Place value is perhaps the most fundamental concept imbedded in the elementary and middle school mathematics curriculum. Correctly solving problems that involve computation of whole and rational numbers is dependent upon understanding and expressing multidigit quantities. “It is absolutely essential that students develop a solid understanding of the base ten numeration system and place-value concepts by the end of grade 2. Students need many instructional experiences to develop their understanding of the systems including how numbers are written (NCTM, 2000, p. 81). Yet, knowing when to exchange groups of ones for tens or how to handle a zero in the hundreds place when subtracting, for example, confuses many students who then struggle with algorithms. Learners can correct these and other misunderstandings by solving real-world problems with hands-on materials and learning aids such as counters, base ten manipulatives, and place value charts. “Understanding and fluency are related . . . and there is some evidence that understanding is the basis for developing procedural fluency” (Kilpatrick, Swafford, & Findell, 2001, p. 197). Correctly recording numerals in the quotient when dividing 348 by 30 is an example of demonstrating procedural fluency.

Research indicates that students’ experience using physical models to represent hundreds, tens, and ones can be effective in dealing with place value issues early in the curriculum. The materials should “help them e think about how to combine quantities and eventually how this process connects with written procedure” (Kilpatrick et al., 2001, p. 198). However, “merely having manipulatives available does not insure that students will think about how to group the quantities and express them symbolically” (NCTM, 2000, p. 80). Rather, students must construct meaning for themselves by using manipulatives to represent groups of tens in classroom discussions and in authentic, cooperative activities.

This article will deal with typical multidigit place value errors of both conceptual understanding and procedure. Appropriate remediation activities will be described for each place value error pattern.

### What Is Place Value?

Place value systems are also termed positional systems because the value of a number is determined in part by the position or place it holds. In a decimal place value system, for example, each digit represents a group or base of 10. Place value “pertains to an understanding that the same numeral represents different amounts depending on which position it is in” (Charlesworth P. & Lind, 2003, pp. 308–309). The place value concept enables us to represent any value using 10 symbols (0–9) and compute using whole numbers. Other positional or place value systems include those based on groups of 12, as seen in clock time for counting hours, or groups of 60, for minutes in the hour.

The following are examples of regrouping in base-10 and base-12 system inwhole-number algorithms. Current literature (Ma, 1999) uses the term “regrouping.” It applies to the exchanges of base groups in the four operations of addition, subtraction, multiplication, and division. For example, in

1   745 + 389 ---------- 1134

10 ones in the sum of the ones column, 14, is regrouped to the tens column as “1 ten.”

Likewise, for the following example:

6       14     7 ft    2 in. -  4 ft    8 in. ------------------  2 ft       6 in.

“7 feet” is renamed to “6 feet 12 inches.” The quantity of “12 inches” is combined with 2 inches to be named as “14 inches” when computing in a place value system based on groups of 12.

#### The Hindu-Arabic Numeration System:

The Hindu-Arabic place value numeration system is based on the principle of collection and exchange of groups of 10. In this system, 10 ones can be traded and represented by one group of 10, 10 groups of 10 each can be exchanged and represented as 100, 10 groups of 100 each can be regrouped and represented as 1,000, and so on. This mechanism of collection and exchange makes possible a system in which only 10 unique symbols are necessary to express any quantity.

The total value of a number is determined by multiplying each quantity by the value of its position or place and then adding all those values together. The following example indicates how the total value is found for 47 and for 385.

(4 X10) + (7 X 1) =  47

(3  x 100) + (8 X 10) + (5 X 1) = 385

Several important properties of the base-10 place value system include:

1. Ten unique symbols (0–9) express any numerical quantity.

2. The value of each base-10 place is multiplied by 10 as the digits move to the left from the ones place.

 Quantity: 4 3 2 Place Value: 100 10 1 Total Value: 400 30 2

3. The decimal point is a symbol that enables the system to express parts of numbers. As one moves to the right of the decimal point in a number, the value of each place is divided by 10 (tenths, hundredths, thousandths, and so on). For example, the value of each place is as follows:

0.       2          3          4             5

.1        .01     .001     .0001

4. The zero symbol (0) is a placeholder that represents a set that has no members or elements and is integral to expressing and computing quantity.