Questions? Call 800-624-6242

| Items in cart [0]

PAPERBACK
price:\$34.95

## Adding It Up: Helping Children Learn Mathematics (2001) Center for Education (CFE)

### Citation Manager

. "3 Number: What Is There to Know?." Adding It Up: Helping Children Learn Mathematics. Washington, DC: The National Academies Press, 2001.

 Page 71

The following HTML text is provided to enhance online readability. Many aspects of typography translate only awkwardly to HTML. Please use the page image as the authoritative form to ensure accuracy.

Adding + It Up: Helping Children Learn Mathematics

## 3NUMBER: WHAT IS THERE TO KNOW?

Seven. What is seven? Seven children; seven ideas; seven times in a row; seventh grade; a lucky roll in dice; seven yards of cotton; seven stories high; seven miles from here; seven acres of land; seven degrees of incline; seven degrees below zero; seven grams of gold; seven pounds per square inch; seven years old; finishing seventh; seven thousand dollars of debt; seven percent alcohol; Engine No. 7; The Magnificent Seven. How can an idea with one name be used in so many different ways, denoting such various senses of quantity? Consider how different a measure of time (seven years) is from one of temperature (seven degrees), how different a measure of length (seven meters) is from a count (seven children), and how different either of these is from a position (finishing seventh or being in seventh grade). Even within measures, some are represented as ratios (seven pounds per square inch, seven percent alcohol) and others as simple units (seven miles, seven liters). Although normally taken for granted, it is remarkable that seven, or any number, can be used in so many ways. That versatility helps explain why number is so fundamental in describing the world.

Its versatility helps explain why number is so fundamental in describing the world.

This chapter surveys the domain of number. It was developed in part in response to the charge to the committee to describe the context of the study with respect to the areas of mathematics that are important as foundations in grades pre-K to 8 for building continued learning. The intent of this chapter is essentially mathematical; learning and teaching are treated elsewhere. The chapter does not set forth a curriculum for students but instead provides a panoramic view of the territory on which the numerical part of the school curriculum is built. Nor is the chapter intended as a curriculum for teachers. Instead, it identifies some of the crucial ideas about number that we think

 Page 71
 Front Matter (R1-R20) Executive Summary (1-14) 1 Looking at Mathematics and Learning (15-30) 2 The State of School Mathematics in the United States (31-70) 3 Number: What Is There to Know? (71-114) 4 The Strands of Mathematical Proficiency (115-156) 5 The Mathematical Knowledge Children Bring to School (157-180) 6 Developing Proficiency with Whole Numbers (181-230) 7 Developing Proficiency with Other Numbers (231-254) 8 Developing Mathematical Proficiency Beyond Number (255-312) 9 Teaching for Mathematical Proficiency (313-368) 10 Developing Proficiency in Teaching Mathematics (369-406) 11 Conclusions and Recommendations (407-432) Biographical Sketches (433-440) Index (441-454)

Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.

OCR for page 71
Adding + It Up: Helping Children Learn Mathematics 3 NUMBER: WHAT IS THERE TO KNOW? Seven. What is seven? Seven children; seven ideas; seven times in a row; seventh grade; a lucky roll in dice; seven yards of cotton; seven stories high; seven miles from here; seven acres of land; seven degrees of incline; seven degrees below zero; seven grams of gold; seven pounds per square inch; seven years old; finishing seventh; seven thousand dollars of debt; seven percent alcohol; Engine No. 7; The Magnificent Seven. How can an idea with one name be used in so many different ways, denoting such various senses of quantity? Consider how different a measure of time (seven years) is from one of temperature (seven degrees), how different a measure of length (seven meters) is from a count (seven children), and how different either of these is from a position (finishing seventh or being in seventh grade). Even within measures, some are represented as ratios (seven pounds per square inch, seven percent alcohol) and others as simple units (seven miles, seven liters). Although normally taken for granted, it is remarkable that seven, or any number, can be used in so many ways. That versatility helps explain why number is so fundamental in describing the world. Its versatility helps explain why number is so fundamental in describing the world. This chapter surveys the domain of number. It was developed in part in response to the charge to the committee to describe the context of the study with respect to the areas of mathematics that are important as foundations in grades pre-K to 8 for building continued learning. The intent of this chapter is essentially mathematical; learning and teaching are treated elsewhere. The chapter does not set forth a curriculum for students but instead provides a panoramic view of the territory on which the numerical part of the school curriculum is built. Nor is the chapter intended as a curriculum for teachers. Instead, it identifies some of the crucial ideas about number that we think

OCR for page 72

OCR for page 73

OCR for page 74
Adding + It Up: Helping Children Learn Mathematics them into parts, and it seriously changes their nature. Thus, joining rods will support an extension of arithmetic into fractional quantities much more easily than counting cats will. Similarly, multiplication has multiple interpretations. We introduced it as adding the same number many times. The set-combination interpretation of multiplication would be to combine several essentially identical collections, such as the packages of cookies mentioned above. If you think of addition in terms of joining rods, then multiplication would amount to joining several rods of the same length end to end. Thus, 4×6 can be visualized by laying four rods of length six end to end, where you can think of each rod as a little row of boxes. A more compact way to arrange the rods would be to lay them side by side rather than end to end. This arrangement produces an array of four rows of boxes with six boxes in each row, which may be called a rectangular array interpretation of multiplication. When the rods have height one, there is an added benefit: The array looks like a rectangle of boxes, and the area of the rectangle (measured in box areas) is just 4×6. This is the area interpretation of multiplication. The multiple interpretations of the basic operations is symptomatic of a general feature of mathematics—the tension between abstract and concrete.3 This tension is a fundamental and unavoidable challenge for school mathematics. On the one hand, as we indicated above, the abstractness of math-

OCR for page 75

OCR for page 76
Adding + It Up: Helping Children Learn Mathematics Commutativity and associativity guarantee that all 12 ways of doing this sum give the same answer—so it does not matter which one I do. (For adding four numbers, there are 120 (!) conceivable different schemes, all of which again give the same result.) This flexibility is very useful when students do computations. For example, 1+8 can be found by thinking of it as 8+1 and then just recalling the next whole number after 8. The standard procedures for doing multidigit arithmetic also heavily exploit commutativity and associativity. However, the flexibility permitted by these rules also greatly increases the challenges of teaching arithmetic. When there are several ways to do a calculation, it is virtually certain that students will produce the answer more than one way. A teacher must therefore have a sufficiently flexible knowledge of arithmetic to evaluate the various student solutions, to validate the correct ones, and to correct errors productively. The commutative and associative laws also hold for multiplication (see Box 3–1). The commutativity of multiplication by 2 is also reflected in the equivalence of the two definitions of even number typically offered by children. The “fair share” definition says that a number is even if it can be divided into two equal parts with nothing left over (which may be written as 2×m); the “pairing” definition says that a number is even if it can be divided into pairs with nothing left over (m×2). In addition to these two laws for each operation, there is a rule, known as the distributive law, connecting the two operations. It can be written symbolically as a×(b+c)=a×b+a×c. An example would be 2×(3+4)=2×7=14=6+8=2×3+2×4. A good way to visualize the distributive law is in terms of the area interpretation of multiplication. Then it says that if I have two rectangles of the same height, the sum of their areas is equal to the area of the rectangle gotten by joining the two rectangles into a single one of the same height but with a base equal to the sum of the bases of the two rectangles: The basic properties of addition and multiplication of whole numbers are summarized in Box 3–1.

OCR for page 77
Adding + It Up: Helping Children Learn Mathematics Box 3–1 Properties of the Arithmetic Operations Commutativity of addition. The order of the two numbers does not affect their sum: 3+5=8=5+3. In general, m+n=n+m. Associativity of addition. When adding three (or more) numbers, it does not matter whether the first pair or the last pair is added first: (3+5)+4=8+4=12=3+ 9=3+(5+4). In general, (m+n)+p=m+(n+p). Commutativity of multiplication. The order of the two numbers does not affect their product: 5×8 produces the same answer as 8×5. In general, m×n=m×n. Associativity of multiplication. When multiplying three or more numbers, it does not matter whether the first pair or the last pair is multiplied first: 3×(5×4) is the same as (3×5)×4. In general, (m×n)×p=m×(n×p).

OCR for page 78

OCR for page 79

OCR for page 80

OCR for page 81
Adding + It Up: Helping Children Learn Mathematics content simply to admire these new creations. You get into situations in which you want to do arithmetic with them also. If I owe Bart two apples and I owe Teresa four apples, how many apples do I owe all together—that is, what is (–2)+(–4)? If on Monday I get into a situation that leaves me two apples short and this happens again on Tuesday and Wednesday, how many apples short am I then—that is, what is 3×(–2)? Besides enlarging their idea of number, people have had to extend the arithmetic operations to this new larger class of numbers. They have needed to create a new, enlarged number system. The new system, encompassing both positive and negative whole numbers, is called the integers. How do people decide what arithmetic in this extended system is (or should be)? How do they create recipes for adding and multiplying integers, and what are the properties of these extended operations? They have two guides: (a) intuition and (b) the rules of arithmetic, as described above and in Box 3–1. Fortunately, the guides agree. Consider first the intuitive approach: Think hard about a lot of different cases and decide what is the right way to add and multiply in each one. To use intuition, you need to think in terms of some concrete interpretation of arithmetic. The yield of financial transactions is a good one for these purposes. Here negative amounts are money you owe, and positive amounts are money that you have or are owed by someone else. If you owe \$2 to Joan and \$3 to Sammy, then you owe \$5 to the two of them together. So (–2)+(–3)=–5. If you owe \$2 to three people, then you owe \$6, so 3×(–2)=–6. If you have a debt of \$2 and someone takes it away, you have gained \$2. So –(–2)=2. If someone takes three \$2 debts away from you, the amount you owe is then \$6 less than before, which means you have \$6 more. Therefore (–3)×(–2)=6. Continuing in this way, you can puzzle out what the sum, difference, or product of any two integers should be. The trouble with this approach is that it is somewhat contrived and depends upon making decisions about how to interpret each case in the particular context.5 Another approach6 is to use an exploratory method to reason how the operations should extend from the whole numbers. By extending the patterns in the table below, you find that (–3)×(–2)=6, just as was shown above in context. 3+2=5 3–2=1 3×2=6 (–3)×2=–6 3+1=4 3–1=2 3×1=3 (–3)×1=–3 3+0=3 3–0=3 3×0=0 (–3)×0=0 3+(–1)= 3–(–1)= 3×(–1)= (–3)×(–2)= 3+(–2)= 3–(–2)= 3×(–2)= (–3)×(–2)=

OCR for page 104
Adding + It Up: Helping Children Learn Mathematics of whole numbers, but the relationship is hard to see, mostly because there is no “carrying,” from the x to the x2 term, for example. The expanded method below shows the relationship a bit more clearly. Box 3–10 Examples of Algorithms The decimal place-value system allows many different algorithms for the four main operations. The following six algorithms for multiplication of two-digit numbers were produced by a class of prospective elementary school teachers. They were asked to show how they were taught to multiply 23 by 15: In Method 6, sometimes called lattice multiplication,* the factors are written across the top and on the right, the products of the pairs of digits are put into the cells (for example, 15 is written ), and the numbers in the diagonals are added to give the product underneath. Note that all of these algorithms produce the correct answer. All except Method 4 are simply methods for organizing the four component multiplications and *   The method is also called gelosia multiplication and is related to the method of Napier’s rods or bones, named after the Scottish mathematician John Napier (1550–1617).

OCR for page 105
Adding + It Up: Helping Children Learn Mathematics adding. The algorithms can be verified by decomposing the factors according to the values of their digits (in this case, 23=20+3 and 15=10+5) and using the distributive law in one of several ways: A more compelling justification uses the area model of multiplication. If the sides of a 23×15 rectangle are subdivided as 20+3 and 10+5, then the area of the whole rectangle can be computed by summing the areas of the four smaller rectangles. Note the correspondence between the areas of the four smaller rectangles and the partial products in Method 3. With more careful examination, it is possible to see the same four partial products residing in the four cells in Method 6. (The 2 in the upper left cell, for example, actually represents 200.) Methods 1,

OCR for page 106
Adding + It Up: Helping Children Learn Mathematics 2, and 5 differ from these only in that they record the areas for one pair of these rectangles at a time. Any of the methods—and, in fact, any of the four justifications that followed— could serve as the standard algorithm for the multiplication of whole numbers because they are all general and exact. Mathematically, these methods are essentially the same, differing only in the intermediate products that are calculated and how they are recorded. These methods, however, are quite different in transparency and efficiency. Methods 3 and 5 and the area model justification are the most transparent because the partial products are all displayed clearly and unambiguously. The three justifications using the distributive law also show these partial products unambiguously, but some of the transparency is lost in the maze of symbols. Methods 1 and 2 are the most efficient, but they lack some transparency because the 23 and the 30 actually represent 230 and 300, respectively. Method 4 takes advantage of the fact that doubling the factor 15 gives a factor that is easy to use. It is quite different from the others. For one thing, the intermediate result is larger than the final answer. This method can also be shown to be correct using the properties of whole numbers, since multiplying one factor by 2 and then dividing the product by 2 has no net effect on the final answer. The usefulness of Method 4 depends on the numbers involved. Doubling 15 gives 30, and 23×30 is much easier to calculate mentally than 23×15. Using this method to find a product like 23×17, on the other hand, would require first calculating 23×34, which is no easier than 23×17. Clearly this method, although completely general, is not very practical. For most factors, it is neither simple nor efficient. Building Blocks The preceding sections have described concepts in the domain of number that serve as fundamental building blocks for the entire mathematics curriculum. Other fundamental ideas—such as those about shape, spatial relationships, and chance—are foundational as well. Students do not need to, and should not, master all the number concepts we have described before they study other topics. Rather, number concepts should serve to support mathematics learning in other domains as students are introduced to them, and, conversely, these other domains should support students’ growing understanding of number.

OCR for page 107
Adding + It Up: Helping Children Learn Mathematics Number is intimately connected with geometry, as illustrated in this chapter by our use of the number line and the area model of multiplication. Those same models of number can, of course, arise when measurement is introduced in geometry. The connection between number and algebra is illustrated in the chapter by our use of algebra to express properties of number systems and other general relationships between numbers. The links from number to geometry and to algebra are forged even more strongly when students are introduced to the coordinate plane, in which perpendicular number lines provide a system of coordinates for each point—an idea first put forward by the French mathematician and philosopher René Descartes (1596–1650), although he did not insist that the number lines were perpendicular. Number is also essential in data analysis, the process of making sense of collections of numbers. Using numbers to investigate processes of variation, such as accumulation and rates of change, can provide students with the numerical underpinnings of calculus. Some of the manifold connections and dependencies between number and other mathematical domains may be illustrated by the so-called handshake problem: If eight people are at a party and each person shakes hands exactly once with every other person, how many handshakes are there? This problem appears often in the literature on problem solving in school mathematics, probably because it can be solved in so many ways. Perhaps the simplest way of getting a solution is just to count the handshakes systematically: The first person shakes hands with seven people; the second person, having shaken the first person’s hand, shakes hands with six people whose hands he or she has not yet shaken; the third person shakes hands with five people; and so on until the seventh person shakes hands with only the eighth person. The number of handshakes, therefore, is 7+6+5+4+3+2+1, which is 28. This method of solution can be generalized to a situation with any number of people, which is what a mathematician would want to do. For a party with 20 people, for example, there would be 19+18+17+16+15+14+13+12+11+10+9+8+7+6+5+4+3+2+1 handshakes, but the computation would be more time consuming. Because mathematicians are interested not only in generalizations of problems but also in simplifying solutions, it would be nice to find a simple way of adding the numbers. In general, for m+1 people at a party, the number of handshakes would be the sum of the first m counting numbers:24

OCR for page 108
Adding + It Up: Helping Children Learn Mathematics 1+2+…+m. Numbers that arise in this way are called triangular numbers because they may be arranged in triangular formations, as shown below. Therefore, 3, 6, 10, 15, 21, and 28 are all triangular numbers. This is a geometric interpretation, but can geometry be used to find a solution to the handshake problem that would simplify the computation? One way to approach geometrically the problem of adding the numbers from 1 to m is to think about it as a problem of finding the area of the side of a staircase. The sum 1+2+3+4+5+6+7, for example, would then be seen as a staircase of blocks in which each term is represented by one layer, as in the diagram on the left below. The diagram on the right below includes a second copy of the staircase, turned upside down. When the two staircases are put together, the result is a 7×8 rectangle, with area 56. So the area of the staircase is half that, or 28. This reasoning, although specific, supports a general solution for the sum of the whole numbers from 1 to m:m(m+1)/2. A closely related numerical approach to the problem of counting handshakes comes from a story told of young Carl Friedrich Gauss (1777–1855), whose teacher is said to have asked the class to sum the numbers from 1 to 100, expecting that the task would keep the class busy for some time. The story goes that almost before the teacher could turn around, Gauss handed in his slate with the correct answer. He had quickly noticed that if the numbers to be added are written out and then written again below but in the opposite

OCR for page 109
Adding + It Up: Helping Children Learn Mathematics order, the combined (double) sum may be computed easily by first adding the pairs of numbers aligned vertically and then adding horizontally. As can be seen below, each vertical sum is 101, and there are exactly 100 of them. So the double sum is 100×101, or 10,100, which means that the desired sum is half that, or 5050. For the original handshake problem, which involves the sum of the blocks in the staircase above, that means taking the double sum 7×8, or 56, and halving it to get 28. The handshake problem can be approached by bringing in ideas from other parts of mathematics. If the people are thought of as standing at the vertices of an eight-sided figure (octagon), then the question again becomes geometric but in a different way: How many segments (sides and diagonals) may be drawn between vertices of an octagon? The answer again is 28, as can be verified in the picture below. As often happens in mathematics, connections to geometry provide a new way of approaching the problem: Each vertex is an endpoint for exactly 7 segments, and there are 8 vertices, which sounds like there ought to be 7×8=56 segments. But that multiplication counts each segment twice (once for each endpoint), so there are really half as many, or 28, segments. In still another mathematical domain, combinatorics—the study of counting, grouping, and arranging a finite number of elements in a collection—the

OCR for page 110
Adding + It Up: Helping Children Learn Mathematics problem becomes how to count the number of ways to choose two items (people shaking hands) from a collection of eight elements. For example, in how many ways can a committee of two be chosen from a group of eight people? This is the same as the handshake problem because each committee of two corresponds to a handshake. It is also the same as the octagon problem because each committee corresponds to a segment (which is identified by its two endpoints). A critically important mathematical idea in the above discussion lies in noticing that these are all the same problem in different clothing. It also involves solving the problem and finding a representation that captures its key features. For students to develop the mathematical skill and ability they need to understand that seemingly different problems are just variations on the same theme, to solve the problem once and for all, and to develop and use representations that will allow them to move easily from one variation to another, the study of number provides an indispensable launching pad. Key Ideas About Number In this chapter, we have surveyed the domain of number with an eye toward the proficiency that students in grades pre-K to 8 need for their future study of mathematics. Several key ideas have been emphasized. First, numbers and operations are abstractions—ideas based on experience but independent of any particular experience. The numbers and operations of school mathematics are organized as number systems, and each system provides ways to consider numbers and operations simultaneously, allowing learners to focus on the regularities and the structure of the system. Despite different notations and their separate treatment in school, these number systems are related through a process of embedding one system in the next one studied. All the number systems of pre-K to grade 8 mathematics lie inside a single system represented by the number line. Second, all mathematical ideas require representations, and their usefulness is enhanced through multiple representations. Because each representation has its advantages and disadvantages, one must be able to choose and translate among representations. The number line and the decimal place-value system are important representational tools in school mathematics, but students should have experience with other useful interpretations and representations, which also are important parts of the content. Third, calculation requires algorithms, and once again there are choices to make because each algorithm has advantages and disadvantages. And finally, the domain of number both supports and is supported by other

OCR for page 111
Adding + It Up: Helping Children Learn Mathematics branches of mathematics. It is these connections that give mathematics much of its power. If students are to become proficient in mathematics by eighth grade, they need to be proficient with the numbers and operations discussed in this chapter, as well as with beginning algebra, measure, space, data, and chance—all of which are intricately related to number. Notes 1.   Some authors (see, e.g., Russell, 1919, p. 3; Freudenthal, 1983, pp. 77ff) call these the natural numbers. We are adopting the common usage of the U.S. mathematics education literature, in which the natural numbers begin 1, 2, 3, and so on, and the whole numbers include zero. 2.   The recognition that zero should be considered a legitimate number—rather than the absence of number—was an important intellectual achievement in the history of mathematics. Zero (as an idea) is present in the earliest schooling, but zero (as a number) is a significant obstacle for some students and teachers. “Zero is nothing,” some people say. “How can we ask whether it is even or odd?” 3.   “To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his [or her] most important tools: analogy, generality, and simplicity” (Stewart, 1989, p. 291). 4.   Although negative numbers are quite familiar today, and part of the standard elementary curriculum, they are quite a recent development in historical terms, having become common only since the Renaissance. Descartes, who invented analytic geometry and after whom the standard Cartesian coordinate system on the plane is named, rejected negative numbers as impossible. (His coordinate axes had only a positive direction.) His reason was that he thought of numbers as quantities and held that there could be no quantity less than nothing. Now, however, people are not limited to thinking of numbers solely in terms of quantity. In dealing with negative numbers, they have learned that if they think of numbers as representing movement along a line, then positive numbers can correspond to movement to the right, and negative numbers can represent movement to the left. This interpretation of numbers as oriented length is subtly different from the old interpretation in terms of quantity, which would here be unoriented length, and gives a sensible and quite concrete way to think about these numbers that Descartes thought impossible. 5.   Freudenthal, 1983, suggests that “negative numbers did not really become important until they appeared to be indispensable for the permanence of expressions, equations, formulae in the ‘analytic geometry’” (p. 436). “Later on arguments of content character were contrived…although some of them are not quite convincing (positive-negative as capital-debt, gain-loss, and so on)” (p. 435). 6.   See Freudenthal, 1983, p. 435. 7.   Although rational numbers seem to present more difficulties for students than negative integers, historically they came well before. The Greeks were comfortable

OCR for page 112