Beyond Counting By Ones: “Thinking Groups”
as a Foundation for Number and Operation Sense
DeAnn Huinker, University of Wisconsin-Milwaukee
C
hildren grow tremendously in their knowledge
of number from birth through first grade. They
begin to understand the meaning of whole
numbers as they learn to quantify their environment,
talk about counting, and represent quantities using
drawings, fingers, or written numerals. One of the
most important accomplishments during these early
years is establishing the foundation for number sense
and operation sense—that is, children’s ability to use
number relationships flexibly, meaningfully, and with
confidence. Key components of this foundation are
encouraging children to “see groups” (Clements,
1999; Risden, 1978) and developing children’s abilities
to decompose numbers into parts and to put parts
together to make a whole (Kline, 1998).
This article examines moving young children beyond
counting by ones to conceptualizing quantities by
“thinking groups.” It begins with snapshots of the
numeric reasoning of two children. Second, we
examine some expectations of the Common Core State
Standards (CCSS) for young learners. Third, the ability
to subitize quantities is discussed. Finally, suggestions
for instruction along with a collection of activities are
included to begin moving students to think groups.
Snapshots of Ethan and Morgan
As you examine the reasoning of these two children,
consider their use of strategies for answering
quantitative questions, including quickly recognizing
the cardinalities of small sets of objects. Ethan and
Morgan are both four years old. They were shown the
dot patterns in Figure 2. Each pattern was shown for a
few seconds and then the child was asked, “How many
dots did you see?” If a child hesitated or was unsure,
the pattern was shown again or given to the child to
examine more closely. Then the child was asked to
explain what he/she saw and how he/she determined
the total number of dots.
(A)
(B)
Ethan was able to identify two dots at a glance (Pattern
A), but needed to count the other sets of dots by ones.
For Pattern C, he guessed “Eight” and for Pattern E
he said, “I call that one eleven.” When he was handed
the dot patterns, he counted the dots by ones and
was able to recite the number names in the correct
order, but often had to re-count the sets as he lost
track of which dots he had counted and which still
needed to be counted. He did eventually demonstrate
a connection between his counting and the cardinality
of each set. In talking with his mother, we discerned
that this was a novel task for Ethan. At home and at
his preschool, he was encouraged to rote count and
sometimes asked to count sets of objects, but rarely if
ever to quickly quantify a set of objects or to talk about
the composition of quantities.
Morgan, on the other hand, was able to quickly identify
the total number of dots in each set without having to
count by ones. For Patterns A, B, and D, the familiar
dice configurations, she very quickly identified the
total number of dots as two, five, and six. When asked,
“How do you know it’s five?” for Pattern B, she replied,
“Cuz there’s two and two and one in the middle. Five!”
For Pattern D, she explained, “Cuz they’re in the right
order.” She hesitated for only a moment with Pattern
C and then said it was five dots and explained, “‘Cuz
these are four and this is five, but not in the middle” as
she pointed to the familiar arrangement of four dots
on the bottom and then pointed to the one dot on
top. It is likely that she recognized the four dots and
counted on to determine the total of five. Morgan was
also successful with Pattern E after she thought about
the pattern for a few seconds. Here is the dialogue that
occurred.
T:
How many dots?
M:
Mmm ... seven.
T:
How do you know that’s seven, Morgan?
(C)
(D)
(E)
Figure 2. Dot Patterns
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M:
‘Cuz I remember it.
T:
You remembered it. What did you remember?
What did you see?
M:
Two and five.
T:
How do you know it’s seven?
M:
This is six and this is seven. [Pointing to the
two dots as she counted on.]
Morgan has been encouraged to work with quantities
beyond rote counting or counting sets of objects by
ones. She has had experiences with dot patterns on
dice, dominoes, cards, and dot plates, and has been
encouraged to talk about how larger quantities are
composed of smaller groups (e.g., seeing four dots
as being composed of two dots and two dots). Ethan
has not yet begun to see and think about quantities
as groups, whereas Morgan is well on her way to
composing (i.e., putting together) and decomposing
(i.e., taking apart) quantities and thinking about
number relationships within ten. She will have a strong
foundation for number and operation sense as she
enters kindergarten.
Common Core Standards Learning
Expectations
Even though the Common Core State Standards for
Mathematics (CCSSM) does not include standards
for prekindergarten, the development of many
mathematical ideas and skills identified as standards
for kindergarten must begin at home and in preschool
programs. The CCSSM identify number as one of two
critical areas in kindergarten; the other critical area is
describing shapes and space.
Children are to use numbers, including written numerals,
to represent quantities and to solve quantitative
problems. In particular, the CCSS states that children
should “choose, combine, and apply effective strategies
for answering quantitative questions, including quickly
recognizing the cardinalities of small sets of objects,
counting and producing sets of given sizes, counting
the number of objects in combined sets, or counting
the number of objects that remain in a set after some
are taken away” (CCSSI, 2010, p. 9).
In Kindergarten, the domain “Counting and
Cardinality” includes standards for learning to count
by ones accurately. The CCSS also includes the domain
“Operations and Algebraic Thinking” (OA). These
Kindergarten standards for the OA domain are listed
in Figure 1. They set expectations for children to
represent, compose, and decompose quantities less
than or equal to ten into pairs as the starting point
for understanding addition and subtraction. Whereas
Ethan is still working on counting by ones accurately,
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WMC 2011 Fall Journal_Inside Pages.indd 8
Morgan is already strong in her ability to decompose
and compose quantities.
Kindergarten Standards
Domain: Operations & Algebraic Thinking (K.OA)
Cluster: Understand addition as putting together
and adding to, and understand subtraction as
taking apart and taking from.
1. Represent addition and subtraction with objects,
fingers, mental images, drawings, sounds (e.g., claps),
acting out situations, verbal explanations, expressions,
or equations.
2. Solve addition and subtraction word problems, and
add and subtract within 10, e.g., by using objects or
drawings to represent the problem.
3. Decompose numbers less than or equal to 10 into
pairs in more than one way, e.g., by using objects or
drawings, and record each decomposition by a drawing
or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
4. For any number from 1 to 9, find the number that
makes 10 when added to the given number, e.g., by
using objects or drawings, and record the answer with
a drawing or equation.
5. Fluently add and subtract within 5.
Figure 1. CCSS Domain:
Operations and Algebraic Thinking
Subitizing
An important aspect of numeracy in young children
is the ability to “subitize” or quickly see and quantify
a group of objects without counting (Clements, 1999,
2004; Young-Loveridge, 2002). Instant recognition of
a quantity is called perceptual subitizing, such as when
Ethan and Morgan identified the number of dots in
Pattern A. This develops into conceptual subitizing,
which involves seeing a collection of objects as
composed of smaller groups and then being able to
quickly combine these groups to find the cardinality of
the entire collection. Morgan demonstrated conceptual
subitizing when she saw five in Pattern C as four and
one.
The Common Core Standards Writing Team (2011)
stated, “Use of conceptual subitizing in adding and
subtracting small numbers progresses to supporting
steps of more advanced methods for adding,
subtracting, multiplying, and dividing single-digit
numbers” (p. 4). This ability to see “groups” sets the
stage for taking children beyond counting by ones to
considering additive relationships between numbers,
Wisconsin Teacher of Mathematics, Fall 2011
9/28/2011 1:50:38 PM
such as part-whole ideas, relationships of one more
and one less, and relationships to five and ten.
This ability to see and think in groups begins with
very young children and needs to be nurtured and
further developed in prekindergarten and kindergarten
programs. Clements (2004, p. 28) identified the
following developmental guidelines for subitizing:
Ages 2–3: Collections of 1 to 3 objects
Ages 4–5: Collections of 1 to 5 objects
Ages 5–6: Collections of 1 to 6 objects, and
patterns up to 10 objects.
Thinking in Groups: Perceptual
Subitizing
Reasoning with quantities starts with perceptual
subitizing. Risden (1978) emphatically stated, “Stop
teaching preschoolers to count ones. Give them
home environments, instead, rich in opportunities to
see and to think twos and threes and groups of twos
and threes—groups, not ones. Send them to school
knowing twoness and threeness” (p. 154). She noted
that her first step with any child of any age was to test
for ability to see twos and threes—to see them with one
flash focus of the eyes. She explained how she would
toss three pennies onto the table and watch the child’s
eyes. Does the child’s eyes swivel from penny to penny?
Do the eyes focus at once upon the group as a whole?
Does the child want to touch and count the pennies by
ones? Risden noted that the first step for any child is
to learn to see twos and threes, because “until they see
them they cannot think them” (p. 155).
The following tasks and activities are suggestions
for assessing and developing students’ ability to
“think groups” and move them beyond counting by
ones. These activities focus on seeing twoness and
threeness. You can also include the quantities zero and
one, and encourage children to begin saying “zero”
by connecting it to their informal understanding of
“none.”
Seeing twoness and threeness. Hide three counters
under a box. As you uncover the counters, ask, “How
many counters are there?” Watch to see if the child
counts by ones, or sees the group as a whole. Repeat
with two counters. Encourage a quick response. The
goal is for the child to tell how many in a one-second
glance.
Making twos and threes. Give each child 10 counters.
Have them break apart the set by making twos. Then
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WMC 2011 Fall Journal_Inside Pages.indd 9
they should put the whole set of objects back together
and repeat until the twos are made more quickly. Next
start with 12 counters and make sets of three. Initially
it may take a minute for the child to complete the task,
but with practice children can accomplish these tasks
in about 10 seconds.
Rolling dot cubes. Make cubes with one, two, or
three dots on the sides. You can draw dots on blank
wooden cubes or cover regular dot cubes with stick on
labels. As children roll the cubes, ask, “How many dots
are there?” and “How did you see them?”
Everyday twos and threes. Ask children to look
around the classroom or go a mathematical walk with
them and ask them to find real-world examples of
things that come in twos and threes. They are likely
to begin noticing two eyes, two ears, two legs, and two
arms on each person. Finding threes is harder, but
children will readily find items, such as stuffed animals
or toys, in groups of three.
Thinking in Groups: Conceptual
Subitizing
Once children demonstrate a sense of twoness
and threeness, they are ready to work with larger
quantities, and develop their sense of numbers within
ten as composites of smaller groups. For example,
developing understanding of fiveness as three and two
or four and one (Novakowski,2007), or developing
tenness as two fives, three threes and a one, or a six
and a four (Risden, 1978).
Dot patterns are a valuable tool to promote conceptual
subitizing. The dot patterns represent quantities
from zero through ten in different configurations.
A suggested set of dot arrangements are shown in
Figure 3. The patterns show the common dice and
domino patterns, as well as other variations that are
combinations of two smaller patterns or a pattern with
one or two additional dots.
In using dot patterns with young children, I create dot
plates by using stick-on dots (3/4 inch diameter) to
show the patterns on dinner-sized white paper plates.
The plates are easy to construct, and children like to
handle and use the plates. The plates are also easy to
use with small groups of children and with the whole
class. It is important to keep the dots close together
in the center of the plate, because spreading out the
dots makes it harder to see the patterns. Some teachers
prefer to use one color of dots and others will use two
colors to more clearly show two parts. For example, to
see eight as four and four you could make four dots
blue and four dots red. I tend to use blue dots for
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9/28/2011 1:50:39 PM
child yells out the total, it stops the other children from
visualizing and reasoning about the groups. Set up
some type of signal and remind children often to “Put
the number in their head and not to yell it out.” Then
you can ask for a unison response from the group, ask
them to raise their hand on your signal, or you can call
on individual children.
(4) Initially, verify the total number of dots in a pattern
by counting the dots by ones. Eventually, as children
gain confidence in their sense of these quantities and
patterns, they will not need to count all the dots by
ones.
(5) The goal of these activities is not just to identify
the total number of dots in a pattern. The purpose
is to discuss with children how they decomposed the
pattern into smaller groups. This discussion is the core
component of all the activities. How are they seeing the
dots? Children should share and compare the different
ways they saw groups within the collection of dots.
It is also important to discuss how they determined
the total number of dots. Are they building fluency
in knowing that five dots and two dots are a total of
Figure 3. Dot Patterns for a Set of Dot Plates
Figure 3. Dot Patterns for a Set of Dot Plates
seven dots? Did they count on from five? You will
likely have some children explain that they just counted
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general guidelines to keep in mind when using the
plates with the more unfamiliar dot arrangements.
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activities that follow:
seconds.
ask the
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thenThen
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to compare their mental image to the actual pattern.
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5 establish the norm that three seconds or longer initially. The children should
(3) It is very important to
children are not to yell out the total amount because it visualize the dots, try to figure out how many total
is important to give everyone some think time. When a dots are in the pattern, and put that number in their
10
WMC 2011 Fall Journal_Inside Pages.indd 10
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9/28/2011 1:50:39 PM
heads. Show the plate again for another three seconds
to let them check or adjust their reasoning. Then ask
for a unison response, “How many dots did you see?”
Then ask a student to explain, “How did you see it?”
Have several students explain how they saw the dots.
There is no right or wrong way to see a pattern and
often students see them in different ways. Begin with
easy patterns and smaller quantities and then add new
patterns when the children are ready for more of a
challenge.
plates, to see the numbers four through ten as made up
of smaller groups. For example, children can observe
that six is two threes or a five and one more. This
thinking involves acquiring a part-whole understanding
of number and is a major advancement in children's
conceptual knowledge of number (Resnick, 1983).
Now children can develop many relationships among
numbers because they can think of a number as both
a whole amount and as being comprised of smaller
groups or parts.
Hold Up. Flash a dot pattern. Then ask students to
“hold up” the number of fingers that is the same as the
number of dots. Discuss how the children saw the dots
and how they showed that amount on their fingers. For
example, did they show four using two fingers on each
hand or four fingers all on the same hand?
We need to devote more time and give children more
learning opportunities to develop the foundation for
reasoning flexibly and meaningfully with quantities. I
encourage every preschool program and kindergarten
classroom to have dot plates, dot cubes, and dominoes
available for children to examine, explore, and discuss.
Give young children the opportunity to learn by
thinking groups together and apart—groups, not just
ones.
Show the Numeral. Have the children lay out numeral
cards from 0 to 10 in front of them. Now flash a dot
plate. Ask the children to hold up the numeral card that
tells how many dots are on the plate. Then discuss how
they saw the pattern.
One More or One Less. This activity presents a
new challenge for children. It focuses on the number
relationships of one more than and one less than. Flash
a dot pattern and ask the students to say the number
that is one more than (or one less than) the number
of dots on the plate. Then ask the children to explain
their reasoning.
Compare. Have children work in pairs. Give each
child his/her own set of dot plates. They should mix
up the plates and place them face down. Each child
turns over a plate and then they figure out, “Which
plate has more dots?” As the children engage in the
activity, ask them to explain their reasoning for some
of the comparisons.
Combine. Working in pairs, each child has a set of
dot plates. Each student displays a dot pattern. The
children then figure out how many total dots are on
both plates combined. Then they share their results
and discuss their reasoning.
Closing Comments
While counting by ones accurately is an essential skill
for children, we need to move beyond just asking
children to count objects. Children can learn to “see
and think groups.” This ability is empowering for
young children and builds their confidence in working
with quantities.
Begin by encouraging children to recognize groups of
two and three—to see twoness and threeness. Then
provide learning opportunities, such as with the dot
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WMC 2011 Fall Journal_Inside Pages.indd 11
References
Clements, D. H. (1999). Subitizing: What is it? Why
teach it? Teaching Children Mathematics, 5 (7), 400405.
Clements, D. H. (2004). Major theme and
recommendations. In D. H. Clements & J. Sarama
(Ed.), Engaging young children in mathematics:
Standards for early childhood mathematics (pp. 7-76).
Mahwah, NJ: Lawrence Erlbaum Associates.
Common Core State Standards Initiative. (2010).
Common core state standards for mathematics.
Washington, DC: National Governors Association and
Council of Chief State School Officers.
Common Core Standards Writing Team. (2011).
Progressions for the common core state standards
in mathematics: Draft K–5 progression on counting
and cardinality and operations and algebraic thinking.
Retreived
from
http://ime.math.arizona.edu/
progressions/#.
Kline, K. (1998). Kindergarten is more than counting.
Teaching Children Mathematics, 5(2), 84-87.
Novakowski, J. (2007). Developing “five-ness”
in kindergarten. Teaching Children Mathematics,
14(November), 226-231.
Resnick, L. B. (1983). A developmental theory of
number understanding. In H. P. Ginsburg (Ed.), The
Development of Mathematical Thinking. New York:
Academic Press.Risden, G. (1978). Words of wisdom
of an 80-year-old tutor. Education, 99(2), 154-156.
Young-Loveridge, J. (2002). Early childhood numeracy:
Building an understanding of part-whole relationships.
Australian Journal of Early Childhood, 27(4), 36-42.
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