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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FOUNDATIONAL MATHEMATICS CONTENT
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 numerosi-
ties), 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 count-
ing 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/Decomposi-
tions 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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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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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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0 MATHEMATICS LEARNING IN EARLY CHILDHOOD
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 op-
erations apply to a wide variety of problems. The middle and bottom sec-
tions 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 col-
lection 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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FOUNDATIONAL MATHEMATICS CONTENT
Visually, we can tell that
there are more white
beads than black beads.
Are there more black beads
or more white beads, or
is it the same number?
Compare by matching:
There are more black beads.
Compare by counting:
We say eight after we say
seven, so eight black beads
“one, two, three, four,
“one, two, three, four, are more than seven white
five, six, seven”
five, six, seven, eight” beads.
8 7
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 col-
Figure 2-4
lection of eight black beads than there are in a collection of seven white
beads because 8 occurs later inR01420
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
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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51
FOUNDATIONAL MATHEMATICS CONTENT
and of multiplication, the associative properties of addition and multipli-
cation, and the distributive property of multiplication over addition. The
commutative properties of addition and multiplication state that
A + B = B + A for all numbers A, B
A × B = B × A for all numbers A, B.
The associative properties of addition and multiplication state that
A + (B + C) = (A + B) + C for all numbers A, B, C
A × (B × C) = (A × B) × C for all numbers A, B, C.
The distributive property states that
A × (B + C) = A × B + A × C for all numbers A, B, C.
Each property can be illustrated by moving and reorganizing objects, some-
times 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 associa-
tive 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 com-
mutative property is especially useful in conjunction with counting on
strategies for solving addition problems (see Chapter 5 for further discus-
sion 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 ad-
dition 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 multiplica-
tion in third and fourth grade. The commutative and associative properties
of multiplication and the distributive property are essential to understand-
ing 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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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.
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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 10-10: Logics, Set
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National Association for the Education of Young Children and National Council of Teachers
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(Eds.). Center for Education, Division of Behavioral and Social Sciences and Education.
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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),
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Sophian, C. (2007). Rethinking the starting point for mathematics learning. In O.N. Saracho
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