Analysis of Arithmetic for Mathematics Teaching

Gaea Leinhardt, Ralph Putnam, Rosemary A. Hattrup

• Publisher: Routledge
• Year: 2020
• ISBN:
  9781135440657
• Edition: 1
• Topic: Pedagogía
• Subtopic: Educación general

This volume emerges from a partnership between the American Federation of Teachers and the Learning Research and Development Center at the University of Pittsburgh. The partnership brought together researchers and expert teachers for intensive dialogue sessions focusing on what each community knows about effective mathematical learning and instruction. The chapters deal with the research on, and conceptual analysis of, specific arithmetic topics (addition, subtraction, multiplication, division, decimals, and fractions) or with overarching themes that pervade the early curriculum and constitute the links with the more advanced topics of mathematics (intuition, number sense, and estimation). Serving as a link between the communities of cognitive researchers and mathematics educators, the book capitalizes on the recent research successes of cognitive science and reviews the literature of the math education community as well.

chapter
1
Making Sense of Numbers in School Mathematics

Judith T. Sowder
San Diego State University

Contents

1. Number Sense and Its Components
   1. Dimensions of Number Sense
   2. Related Issues
2. Developing Number Sense and Estimation Skills: Analysis of School Programs
   1. Overview
   2. Foundations for Whole Number Sense
   3. Foundations for Rational Number Sense
   4. Mental Computation
   5. Computational Estimation
   6. A Caveat
3. Conclusion
4. Acknowledgment
5. References

NUMBER SENSE AND ITS COMPONENTS

The Curriculum and Evaluation Standards for School Mathematics (National Council of Teachers of Mathematics, 1989) is a response to the call for reform of school mathematics. The curriculum standards attempt to provide a guide to what mathematics students need to know. The Standards has been well received by the education, scientific, and business communities, and has already begun to influence school curricula at the local and state levels and text planning by all major publishers of school mathematics texts.

This preamble on the Standards is intended to show that its recommendations are being taken quite seriously by the people who decide what is included in school curricula at the state and local levels. Among its recommendations are that the K-4 curriculum should include whole number concepts and skills so that children can develop number sense, and that the 5-8 curriculum should include continued development of number and number relationships so that students can develop number sense for whole numbers, fractions, decimals, integers, and rational numbers. The conscious integration of number sense into the elementary school mathematics curriculum is quite a recent phenomenon. I say conscious, because I believe insightful teachers of mathematics have always taught in a manner that encourages the development of number sense, even though they may not consciously have had number sense as an instructional goal. "Instruction that promotes number sense is no doubt the same kind of instruction that promotes meaningful mathematics learning across the board" (Hiebert, 1989, p. 83). As I show later in this chapter, some text series, through the philosophy of teaching mathematics that they impart to teachers, are likely to lead students to develop better number sense than other texts, even though it is not yet possible to find number sense specifically addressed in the texts' program descriptions, lists of objectives, or scope-and-sequence charts. A text program's philosophy can be discerned through the manner and extent to which it focuses on conceptual development and encourages teachers to discuss and explore diverse problems, and through the types of exercises it provides for students.

As textbook companies take upon themselves the responsibility consciously to include number sense in their respective curricula, there is concern on the part of many (see Sowder & Schappelle, 1989) that number sense will be treated in a piecemeal fashion, with activities designed to promote number sense strategically placed on student pages and listed in scope-and-sequence charts, rather than promoted as a way of thinking about numbers that should encompass all of number learning. Textbooks cannot teach number sense, even though they can help the teacher to do so through many reminders and examples.

Number sense . . . develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms. [T]extbooks ... can only suggest ideas to be investigated, they cannot replace the "doing of mathematics" that is essential for the development of number sense. No substitute exists for a skillful teacher and an environment that fosters curiosity and exploration at all grade levels. (Howden, 1989, p. 11)

The purpose of this chapter is to attempt to clarify the many dimensions of number sense, and to examine the ways in which number sense develops and is exhibited. In this first section, I explore four curricular topics that are closely related to number sense: numeration, number magnitude, mental computation, and computational estimation. Understanding our numeration system, according to the Standards, requires relating counting, grouping, and place value. Understanding of number magnitude, exhibited by the ability to order and compare numbers, is a defining characteristic of number sense. Acquiring skill at mental computation and computational estimation calls on number sense and at the same time provides opportunities for expanding one's number sense. Acquisition of skills with these alternate types of computation is also a focus of the K-4 and 5-8 Standards. Sowder and Markovits (1991) argued that instruction in each of the four areas, when the approach used is one that encourages discussion and exploration of alternative strategies, will help students develop number sense. Certainly, proficiency within these four skill areas is indicative of the possession of a high-degree of number sense. Although these four areas of mathematical learning are admittedly interrelated, it should be noted that all have distinct and unique characteristics that justify their treatment as curriculum topics and arenas for research apart from number sense (R. Reys, 1989).

In this section, I also examine some other areas of learning that should be addressed when considering the role of instruction in developing number sense. Finally, I briefly describe some current, innovative programs where number sense happily appears to be a byproduct of the instruction. Against this backdrop, the second section contains a survey of two current elementary mathematics textbook programs and contrasts their approaches to the topics discussed in the first part of the chapter. To provide perspective to the discussion of these programs, the approaches taken by an older, out-of-print-series, and by the state-mandated Soviet elementary school mathematics programs also are considered.

Dimensions of Number Sense

Attempting to define number sense is in many ways analogous to attempting to define problem solving. We have not reached consensus on a definition of problem solving, although most teachers and scholars believe they have an understanding of what problem solving encompasses. This lack of a uniformly accepted definition has not dampened enthusiasm for teaching or for studying problem solving. Another way that number sense is like problem solving is that both represent a certain way of thinking rather than a body of knowledge that can be transmitted to others. Trafton (conference notes, in Sowder & Schappelle, 1989) spoke of number sense as something that "unfolds" rather than something that is taueht "directly."

Resnick (conference notes, in Sowder & Schappelle, 1989) has suggested that perhaps we need a different type of definition for number sense, more like Rosch's (1973) idea of central prototypes that one carries around and uses to judge particular cases as being closer to or farther away from the prototype. One way to begin to formulate a central prototype, or an understanding of the various dimensions of number sense, is to consider the attempts of others to define and characterize number sense. The Standards (NCTM, 1989) claims that "children with good number sense (1) have well-understood number meanings, (2) have developed multiple relationships among numbers, (3) recognize the relative magnitudes of numbers, (4) know the relative effect of operating on numbers, and (5) develop referents for measures of common objects and situations in their environment" (p. 38). Many of these same components appear in a characterization of number sense that I have frequently used: Number sense (a) is a well-organized conceptual network that enables a person to relate number and operation properties; (b) can be recognized in the ability to use number magnitude, relative and absolute, to make qualitative and quantitative judgments necessary for (but not restricted to) number comparison, the recognition of unreasonable results for calculations, and the use of nonstandard algorithmic forms for mental computation and estimation; (c) can be demonstrated by flexible and creative ways to solve problems involving numbers; and (d) is neither easily taught nor easily measured. This characterization hints at some assessable aspects of number sense while at the same time indicating that any assessment items we might develop could never fully measure number sense.

From these attempts to define numbers sense, and from the characteristics of number sense noted by others (Behr, 1989; Greeno, 1989; Hiebert, 1989; Howden, 1989; Markovits, 1989; Resnick, 1989a; Silver, 1989; Trafton, 1989), it is possible to compile a list of behaviors that demonstrate some presence of number sense. These behaviors are, admittedly, not without some overlap.

1. Ability to compose and decompose numbers; to move flexibly among different representations; to recognize when one representation is more useful than another. Examples would include the case of a second grader noting that sharing 24¢ among four children is easy if 24¢ is thought of as four nickels and four pennies, or the sixth grader thinking of 25 as 100/4 before mentally multiplying 12 x 25. The ability to decompose and recompose numbers is closely related to understanding place value concepts because most compositions and decompositions are place value related, such as when thinking of 53 as four tens and 13 ones in order to subtract 28 from 53.

2. Ability to recognize the relative magnitude of numbers. This includes the abilities both to compare and to order numbers, such as recognizing that 89 is smaller than 91, and that 5 is larger than 4 which in turn is larger than $\frac{1}{8}$; the ability to understand and use the density property of rational numbers (without the terminology), such as in finding a number between ⅓ and $\frac{1}{4}$; and the ability to compare differences relatively, such as in noting that the difference between 3 and 5 and the difference between 123 and 125 are absolutely the same but relatively very different.

3. Ability to deal with the absolute magnitude of numbers; for example, to realize that 1 cannot hold 200 pennies in my hand at one time, or that 1 million people cannot attend a rock concert.

4. Ability to use benchmarks. For example, using 1 as a benchmark, the sum of $\frac{7}{8}$ and should be a little under 2, because each fraction is a little under 1. Resnick (1989a) rephrased this as using well-known number facts to figure out facts of which one is not so sure.

5. Ability to link numeration, operation, and relation symbols in meaningful ways. Hiebert (1989) noted that written symbols function both as records of things already known and as tools for thinking, and that number sense requires symbols to function in both ways. Symbols that act as records can produce anchors of reasonableness that can then be used to monitor actions on symbols. Trafton (1989) made a distinction between manipulation of quantities and the manipulation of symbols. Children with number sense often think of symbols as quantities, as shown in their use of expressions such as "knocking off" and "tacking a little on here." He gave as an example a boy asked to find $6.00 - $2.85 mentally: "If you take away the two dollars, that only leaves four dollars, and you knock off eighty-five cents, and that leaves $3.15" (p. 16). Similarly, Greeno (1989) noted that operations on quantities have counterparts in operations on numbers.

6. Understanding the effects of operations on numbers. Behr (1989) noted two aspects of this characteristic: (a) Recognizing how to compensate (if necessary) when one or more operands are changed in a computational problem; for example, if 348 - 289 is 59, then what is 358 - 289? (b) Recognizing when a result of a computation remains the same after changing the original numbers operated upon; for example, how can knowing 350 - 291 = 59 be used to find 348 - 289?

7. The ability to perform mental computation through "invented" strategies that take advantage of numerical and operational properties. A student faced with finding 1,000 - 784 might try counting up, 16 to 800, 200 more to 1,000, making the problem much easier than it would be if the Student attempted to mentally apply the standard paper-and-pencil algorithm.

8. Being able to use numbers flexibly to estimate numerical answers to computations, and to recognize when an estimate is appropriate. The level of resolution with which estimation can be performed will vary with age. Young children may only be able to say whether the sum of two-di...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Contents
• ACKNOWLEDGMENTS
• PREFACE
• 1 MAKING SENSE OF NUMBERS IN SCHOOL MATHEMATICS
• 2 RESEARCH ON LEARNING AND TEACHING ADDITION AND SUBTRACTION OF WHOLE NUMBERS
• 3 SOLVING MULTIPLICATION WORD PROBLEMS
• 4 TEACHING AND LEARNING LONG DIVISION FOR UNDERSTANDING IN SCHOOL
• 5 MATHEMATICAL, COGNITIVE, AND INSTRUCTIONAL ANALYSES OF DECIMAL FRACTIONS
• 6 RATIONAL AND FRACTIONAL NUMBERS AS MATHEMATICAL AND PERSONAL KNOWLEDGE: IMPLICATIONS FOR CURRICULUM AND INSTRUCTION
• 7 FROM PROTOQUANTITIES TO OPERATORS: BUILDING MATHEMATICAL COMPETENCE ON A FOUNDATION OF EVERYDAY KNOWLEDGE
• EPILOGUE
• AUTHOR INDEX
• SUBJECT INDEX