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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



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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

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Adding + It Up: Helping Children Learn Mathematics teachers should know. Many of these ideas are treated in more detail in textbooks intended for prospective elementary school teachers. A major theme of the chapter is that numbers are ideas—abstractions that apply to a broad range of real and imagined situations. Operations on numbers, such as addition and multiplication, are also abstractions. Yet in order to communicate about numbers and operations, people need representations—something physical, spoken, or written. And in order to carry out any of these operations, they need algorithms: step-by-step procedures for computation. The chapter closes with a discussion of the relationship between number and other important mathematical domains such as algebra, geometry, and probability. Number Systems At first, school arithmetic is mostly concerned with the whole numbers: 0, 1, 2, 3, and so on.1 The child’s focus is on counting and on calculating— adding and subtracting, multiplying and dividing. Later, other numbers are introduced: negative numbers and rational numbers (fractions and mixed numbers, including finite decimals). Children expend considerable effort learning to calculate with these less intuitive kinds of numbers. Another theme in school mathematics is measurement, which forms a bridge between number and geometry. Mathematicians like to take a bird’s-eye view of the process of developing an understanding of number. Rather than take numbers a pair at a time and worry in detail about the mechanics of adding them or multiplying them, they like to think about whole classes of numbers at once and about the properties of addition (or of multiplication) as a way of combining pairs of numbers in the class. This view leads to the idea of a number system. A number system is a collection of numbers, together with some operations (which, for purposes of this discussion, will always be addition and multiplication), that combine pairs of numbers in the collection to make other numbers in the same collection. The main number systems of arithmetic are (a) the whole numbers, (b) the integers (i.e., the positive whole numbers, their negative counterparts, and zero), and (c) the rational numbers—positive and negative ratios of whole numbers, except for those ratios of a whole number and zero. Thinking in terms of number systems helps one clarify the basic ideas involved in arithmetic. This approach was an important mathematical discovery in the late nineteenth and early twentieth centuries. Some ideas of arithmetic are fairly subtle and cause problems for students, so it is useful to have a viewpoint from which the connections between ideas can be surveyed.

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Adding + It Up: Helping Children Learn Mathematics The Whole Numbers One of the starting points of arithmetic is counting. Children can find out how many objects are in a collection by counting them: one, two, three, four, five. They also need zero to say that there is not any of some type of thing.2 Addition arises to simplify counting. When children join two collections, instead of recounting all the objects in the combined set, they add the numbers of objects in each of the original sets. (I have five apples, and Dave has three apples. How many apples do we have together?) Multiplication provides a further shortcut when children want to add many copies of the same number. (I have 10 boxes of cookies, with 12 cookies in each box. How many cookies do I have?) The whole numbers, with the two operations of addition and multiplication, form the whole number system, the most basic number system. It is important to take note that, although the whole numbers with their operations are very familiar, they are already abstract. Although counting is usually done with some particular kind of things (apples or cats or dollars), arithmetic can be independent of the things counted. Five apples plus three apples makes eight apples; five cats plus three cats makes eight cats; five dollars plus three dollars makes eight dollars. (A word of caution: when adding, you must combine units of the same kind: five dollars plus three cats does not make eight of anything in particular.) This independence of the results from whatever is being counted leads to the abstract operation called addition. It is similar with multiplication. Note that the abstract nature of the arithmetic operations is exactly what makes them useful. If addition of apples, of cats, and of dollars each required its own peculiar set of rules, people would probably have no general concept of addition, just ideas about combining each type of object in its own individual way. Mathematics itself might not exist. Certainly, it would require a lot more work. Although the whole numbers with their operations are very familiar, they are already abstract. Appropriate to the abstract nature of arithmetic, each operation has several concrete interpretations. We introduced addition by means of its interpretation in terms of combining sets of like objects. Other interpretations are often used. One is the joining of segments of various lengths. If Jane has a rod 3 inches long, and another rod 5 inches long, she can lay them end to end (or perhaps even attach them together) to get a rod 8 inches long. This interpretation may seem the same, or almost the same, as the combining-sets interpretation. Indeed, it must be somewhat similar, since it is a representation of addition. But it differs in perhaps subtle ways. For example, inches can be subdivided into parts, which are hard to tell from the wholes, except that they are shorter; whereas it is painful to cats to divide

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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-

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Adding + It Up: Helping Children Learn Mathematics ematics is an important reason for its usefulness: A single idea can apply in many circumstances. On the other hand, it is difficult to learn an idea in a purely abstract setting; one or another concrete interpretation must usually be used to make the idea real. But having been introduced to a mathematical concept by means of one interpretation, children then need to pry it away from only that interpretation and take a more expansive view of the abstract idea. That kind of learning often takes time and can be quite difficult. Sometimes the way in which a concept is first learned creates obstacles to learning it in a more abstract way. At other times, overcoming such obstacles seems to be a necessary part of the learning process. Properties of the Operations Experience with the operations of addition and multiplication leads to the observation of certain regularities in their behavior. For example, it does not matter in what order two numbers are added. If I dump a basket of three apples into a basket with five apples already in it, there will be eight apples in the basket; and if I dump the basket of five apples into the basket with three, I will also have eight. Thus 5+3=8=3+5. The similar fact is true for any two numbers. Thus, I know that 83,449+173,248,191=173,248,191+83,449 without actually doing either addition. I have used what is known as the commutative law of addition. When three numbers are to be added, there are several options. To add 1 and 2 and 3, I can add 1 and 2, giving 3, and then add the original 3 to this, to get 6. Or I can add 1 to the result of adding the 2 and the 3. This process again gives 6. These two ways of adding give the same final answer, although the intermediate steps look quite different: (1+2)+3=3+3=6=1+5=1+(2+3). This statement of equality uses what is known as the associative law. Again, it holds for any three numbers. I know that (83,449+173,248,191)+417=83,449+(173,248,191+417) without doing either sum. The commutative and associative laws in combination allow tremendous freedom in doing arithmetic. If I want to add three numbers, such as 1, 2, and 3, there are potentially 12 ways to do it: (1+2)+3 (2+1)+3 (1+3)+2 (3+1)+2 (2+3)+1 (3+2)+1 1+(2+3) 2+(1+3) 1+(3+2) 3+(1+2) 2+(3+1) 3+(2+1)

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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.

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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).

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Adding + It Up: Helping Children Learn Mathematics Distributivity of multiplication over addition. When multiplying a sum of two numbers by a third number, it does not matter whether you find the sum first and then multiply or you first multiply each number to be added and then add the two products: 4×(3+2)=(4×3)+(4×2). In general, m×(n+p)=(m×n)+(m×p). Question: Is subtraction commutative? Answer: No. For example, 6–2=4, but 2–6=–4. Question: Is subtraction associative? Answer: No. For example, (7–4)–2=3–2=1, but 7–(4–2)=7–2=5. Subtraction and Division So far we have talked only about addition and multiplication. It is traditional, however, to list four basic operations: addition and subtraction, multiplication and division. As implied by the usual juxtapositions, subtraction is related to addition, and division is related to multiplication. The relation is in some sense an inverse one. By this, we mean that subtraction undoes addition, and division undoes multiplication. This statement needs more explanation. Just as people sometimes want to join sets, they sometimes want to break them apart. If Eileen has eight apples and eats three, how many does she have left? The answer can be pictured by thinking of eight apples as composed of two groups, a group of five apples and a group of three apples. When the three are taken away, the five are left. In this solution, you think of eight as 5+3, and then when you subtract the three, you are again left with five. Thus subtracting three undoes the implicit addition of three and leaves you with the original amount. It is the same no matter what amount you start

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Adding + It Up: Helping Children Learn Mathematics with: 5+3–3=5; 9+3–3=9; 743+3–3=743. More formally, subtracting 3 is the inverse of adding 3. It is similar with division and multiplication. Just as people sometimes want to form sets of the same size into one larger set, they sometimes want to break up a large set into equal-sized pieces. If you think of 15 as 5×3, then when you divide 15 by 3, you are again left with 5. Thus division by 3 undoes implicit multiplication by 3 and leaves you with the original amount. It is the same no matter what amount you start with: 5×3÷3=5; 9×3÷3=9; 743×3÷3=743. More formally, dividing by 3 is the inverse of multiplying by 3. Two interpretations of division deserve particular mention here. If I have 20 cookies, and want to sort them into 5 bags, how many go in each bag? This is the so-called sharing model of division because I know in how many ways the cookies are to be shared. I can find the answer by picturing the 20 cookies arranged in 5 groups of 4 cookies, which will be the contents of 1 bag. If the cookies originally came out of 5 bags of 4 each, when I put them back into those bags, I will again have 4 in each. Thus, division by 5 undoes multiplication by 5, or division by 5 is the inverse of multiplication by 5. The picture below shows the sharing model for this situation. To think about 20÷5, you could also use the measurement model: If I have 20 cookies that are to be packaged in bags of 5 each, how many bags will I get? In the sharing model (also called the partitioning model or partitive division), you know the number of groups and seek the number in a group. In the measurement model (also called quotative division), you know the size of the groups and seek the number of groups. The circled numbers in the figures above and below illustrate a crucial difference between the two models: the order in which the cookies are placed in bags. In the sharing

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Adding + It Up: Helping Children Learn Mathematics model, the cookies are dealt into the bags one at a time; in the measurement model, the cookies are counted out by complete bags. When you deal with actual cookies, the processes are quite different, but abstractly they are both 20÷5. Note that because multiplication is commutative, 5 bags of 4 cookies each is the same total number of cookies as 4 bags of 5 cookies each. Eventually students come to see the two kinds of division as interchangeable and use whichever model helps them with a particular division problem. Subtraction and the integers We might summarize the story so far by saying that there are two pairs of operations—addition and subtraction, and multiplication and division—and these are inversely related in the sense described above. However, this summary would not quite be correct. In fact, subtraction is not actually an operation on whole numbers in the same sense that addition is. You can add any pair of whole numbers together, and the result is again a whole number. Sometimes, however, you cannot subtract one whole number from another. If I have three apples, and Bart asks for five, I can’t give them to him. I just don’t have five apples. If I’m really supposed to give him five apples (maybe he left five apples in my care, I ate two, and then he came back to reclaim his apples), then I am in trouble. This situation can be described by using negative numbers: I have negative-two apples, meaning that after I give Bart all the apples I have, I still owe him two. What is happening mathematically is that I have bumped up against a subtraction problem, 3–5, for which there is no solution (in whole numbers). Mathematicians respond by inventing a solution for it, and they call the solution –2. Thus, the desire to describe solutions for certain “impossible” subtraction problems leads to the invention of new numbers, the negative integers.4 Thanks to the negative integers, you can solve all whole number subtraction problems. But your problems are not over. You soon find that you cannot be

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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)=

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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).

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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,

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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.

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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

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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

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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

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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

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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

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Adding + It Up: Helping Children Learn Mathematics     with positive rational numbers over 2000 years before negative numbers became accepted. See also Behr, Harel, Post, and Lesh, 1992. 8.   The rules are in a sense guided by the fractional notation, a/b. In other notational systems, such as decimal representation, the rules will look somewhat different, although they will be equivalent. 9.   These numbers (and many others) are not rational because they cannot be expressed as fractions with integers in the numerator and denominator. 10.   In the number-line illustrations throughout this chapter, the portion displayed and the scale vary to suit the intent of the illustration. That is reasonable not just because one can imagine moving a “lens” left and right and zooming in and out, but also because the ideas are independent of the choice of origin and unit. 11.   Bruner, 1966 (pp. 10–11), suggests three ways of transforming experience into models of the world: enactive, iconic, and symbolic representations. Enactively, addition might be the action of combining a plate of three cookies with a plate of five cookies; iconically, it might be represented by a picture of two plates of cookies; symbolically, it might be represented as 5 cookies plus 3 cookies, or merely 5+3. 12.   Greeno and Hall, 1997. 13.   Pimm, 1995, suggests that people seek representational systems in which they can operate on the symbols as though the symbols were the mathematical objects. 14.   Duvall, 1999. 15.   Kaput, 1987, argues that much of elementary school mathematics is not about numbers but about a particular representational system for numbers. See Cuoco, 2001, for detailed discussions of various ways representations come into play in school mathematics. 16.   See Lakoff and Núñez, 1997, and Sfard, 1997, for detailed discussion of the metaphoric nature of mathematics. 17.   Sfard, 1997, p. 36, emphasis in original. 18.   “I remember as a child, in fifth grade, coming to the amazing (to me) realization that the answer to 134 divided by 29 is 134/29 (and so forth). What a tremendous labor-saving device! To me, ‘134 divided by 29’ meant a certain tedious chore, while 134/ 29 was an object with no implicit work. I went excitedly to my father to explain my discovery. He told me that of course this is so, a/b and a divided by b are just synonyms. To him it was just a small variation in notation” (Thurston, 1990, p. 847). 19.   Grouping is a common approach in measurement activities. For example, in measuring time, there are 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, approximately 30 days in a month, 12 months in a year, and so on. For distance, the customary U.S. system uses inches, feet, yards, and miles, and the metric system uses centimeters, meters, and kilometers. 20.   For example, IX means nine (that is, one less than ten), whereas XI means eleven (one more than ten). 21.   This generality was a significant accomplishment. In the third century B.C. in Greece, with its primitive numeration system, a subject of debate was whether there even existed a number large enough to describe the number of grains of sand in the universe. The issue was serious enough that Archimedes, the greatest mathematician

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Adding + It Up: Helping Children Learn Mathematics     of classical times, wrote a paper in the form of a letter to the king of his city explaining how to write such very large numbers. Archimedes, however, did not go so far as to invent the decimal system, with its potential for extending indefinitely. 22.   Knuth, 1974, p. 323. 23.   Steen, 1990. See Morrow and Kenney, 1998, for more perspectives on algorithms. 24.   The ellipsis points “…” in the expression are a significant piece of abstract mathematical notation, compactly designating the omission of the terms needed (to reach m, in this case). References Behr, M.J., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio, and proportion. In D.A.Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 296–333). New York: Macmillan. Bruner, J.S. (1966). Toward a theory of instruction. Cambridge, MA: Belknap Press. Cuoco, A. (Ed.). (2001). The roles of representation in school mathematics (2001 Yearbook of the National Council of Teachers of Mathematics). Reston, VA: NCTM. Duvall, R. (1999). Representation, vision, and visualization: Cognitive functions in mathematical thinking. Basic issues for learning. In F.Hitt & M.Santos (Eds.), Proceedings of the twenty-first annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (vol. 1, pp. 3–26). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education. (ERIC Document Reproduction Service No. ED 433 998). Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht, The Netherlands: Reidel. Greeno, J.G., & Hall, R. (1997). Practicing representation: Learning with and about representational forms. Phi Delta Kappan, 78, 1–24. Available: http://www.pdkintl.org/kappan/kgreeno.htm. [July 10, 2001]. Kaput,]. (1987). Representation systems and mathematics. In C.Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 19–26). Hillsdale, NJ: Erlbaum. Knuth, D.E. (1974). Computer science and its relation to mathematics. American Mathematical Monthly, 81, 323–343. Lakoff, G., & Núñez, R.E. (1997). The metaphorical structure of mathematics: Sketching out cognitive foundations for a mind-based mathematics. In L.D.English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 21–89). Mahwah, NJ: Erlbaum. Morrow, L.J., & Kenney, M.J. (Eds.). (1998). The teaching and learning of algorithms in school mathematics (1998 Yearbook of the National Council of Teachers of Mathematics). Reston, VA: NCTM. Pimm, D. (1995). Symbols and meanings in school mathematics. London: Routledge. Russell, B. (1919). Introduction to mathematical philosophy. New York: Macmillan. Sfard, A. (1997). Commentary: On metaphorical roots of conceptual growth. In L.D. English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 339–371). Mahwah, NJ: Erlbaum.

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Adding + It Up: Helping Children Learn Mathematics Steen, L. (1990). Pattern. In L.Steen (Ed.), On the shoulders of giants: New approaches to numeracy (pp. 1(10). Washington, DC: National Academy Press. Available: http://books.nap.edu/catalog/1532.html. [July 10, 2001]. Stevin, S. (1959). On decimal fractions (V.Sanford, Trans). In D.E.Smith (Ed.), A source book in mathematics (pp. 20–34). New York: Dover. (Original work published 1585) Stewart, I. (1989). Does God play dice? The mathematics of chaos. Oxford, England: Blackwell. Thurston, W.P. (1990). Mathematical education. Notices of the American Mathematical Society, 37, 844–850.