might want to try making up some hypothetical problems yourself to get a feel for how this approach works.

In the past, too many children ended up disliking mathematics and believing wrongly that they were not good at it. We need to turn that perception around. Mathematics is all about making sense, so we need to teach it in such a way that sense-making is always apparent. If young children have the opportunity to build a firm foundation of understanding in the realm of numbers, they discover that they can achieve mastery over an important part of our world.

*In addition, please see the videos on the Video Resources tab in the Math section- "How (not why) Borrowing Works" and "Why Math is Taught Differently Now."