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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity 2 Foundational Mathematics Content Mathematics provides a powerful means for understanding and analyzing the world. Mathematical ways of describing and representing quantities, shapes, space, and patterns help to organize people’s insights and ideas about the world in systematic ways. Some of these mathematical systems have become such a fundamental part of people’s everyday lives—for example, counting systems and methods of measurement—that they may not recognize the complexity of the ideas underpinning them. In fact, the mathematical ideas that are suitable for preschool and the early grades reveal a surprising intricacy and complexity when they are examined in depth. At the deepest levels, they form the foundations of mathematics that have been studied extensively by mathematicians over centuries (e.g., see Grattan-Guinness, 2000) and remain a current research topic in mathematics. In this chapter, we provide an overview of the mathematical ideas that are appropriate for preschool and the early grades and discuss some of the more complex mathematical ideas that build on them. These foundational ideas are taken for granted by many adults and are not typically examined in high school or college mathematics classes. Thus, many people with an interest in early childhood education may not have had adequate opportunities in their preparation to examine these ideas. Chapters 5 and 6 examine these ideas again in some detail, from the perspective of how children come to understand them and the conceptual connections they make in doing so. This chapter has four sections. The first two describe mathematics for young children in two core areas: (1) number and (2) geometry and mea-
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity surement. These ideas, which are important preparation for school and for life, are also genuinely mathematical, with importance from a mathematician’s perspective. Moreover, they are interesting to children, who enjoy engaging with these ideas and exploring them. The third section describes mathematical process goals, both general and specific. The general process goals are used throughout mathematics, in all areas and at every level, including in the mathematics for very young children. The specific process goals are common to many topics in mathematics. These process goals must be kept in mind when considering the teaching and learning of mathematics with young children. The fourth section describes connections across the content described in the first two sections as well as to important mathematics that children study later in elementary school. These connections help to demonstrate the foundational nature of the mathematics described in the first two sections. NUMBER CONTENT Number is a fundamental way of describing the world. Numbers are abstractions that apply to a broad range of real and imagined situations—five children, five on a die, five pieces of candy, five fingers, five years, five inches, five ideas. Because they are abstract, numbers are incredibly versatile ways of explaining the world. “Yet, in order to communicate about numbers, people need representations—something physical, spoken, or written” (National Research Council, 2001, p. 72). Understanding number and related concepts includes understanding concepts of quantity and relative quantity, facility with counting, and the ability to carry out simple operations. We group these major concepts into three core areas: number, relations, and operations. Box 2-1 summarizes the major ideas in each core area. Developing an understanding of number, operations, and how to represent them is one of the major mathematical tasks for children during the early childhood years. The Number Core The number core concerns the list of counting numbers 1, 2, 3, 4, 5, … and its use in describing how many things are in collections. There are two distinctly different ways of thinking about the counting numbers: on one hand, they form an ordered list, and, on the other hand, they describe cardinality, that is, how many things are in a set. The notion of 1-to-1 correspondence bridges these two views of the counting numbers and is also central to the notion of cardinality itself. Another subtle and important aspect of numbers is the way one writes (and says) them using the base 10
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity BOX 2-1 Overview of Number, Relations, and Operations Core The Number Core: Perceive, Say, Describe/Discuss, and Construct Numbers Cardinality: giving a number word for the numerosity of a set obtained by perceptual subitizing (immediate recognition of 1 through 3) or conceptual subitizing (using a number composition/decomposition for larger numerosities), counting, or matching. Number word list: knowing how to say the sequence of number words. 1-to-1 counting correspondences: counting objects by making the 1-to-1 time and spatial correspondences that connect a number word said in time to an object located in space. Written number symbols: reading, writing, and understanding written number symbols (1, 2, 3, etc.). Coordinations across the above, such as using the number word list in counting and counting to find the cardinality of a set. The Relations Core: Perceive, Say, Describe/Discuss, and Construct the Relations More Than, Less Than, and Equal To on Two Sets by Using general perceptual, length, density strategies to find which set is more than, less than, or equal to another set, and then later. Using the unitizing count and match strategies to find which set is more than, less than, or equal to another set, and then later. Seeing the difference between the two sets, so the relational situation becomes the additive comparison situation listed below. The Operations Core: Perceive, Say, Describe/Discuss, and Construct the Different Addition and Subtraction Operations (Compositions/Decompositions of Numbers) Change situations: addition change plus situations (start + change gives the result) and subtraction change minus situations (start − change gives the result). Put together/take apart situations: put together two sets to make a total; take apart a number to make two addends. Compose/decompose numbers: Move back and forth between the total and its composing addends: “I see 3. I see 2 and 1 make 3.” Embedded number triads: Experience a total and addends hiding inside it as a related triad in which the addends are embedded within the total. Additive comparison situations: Comparing two quantities to find out how much more or how much less one is than the other (the Relations Core precedes this situation).
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity system. The top section of Box 2-1 provides an overview of the number core from the perspective of children’s learning; this is discussed in more detail in Chapter 5. Here we discuss the number core from a mathematical perspective, as a foundation for the discussion of children’s learning. Numbers Quantify: They Describe Cardinality Numbers tell “how many” or “how much.” In other words, numbers communicate how many things there are or how much of something there is. One can use numbers to give specific, detailed information about collections of things and about quantities of stuff. Initially, some toy bears in a basket may just look like “some bears,” but if one knows there are seven bears in the basket, one has more detailed, precise information about the collection of bears. Numbers themselves are an abstraction of the notion of quantity because any given number quantifies an endless variety of situations. We use the number 3 to describe the quantity of three ducks, three toy dinosaurs, three people, three beats of a drum, and so on. We can think of the number 3 as an abstract, common aspect that all these limitless examples of sets of three things share. How can one grasp this common aspect that all sets of three things share? At the heart of this commonality is the notion of 1-to-1 correspondence. Any two collections of three things can be put into 1-to-1 correspondence with each other. This means that the members of the first collection can be paired with the members of the second collection in such a way that each member of the first collection is paired with exactly one member of the second collection, and each member of the second collection is paired with exactly one member of the first collection. For example, each duck in a set of three ducks can be paired with a single egg from a set of three eggs so that no two ducks are paired with the same egg, no two eggs are paired with the same duck, and no ducks or eggs remain unpaired. The Number List The counting numbers can be viewed as an infinitely long and ordered list of distinct numbers. The list of counting numbers starts with 1, and every number in the list has a unique successor. This creates a specific order to the counting numbers, namely 1, 2, 3, 4, 5, 6, …. It would not be correct to leave a number out of the list, nor would it be correct to switch the order in which the list occurs. Also, every number in the list of counting numbers appears only once, so it would be wrong to repeat any of the numbers in the list. The number list is useful because it can be used as part of 1-to-1 ob-
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity ject counting to tell how many objects are in a collection. Although the number of objects in small collections (up to 3 or 4) can be recognized immediately—this is called subitizing—in general, one uses the number list to determine the number of objects in a set by counting. Counting allows one to quantify exactly collections that are larger than can be immediately recognized. To count means to list the counting numbers in order, usually starting at 1, but sometimes starting at another number, as in 5, 6, 7, …. (Other forms of counting include “skip counting,” in which one counts every second, or third, or fourth, etc., number, such as 2, 4, 6, …, and counting backward, as in 10, 9, 8, 7, ….) Although adults take it for granted because it is so familiar, the connection between the list of counting numbers and the number of items in a set is deep and subtle. It is a key connection that children must make. There are also subtleties and deep ideas involved in saying and writing the number list, which adults also take for granted because their use is so common. Because of the depth and subtlety of ideas involved in the number list and its connection to cardinality, and because these ideas are central to all of mathematics, it is essential that children become fluent with the number list (see Box 2-2). Connecting the number list with cardinality. In essence, counting is a way to make a 1-to-1 correspondence between each object (in which the BOX 2-2 The Importance of Fluency with the Number List All of the work on the relations/operation core in kindergarten serves a double purpose. It helps children solve larger problems and become more fluent in their Level 1 solution methods. It also helps them reach fluency with the number word list in addition and subtraction situations, so that the number word list can become a representational tool for use in the Level 2 counting of solution methods. To get some sense of this process, try to add or subtract using the alphabet list instead of the number word sequence. For counting on, you must start counting with the first addend and then keep track of how many words are counted on. Many adults cannot start counting within the alphabet from D or from J because they are not fluent with this list. Nor do they know their fingers as letters (How many fingers make F?), so they cannot solve D + F by saying D and then raising a finger for each letter said after D until they have raised F fingers. It is these prerequisites for counting on that kindergarten children are learning as they count, add, and subtract many, many times. Of course as they do this, they will also begin to remember certain sums and differences as composed/decomposed triads (as number facts).
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity objects can be any discrete thing, from a doll, to a drumbeat, to the idea of a unicorn) and a prototypical set, namely a set of number words. For example, when a child counts a set of seven bears, the child makes a 1-to-1 correspondence between the list 1, 2, 3, 4, 5, 6, 7 and the collection of bears. To count the bears, the child says the number word list 1, 2, 3, 4, 5, 6, 7 while pointing to one new bear for each number. As a result, each bear is paired with one number, each number is paired with one bear, and there are no unpaired numbers or bears once counting is completed. The pairing could be carried out in many different ways (starting with any one of the bears and proceeding to any other bear next, and so on), but any single way of making such a 1-to-1 correspondence by counting establishes that there are seven bears in the set. A key characteristic of object counting is that the last number word has a special status, as it specifies the total number of items in a collection. For example, when a child counts a set of seven bears, the child counts 1, 2, 3, 4, 5, 6, 7, pointing to one bear for each number. The last number that is said, 7, is not just the last number in the list, but also indicates that there are seven bears in the set (i.e., cardinality of the set). Thus when counting the 7 bears, the counter shifts from a counting reference (to 7 as the last bear when counting) to a cardinal reference when referring to 7 as the number of bears in all. Counting therefore provides another way to grasp the abstract idea that all sets of a fixed number of things share a common characteristic—that when one counts two sets that have the same number of objects, the last counting word said will be the same for both. Another key observation about counting is that, for any given number in the list of counting numbers, the next number in the list tells how many objects are in a set that has one more object than do sets of the given number of objects. For example, if there are five stickers in a box and one more sticker is put into the box, then one knows even without counting them all again that there will now be six stickers in the box, because 6 is the next number in the counting list. Generally each successive counting number describes a quantity that is one more than the quantity that the previous number describes. In a sense, then, counting is adding: Each counting number adds one more to the previous collection (see Figure 2-1). Of course, if one counts backward, then one is subtracting. These observations are essential for children’s early methods of solving addition and subtraction problems. Also, each step in the counting process can be thought of as describing the total number of objects that have been counted so far. The number word list and written number symbols in the base 10 place-value system. Each number in the number list has a unique spoken name and can be represented by a unique written symbol. The names and symbols for the initial numbers in the list have been passed along by tradition, but
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity FIGURE 2-1 Each counting number describes a quantity that is one more than the previous number describes. the English names of the first 10 (or so) counting numbers and the symbols of the first 9 counting numbers are arbitrary and could have been different. For example, instead of the English word “three,” one could be using “bik” or “Russell” or any other word, such as the words for “three” in other languages. Instead of the symbol 3, one could use a symbol that looks completely different. The list of counting numbers needs to go on and on in order to count ever larger sets. So the problem is how to give a unique name to each number. Different cultures have adopted many different solutions to this problem (e.g., Menninger, 1958/1969; see Chapter 4 of this volume for a discussion of counting words in different languages). The present very efficient solution to this problem was not obvious and was in fact a significant achievement in the history of human thought (Menninger, 1958/1969). Even though the first nine counting numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, are represented by distinct, unrelated symbols, some mechanism for continuing to list numbers without resorting to creating new symbols indefinitely is desirable. The decimal system (or base 10 system) is the ingenious system used today to write (and say) counting numbers. The decimal system allows one to use only the 10 digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to write any counting number as a string of digits (such a written representation of a number is often called a numeral).
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity The system is called a base 10 system because it uses 10 distinct digits and is based on repeated groupings by 10. The use of only 10 digits to write any counting number, no matter how large, is achieved by using place value. That is, the meaning of a digit in a written number depends (in a very specific way) on its placement. The details about using the decimal system BOX 2-3 Using the Decimal System to Write the List of Counting Numbers Each of the first nine counting numbers (or number words) “one, two, …, nine,” requires only one digit to write, 1, 2, …, 9.Each digit stands for that many things—in other words, that many “ones,” as indicated at the top of Figure 2-2. Each of these digits is viewed as being in the “ones place.” FIGURE 2-2 Decimal system 1 The next counting number, ten, requires two digits to write. The 1 stands for 1 ten and the 0 stands for 0 ones, and 10 stands for the combined amount in 1 ten and 0 ones. This way of describing and writing the number ten requires thinking of it as a single group of ten—in other words, as a new entity in its own right, which is created by joining 10 separate things into a new coherent whole, as indicated in the figure by the way 10 dots are shown grouped to form a single unit of 10.
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity to write the list of counting numbers are given in Box 2-3: A key idea is to create larger and larger units, which are the values of places farther and farther to the left, by taking the value of each place to be 10 times the value of the previous place to its right. One can think of doing this by bundling together 10 of the previous place’s value. The greater and greater values In each of the next two-digit counting numbers, 11, 12, 13, 14, 15, …, 20, 21, 22, … , 30, 31,, 97, 98, 99, the digit on the right stands for that many ones, so one says this digit is in the “ones place,” and the digit on the left stands for that many tens, so one says it is in the “tens place”; the number stands for the combined amount in those tens and ones. For example, in 37, the 3 stands for 3 tens, the 7 stands for 7 ones, and 37 stands for the combined amount in 3 tens and 7 ones. Notice that from 20 on, the way one says number words follows a regular pattern that fits with the way these numbers are written. But the way one says 11 through 19 does not fit this pattern. In fact, 13 through 19 are said backward, because the ones digit is said before the tens digit is indicated. The number 99 is the last two-digit counting number, and it stands for the combined amount in 9 tens and 9 ones (see Figure 2-3). The next counting number will be the number of dots there are when one more dot is added to the dots on the left of the figure. This additional dot “fills up” a group of ten, as indicated in the middle of the figure. Now there are 10 tens, but there isn’t a digit that can show this many tens in the tens place. So the 10 tens are bundled together to make a new coherent whole, as indicated on the right in Figure 2-3, which is called a hundred. From 0 to 9 hundreds can be recorded in the place to the left of the tens place, which is called the hundreds place. So the next counting number after 99 is written as 100, in which the 1 stands for 1 hundred, and the 0s stand for 0 tens and 0 ones. FIGURE 2-3 Decimal system 2. The decimal system has a systematic way to make new larger units by bundling 10 previously made units and recording the new unit one place to the left of the given unit’s place. Just as 10 ones make a new unit of 10, which is recorded to the left of the ones place, 10 tens make a new unit of a hundred, which is recorded to the left of the tens place, and 10 hundreds make a new unit of a thousand, which is recorded to the left of the hundreds place. This pattern continues on and on to new places on the left.
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity of the places allow any number, no matter how large, to be expressed as a combination of between 0 and 9 of each place’s value. In this way, every counting number can be expressed in a unique way as a numeral made of a string of digits. (See Howe, 2008, for a further discussion of the decimal system and place value.) Even though most countries around the world now use this system of written numerals, they still use their own list of counting words that relate closely, or not so closely, to the written system of numerals. English and other European lists of counting words have various aspects that do not fit the decimal system so well and that create difficulties in learning the system. These, and ways to compensate for these difficulties, are discussed in Chapter 4. The Relations/Operations Core Numbers do not exist in isolation. They make up a coherent system in which numbers can be compared, added, subtracted, multiplied, and divided. Just as numbers are abstractions of the notion of quantity, the relations “less than,” “greater than,” and “equal to” and the operations of addition, subtraction, multiplication, and division are abstractions of comparing, combining, and separating quantities. These relations and operations apply to a wide variety of problems. The middle and bottom sections of Box 2-1 are an overview of the relations core and the operations core for young children (which concerns only addition and subtraction, not multiplication or division). Comparing In some cases it is visually evident that there are more things in one collection than in another, such as in the case of the two sets of beads shown at the top of Figure 2-4. But in other cases it is not immediately clear which collection (if either) has more items in it. A basic way to compare two collections of objects is by direct matching (as in the middle of Figure 2-4). If a child has a collection of black beads and another collection of white beads, and if these collections are placed near each other, the child can place each black bead with one and only one white bead. If there is at least one extra white, then there are more whites; if at least one extra black, then more blacks. And if none is left over, then the two groups have the same number (although one may not know and does not need to know exactly what number it is). When direct matching is not possible, a child can count the number of beads in two collections to determine which collection (if either) has more beads or if they both have the same number of beads. A key observation
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity FIGURE 2-4 Comparing. about using counting to compare is that a number that is said later in the counting word list corresponds to a collection that has a greater number of objects than does a collection corresponding to a number earlier in the sequence. For example, one knows that there are more beads in a collection of eight black beads than there are in a collection of seven white beads because 8 occurs later in the counting list than 7 (see the bottom of Figure 2-4). Counting thus provides a more advanced way to compare sets of things than direct matching because it relies on knowledge about how numbers compare. Counting is also a more powerful way to compare sets of things than direct matching because it allows sets that are not in close proximity to be compared. A key point about comparing collections of objects is that counting can be used to do so, and it relies on the link between the number list and cardinality: Numbers later in the list describe greater cardinalities than do numbers earlier in the list. Finding out which collection is more than another collection is easier than determining exactly how many more that collection has than the other, which can be formulated as an addition or subtraction problem. This more specific version of comparison is discussed in the next section.
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity MATHEMATICAL CONNECTIONS In this section we discuss some of the main connections across content areas of early childhood mathematics and into later mathematics. Mathematics as a whole is a web of interconnected ideas, and the mathematics of early childhood is no exception. Mathematics is also deep, in that every mathematical idea, including those of early childhood, is embedded in long chains of related ideas. As this section shows, the foundational and achievable mathematical ideas discussed in the previous sections are tightly interwoven with each other and with other important ideas that are studied later in mathematics. Connections in Structuring Numbers, Shapes, and Space Throughout mathematics, structure is found and analyzed by composing and decomposing. A group of objects can be joined to form a new composite object. An object can be decomposed to reveal its finer structure. Some of the most important connections in elementary mathematics concern structuring of numbers and space via composition and decomposition. We now discuss several of these connections. Making Units by Grouping Numbers are structured by composition because the decimal place-value system relies on grouping by tens. In the realm of number, 10 individual counters are viewed as forming a single composite unit of 10. A geometric version of this grouping idea occurs when several shapes are put together to form another larger shape, which is then viewed as a unified shape in its own right, such as if the unified shape is seen as a possible substitute for another shape or as able to fill a space in a puzzle. When children (or adults) make a repeating pattern, they might focus mainly on maintaining a certain order. But repeating patterns can also be viewed as made from a single composite unit that is copied over and over. This is not unlike viewing the counting numbers as a sequence that is structured in groups of 10 (see Figure 2-8). Repeating patterns and, more generally, making groups of equal size are the basis for multiplication and division. Later in elementary school, when children skip count by fives, by counting 5, 10, 15, 20, … to list the multiples of 5, this pattern can be viewed as a growing pattern, but it can also be viewed as counting every fifth entry in a repeating pattern of 5. When children study division with remainders (in around fourth or fifth grade), they may observe a repeating pattern in the remainders. For example, when dividing successive counting numbers by 5, say, the remainders cycle through 0, 1, 2, 3, and 4.
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity FIGURE 2-8 A repeating pattern is formed by repeating a unit. In counting, the ones digits form a repeating pattern. Groups of Groups: Numbers, Shapes, and 2-D Space The compositional structure of the decimal system is more complex than just making groups of 10 from 10 ones, since every 10 groups of 10 are composed into a unit of 100. A geometric version of this group’s idea occurs when shapes are put together to form a new, composite shape, and composite shapes are then put together to make another composite shape—a composite of the composite shapes. An especially important case of geometric structuring as composites of composites occurs when analyzing rectangles and their areas. When considering the area of a rectangle, one views the rectangle as composed of identical square tiles that cover the rectangle without gaps or overlaps. Each square tile has area one square unit. The area of the rectangle (in square units) is the number of squares that cover the rectangle. Although these squares can be counted one by one, to develop and understand the length × width formula for the area of a rectangle, the squares must be seen as grouped, either into rows or into columns (see Figure 2-6). Each row has the same number of squares in it, and the number of rows in the rectangle is equal to the number of squares in a column (likewise, each column has the same number of squares in it, and the number of columns is the number of squares in a row). Because of this grouping structure, the area of the rectangle is # rows × # in each row or length × width (square units). Similarly, the decimal system has a multiplicative structure because 100 is formed (by definition) by making 10 groups of 10, and so 100 = 10 × 10. The idea of structuring rectangles as arrays of squares can be extended to structuring an entire infinite plane (in the imagination) as an infinite array of squares. This idea of a plane structured by an infinite array is essentially the idea of the Cartesian coordinate plane, in which each point in the plane is described by a pair of numbers that indicate its location relative to two coordinate lines (axes) (see Figure 2-9).
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity FIGURE 2-9 The coordinate plane. Groups of Groups of Groups: Numbers, Shapes, and 3-D Space The compositional structure of the decimal system consists not only of making groups of 10 from 10 ones and groups of 100 from 10 groups of 10, but also groups of 1,000 from 10 groups of 100, so that 1,000 = 10 × 10 × 10. The grouping structure of the decimal system continues in such a way that all successive groupings are obtained by repeatedly grouping by 10. The geometric counterpart of this grouping structure of the decimal system takes one into 3-D space and then higher dimensional space. Just as 2-D rectangles can be structured as 2-D arrays of squares, so, too, 3-D rectangular prisms (box shapes) can be structured as 3-D arrays of cubes. As in the case of rectangles, the multiplicative structure of a 3-D array of cubes explains why one multiplies the three dimensions of length, width, and height of a box to find its volume. Box shapes can be built as layers of identical cubes, as in Figure 2-12, and each layer can be viewed as groups of rows, so a box built from cubes can be viewed as a group of a group of cubes in the same way that 1,000 is 10 groups of 10 groups of 10. When one extends the array structure of rectangular prisms to all of 3-D space, one gets essentially the idea of coordinate space, in which the location of each point in space is described by a triple of numbers that indicate its location relative to three coordinate lines. Motion, Decomposing and Composing, Symmetry, and Properties of Arithmetic The properties (or laws) of arithmetic are the fundamental structural properties of addition and multiplication from which all of arithmetic is derived. These properties include the commutative properties of addition
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity and of multiplication, the associative properties of addition and multiplication, and the distributive property of multiplication over addition. The commutative properties of addition and multiplication state that The associative properties of addition and multiplication state that The distributive property states that Each property can be illustrated by moving and reorganizing objects, sometimes also by decomposing and recomposing a grouping, and sometimes even in terms of symmetry. The report Adding It Up: Helping Children Learn Mathematics has a good discussion and an illustration of the commutative and associative properties of addition, the commutative and associative properties of multiplication, and the distributive property (National Research Council, 2001, Chapter 3 and Box 3-1). The commutative property of addition is illustrated by switching the order in which two sets are shown. The commutative property is especially useful in conjunction with counting on strategies for solving addition problems (see Chapter 5 for further discussion of children’s problem-solving strategies for addition and subtraction). For example, instead of counting on 6 from 2 to calculate 2 + 6, a child can switch the problem to 6 + 2 and count on 2 from 6. The associative property involves starting with three separate sets, two of which are close together, separating the two that are close together, and moving one of those sets to reassociate with the other set. The associative property of addition is used in make-a-ten methods, when one number is decomposed so that one of the pieces can be recomposed with another number to make a group of 10. Early experiences with properties of addition then extend to multiplication in third and fourth grade. The commutative and associative properties of multiplication and the distributive property are essential to understanding relationships among basic multiplication facts and to understanding multidigit multiplication and division. For example, knowing that 3 × 5 = 5 × 3 and that 7 × 8 can be obtained by adding 5 × 8 and 2 × 8 lightens the load in learning the multiplication tables. The commutative property of multiplication is illustrated by decomposing a rectangular array in two different ways: by the rows or by the columns (as shown in Figure 2-6)
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity or in terms of a rotation (see National Research Council, 2001, Box 3-1). The associative property of multiplication can be illustrated by decomposing a 3-D array (or box shape built of blocks) in different ways (one way is shown in Figure 2-7). The distributive property is illustrated by viewing objects as grouped in two different ways (see National Research Council, 2001, Box 3-1). The properties of multiplication can be illustrated with arrays and rectangles, and they are also visible in the multiplication tables, which contain many relationships and have important structure. One structural aspect of the multiplication tables is their diagonal symmetry. This diagonal symmetry corresponds with the commutative property of multiplication, namely that a × b = b × a for all numbers a and b. Recognizing this symmetry allows children to learn multiplication facts more efficiently. In other words, once they know the upper right-hand triangular portion of the multiplication tables in around third grade, they can fill in the rest of the table by reflecting across the diagonal (see Figure 2-10). Patterns associated with horizontal or vertical shifts (slides) can also be seen in the multiplication tables. For example, the entries in two columns are related by the column that is associated with the amount of shift between the columns (see Figure 2-10). This structural relationship corresponds with the distributive property. Connections in Measurement and Number: Fractions Once children encounter measurement situations, the possibility of fractions arises naturally. Fractions can be shown well in the context of FIGURE 2-10 Symmetry and relationships in the multiplication table.
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity length and on number lines (in around second or third grade). A number line is much like an infinitely long ruler, so number lines can be viewed as unifying measurement and number in a one-dimensional space. A number on a number line can be thought of as representing the length from 0 to the number (see Figure 2-11). Because of the close connection between number lines and length, number lines are difficult for children below about second grade. In contrast, the number paths on most number board games used for preschoolers are a count model, not a number line. There is a path of squares, circles, or rocks, each has a number on it, and players move along this path by counting the squares or other objects or saying the number on them as they move. These are appropriate for younger children because they can support their knowledge of counting, cardinality, comparing, and number symbols. In measurement, there is an important relationship between the size of a unit and the number of units it takes to make a given, fixed quantity. For example, if the triangle in Figure 2-5 is designated to have 1 unit of area, then the hexagon has an area of 6 units. But if one picks a new unit of area, such as designating the area of the rhombus in Figure 2-5 to be 1 unit, which is twice the size of the triangle, then the hexagon has an area of only 3 units. Later in elementary school (in around second grade), children see this inverse relationship between the size of a unit of measurement and the number of units it takes to make a given quantity reflected in the inverse relationship between the ordering of the counting numbers and the ordering of the unit fractions (see Figure 2-12). Connections in Data Analysis, Number, and Measurement To use data to answer (or address) a question, one must analyze the data, which often involves classifying the data into different categories, FIGURE 2-11 Number lines relate numbers to lengths.
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity FIGURE 2-12 1>½ >⁄3 >¼. displaying the categorized data graphically, and describing or comparing the categories. Because the process of describing or comparing categories usually involves number or measurement, number and measurement are central to data analysis, and data analysis provides a context to which number and measurement can be applied. The collection of data should ideally start with a question of interest to children. For example, children in a class might be interested in how everyone got to school in the morning and might wonder what way was most popular. To answer this question, children might divide themselves into different groups according to how they got to school in the morning (by bus, by car, by walking, or by bike). The children could then make “real graphs” (graphs made of real objects) either by lining up in their categories or by each placing a small toy or token to represent a bus, a car, a pair of shoes, or a bike into predrawn squares, as shown on the left in Figure 2-13 (the predrawn squares ensure that each object occupies the same amount of space in the graph). Instead of a real graph, children could display the data somewhat more abstractly in a pictograph by lining up sticky notes in categories, as on the right in the figure. Each child places a sticky note in the category for how the child got to school. In general, pictographs use small, identical pictures to represent data. In this case, each sticky note stands for a single piece of data and functions as a small picture in a pictograph. Children can then use these real graphs or pictographs to answer such questions as “How many children rode a bus to get to school today?” or “Did more children ride in a car or walk to school today?” or even “If it were raining today, how do you think the graph might be different?” Data displays that are used in posing and answering such quantitative questions serve a purpose and help children mathematize their daily experiences. In contrast, data displays that are only
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity FIGURE 2-13 A template for a “real graph” and a pictograph made with sticky notes. made but not discussed are not likely to help children develop or extend their mathematical thinking. In around second or third grade, once children have worked with linear measurement, they can begin to work with bar graphs. One can think of bar graphs as arising from pictographs by fusing the separated entries in a pictograph to make the bars in a bar graph. In this way, the discrete counting of separate entries in a pictograph gives way to the length measurement of a bar in a bar graph. In third grade or so, once children have begun to skip count and to multiply, the entries in a pictograph can be used to represent more than one single piece of data. For example, each picture might represent 2 pieces of data or 10 pieces of data. SUMMARY This chapter describes the foundational and achievable mathematics content for young children. The focus of this chapter is on the mathematical ideas themselves rather than on the teaching or learning of these ideas. These mathematical ideas are often taken for granted by adults, but they are surprisingly deep and complex. There are two fundamental areas of mathematics for young children: (1) number and (2) geometry and measurement as identified in NCTM’s Curriculum Focal Points and outlined by this committee. There are also important mathematical reasoning processes that children must engage in. This chapter also describes some of the most important connections of the mathematics for young children to later mathematics.
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity In the area of number, a fundamental idea is the connection between the counting numbers as a list and for describing how many objects are in a set. We can represent arbitrarily large counting numbers in an efficient, systematic way by means of the remarkable decimal system (base 10). We can use numbers to compare quantities without matching the quantities directly. The operations of addition and subtraction allow us to describe how amounts are related before and after combining or taking away, how parts and totals are related, and to say precisely how two amounts compare. In the area of geometry and measurement, a fundamental idea is that geometric shapes have different parts and aspects that can be described, and they can be composed and decomposed. To measure the size of something, one first selects a specific measurable attribute of the thing, and then views the thing as composed of some number of units. The shapes of geometry can be viewed as idealized and simplified approximations of objects in the world. Space has structure that derives from movement through space and from relative location within space. An important way to think about the structure of 2-D and 3-D space comes from viewing rectangles as composed of rows and columns of squares and viewing box shapes as composed of layers of rows and columns of cubes. REFERENCES AND BIBLIOGRAPHY Clements, D.H., and Sarama, J. (2007). Early childhood mathematics learning. In F.K. Lester, Jr. (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 461-555). New York: Information Age. Clements, D.H., Battista, M.T., Sarama, J., Swaminathan, S., and McMillen, S. (1997). Students’ development of length measurement concepts in a logo-based unit on geometric paths. Journal for Research in Mathematics Education, 28(1), 70-95. Grattan-Guinness, I. (2000). The Search for Mathematical Roots 1870-1940: Logics, Set Theories, and the Foundations of Mathematics from Cantor Through Russell to Gödel. Princeton, NJ: Princeton University Press. Howe, R. (2008). Taking Place Value Seriously: Arithmetic, Estimation and Algebra. Available: http://www.maa.org/pmet/resources/PlaceValue_RV1.pdf [accessed September 2008]. Menninger, K. (1958/1969). Number Words and Number Symbols: A Cultural History of Numbers. (P. Broneer, Trans.). Cambridge, MA: MIT Press. (Original work published 1958.) National Association for the Education of Young Children and National Council of Teachers of Mathematics. (2002). Early Childhood Mathematics: Promoting Good Beginnings. A joint position statement of the National Association for the Education of Young Children and National Council of Teachers of Mathematics. Available: http://www.naeyc.org/about/positions/pdf/psmath.pdf [accessed August 2008]. National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: Author. National Research Council. (2001). Adding It Up: Helping Children Learn Mathematics. Mathematics Learning Study Committee. J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity Reynolds, A., and Wheatley, G.H. (1996). Elementary students’ construction and coordination of units in an area setting. Journal for Research in Mathematics Education, 27(5), 564-581. Sophian, C. (2007). Rethinking the starting point for mathematics learning. In O.N. Saracho and B. Spodek (Eds.), Contemporary Perspectives in Early Childhood Education: Mathematics, Science, and Technology in Early Childhood Education (pp. 21-44). New York: Information Age. Wheatley, G.H. (1990). Spatial sense and mathematics learning. Arithmetic Teacher, 37(6), 10-11.
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