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5
The Teaching-Learning Paths for
Number, Relations, and Operations
In this chapter we describe the teaching-learning paths for number,
relations, and operations at each of the four age/grade steps (2- and 3-year-
olds, 4-year-olds [prekindergarten], kindergarten, and Grade 1). As noted,
the four steps are convenient age groupings, although, in fact, children’s
development is continuous. There is considerable variability in the age at
which children do particular numerical tasks (see the reviews of the litera-
ture in Clements and Sarama, 2007, 2008; Fuson, 1992a, 1992b; also see
Chapter 4). However, a considerable amount of this variability comes from
differences in the opportunities to learn these tasks and the opportunity to
practice them with occasional feedback to correct errors and extend the
learning. Once started along these numerical learning paths, children be-
come interested in consolidating and extending their knowledge, practicing
by themselves and seeking out additional information by asking questions
and giving themselves new tasks. Home, child care, and preschool and
school environments need to support children in this process of becoming a
self-initiating and self-guiding learner and facilitate the carrying out of such
learning. Targeted learning path time is also needed—time at home or in an
early childhood learning center—that will support children in consolidating
thinking at one step and moving along the learning path to the next step.
Although we consider the mathematics goals described in this and the
next chapter foundational and achievable for all children in the designated
age range for that step, we recognize that some children’s learning will be
advanced while others’ functioning will be significantly behind. Children at
particular ages/grades may be able to work correctly with larger numbers or
more complex geometric ideas than those we specify in the various tables
127

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128 MATHEMATICS LEARNING IN EARLY CHILDHOOD
and text. Each subsequent step assumes that children have had sufficient
experiences with the topics in the previous step to learn the earlier content
well. (See Box 5-1 for a discussion of what it means to learn something
well.) However, many children can still learn the content at a given step
without having fully mastered the previous content if they have sufficient
time to learn and practice the more challenging content. Of course, some
children have difficulty in learning certain kinds of mathematical concepts,
and a few have really significant difficulties. But most children are capable
of learning the foundational and achievable mathematics content specified
in the learning steps outlined here.
In both the number and operations and the geometry and measurement
core areas, children learn about the basic numerical or geometric con-
cepts and objects (numbers, shapes), and they also relate those objects and
c
ompose/decompose (operate on) them. Therefore, each core area begins by
discussing the basic objects and then moves to the relations and operations
on them. In all of these, it is important to consider how children perceive,
say, describe/discuss, and construct these objects, relations, and operations.
The development of the elements of the number core across ages is de-
scribed first, and then the development of the relations and operations core
BOX 5-1
Learning Something Well
In most aspects of the number and the relations/operation core, children need
a great deal of practice doing a task, even after they can do it correctly. The rea-
sons for this vary a bit across different aspects, and no single word adequately
captures this need, because the possible words often have somewhat different
meanings for different people.
Overlearning can capture this meaning, but it is not a common word and
might be taken to mean something learned beyond what is necessary rather
than something learned beyond the initial level of correctness. Automaticity is a
word with technical meaning in some psychological literature as meaning a level
of performance at which one can also do something else. But to some people it
carries only a sense of rote performance. Fluency is the term used by several
previous committees, and we have therefore chosen to continue this usage. Flu-
ency also carries for some a connotation of flexibility because a person knows
something well enough to use it adaptively. We find this meaning useful as well
as the usual meaning of doing something rapidly and relatively effortlessly. Re-
search on reading in early childhood has recently used fluency only in the latter
sense as measured by performance on standardized tests of reading, such as
the Dynamic Indicators of Basic Early Literacy Skills (DIBELS). We do not mean
fluency to be restricted to this rote sense. By fluent we mean accurate and (fairly)
rapid and (relatively) effortlessly with a basis of understanding that can support
flexible performance when needed.

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PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 129
is summarized. These cores are quite related, and their relationships are
discussed. Box 5-2 summarizes the steps along the teaching-learning paths
in the core areas. As children move from age 2 through kindergarten, they
learn to work with larger and more complicated numbers, make connec-
tions across the mathematical contents of the core areas, learn more com-
plex strategies, and move from working only with objects to using mental
representations. This journey is full of interesting discoveries and patterns
that can be supported at home and at care and education centers.
THE NUMBER CORE
The four mathematical aspects of the number core identified in Chap-
ter 2 involve culturally specific ways that children learn to perceive, say,
describe/discuss, and construct numbers. These involve
1. Cardinality: Children’s knowledge of cardinality (how many are in a
set) increases as they learn specific number words for sets of objects
they see (I want two crackers).
2. Number word list: Children begin to learn the ordered list of number
words as a sort of chant separate from any use of that list in count-
ing objects.
3. 1-to-1 counting correspondences: When children do begin counting,
they must use one-to-one counting correspondences so that each
object is paired with exactly one number word.
4. Written number symbols: Children learn written number symbols
through having such symbols around them named by their number
word (That is a two).
Initially these four aspects are separate, and then children make vital con-
nections. They first connect saying the number word list with 1-to-1 cor-
respondences to begin counting objects. Initially this counting is just an
activity without an understanding of the total amount (cardinality). If asked
the question How many are there? after counting, children may count again
(repeatedly) or give a number word different from the last counted word.
Connecting counting and cardinality is a milestone in children’s numerical
learning path that coordinates the first three aspects of the number core.
As noted, we divide the teaching-learning path into four broad steps.
In Step 1, for 2- and 3-year-olds, children learn about the separate aspects
of number and then begin to coordinate them. In Step 2, for approximately
4-year-olds/prekindergartners, children extend their understanding to larger
numbers. In Step 3, for approximately 5-year-olds/kindergartners, children
integrate the aspects of number and begin to use a ten and some ones in
teen numbers. In Step 4, approximately Grade 1, children see, count, write,
and work with tens-units and ones-units from 1 to at least 100.

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130 MATHEMATICS LEARNING IN EARLY CHILDHOOD
BOX 5-2
Overview of Steps in the Number,
Relations, and Operations Core
Steps in the Number Core
Step 1 (ages 2 and 3): Beginning 2- and 3-year-olds learn the number core
components for very small numbers: cardinality, number word list, 1-1 counting
correspondences, and written number symbols; later 2- and 3-year-olds coordi-
nate these number core components to count n things and, later, say the number
counted.
Step 2 (age 4/prekindergarten): Extend all four core components to larger
numbers and also use conceptual subitizing if given learning opportunities
to do so.
Step 3 (age 5/kindergarten): Integrate all core components, see a ten and
some ones in teen numbers, and relate ten ones to one ten and extend the core
components to larger numbers.
Step 4 (Grade 1): See, say, count, and write tens-units and ones-units from 1
to 100.
Steps in the Relations (More Than/Less Than) Core
Step 1 (ages 2 and 3): Use perceptual, length, and density strategies to find
which is more for two numbers ≤ 5.
Step 2 (age 4/prekindergarten): Use counting and matching strategies to find
which is more (less) for two numbers ≤ 5.
Step 3 (age 5/kindergarten): Kindergartners show comparing situation with
objects or in a drawing and match or count to find out which is more and which is
less for two numbers ≤ 10.
Step 4 (Grade 1): Solve comparison word problems that ask, “How many more
(less) is one group than another?” for two numbers ≤ 18.
Steps in the Addition/Subtraction Operations Core
Step 1 (ages 2 and 3): Use subitized and counted cardinality to solve situation
and oral number word problems with totals ≤ 5; these are much easier to solve if
objects present the situation rather than the child needing to present the situation
and the solution.
Step 2 (age 4/prekindergarten): Use conceptual subitizing and cardinal
counting of objects or fingers to solve situation, word, and oral number word
problems with totals ≤ 8.
Step 3 (age 5/kindergarten): Use cardinal counting to solve situation, word,
oral number word, and written numeral problems with totals ≤ 10.
Step 4 (Grade 1): Use counting on solution procedures to solve all types of
addition and subtraction word problems: Count on for problems with totals ≤ 18
and find subtraction as an unknown addend.

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PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 131
Step 1 (Ages 2-3)
At this step, children first begin to learn the core components of num-
ber: cardinality, the number word list, 1-to-1 correspondences, and written
number symbols (see Box 5-3).
BOX 5-3
Step 1 in the Number Core
Children at particular ages/grades may exceed the specified numbers and be
able to work correctly with larger numbers. The numbers for each age/grade are
the foundational and achievable content for children at this age/grade. The major
types of new learning for each age/grade are given in italics. Each level assumes
that children have had sufficient learning experiences at the lower level to learn
that content; many children can still learn the content at a level without having
fully mastered the content at the lower level if they have sufficient time to learn
and practice.
Beginning 2- and 3-Year-Olds Learn the Number Core Components
Cardinality: How many animals (crackers, fingers, circles, . . . )? uses perceptual
subitizing to give the number for 1, 2, or 3 things.
Number word list: Count as high as you can (no objects to count) says 1 to 6.
1-to-1 counting correspondences: Count these animals (crackers, fingers,
circles, . . . ) or How many animals (crackers, fingers, circles, . . . )? counts ac-
curately 1 to 3 things with 1-1 correspondence in time and in space.
Written number symbols: This (2, 4, 1, etc.) is a______? knows some symbols;
will vary.
Later 2- and 3-Year-Olds Coordinate the Number Core Components
Cardinality: Continues to generalize perceptual subitizing to new configurations
and extends to some instances of conceptual subitizing for 4 and 5: can give
number for 1 to 5 things.
Number word list: Continues to extend and may be working on the irregular
teen patterns and the early decade twenty to twenty-nine, etc., pattern: says
1 to 10.
1-to-1 counting correspondences: Continues to generalize to counting new
things, including pictures, and to extend accurate correspondences to larger
sets (accuracy will vary with effort): counts accurately 1 to 6 things.
Written number symbols: Continues to learn new symbols if given such learning
opportunities.
Coordinates counting and cardinality into cardinal counting in which the
last counted word tells how many and (also or later) tells the cardinality (the
number in the set).

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132 MATHEMATICS LEARNING IN EARLY CHILDHOOD
Cardinality
The process of identifying the number of items in a small set (cardinal-
ity) has been called subitizing. We will call it perceptual subitizing to differ-
entiate it from the more advanced form we discuss later for larger numbers
called conceptual subitizing (see Clements, 1999). For humans, the process
of such verbal labeling can begin even before age 2 (see Chapter 3). It first
involves objects that are physically present and then extends to nonpresent
objects visualized mentally (for finer distinctions in this process, see Benson
and Baroody, 2002). This is an extremely important conceptual step for
attaching a number word to the perceived cardinality of the set. In fact,
there is growing evidence that the number words are critical to toddlers’
construction of cardinal concepts of even small sets, like three and four and
possibly one and two (Benson and Baroody, 2002; Spelke, 2003; also see
Baroody, Lai, and Mix, 2006; and Mix, Sanhofer, and Baroody, 2005).
Children generally learn the first 10 number words by rote first and
do not recognize their relation to quantity (Fuson, 1988; Ginsburg, 1977;
Lipton and Spelke, 2006; Wynn, 1990). They do, however, begin to learn
sets of fingers that show small amounts (cardinalities). This is an important
process, because these finger numbers will become tools for adding and
subtracting (see research literature summarized in Clements and Sarama,
2007; Fuson, 1992a, 1992b). Interestingly, the conventions for counting on
fingers vary across cultures (see Box 5-4).
In order to fully understand cardinality, children need to be able to both
generalize and extend the idea. That is, they need to generalize from a spe-
cific example of two things (two crackers), to grasp the “two-ness” in any
set of two things. They also need to extend their knowledge to larger and
larger groups—from one and two to three, four, and five, although these
are more difficult to see and label (Baroody, Lai, and Mix, 2006; Ginsburg,
1989). Children’s early notions of cardinality and how and when they learn
to label small sets with number words are an active area of research at
present. The timing of these insights seems to be related to the grammatical
structure of the child’s native language (e.g., see the research summarized
in Sarnecka et al., 2007).
Later on, children can learn to quickly see the quantity in larger sets
if these can be decomposed into smaller subitized numbers (e.g., I see two
and three, and I know that makes five). Following Clements (1999), we call
such a process conceptual subitizing because it is based on visually appre-
hending the pair of small numbers rather than on counting them. Concep-
tual subitizing requires relating the two smaller numbers as addends within
the conceptually subitized total. With experience, the move from seeing the
smaller sets to seeing and knowing their total becomes so rapid that one can
experience this as seeing 5 (rather than as seeing 2 and 3). Children may
also learn particular patterns, such as the 5 pattern on a die. Because these

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PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 133
BOX 5-4
Using Fingers to Count: Cultural Differences
Around the world, most children learn from their family one of the three major
ways of raising (or in some cultures, lowering) fingers to show numbers. All of
these methods can be seen in centers or schools with children coming from differ-
ent parts of the world, as well as some less frequent methods (the Indian counting
on cracks of fingers with the thumb, Japanese lowering and raising fingers). The
most common way is to raise the thumb first and then the fingers in order across
to the small finger. Another way is to raise the index finger, then the next fingers in
order to the smallest finger, and then the thumb. The third way is to begin with the
little finger and move across in order to the thumb. The first way is very frequent
throughout Latin America, and the third way also is used by some children coming
from Latin America. The second way is the most usual in the United States. It is
the common way to show ages (for example, I am two years old by holding up
the index and largest finger). This method allows children to hold down unused
fingers with their thumb. But the other two methods show numbers in a regular
pattern going across the fingers. Children in a center or school where children
show numbers on fingers in different ways may come to use multiple methods.
Because fingers are such an important tool for numerical problem solving, it is
probably best not to force a child to change his or her method of showing numbers
on fingers if it is well established. It is important for teachers to be aware and ac-
cepting of these differences.
kinds of patterns can also be considered in terms of addends that compose
them, they are included in conceptual subitizing. Such patterns can help
older children learn mathematically important groups, such as five and ten;
these are discussed in the later levels and in the relations and operation core
discussion of addition and subtraction composing/decomposing.
Children also learn to assign a number to sets of entities they hear but
do not see, such as drum beats or ringing bells. There is relatively little
research on auditory quantities, and they play a much smaller role in ev-
eryday life or in mathematics than do visual quantities. For these reasons,
and because auditory quantities relate to music and rhythm and body move-
ments, it seems sensible to have some activities in the classroom in which
children repeat simple or complex sets they hear (clap clap or, later, clap
clap clap pause clap clap), tell the number they hear (of bells, drumbeats,
feet stamping, etc.), and produce sounds with body movements for particu-
lar quantities (Let me hear three claps).
In home and care/educational settings, it is important that early experi-
ences with subitizing be provided with simple objects or pictures. Textbooks
or worksheets often present sets that discourage subitizing and depict col-
lections of objects that are difficult to count. Such complicating factors
include embedded or overlapping pictures, complex noncompact things

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134 MATHEMATICS LEARNING IN EARLY CHILDHOOD
or pictures (e.g., detailed animals of different sizes rather than circles or
squares), lack of symmetry, and irregular arrangements (Clements and
Sarama, 2007).
The importance of facilitating subitizing is underscored by a series of
studies, which first found that children’s spontaneous tendency to focus on
numerosity was related to counting and arithmetic skills, then showed that
it is possible to enhance such spontaneous focusing, and then found that
doing so led to better competence in cardinality tasks (Hannula, 2005).
Increasing spontaneous focusing on numerosity is an example of helping
children mathematize their environment (seek out and use the mathemati-
cal information in it). Such tendencies can stimulate children’s self-initiated
practice in numerical skills because they notice those features and are in-
terested in them.
Number Word List
A common activity in many families and early childhood settings is
helping a child learn the list of number words. Children initially may say
numbers in the number word list in any order, but rapidly the errors take
on a typical form. Children typically say the first part of the list correctly,
and then may omit some numbers in the next portion of the list, or they say
a lot of numbers out of order, often repeating them (e.g., one, two, three,
four, five, eight, nine, four, five, two, six) (Fuson, 1988; Fuson, Richards,
and Briars, 1982; Miller and Stigler, 1987; Siegler and Robinson, 1982).
Children need to continue to hear a correct number list to begin to include
the missing numbers and to extend the list.
Children can learn and practice the number word list by hearing and
saying it without doing anything else, or it can be heard or said in coordina-
tion with another activity. Saying it alone allows the child to concentrate on
the words, and later on the patterns in the words. However, it is also helpful
to practice in other ways to link the number words to other aspects of the
number core. Saying the words with actions (e.g., jumping, pointing, shak-
ing a finger) can add interest and facilitate the 1-to-1 correspondences in
counting objects. Raising a finger with each new word can help in learning
how many fingers make certain numbers, and flashing ten fingers at each
decade word can help to emphasize these words as made from tens.
Counting: 1-to-1 Correspondences
In order to count a group of objects the person counting must use some
kind of action that matches each word to an object. This often involves
moving, touching or pointing to each object as each word is said. This
counting action requires two kinds of correct matches (1-to-1 correspon-
dences): (1) the matching in a moment of time when the action occurs and a

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PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 135
word is said, and (2) the matching in space where the counting action points
to an object once and only once. Children initially make errors in both of
these kinds of correspondences (e.g., Fuson, 1988; Miller et al., 1995). They
may violate the matching in time by pointing and not saying a word or by
pointing and saying two or more words. They may also violate the match-
ing in space by pointing at the same object more than once or skipping an
object; these errors are often more frequent than the errors in time.
Four factors strongly affect counting correspondence accuracy:
(1) amount of counting experience (more experience leads to fewer errors),
(2) size of set (children become accurate on small sets first), (3) arrange-
ment of objects (objects in a line make it easier to keep track of what has
been counted and what has not), and (4) effort (see research reviewed in
C
lements and Sarama, 2007, and in Fuson, 1988). Small sets (initially up
to three and later also four and five) can be counted in any arrangement,
but larger sets are easier to count when they are arranged in a line. Children
ages 2 and 3 who have been given opportunities to learn to count objects
accurately can count objects in any arrangement up to 5 and count objects
in linear arrangements up to 10 or more (Clements and Sarama, 2007;
Fuson, 1988).
In groundbreaking research, Gelman and Gallistel (1978) identified five
counting principles that stimulated a great deal of research about aspects
of counting. Her three how-to-count principles are the three mathematical
aspects we have just discussed: (1) the stable order principle says that the
number word list must be used in its usual order, (2) the one-one principle
says that each item in a set must be tagged by a unique count word, and
(3) the cardinality principle says that the last number word in the count list
represents the number of objects in the set. Her two what-to-count prin-
ciples are mathematical aspects we have also discussed: (1) the abstraction
principle states that any combination of discrete entities can be counted
(e.g., heterogeneous versus homogeneous sets, abstract entities, such as the
number of days in a week) and (2) the order irrelevance principle states that
a set can be counted in any order and yield the same cardinal number (e.g.,
counting from right to left versus left to right).
Gelman took a strong position that children understand these count-
ing principles very early in counting and use them in guiding their count-
ing activity. Others have argued that at least some of these principles are
understood only after accurate counting is in place (e.g., Briars and Siegler,
1984). Still others, taking a middle ground between the “principles before”
view and the “principles after” view, suggest that there is a mutual (e.g.,
iterative) relation between understanding the count principles and count-
ing skill (e.g., Baroody, 1992; Baroody and Ginsburg, 1986; Fuson, 1988;
Miller, 1992; Rittle-Johnson and Siegler, 1998).
Each of these aspects of counting is complex and does not necessarily
exist as a single principle that is understood at all levels of complexity at

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136 MATHEMATICS LEARNING IN EARLY CHILDHOOD
once. Children may initially produce the first several number words and not
even separate them into distinct words (Fuson, Richards, and Briars, 1982).
They may think that they need to say the number word list in order as they
count, but early on they cannot realize the implication that they need a
unique last counted word, or they would not repeat words so frequently as
they say the number word list.
The what-to-count principles also cover a range of different under-
standings. It takes some time for children to learn to count parts of a thing
(Shipley and Shepperson, 1990; Sophian and Kailihiwa, 1998), a later use
of the abstraction principle. And the order irrelevance principle (counting
in any order will give the same result) seems to be subject to expectations
about what is conventional “acceptable” counting (e.g., starting at one end
of a row rather than in the middle) as well as involving, later on, a deeper
understanding of what is really involved in 1-to-1 correspondence: Count-
BOX 5-5
Common Counting Errors
There are some common counting errors made by young children as they learn
the various principles that underpin successful counting. Counting requires effort
and continued attention, and it is normal for 4-year-olds to make some errors and
for 5-year-olds to make occasional errors, especially on larger sets (of 15 or more
for 4-year-olds and of 25 or more for 5-year-olds). Younger children may initially
make quite a few errors. It is much more important for children to be enthusiastic
counters who enjoy counting than for them to worry so much about errors that they
are reluctant to count. If one looks at the proportion of objects that receive one
word and one point, children’s counting often is pretty accurate. Letting errors go
sometimes or even somewhat frequently if children are trying hard and just mak-
ing the top four kinds of errors is fine as long as children understand that correct
counting requires one point and one word for each object and are trying to do that.
As with many physical activities, counting will improve with practice and does not
need to be perfect each time. Teachers do not have to monitor children’s counting
all of the time. It is much more important for all children to get frequent counting
practice and watch and help each other, with occasional help and corrections from
the teacher.
Very young children counting small rows with high effort make more errors in
which their say-point actions do not correspond than errors in the matching of the
points and objects. Thus, they may need more practice coordinating their actions
of saying one word and pointing at an object. Energetic collective practice in which
children rhythmically say the number word list and move down their hand with a
finger pointed as each word is said can be helpful. To vary the practice, the words
can sometimes be said loudly and sometimes softly, but always with emphasis (a
regular beat). The points can involve a large motion of the whole arm or a smaller
motion, but, again, in a regular beat with each word. Coordinating these actions

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PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 137
ing is correct if and only if each object receives one number word (LeFevre
et al., 2006). An aspect of the 1-to-1 principle that is difficult even for high
school students or adults to execute is remembering exactly which objects
they have already counted with a large fixed set of objects scattered irregu-
larly around (such as in a picture) (Fuson, 1988).
The principles are useful in understanding children’s learning to count,
but they should not be taken as simplistic statements that describe knowl-
edge that is all-or-nothing or that has a simple relationship to counting
skill. It can be helpful for teachers or parents to make statements of vari-
ous aspects of counting (e.g., Remember that each object needs one point
and one number word, You can’t skip any, Remember where you started in
the circle so you stop just before that.). But children will continue to make
counting errors even when they understand the task, because counting is a
complex activity (see Box 5-5).
of saying and pointing is the goal for overcoming this type of error. For variety,
these activities can involve other movements, such as marching around the room
with rhythmic arm motions or stamping a foot saying a count word each time.
Counting an object twice or skipping over an object are errors made occasion-
ally by 4-year-olds and even by 5-year-olds on larger sets. These seem to stem
from momentary lack of attention rather than lack of coordination. Trying hard or
counting slowly can reduce these errors. However, when two counts of the same
set disagree, many children of this age think that their second count is correct,
and they do not count again. Learning the strategy of counting a third time can
increase the accuracy of their counts. If children are skipping over many objects,
they need to be asked to count carefully and don’t skip any.
Young children sometimes make multiple count errors on the last object. They
either find it difficult to stop or think they need to say a certain number of words
when counting and just keep on counting so they say that many. When they say
the number word list, more words are better, so they need to learn that saying the
number word list when counting objects is controlled by the number of objects.
Reminding them that even the last object only gets one word and one point can
help. They also may need the physical support of holding their hand as they reach
to point to the last object so that the hand can be stopped from extra points and
the last word is said loudly and stretched out (e.g., fii-i-i-ve) to inhibit saying the
next word.
Regularity and rhythmicity are important aspects of counting. Activities that
increase these aspects can be helpful to children making lots of correspondence
errors. Children who are not discouraged about their counting competence gener-
ally enjoy counting all sorts of things and will do so if there are objects they can
count at home or in a care or education center. Counting in pairs to check each
other find and correct errors is often fun for the pairs. Counting in other activities,
such as building towers with blocks, should also be encouraged.

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164 MATHEMATICS LEARNING IN EARLY CHILDHOOD
siiiixxx 7 8 9 count on like this, a child must shift from the
To
cardinal meaning above to the count meaning of
six and then keep counting 7, 8, 9.
Trying this with different problems enables many children to see this
general pattern and begin counting on. Transition strategies, such as count-
ing 1, 2, 3, 4, 5, 6 very quickly or very softly or holding the 6 (siiiiiiixxxxxx),
have been observed in students who are learning counting on by themselves;
these can be very useful in facilitating this transition to counting on (e.g.,
Fuson, 1982; Fuson and Secada, 1986; Secada, Fuson, and Hall, 1983).
Some weaker students may need explicit encouragement to trust the six
and to let go of the initial counting of the first addend, and they may need
to use these transitional methods for a while.
Counting on has two parts, one for each addend. The truncation of the
final counting all by starting with the cardinal number of the first addend
was discussed above. Counting on also requires keeping track of the second
addend—of how many you count on so that you count on from the first ad-
dend exactly the number of the second addend. When the number is small,
such as for 6 + 3, most children use perceptual subitizing to keep track of
the 3 counted on. This keeping track might be visual and involve actual
objects, fingers, or drawn circles. But it can also use a mental visual image
(some children say they see 3 things in their head and count them). Some
children use auditory subitizing (they say they hear 7, 8, 9 as three words).
For larger second addends, children use objects, fingers, or conceptual subi-
tizing to keep track as they count on. For 8 + 6, they might think of 6 as
3 and 3 and count with groups of three: 8 9 10 11 12 13 14 with a pause
after the 9 10 11 to mark the first three words counted on. Other children
might use a visual (I saw 3 circles and another 3 circles) or an auditory
rhythm to keep track of how many words they counted on. So here we see
how the perceptual subitizing and the conceptual subitizing, which begin
very early, come to be used in a more complex and advanced mathematical
process. This is how numerical ideas build, integrating the levels of thinking
visually/holistically and thinking about parts into a complex new concep-
tual structure that relates the parts and the whole. Children can discuss the
various methods of keeping track, and they can be helped to use one that
will work for them. Almost all children can learn to use fingers successfully
to keep track of the second addend.
Many experiences with composing/decomposing (finding partners hid-
ing inside a number) can give children the understanding that a total is any
number that has partners (addends) that compose it. When subtracting,
they have been seeing that they take away one of those addends, leav-
ing the other one. These combine into the understanding that subtracting
means finding the unknown addend. Therefore, children can always solve

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PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 165
subtraction problems by a forward method that finds the unknown addend,
thus avoiding the difficult and error-prone counting down methods (e.g.,
Baroody, 1984; Fuson, 1984, 1986b). So 14 − 8 = ? can be solved as 8 + ?
= 14, and students can just count on from 8 up to 14 to find that 8 plus 6
more is 14.
Some first graders will also move on to Level 3 derived fact solution
methods (see Box 5-11) such as doubles plus or minus one and the gen-
eral method that works for all teen totals: the make-a-ten methods taught
in East Asia (see Chapter 4 and, e.g., Geary et al., 1993; Murata, 2004).
These make-a-ten methods are particularly useful in multidigit addition
and subtraction, in which one decomposes a teen number into a ten to give
to the next column while the leftover ones remain in their column. More
children will be able to learn make-a-ten methods if they have learned the
prerequisites for them in kindergarten or even in Grade 1.
The comparison situations compare a large quantity to a smaller quan-
tity to find the difference. These are complex situations that are usually not
solvable until Grade 1. The third quantity, the difference, is not physically
present in the situation, and children must come to see the differences as
the extra leftovers in the bigger quantity or the amount the smaller quantity
needs to gain in order to be the same as the bigger quantity. The language
involved in comparison situations is challenging, because English gives two
kinds of information in the same sentence. Consider, for example, the sen-
tence Emily has five more than Tommy. This says both that Emily has more
than Tommy and that she has five more. Many children do not initially hear
the five. They will need help and practice identifying and using the two
kinds of information in this kind of sentence (see the research reviewed in
Clements and Sarama, 2007, 2008; Fuson, 1992a, 1992b; Fuson, Carroll,
and Landis, 1996).
Learning to mathematize and model addition and subtraction situa-
tions with objects, fingers, and drawings is the foundation for algebraic
problem solving. More difficult versions of the problem situations can be
given from Grade 1 on. For example, the start or change number can be the
unknown in change plus problems: Joey drew 5 houses and then he drew
some more. Now he has 9 houses. How many more houses did he draw?
Children naturally model the situation and then reflect on their model
(with objects, fingers, or a drawing) to solve it (see research summarized in
C
lements and Sarama, 2007; Fuson, 1992a, 1992b). From Grade 2 on they
can also learn to represent the situation with a situation equation (e.g., 5 +
? = 9 as in the example above, or ? + 4 for an unknown start number) and
then reflect on that to solve it. This process of mathematizing (including
representing the situation) and then solving the situation representation is
algebraic problem solving.

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166 MATHEMATICS LEARNING IN EARLY CHILDHOOD
Issues in Learning Relations and Operations
The Extensive Learning Path for Addition and Subtraction
The teaching-learning path we describe shows that even the most ad-
vanced solution strategies for adding and subtracting single-digit numbers
have their roots before age 2 and may not culminate until Grade 1 or even
Grade 2. The paths also illustrate how children coordinate several differ-
ent complex kinds of understandings and skills beginning with perceptual
subitizing through conceptual subitizing and then counting and matching
to employ more sophisticated problem-solving strategies. This makes it
clear that one cannot characterize the learning of single-digit addition and
subtraction as simply “memorizing the facts” or “recalling the facts,” as if
children had been looking at an addition table of numbers and memoriz-
ing these. Children do remember particular additions and subtractions as
early as age 2, but each of these has some history as perceptually or con-
ceptually subitized situations, counted situations over many examples, or
additions/subtractions derived from other known additions/subtractions. It
is therefore much more appropriate to set learning goals that use the ter-
minology fluency with single-digit additions and their related subtractions
rather than the terms recalled or memorized facts. The latter terms imply
simplistic rote teaching/learning methods that are far from what is needed
for deep and flexible learning.
The Mental Number Word List as a Representational Tool
We have demonstrated how children come to use the number word
list (the number word sequence) as a mental tool for solving addition and
subtraction problems. They are able to use increasingly abbreviated and
abstract solution methods, such as counting on and the make-a-ten meth-
ods. The number words themselves have become unitized mental objects
to be added, subtracted, and ordered as their originally separate sequence,
counting, and cardinal meanings become related and finally integrated
over several years into a truly numerical mental number word sequence.
Each number can be seen as embedded within each successive number and
as seriated: related to the numbers before and after it by a linear ordering
created by the order relation less than applied to each pair of numbers
(see Box 5-12). This is what Piaget (1941/1965) called truly operational
cardinal number: Any number in the sequence displays both class inclusion
(the embeddedness) and seriation (see also Kamii, 1985). But this fully
Piagetian integrated sequence will not be finished for most children until
Grade 1 or Grade 2, when they can do at least some of the Step 3 derived
fact solution methods, which depend on the whole teaching-learning path
we have discussed.

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PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 167
BOX 5-12
Ordering and Ordinal Numbers
There is frequent confusion in the research literature in the use of the terms
ordered or ordering, ordinal number, and order relation. Some of this confusion
stems from the fact that adults can flexibly and fluently use the counting, cardinal,
and ordinal meaning of number words without needing to consciously think about
the different meanings. As a result, they may not be able to differentiate the mean-
ings very clearly. But young children learn the meanings separately and need to
connect them.
When counting to find the total number in a set, the order for connecting each
number word to objects is arbitrary and could be done in any order. As noted
previously, the last number takes on a cardinal meaning and refers to the total
numbers of items counted. Thus, the cardinal meaning of a number refers to a
set with that many objects. Cardinal numbers can be used to create an order rela-
tion. That is the idea that one set has more members than another set. An order
relation (one number or set is less than or more than another number or set) tells
how two quantities are related. This order relation produces a linear ordering on
these numbers or sets. An ordinal number tells where in the ordering a particular
number or set falls. A child can subitize for the small ordinal numbers (see whether
an object in an ordered set is first, second, or third), but needs to count for larger
ordinal numbers and shift from a count meaning to an ordinal meaning (e.g., count
one, two, three, four, five, six, seven [count meaning]. That person is seventh
[ordinal meaning and ordinal work] in the line to buy tickets.).
We have not emphasized ordinal words in this chapter because they are so
much more difficult than are cardinal words, and children learn them much later
(e.g., Fuson, 1988). Although 4- and 5-year-olds could learn to use the ordinal
words first, second, and last, it is not crucial that they do so. The ordinal words
first through tenth could wait until Grade 1.
Many researchers have noted how the number word list turns into a
mental representational tool for adding and subtracting. A few researchers
have called this a mental number line. However, for young children this is
a misnomer, because children in kindergarten and Grade 1 are using the
number word list (sequence) as a count model: Each number word is taken
as a unit to be counted, matched, added, or subtracted. In contrast, a num-
ber line is a length model, like a ruler or a bar graph, in which numbers
are represented by the length from zero along a line segmented into equal
lengths. Young children have difficulties with the number line representa-
tion because they have difficulty seeing the units—they need to see things,
so they focus on the numbers instead of on the lengths. So they may count
the starting point 0 and then be off by one, or they focus on the spaces
and are confused by the location of the numbers at the end of the spaces.
The report Adding It Up: Helping Children Learn Mathematics (National

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168 MATHEMATICS LEARNING IN EARLY CHILDHOOD
Research Council, 2001a) recognized the difficulties of the number line
representation for young children and recommended that its use begin at
Grade 2 and not earlier.
The number line is particularly important when one wants to show
parts of one whole, such as one-half. In early childhood materials, the term
number line or mental number line often really means a number path, such
as in the common early childhood games in which numbers are put on
squares and children move along a numbered path. Such number paths are
count models—each square is an object that can be counted—so these are
appropriate for children from age 2 through Grade 1. Some research sum-
marized in Chapter 3 did focus on children’s and adult’s use of the analog
magnitude system to estimate large quantities or to say where specified
larger numbers fell along a number line. Again, it is not clear, especially for
children, whether they are using a mental number list or a number line; the
crucial research issue is the change in the spacing of the numbers with age,
and this could come either from children’s use of a mental number list or
a number line. The use of number lines, such as in a ruler or a bar graph
scale, is an important part of measurement and is discussed in Chapter 6.
But for numbers, relations, and operations, physical and mental number
word lists are the appropriate model.
Variability in Children’s Solution Methods
The focus of this chapter is on how children follow a learning path
from age 2 to Grade 1 in learning important aspects of numbers, relations,
and operations. We continually emphasize that there is variability within
each age group in the numbers and concepts with which a given child can
work. As summarized in Chapter 3, much of this variability stems from
differences in opportunities to learn and to practice these competencies, and
we stress how important it is to provide such opportunities to learn for all
children. We close with a reminder that there is also variability within a
given individual at a given time in the strategies the child will use for a given
kind of task. Researchers through the years have shown that children’s
strategy use is marked by variability both within and across children (e.g.,
Siegler, 1988; Siegler and Jenkins, 1989; Siegler and Shrager, 1984). Even on
the same problem, a child might use one strategy at one point in the session,
and another strategy at another point. As children gain proficiency, they
gradually move to more mature and efficient strategies, rather than doing
so all at once. The variability itself is thought to be an important engine
of cognitive change. Similarly, as discussed above, accuracy can vary with
effort, particularly with counting. The variability in the use of strategies
within or across children can provide important opportunities to discuss
different methods and extend understandings of all participants. The vari-

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PATHS FOR NUMBER, RELATIONS, AND OPERATIONS 169
ability in results with different levels of effort can lead to discussions about
how learning mathematics depends on effort and practice and that everyone
can get better at it if they practice and try hard. Effort creates competen-
cies that are the building blocks for the next steps in the learning path for
numbers, relations, and operations.
SUMMARY
The teaching-learning path described in this chapter shows how young
children learn, integrate, and extend their knowledge about cardinality, the
number word list, 1-to-1 counting correspondences, and written number
symbols in successive steps from age 2 to 7. Much of this knowledge re-
quires specific cultural knowledge—for example, the number word list in
English, counting, matching, vocabulary about relations and operations.
Children require extensive, repeated experiences with small numbers and
then similar experiences with larger and larger numbers. Counting must
become very fluent, so that it can become a mental representational tool for
problem solving. As we have shown, even young children can have experi-
ences in the teaching-learning path that support later algebraic learning.
To move through the steps in the teaching-learning path, children require
teaching and interaction in the context of explicit, real-world problems
with feedback and opportunities for reflection provided. They also require
accessible situations in which they can practice (consolidate), deepen, and
extend their learning and their own.
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