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From page 71... ...
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.

From page 72... ...
Another theme in school mathematics is measurement, which forms a bridge between number and geometry. Mathematicians like to take a bird'seye view of the process of developing an understanding of number.

From page 73... ...
, 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.

From page 74... ...
This is the area interpretation of multiplication. 6 ~ 6 ~ 6 ~ 6 I T I T I T 11 I T I T I 1I T I T I T If I T T I T I  `':TTTTT1 ~ 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.

From page 75... ...
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.

From page 76... ...
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.

From page 77... ...
3 NUMBER WHAT 15 THERE TO KNOWN 77 Box 31 Properties of the Arithmetic Operations Commutativity of addition. The order of the two numbers does not affect their sum: 3 + 5 = 8 = 5 + 3.

From page 78... ...
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.

From page 79... ...
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)

From page 80... ...
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.

From page 81... ...
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 numtersystem.

From page 82... ...
Recipes laboriously constructed by means of some sort of mathematics: basing a concrete interpretation of negative numbers are all completely dictated by mathematical this shortlistof rules of arithmetic. This uniquenessisa striking exhibition system on a of the power of these rules that they capture in a few general statements a short list large chunk of people's intuition about arithmetic.

From page 83... ...
Now, in the integers, subtraction is a true operation in the sense that you can subtract any integer from any other. As described in the rule on additive inverses in Box 32, for every integer, there is another integer, called its opposite or additive inverse, that counterbalances it: the two sum to zero.

From page 84... ...
Although this definition suffices to specify fractions as mathematical objects, fractions have many concrete interpretations. We refer the reader to the section "Discontinuities in Proficiency" in chapter 7 for a list of such interpretations.

From page 85... ...
The end result, however, is as elegant as one could wish. It turns out that either procedure produces a system in which all operations are possible, with additive inverses for all numbers and multiplicative inverses for all numbers except zero.

From page 86... ...
Adding fractions requires that they have a common denominator, which often requires conversion to equivalent fractions. When fractions have a common denominator, their sum is the fraction whose numerator is the sum of their numerators and whose denominator is the common denominator.

From page 87... ...
This choice amounts to specifying which side of the origin will be the positive half of the line; the other side is then the negative half. Finally, a unit of length is chosen.

From page 88... ...
The finite decimal system is intermediate between the integers and the rational numbers. The advantage of working with finite decimals rather than all the rational numbers is that the usual arithmetic for integers extends almost without change.

From page 89... ...
It is that arithmetic can remain procedurally similar to the arithmetic of whole numbers, and yet finite decimals can be arbitrarily small and, as a consequence, can approximate any number as closely as you wish. This process is best illustrated by using the number line.

From page 90... ...
With a number that is not a finite decimal, the process would go on forever, with each successive digit giving the number 10 times more precision. Thus, the finite decimals give you a systematic method for approximating any number to any desired accuracy.

From page 91... ...
. 1 1 1 0 O O rational numbers, although in the latter case it is hard to interpret the answer in simple form without dividing the intervals according to a common denominator.

From page 92... ...
. A deep understanding of number and operations on the number line requires flexibility in using each interpretation.

From page 93... ...
Multiplication by 1 takes positive numbers to their negative counterparts and vice versa, which amounts to flipping the line about the origin. These geometric interpretations of addition and multiplication as transformations of the line are quite sophisticated despite their pictorial nature.

From page 94... ...
Representations In this chapter we are concerned primarily with the physical representations for number, such as symbols, words, pictures, objects, and actions.ll Physical representations serve as tools for mathematical communication, thought, and calculation, allowing personal mathematical ideas to be externalized, shared, and preserved.l2 They help clarify ideas in ways that support reasoning and build understanding. These representations also support the development of efficient algorithms for the basic operations.l3 Mathematics requires representations.

From page 95... ...
Because many mathematical representations are suggestive of the corresponding metaphors, mathematical ideas are enhanced through multiple representations, which serve not merely as illustrations or pedagogical tricks but form a significant part of the mathematical content and serve as a source of mathematical reasoning. Even the numeral "729" is a representation that embodies a significant amount of mathematical thinking and interpretation.

From page 96... ...
It should also be emphasized that a representation system discussed previously, the number line, also deserves significant attention. In fact, the main unifying and synthesizing point of the previous section was that the number systems of school mathematics, which remain often fragmented and disjointed in the perceptions conveyed by school curricula, are in fact all subsystems of a single system, which has a geometric model that is the foundation of later analysis and geometry.

From page 97... ...
Thus, in 729 the "7" represents seven hundreds, whereas in 174 the "7" means seven tens. Some pictorial and physical representations can be helpful in understanding the decimal placevalue system.

From page 98... ...
Furthermore, it is quite concise, requiring only nine digits to represent the population of the United States, and only 10 digits to represent the population of the entire earth. This conciseness, however, presents a challenge to young learners as they try to understand this compact notational system.

From page 99... ...
Fractional values are often represented with pictures, and relationships between quantities are often represented with graphs or tables. Communicating about mathematical ideas, therefore, requires that one choose representations and translate among them.

From page 100... ...
It does not mean 3 X 5 . Fu rthermore, juxtaposing sym bols to indicate multiplication creates confusion in high school mathematics with the introduction of function notation, where f(4)

From page 101... ...
(See Box 39 for an example.) Box 39 Translating Among Representations: An Example Perhaps the deepest translation problem in preK to grade 8 mathematics concerns the translation between fractional and decimal representations of rational numbers.

From page 102... ...
So the decimal representation of a rational number must be either a repeating or a terminating decimal. Thus a nonrepeating decimal cannot be a rational number and there are many such numbers, such as ~ and 5.

From page 103... ...
ltles. Algorithms are important in school mathematics because they can help students understand better the fundamental operations of arithmetic and important concepts such as place value and also because they pave the way for learning more advanced topics.

From page 104... ...
Multiplication Expanded method Multiplying polynomials 23 23 = 20 + 3 2x + 3 x15 X 15 = 10 + 5 x + 5 115 100 + 15 10x + 15 23 200 + 30 2x2 + 3x 345 200 + 130 + 15 = 345 2x2 + 1 3x + 15 Box 310 Examples of Algorithms The decimal placevalue system allows many different algorithms for the four main operations. The following six algorithms for multiplication of twodigit numbers were produced by a class of prospective elementary school teachers.

From page 105... ...
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.)

From page 106... ...
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.

From page 107... ...
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.

From page 108... ...
_~ _ __ A closely related numerical approach to the problem of counting handshakes comes from a story told of young Carl Friedrich Gauss (17771855) , 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.

From page 109... ...
100 + 99 + ~ + 2 + 98 + L 3 + L + 98 + 99 + 100 + 3 + 2 + 101 + 101 + 101 + L + 101 + 101 + ~ 01 . For the original handshake problem, which involves the sum of the blocks in the staircase above, that means taking the double sum 7 x 8, or 56, and halving it to get 28.

From page 110... ...
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.

From page 111... ...
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.)

From page 112... ...
15. Kaput, 1987, argues that much of elementary school mathematics is not about numbers but about a particular representational system for numbers.

From page 113... ...
(1998~. The teaching and learning of algorithms in school mathematics ~ 1998 Yearbook of the National Council of Teachers of Mathematics)

From page 114... ...
Notices of the American Mathematical Society, 37, 844850.

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