Place Value: Problem Solving and Written Assessment. (Research, Reflection, Practice)

Article excerpt

Elementary school children have traditionally found place value to be difficult to learn, and their teachers have found it difficult to teach. Understanding place value requires a child to make connections among and sense of a highly complex system for symbolizing quantities. Our numeration system is characterized by the following four mathematical properties:

1. Additive property. The quantity represented by the whole numeral is the sum of the values represented by the individual digits.

2. Positional property. The quantities represented by the individual digits are determined by the positions that they hold in the whole numeral.

3. Base-ten property. The values of the positions increase in powers of ten from right to left.

4. Multiplicative property. The value of an individual digit is found by multiplying the face value of the digit by the value assigned to its position.

An understanding of place-value numeration occurs on many levels; it may include an understanding of tens and ones, computational procedures, decimal numerals, binary numerals, or scientific notation. For example, Principles and Standards for School Mathematics recommends that children in grades pre-K-2 should "use multiple models to develop initial understandings of place value and the base-ten system" (NCTM 2000, p. 78). In grades 3-5, children should "understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers and decimals" and "recognize equivalent representations for the same number and generate them by decomposing and composing numbers" (p. 148).

Researchers have suggested that because young children's conceptualization of number and logical-class inclusion are under construction, their understanding of place value may be undeveloped in the early grades (cf. Cobb and Wheatley [1988]; Fuson et al. [1997]; Kamii and Joseph [1988]; Ross [1989]).

Lack of linguistic support may contribute to the challenges young children face in constructing their knowledge of our numeration system in the early grades (Miura et al. 1993). European languages do not explicitly name the tens in two-digit numbers. In contrast, several Asian languages (Chinese, Japanese, and Korean) do name the tens; that is, 12 is said as "ten two" and 52 is said as "five ten two." Kamii and colleagues have argued that in non-Asian-language cultures, young students' emerging understanding of place value is eroded by traditional algorithmic instruction in addition and subtraction, in which individual digits are all treated as "ones" (Kamii, Lewis, and Livingston 1993). First- and second-grade children have demonstrated significant gains in conceptual understanding of place value by participating in full-year studies that treat the learning of multi-digit concepts and procedures as a conceptual problem-solving activity rather than as the transmission of established rules and procedures and t hat give students a variety of conceptual supports (cf. Fuson et al. [1997]; Kamii [1989]).

A Classroom Study

The purpose of this study was to explore the effects of digit-correspondence lessons on students' understanding of two- and three-digit numeration (Ross 1999). In digit-correspondence tasks, students are asked to construct meaning for the individual digits in a multidigit numeral by matching the digits to quantities in a collection of objects. Earlier studies using such tasks have shown that even in the fourth and fifth grades, no more than half the students interviewed demonstrated an understanding that the 5 in 25 represents five of the objects and the 2, the remaining twenty objects (Ross 1989).

In earlier studies, when our research team interviewed students individually using digit-correspondence tasks, we often wondered what would happen if children had an opportunity to interact. What would students learn from their peers if they shared their thinking about the meanings of the digits in digit-correspondence tasks? …