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### 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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2 Foundational Mathematics Content Mathematics provides a powerful means for understanding and analyz- ing 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 ex- ample, 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 mathemat- ics 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 op- portunities 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- 21

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22 MATHEMATICS LEARNING IN EARLY CHILDHOOD surement. These ideas, which are important preparation for school and for life, are also genuinely mathematical, with importance from a mathema- tician’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 math- ematics. 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 ver- satile 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 num- ber 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 cor- respondence 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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2 MATHEMATICS LEARNING IN EARLY CHILDHOOD 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 collec- tions 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 be- cause 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 corre- spondence. 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 or- der 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 count- ing 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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25 FOUNDATIONAL MATHEMATICS CONTENT 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 con- nection 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 com- mon. 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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26 MATHEMATICS LEARNING IN EARLY CHILDHOOD 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 num- ber 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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2 FOUNDATIONAL MATHEMATICS CONTENT 1: “one” +1 2: “one” “two” +1 3: “one” “two” “three” +1 4: “one” “two” “three” “four” 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 Figure 2-1 of the first 9 counting numbers are arbitrary and could have been differ- ent. For example, instead of the R01420 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 ef- ficient 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 continu- ing 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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2 MATHEMATICS LEARNING IN EARLY CHILDHOOD 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 alue. 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.” 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Developing shor thand pictures for a group of ten: 20 21 22 23 24 25 26 27 28 29 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 Figure 2-2 is created by joining 10 separate things into a new coherent whole, as indicated R01420 in the figure by the way 10 dots are shown grouped to form a single unit of 10.

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2 FOUNDATIONAL MATHEMATICS CONTENT 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 com- bined 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. +1 99 How to write 10 tens? 100 FIGURE 2-3 Decimal system 2. The decimal system has a systematic way to make new larger units by bun- dling 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 Figure 2-3 the left of the ones place, 10 tens make a new unit of a hundred, which is recorded R01420 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 MATHEMATICAL CONNECTIONS In this section we discuss some of the main connections across content areas of early childhood mathematics and into later mathematics. Math- ematics 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 compos- ing 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 con- cern 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 struc- tured 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 ex- ample, when dividing successive counting numbers by 5, say, the remainders cycle through 0, 1, 2, 3, and 4.

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 FOUNDATIONAL MATHEMATICS CONTENT 0123456789 10 11 12 13 14 15 16 17 18 19 2 0 2 1 2 2 2 3 2 4 2 5 26 27 28 29 30 31 32 33 34 35 36 37 38 39 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,2-8 2-D Space and Figure The compositional structure R01420 decimal system is more complex of the 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 rect- angle 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 es- sentially 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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50 MATHEMATICS LEARNING IN EARLY CHILDHOOD 5 4 (5, 3) 3 2 1 0 1 2 3 4 5 6 7 FIGURE 2-9 The coordinate plane. Groups of Groups of Groups:Figure 2-9 Numbers, Shapes, and -D Space R01420 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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52 MATHEMATICS LEARNING IN EARLY CHILDHOOD or in terms of a rotation (see National Research Council, 2001, Box 3-1). The associative property of multiplication can be illustrated by decompos- ing 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 rect- angles, 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 symme- try 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 reflect- ing 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 col- umns are related by the column that is associated with the amount of shift between the columns (see Figure 2-10). This structural relationship corre- sponds 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 shifting over by 2 adds the 2× column × 1 2 3 4 5 6 7 8 9 10 × 1 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10 2 2 4 6 8 10 12 14 16 18 20 2 2 4 6 8 10 12 14 16 18 20 3 3 6 9 12 15 18 21 24 27 30 3 3 6 9 12 15 18 21 24 27 30 4 4 8 12 16 20 24 28 32 36 40 4 4 8 12 16 20 24 28 32 36 40 5 5 10 15 20 25 30 35 40 45 50 5 5 10 15 20 25 30 35 40 45 50 6 6 12 18 24 30 36 42 48 54 60 6 6 12 18 24 30 36 42 48 54 60 7 7 14 21 28 35 42 49 56 63 70 7 7 14 21 28 35 42 49 56 63 70 8 8 16 24 32 40 48 56 64 72 80 8 8 16 24 32 40 48 56 64 72 80 9 9 18 27 36 45 54 63 72 81 90 9 9 18 27 36 45 54 63 72 81 90 10 10 20 30 40 50 60 70 80 90 100 10 10 20 30 40 50 60 70 80 90 100 The diagonal symmetry in the multiplication The relationship among the columns is due to table is due to the commutative proper ty. the distributive proper ty, e.g., 6 × 7 = 6 × 5 + 6 × 2. FIGURE 2-10 Symmetry and relationships in the multiplication table. Figure 2-10

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5 FOUNDATIONAL MATHEMATICS CONTENT 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, num- ber 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 count- ing 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, The number 12 length is 12 units away from 0. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 A number line is like an infinitely long ruler. A number on a number line tells its distance from 0 or the length between 0 and the number. FIGURE 2-11 Number lines relate numbers to lengths.

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5 MATHEMATICS LEARNING IN EARLY CHILDHOOD 1 1 2 1 3 1 4 FIGURE 2-12 1 > ½ > ¹⁄3 > 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 Figure 2-12 number and measurement can be applied. The collection of data shouldR01420 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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55 FOUNDATIONAL MATHEMATICS CONTENT How we got to school this mor ning How we got to school this mor ning Each student Each student places a places a sticky note small item (possibly (e.g., toy car) with their in a box to name on it) indicate the in the method of appropriate transportation. category. bus car bike walking bus car bike walking Connect to math by asking questions such as: • How many students walked to school this morning? • Did more students walk or ride a car? • How many more students rode a bus than rode in a car? • How many sticky notes are on our graph? FIGURE 2-13 A template for a “real graph” and a pictograph made with sticky notes. Figure 2-13 R01420 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 count- ing 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 mathemati- cal 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 mea- surement as identified in NCTM’s Curriculum Focal Points and outlined by this committee. There are also important mathematical reasoning pro- cesses 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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56 MATHEMATICS LEARNING IN EARLY CHILDHOOD 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 di- rectly. 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). Stu- dents’ development of length measurement concepts in a logo-based unit on geometric paths. Journal for Research in Mathematics Education, 2(1), 70-95. Grattan-Guinness, I. (2000). The Search for Mathematical Roots 10-10: 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 Chil- dren 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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5 FOUNDATIONAL MATHEMATICS CONTENT Reynolds, A., and Wheatley, G.H. (1996). Elementary students’ construction and coordina- tion of units in an area setting. Journal for Research in Mathematics Education, 2(5), 564-581. Sophian, C. (2007). Rethinking the starting point for mathematics learning. In O.N. Saracho and B. Spodek (Eds.), Contemporary Perspecties in Early Childhood Education: Math- ematics, 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, (6), 10-11.

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