Parts, Wholes, and Place Value: A Developmental ViewAuthor(s): Sharon H. RossReviewed work(s):Source: The Arithmetic Teacher, Vol. 36, No. 6, FOCUS ISSUE: NUMBER SENSE (February1989), pp. 47-51Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41194463 .Accessed: 10/09/2012 19:33

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Parts, Wholes, and Place Value: A Developmental View

By Sharon H. Ross

When a student mentally computes 32 + 59 by thinking 30 4- 50 is 80, 9 + 2 is 1 1 , and 80 + 1 1 is 91 , that student has had to call on some well-devel- oped concepts of numerical part- whole relationships and place value. People with good number sense make frequent and flexible use of these two related concepts to perform mental computations and numeric estimates. Students find these concepts difficult; their understanding develops slowly over a period of several years. To be successful at teaching number sense, we must design instruction that re- spects students' need to construct their own knowledge.

Part-Whole Relationships A pupil's numerical awareness devel- ops gradually. First graders explore the variety of ways in which small numbers can be partitioned; 7 can be partitioned as 3 + 4 as well as 1 + 6 and 2 + 5. As young pupils mature in number sense and skills, they become more flexible. They can solve a wide variety of verbal addition and subtrac- tion problems (Riley, Greeno, and Heller 1983) and use related-fact strat- egies to recall basic facts (Cobb and Merkel, in press; Thornton 1978). Eventually their thinking allows them

Sharon Ross is an assistant professor in the department of mathematics and statistics, Cal- ifornia State University, Chico, Chico, CA 95929-0525. She teaches courses for pre service and in-service mathematics educators. She is also codirector of the Chico site of the Califor- nia Mathematics Project, a summer institute and leadership training program for teachers.

February 1989

mentally to compose wholes from their component parts, decompose whole quantities into parts, and per- haps rearrange the parts and recom- pose the whole quantity, confident all the while that the quantity of the whole has not changed.

Place value and part-whole relationships A natural partitioning of whole quan- tities that is frequently used by those with good number sense is that of "tens and ones" - or higher powers of 10 with numbers larger than 99. To use the kind of thinking described in the example of 32 + 59, a student must have mentally constructed a firm understanding of the concept of place value. Yet classroom teachers and researchers repeatedly observe that students' understanding of place value is poor.

Understanding place value requires an elaboration of the student's emerg- ing understanding of a part-whole concept. Many ways can be found to form a two-way partitioning of a num- ber as large as 52: 51 + 1, 50 + 2, 49 + 3, 48 + 4, .... Only one of these, 50 + 2, is directly linked to the mean- ings of the individual digits in our conventional notational system. This representation is known as a standard place- value partitioning. Nonstandard tens-and-ones partitionings like 40 + 12 are useful in understanding the regrouping aspect of computational algorithms.

Our relatively elaborate numeration system is characterized by the follow- ing four properties:

1. Positional property. The quanti-

ties represented by the individual dig- its are determined by the position they hold in the whole numeral.

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

3. 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.

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

To understand place value the stu- dent must coordinate and synthesize a variety of subordinate knowledge about our culture's notational system for numbers and about numerical part-whole relationships. A student who understands place value knows not only that the numeral 52 can be used to represent "how many" for a collection of fifty-two objects but also that the digit on the right represents two of them, the digit on the left represents fifty of them (five sets of ten), and that 52 is the sum of the quantities represented by the individ- ual digits.

A Research Study Using Digit-Correspondence Tasks We expect students with good number sense to have constructed meanings for numerals in the initial years of school. In a study I conducted of the meanings that students attribute to two-digit numerals, however, I found that many of them were still con- structing meanings for the individual

47

digits as late as fifth grade (Ross 1985, 1986). In my study of sixty students in second through fifth grade, randomly selected from five diverse elementary schools, students were individually administered the following task:

The student was instructed to spill a collection of twenty-five sticks out of a bag, and I asked, "How many sticks are there?" After counting, nearly all gave a correct oral re- sponse of "twenty-five." I then

Many students are still constructing meaning for place value in fifth grade.

asked the child to write down how many sticks had been counted. Nearly all correctly wrote the nu- meral 25. 1 then encircled first the 5 and then the 2 and each time asked, "Does this part of your twenty-five have anything to do with how many sticks you have?"

Of the sixty students interviewed, twenty-six were successful - they ex- plained in a variety of ways that the 5 in 25 represented five of the sticks and that the 2 represented the other twen- ty. Twelve students thought the indi- vidual digits had nothing to do with how many sticks were in the collec- tion; fourteen described invented nu- merical meanings, such as that the 5 meant "half of ten," that the 5 meant that groups contained five sticks, or that the 2 meant "count by twos." Eight students thought that the 2 meant two sticks and that the 5 meant either five sticks or had nothing to do with the number of sticks in the col- lections.

In the sticks task just described, the collection of sticks was not grouped in any way. When the task was altered by representing the number 52 with a standard place-value partitioning of base-ten blocks (five ten-blocks and two unit-blocks) many more of the students (44 out of the 60) were suc- cessful. But when 52 was represented using four ten-blocks and twelve unit- blocks, the number of successful stu- dents dropped to twenty. Similar re-

sults were found in tasks in which a collection of forty-eight beans was partitioned into cups. Figure 1 dis- plays the materials used in the six digit-correspondence tasks used in the study.

Why did children find the tasks that used a standard place-value partition- ing to be so much easier? In both standard tasks, the representations of the groups of ten were very promi- nent. When, for example, I asked if the 2 and 5 in the numeral 52 had anything to do with how many base- ten blocks were on the table, the stu- dent was looking at five long purple

ten-blocks and two tan unit-blocks. It is easy to see how the student might have proposed that the 5 represented the five purple blocks and yet have had no thought of tens or fifty in mind - just five purple blocks.

In a follow-up study designed to examine whether some students did in fact use this face-value interpretation to assign meaning to the individual digits, thirty third-grade students were individually asked to count a collection of twenty-six objects and to "write down how many." They were then asked to sort the objects into sets of four ("candies into party-favor

48 Arithmetic Teacher

Fig. 1 Digit-correspondence task materials

Fig. 2 Twenty-six objects partitioned ^ ^

^ into sets of four ä /~' f~' ^ ^^ f

' QOJ 00 AX)

cups" or "wheels to make toy cars"). The resulting partitioned collection, pictured in figure 2, contained six sets of four objects and two "left over." The interviewer then encircled the individual digits in 26 and asked, for each digit, "Does this part of your 26 have anything to do with how many you have?"

The prominent groupings in this task, instead of facilitating a correct response, facilitated an incorrect re- sponse, where students reversed the meanings of the digits and where the groupings were not in base ten. The incorrect response, however, was con- sistent with the face-value interpreta- tion. Nearly half of the third graders incorrectly responded that the 2 in 26 stood for two of the objects, whereas the 6 in the 26 represented the "six cups of candies" or "the six cars."

Stages of Interpreting Two-Digit Numerals A five-stage model of the interpreta- tions children assign to two-digit nu- merals was developed based on data from the original study and findings from related research (Ashlock 1978; Baroody et al. 1983; Barr 1978; Bednarz and Janvier 1982; Flournoy 1967; Heibert and Wearne 1983; M. Kamii 1980, 1982; National Assess- ment of Educational Progress 1983; Rathmell 1972; Resnick 1983; Rick- man 1983; Scrivens 1968; C. Smith 1969; R. Smith 1973). Descriptions of the stages follow.

Stage 1, whole numeral. As pupils in our culture construct their knowl- edge about quantities up to ninety- nine and their symbolic representa- tion as two-digit numerals, their cognitive construction of the whole comes first - the numeral 52 repre- sents the whole amount. They assign no meaning to the individual digits. For related work that reports ob- served differences between pupils in the U. S. and those from Asian cul- tures, see the work of Miura (1987) and of Miura et al. (in press).

Students in their early stages of understanding of place value appear to understand more than they actually do.

Stage 2, positional property. Pupils know that in a two-digit numeral the digit on the right is in the "ones place" and the digit on the left is in the "tens place." Their knowledge of the individual digits is limited, howev- er, to the position of the digits and does not encompass the quantities to which each corresponds.

Stage 3 y face value. Students inter- pret each digit as representing the number indicated by its face value. The set of objects represented by the tens digit, however, may be different from the objects represented by the ones digit. They may verbally label as "tens" the objects that correspond to the tens digit, but these objects do not

Table 1 Stages in the Development of Students' Understanding of Two-Digit Numeration by Grade in School

Stage of development Grade* 1 Π Ш IV V

2 8 0 3 4 0 3 0 2 5 6 2 4 10 6 17 5 0 12 5 7

Total 9 3 16 16 16

*n = 15 for each grade level.

truly represent groups of ten units to students in stage 3; students do not recognize that the number repre- sented by the tens digit is a multiple of ten.

Stage 4, construction zone. Stu- dents know that the left digit in a two-digit numeral represents sets of ten objects and that the right digit represents the remaining single ob- jects, but this knowledge is tentative and is characterized by unreliable task performances.

Stage 5, understanding. Students know that the individual digits in a two-digit numeral represent a parti- tioning of the whole quantity into a tens part and a ones part. The quan- tity of objects corresponding to each digit can be determined even for col- lections that have been partitioned in nonstandard ways.

Ages and Stages Each of the sixty students in the re-

February 1989 49

ported study was assigned to one of the five stages according to perfor- mances on six digit-correspondence tasks and a positional-knowledge task in which students were asked to iden- tify, in a two-digit numeral, which digit was in the "tens place" and which was in the "ones place." The number of students in each stage is shown, by grade level, in table 1.

Although all the students in this study were at least at stage 1 - able to count collections of as many as fifty- two objects and write the two-digit numeral that corresponded to the count - twelve of them demonstrated no quantitative interpretation of the individual digits.

Sixteen students succeeded only on those digit-correspondence tasks in which numbers were represented us- ing a standard place-value partitioning of a collection of objects - that is, those tasks in which they could be successful using a stage-3 face-value interpretation.

Sixteen students succeeded on all six digit-correspondence tasks, dem- onstrating a stage-5 understanding of the numbers represented by the indi- vidual digits in two-place numerals. No second-grade pupil demonstrated stage-5 understanding. Even among the fifth graders, only half were at stage 5.

Implications for the Classroom More instructional support focusing on two-digit numeration is needed in the middle grades. Students do not all develop at the same rate, nor do they all have identical experiences with numbers and numerals. The data dis- played in table 1 show that even in fourth and fifth grades, only half the students interviewed demonstrated good understanding of the individual digits in two-digit numerals. Others who have used digit-correspondence tasks to assess students' understand- ing of place value report similar re- sults (C. Kamii 1986; M. Kamii 1980, 1982). The typical place- value instruc- tion in the middle grades, which fo- cuses on symbolic expressions for much larger numbers and on decimal

fractions, is inappropriate for many students.

How do so many students reach the middle grades with so little under- standing of two-digit numbers? One reason is that pupils in stages 2 and 3 may appear to understand more than they actually do. With stage-2 knowl- edge of the left-right positions of the

Each pupil must be provoked into constructing his or her own knowledge of numbers and the relations among them.

digits, pupils are able to succeed on a variety of tasks typically found in their textbooks and on standardized tests, such as the following:

In 27, which digit is in the tens place?

How many tens are in 84? 35 = tens and ones. 7 tens + 5 ones = Students who use a stage-3, face-

value interpretation of digits succeed on an even wider variety of tasks, including many that use manipulative materials. In many instructional tasks students are asked to make corre- spondences between digits and mate- rials. If a collection is already grouped into a standard place-value partition- ing of tens and ones, a student who is asked to make correspondences for the digits in 52, for example, need only look for "five of something and

When a teacher shows students something, students do not have to think; they simply follow directions.

two of something else." Using this face- value strategy, a student easily adapts to new materials - beans in cups, linking cubes, base-ten blocks, or colored chips, for example. Only when the collections are grouped into nonstandard partitionings does stu- dents' face- value interpretation fail

them; when they are confronted with regrouping tasks, students' faulty un- derstanding becomes apparent.

Further study needs to be done, but even extensive experience with em- bodiments like base-ten blocks and other place-value manipulatives does not appear to facilitate an understand- ing of place value as measured by the digit-correspondence tasks (Ross 1988). If we introduce materials that have been designed to embody base- ten groupings before students have constructed appropriate quantitative meanings for the individual digits, we may unwittingly provoke or reinforce a stage-3 (face- value) interpretation of digits. With these materials the teacher and manufacturer may have "embodied the ten," but the student need not.

Manipulative materials can serve as a useful communication tool between teacher and student - they give us something to talk about. Through the use of such materials we can often learn a great deal about students' thinking, including their misconcep- tions (Labinowicz 1985). We can also use the materials to show and explain what we want students to do. Careful instruction with place- value manipula- tives can facilitate the acquisition of the procedural knowledge required for facility with computational algo- rithms (Fuson 1986; Resnick 1983). We should not mislead ourselves, however, that as a result of such in- struction students construct under- standing of either the complex place- value numeration system or the algorithm. If understanding is the goal, it probably doesn't matter if the teacher shows children procedures with paper and pencil, beans and cups, or base-ten blocks. When a teacher shows students something, students do not have to think; they simply follow directions.

Understanding comes only from thinking; each pupil must be provoked into constructing his or her own knowledge of numbers and the rela- tions among them. Pupils need to en- gage in problem-solving tasks that challenge them to think about useful ways to partition and compose num- bers. The addition and subtraction of numbers through ninety-nine are ap-

50 Arithmetic Teacher

propriate topics for second graders, but we should encourage pupils to find sums and differences in their own ways.

For students who have not been taught a conventional paper-and- pencil procedure, finding the sum of 32 + 59 is a nonroutine problem whose solution strategy is not readily apparent. The problem can be solved using a variety of strategies and a variety of materials - counters, fin- gers, or paper and pencil. If they are working in cooperative groups, stu- dents can be encouraged to compare methods and try to convince one an- other that a strategy does or doesn't "work"; they might use a calculator to verify the solution. In the larger group discussion, students can be asked to demonstrate their successful strategies.

Teaching place-value concepts separately as a prerequisite to double- digit addition and subtraction is inef- fective and unnecessary. Experimen- tal instructional studies have shown that pupils in first and second grade can invent their own efficient algo- rithms, and do so without place- value manipulatives (Kamii and Joseph 1988; Cobb and Merkel [in press]). In fact, manipulative materials may actu- ally detract from thinking because tasks are too easy to do with the materials.

As teachers of mathematics we need to provide opportunities for all students to develop a strong sense of number. To accomplish that goal we can help each student to construct mentally, as she or he matures, in- creasingly elaborate concepts of nu- merical part-whole relations and place value. If we challenge students with more opportunities for making esti- mates and mental computations and show them the conventional algo- rithms only after they have experi- enced a fertile period of inventing their own efficient procedures for solving problems like 32 + 59, we can expect more young students to dem- onstrate the numeric partitioning flex- ibility that is characteristic of those with good number sense. If students then learn a conventional algorithm, they will view it as one of many ways to find a sum or difference; they'll be

able to choose from and use, as ap- propriate, estimation, mental compu- tation, and calculators, as well as in- vented and conventional algorithms. References

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February 1989 51