Running head: COUNTING-ON
Counting-on as a Strategy for Addition
Jacqueline Kreiner and Jessica Bacon
Augustana College
Running head: COUNTING-ON
Counting-on as a Strategy for Addition
The Number Sense Project stems from Augustana’s partnership with Longfellow Elementary
School. Each year, Augustana elementary education students collaborate with Longfellow’s kindergarten
teachers and classes, as well as the Augustana education faculty in order to conduct an action research
project. The goals of this project are to provide kindergarten students with differentiated small group
instruction and to enhance the Augustana teacher candidates’ understanding of developing number sense
in kindergarteners. Throughout the duration of the Number Sense Project, the Augustana teaching
candidates have developed research questions that have been explored through their work with the
Longfellow students. The area of content that we have chosen to focus on is counting-on in addition and
the skills that students must possess in order to do so. We chose this area of study because during our
work with the kindergarteners we found that a select few could solve addition problems by using the
concept of counting-on, while others required the use of concrete manipulatives to solve the problem.
After witnessing this, we started to question how the students who were able to count-on had developed
this skill and whether it had been accomplished through direct instruction at school, direct instruction at
home, self-teaching, or through teacher-monitored exploration of addition strategies. Further questions
arose about whether students had to have certain background knowledge in order to count-on, if students
needed to demonstrate specific subskills related to counting on before utilizing the strategy, or if any
student who was working on addition could be taught this skill.
The areas of focus that we researched were: what general background knowledge did students
who knew how to count-on have, what subskills are directly related to counting-on, and what are the best
strategies to help students reach the point of counting-on.
What background knowledge did students who knew how to count-on have?
After our work with the kindergarteners, we have come to the conclusion that students need
specific background knowledge of addition and beginning numeracy skills before the strategy of
counting-on can be addressed. Based on our time with the students, we have identified cardinality,
counting, the concept of greater than and less than, the ability to identify an addition problem, and part-
Running head: COUNTING-ON
part whole relationships as skills that should be addressed before counting-on is introduced. We feel that
these skills are important because they are the first steps in the development of number sense. In the Van
de Walle, Karp, & Bay-Williams (2009) text, an early number concept that is addressed is the ability to
understand a number and see that number relatively. Students must see that a number can be greater than
another number or that the number seven also represents how many days are in a week. The ideas that are
held about numbers begin to grow as new these connections are formed.
One of the connections that must be formed about numbers is knowledge of the counting
sequence. According to Van de Walle, Karp, & Bay-Williams (2009), one of the biggest milestones in
terms of counting is not simply rote counting, but also realizing that the numbers that are said during the
sequence are representative of quantity. For this reason, we have identified both counting and cardinality
as background knowledge that should be held before counting on can be utilized. In a study done by
Secada, Fusion, & Hall (1983), one of the subskills that they identify as related to counting on is the
ability to start at a random number in the counting sequence and continue the count. Since Van de Walle,
Karp, & Bay-Williams (2009) states that counting is a dichotomous act because it involves both saying
the number and understanding its quantity, we conclude that both counting and cardinality should be in
the students repertoire before they can work on the subskills, and furthermore before they can count-on.
Another concept that is linked closely to the counting sequence is the understanding of numbers
in relation to each other. Van de Walle, Karp, & Bay-Williams (2009) states that one of the most basic
concepts of number that begins to form before school age is the concept of more, less, and the same. In
order for students to count on, they must understand that we are adding “x” amount more to an existing
quantity. Thus, we deem the concept of more as particularly crucial to the counting-on strategy. Van de
Walle, Karp, & Bay-Williams (2009) talks about how these relationships can first be addressed with the
use of counters so that students can visually see the concept of “more.” We believe that before students
can count-on, they must be able to understand “more” without using a visual representation.
The final skills we have identified as necessary background knowledge are identifying addition
problems and understanding part-part whole relationships. These last two skills have been grouped
Running head: COUNTING-ON
together because they are very similar in concept. We have seen students who are presented with two
quantities but then must prompted to take the first addend and the second addend and put them together in
order to decide where to begin when forming an addition problem. We believe that in order to count-on,
students must be able to identify and begin solving an addition problem without scaffolding. Van de
Walle, Karp, & Bay-Williams (2009) states that an understanding of part-part whole relationships is one
of the critical developments that occurs in terms of understanding numbers. This idea that two parts can
be individual sets and also come together to create one whole set is critical to the concept of addition.
Since counting-on is a strategy that is used to make addition more efficient, we believe that students must
have a complete understanding of addition as an entity before they can can grasp strategies that are related
to it and have a full understanding of what these strategies imply.
What subskills are required in order for students to count-on?
We are aware that counting-on is a skill that not all kindergarteners demonstrate and it takes a
certain skill set and a lot of practice in order for students to fluently be able to use it. One aspect of that
skill set include being able to start at a random number and continue to state the counting sequence
without having to begin at the number one. This is different from simply being able to rote count in that
students will not be able to rely on the memorized rhythm of starting the counting sequence at one and
continuing on mindlessly. Some other aspects of the skill set include understanding that the first addend
that is dealt with does not need to be counted again (cardinality), and lastly an understanding that the final
addend begins as the first addend plus one more (Secada, Fusion, & Hall, 1983). Even in cases of students
where all of these subskills are present, not all of the students were able to demonstrate the strategy of
counting on. This is what led us to believe that there may be other factors that determine whether a
student is able to count-on or not. In a study done by Carpenter, Hiebert, & Moser (1981), a group of
students who had at one point successfully utilized counting-on were observed using other less effective
addition problem-solving strategies such as concrete counting. During the discussion of the consistency of
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students’ responses, it was said that, “when children have several strategies available, they often use them
interchangeably rather than exclusively using the most efficient one” (Carpenter, Hiebert, & Moser,
1981). This variability can be contributed to the students’ lack of practice with counting-on. They have
learned concrete- counting skills early on and they fall back on those strategies when concrete
manipulatives are available to them. We have seen an example of this during our work with the
Longfellow students. A particular student, M, has demonstrated her ability to count-on since early
December, but when working with her we have noticed that she will use manipulatives when they are
made available to her instead of counting-on like we have seen her do before. Given what we know about
students using counting-all strategies when manipulatives are available, we further examined the design
of the Secada, Fusion, & Hall (1983) study. During their work with the children, they had specific tasks
that related back to each of the three subskills that we identified earlier. The main purpose of the set-up
and prompting questions that the researchers used was to get students to count-on. If the students veered
off course, the instructor responded in a specific way to lead students back to the concept of counting-on
without explicitly telling students to do so. As a result, students were constantly using the subskills that
they had acquired and in turn they were solving all of their addition problems with counting-on
procedures. This process led students to discover that counting-on was an efficient strategy, and they did
not fall back on other strategies. This research has suggested that in order for students to apply the
strategy of counting-on to a variety of structures of addition problems, that they need guided practice
where they are encouraged to use counting-on and discover that it’s the most effective strategy in their
arsenal.
What are the best strategies to help students reach the point of counting-on?
As we mentioned earlier, guided instruction was a strategy that researchers employed in order to
have students discover how effective counting-on can be. In a study by Secada, Fusion, & Hall (1983),
when students were ready to utilize the strategy of counting-on they were put through a series of tasks that
reinforced the subskills that we mentioned earlier. During the tasks, a teacher was present to ensure that
the students were grasping the concept, and if they weren’t they were there to ask questions that they felt
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would refocus the students on the subskill that they were testing. In each of the tasks, the teacher never
told the students that the ultimate goal of the tasks was to have them learn to count-on, instead they
focused primarily on the subskills and hypothesized that once they were mastered they would lead to
counting-on. After working on these subskills with students, seven of the eight students that completed
the tasks were able to demonstrate counting-on, whereas they had been able to prior to the subskill tasks.
On the other hand, there was a control group of eight students that were not exposed to these tasks, and as
a result only one of the eight was able to demonstrate counting-on. In a similar study done by Carpenter,
Hiebert, & Moser (1981), they came to the conclusion that explicitly telling students to start at one
addend and count-on may not be the most effective way to have students acquire these skills. In order for
a method to be declared effective, the students must not simply be able to perform the addition strategy
but also must be able to explain why it works the same as counting-all and why it should be used instead
of counting all. Explicitly telling students to count-on does not allow students to explore addition
strategies and make their own connections. This further reinforces the work that was done in the Secada,
Fusion, & Hall (1983) study. All of these researchers believe that after being exposed to tasks that
emphasize the subskills, students will be able to apply it when solving addition problems, and in the end
determine on their own that it is the most effective addition solving strategy.
Another strategy that we found to be support instruction on counting-on was one proposed by
Baroody (1987). He pointed out that most kindergarten-aged students have the ability to solve problems
that follow the equation n+1 without the use of concrete manipulatives. Using this knowledge, teachers
can rephrase addition problems with the goal of counting-on. For example, when a problem is five plus
one and students don’t know how to solve it, they are usually prompted by saying, “what is one more than
five?” In our experience with the kindergarteners at Longfellow, most of them are able to answer that
addition problem when it has been rephrased. This implies that students have already have a sense of what
counting on is, but on a small scale and may not be aware that it applies to addition problems with larger
addends than one. We would be interested to see if an addition problem such as four plus three rephrased
as “what is three more than four”, would elicit the counting-on process in more cases. We are especially
Running head: COUNTING-ON
curious about this because of a previous study done by Carpenter, Hiebert, & Moser (1981) that showed
that the way that an addition problem was phrased, changed the strategy that the child used to solve the
problem. Students who had not yet discovered that counting-on was more efficient than counting-all,
changed strategies depending on the phrasing of the problem more often than students who were decided
on one strategy in particular. Rephrasing the way we state addition problems could be especially useful
for our work with the Longfellow kindergarteners, because most of them have yet to solely solve addition
problems with counting-on.
Our Findings
In order to further our research on counting-on, we took what we had learned from our literature
review and applied it to our work with the Longfellow kindergarteners. Since our research focuses on
background knowledge, subskills of counting-on, and ways to teach counting-on, we wanted to first
identify students in our math groups who were already demonstrating the counting-on approach. In our
groups, we were able to identify students who demonstrated counting-on as a strategy for the majority of
addition problems they solved, students who demonstrated counting-on after it had been modeled or only
in certain situations, and students who did not demonstrate a complete understanding of addition in terms
of counting-all and therefore did not count-on.
The group of students who utilized counting-on a majority of the time did so without having
counting-on modeled or suggested to them. Counting-on was a strategy they chose to use themselves, and
often on the first addition problem they were presented. Our research on background knowledge
addresses the need for knowledge of part-part-whole relationships before counting-on can be utilized.
Additionally, a subskill that our research claimed was an understanding that the first addend that was
stated did not need to be counted again. During our work with the kindergarteners, one of the students, T,
utilized counting-on and also demonstrated an understanding of part-whole relationships and the fact that
the first addend had already been counted. T demonstrated this when he was asked to show his thinking
when he counted-on for an addition problem where the first addend was 10. T first flashed 10 fingers on
his hands, did not count them but said “10.” He then held up 7 fingers, which was the second addend, and
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proceeded to count up until he reached the correct answer of 17. By thinking out loud in that way, T
demonstrated his understanding that 10 was not only an arbitrary number to start at, but also represented a
quantity of 10 that did not need to be counted out one-by-one. Another student, M, also demonstrated this
subskill. When asked what 5+4 was, she immediately put up five fingers and then stated the number
sequence starting at 6, until she had four fingers up on her right hand. Noticing this, we started giving M
addition problems where the second addend was greater than the first. To our surprise, the student began
with the larger of the two addends and still utilized the counting-on strategy. When we inquired why she
started with the second addend instead of the first, she told us, “You always have to start with the bigger
number. That way you don’t have to count as much.” When we asked M how she knew this, she told us
that her older brother had taught her at home. During our work with the kindergarteners, we have noticed
a trend where the students who are using advanced strategies, typically tell us that they learned that
strategy at home from a sibling or a parent.
On the other hand, another student, TA, began utilizing the strategy of counting-on after it was
modeled for her on the Number Line app she was using. TA did not choose to count-on originally, but
after seeing the app start at the first addend and then count up, she began to do so on the next problem.
TA counted-on, however, by starting at the first addend and then counting up the number of the first
addend instead of the second. TA knew that the quantity represented by the first addend had to be
incorporated into the sum, but did not realize that by starting at the first addend she was showing that she
had already accounted for that quantity and did not have to count it again. Another student, C, was able to
utilize counting-on when a number line was present, but when the visual aid was no longer in front of
him, he reverted to using his fingers or counters and using the count-all strategy. A similar observation
was made when a student, W, began to use counting-on after seeing it modeled. W started at the first
addend, counted up the second, and then continued to count up the amount of the first addend. This shows
that the understanding of counting-on as adding two quantities together has not yet been formed, and the
students are following a procedure they have been shown but do not yet understand the reasoning behind
it. Following a similar trend, a student, K, started at the first addend, 6, and counted up to the number of
Running head: COUNTING-ON
the second addend, 7, instead of adding 7 more. When K solved addition problems using the count-all
method, however, she knew that she had to count out the entire first addend and the quantity of the entire
second addend in order to find the whole.
Another interesting finding that came out of the group of students who counted-on after the
modeling of the strategy or only in certain situations, was that a number of students counted-on when the
first addend was 10 and the problem was stated as 10 and “x” more. The number 10 seemed to be a useful
anchor number, and one that many students could use to help them visualize “x” amount more. The only
students who did not deal with counting-on in some form were those that did not demonstrate an
understanding of addition as counting-all.
With some of the students it was clear that they were not prepared to be taught the counting-on
strategy. When asked to add a problem such as 4+2, two students said that they were unsure of how to
solve it. After being shown that they could use their fingers, counters, or a number line to solve the
problem, the two students chose to use their fingers, but still needed assistance in order to solve the
problem. In this case, these students still need practice with addition and they need to be engaged in
activities that teach them the meaning behind addition.
For the group of students who was able to add, but did not utilize counting-on as a strategy, we
began to introduce them to activities that emphasized that the first addend represented an already known
quantity, and therefore did not have to be counted again. In order to show this to our students we engaged
them in an activity where they were asked to count out counters to represent each of the addends. After
the students completed this, we would place the counters for the first addend into a plastic bag and then
ask the students to tell us how many counters there were without counting again. Some students were
confused as to how they were supposed to answer this question for us without counting the counters
again. If this happened, we would point to the first addend and say, “look it tell us right here how many
counters there are.” Once students determined how many counters were in the first group we had them
solve the problem by counting up from that number using the counters that were laid out for the second
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addend. Repetition of this activity eventually resulted in most of our students displaying counting-on as a
strategy for addition.
From the work with our kindergarteners we have concluded that the most effective way to teach
counting-on as a strategy for addition is to address each of the subskills that we have found necessary for
the strategy. We witnessed students who could count-on when they were told to do so, but were hesitant
to count-on from the first addend without starting back at one and counting-all. These students did not
demonstrate an understanding of counting-on, although they could follow the steps if they were directly
instructed to do so. Therefore, we believe that exposure to activities that reinforce the fact that the first
addend, or the bigger addend, are already known quantities and do not need to be recounted, is the most
effective strategy to teach counting-on. We witnessed student growth in their addition abilities each week
after these activities were introduced. With continued exposure, we believe that all of the kindergarteners
that we worked with would be able to utilize counting-on as a strategy for addition.
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References
Baroody, A.J. (1987). The development of counting strategies for single-digit addition. Journal for
Research in Mathematics Education, 18(2), 141-157.
Carpenter, T.P., Hiebert, J., & Moser, J.M. (1981). Problem structure and first-grade children’s initial
solution
processes for simple addition and subtraction problems. Journal for Research in Mathematics
Education, 12(1), 27-39.
Secada, W.G., Fuson, K.C., & Hall, J.W. (1983). The transition from counting-all to counting-on in
addition.
Journal for Research in Mathematics Education, 14(1), 47-57.
Van de Walle, J., Karp, K.S., Bay-Williams, J.M. (2009). Elementary and Middle School Mathematics:
Teaching Developmentally (7th Ed.). Canada: Pearson.