Fostering Multiplication Fluency Skills Through Skip Counting
Matthias Grünke1
1
University of Cologne, Department of Special Education and Rehabilitation, Klosterstr. 79b, 50931
Cologne, Germany
[email protected]
Abstract – The purpose of this single-case study was to implement and evaluate a
technique to teach multiplication fluency skills (skip counting) to a 7-year old girl
who had severe difficulties with multiplication tables. A multiple-baseline design
across two fact sets was applied to determine the effectiveness of the approach.
Assessment prior to the intervention showed very low fluency skills. During the
course of the treatment, the student’s performance improved, reaching mastery.
Overall, the skip-counting approach turned out to be very successful. The paper
ends with a discussion of the limitations of the study and suggestions for future
research directions.
Key Words – Math fact fluency; Math problems; Single-case research; Skip
counting; Special education.
1
Introduction
Fact fluency in addition, subtraction, multiplication, and division is essential for access to and success
with higher-level mathematical concepts. It is important to reduce working memory overload to
increase the amount of energy available for problem solving. In order to master complex tasks, it is
necessary to execute basic arithmetic operations fast and accurately (Codding, Burns, & Lukito, 2011;
Nelson, Burns, Kanive, & Ysseldyke, 2013). Thus, if students do not acquire sufficient math fluency
during their elementary education, they will very likely continue to demonstrate difficulties in this
respect throughout their lives (Gersten, Jordan, & Flojo, 2005). Automaticity of multiplication facts
seems to be especially crucial in this context (National Mathematics Advisory Panel, 2008), as the
inability to master multiplication tables has been found to have a very negative effect on long-term
performance in mathematics instruction (Jordan, Hanich, & Kaplan, 2003). For example, students with
problems in this domain will, in all likelihood, struggle with ratio and proportion, fraction concepts,
and measurement conversions (National Mathematics Advisory Panel, 2008).
Unfortunately, a great proportion of children are not adequately prepared in mathematics by the end of
their elementary education (Duncan et al., 2007), primarily because many teachers do not incorporate
sufficient opportunities in their daily instruction for students to practice basic multiplication skills
(Daly, Martens, Barnett, Witt, & Olson, 2007). However, several promising approaches have been
found to remedy the problem of insufficient fluency. For example, in their meta-analysis, Codding et
al. (2011) present a number of methods that have proven to be helpful in building automaticity of
multiplication facts, including cover copy compare, interspersal techniques, self-management, taped
problems, peer-delivered feedback, positive practice overcorrection, and various flashcard procedures.
One instructional approach that is not featured in this meta-analysis is skip counting (also known as
count-bys strategy). Skip counting involves counting by a number that is not 1. For example, if you
skip counts by 4 up to 20, you would count in the following order: 4, 8, 12, 16, 20. Combining the
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base (4, in this example) with the number of groups (5, in this example) produces the standard
multiplication equation: 4 multiplied by 5 equals 20 (Wagganer, 2015). Most children find
multiplication by 2, 5, and 10 to be easier than other multiplication facts (Baroody & Dowker, 2003).
Skip counting is a means to help students understand the concept of repeated addition and to memorize
multiplication tables other than the x2s, x5s, and x10s. It has been used as a way to teach
multiplication for a long time. For example, skip counting is part of Investigations in Numbers, Data,
and Space, a widely used K-5 mathematics curriculum, developed between 1990 and 1998 (Budak,
2015).
Surprisingly, as of February 2016, the database PsycINFO lists only six publications that contain the
term "skip counting" and only two publications that contain the term "count bys" in their titles.
Besides, only three of those studies are empirical (i.e., Duvall, McLaughlin, & Cooke-Sederstrom,
2003; Grünke & Calder Stegemann, 2014; McIntyre, Test, Cooke, & Beattie, 1991). Using a singlecase design, the results of each experiment provide impressive evidence for the effectiveness of the
approach. In every instance, a skip-counting intervention of as little as 10 sessions produced
remarkable improvements in the multiplication fluency skills of the participants.
As mentioned, to date, skip counting has generally gone unnoticed by the research community. The
purpose of the present study was to extend the body of empirical literature about this technique to
teaching multiplication fact fluency to struggling learners. As in the previous experiments, a singlecase design was applied to answer the underlying research question concerning how much an
elementary school student’s multiplication skills would improve with not yet mastered fact sets using a
simple skip-counting intervention.
2
Method
2.1
Participant
The participant of this study, Belma, was a 7-year-old girl enrolled in the second grade in a large
elementary school in a major city in Western Germany. Her parents had moved from Turkey before
Belma, the third of four children, was born. She spoke both Turkish and German fluently and was able
to master addition and subtraction facts without major difficulty. However, when her teacher
introduced the concept of multiplication in the middle of second grade, she did not seem to grasp it.
She was mostly unable to solve problems dealing with multiplication tables that her teacher presented
to the students in her class through worksheets. According to her teacher, Belma especially failed with
8s and 9s fact sets. Otherwise, she did not demonstrate any serious academic problems.
2.2
Experimental Design
An AB multiple-baseline design (Horner & Odom, 2014) across two fact sets (8s and 9s) was applied.
During the whole study, 20 weekdaily probes were collected. The beginning of the instruction to
practice each fact set was randomly determined within certain specifications: In accordance with the
single-case intervention research design standards proposed by Alberto and Troutman (2012), the
minimum number of data points during the baseline was set at five. Therefore, the instruction had to
last at least five daily sessions. Thus, the intervention could have started any time between the 6 th and
the 16th probe. A drawing of all 10 possible options for each fact set resulted in an arrangement
whereby the intervention to practice the 8s fact set started before the sixth probe (i.e., after the fifth
baseline measurement point), the intervention to practice the 9s fact set started before the 11th probe
(i.e., after the 10th baseline measurement point).
2.3
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Intervention
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Weekdaily training of each fact set lasted for 15 minutes. A 25-year-old female graduate student in
special education from a large German university served as the teacher. She instructed Belma in a
quiet corner of the classroom, while the rest of the children were engaged in independent seatwork.
The student teacher presented Belma with large index cards that portrayed the counting sequence for
the 8s (8, 16, 24, 32, 40, 48, 56, 64, 72, 80) and for the 9s (9, 18, 27, 36, 45, 54, 63, 72, 81, 90),
respectively. At the start of the intervention, the student teacher repeatedly read the respective
sequence out loud. Then, she encouraged Belma to join her. In order to facilitate memorization, they
reiterated the sequences in form of a rap song.
After Belma was able to recite the order of the numbers by heart, the student teacher presented her
with rather simple multiplication problems (e.g., 1 x 8, 2 x 8, and 10x 8) and later moved on to more
challenging ones (e.g., 3 x 9, 4 x 9, 5 x 9, 6 x 9, 7 x 9, 8 x 9, and 9 x 9). The problems were solved by
taking a certain number of steps of the sequence for the 8s or 9s. For example, if Belma needed to
calculate 4 x 9, she had to count the first four numbers of the counting order for the 9s (9, 18, 27, 36).
The teacher scaffolded the process and fell back on the index cards as necessary.
2.4
Dependent Variable
The number of correctly solved multiplication problems in response to worksheets containing all 10
tasks for each fact set served as the dependent variable. Thus, each worksheet consisted of 20
problems, which were presented in random order. Belma was presented with a different worksheet
each day. During the intervention phase, she worked on the problems after the daily training. The time
limit for finishing the daily assignment was 5 minutes.
3
Results
As shown in Figure 1, the number of correctly written digits in response to the presented worksheets
increased drastically from initial levels. Specifically, the means of Belma’s baseline and intervention
phase measures grew from 0.40 to 7.87 for the 8s, and from 0.50 to 7.10 for the 9s. By the time the
intervention ended, she had reached mastery on both fact sets. Visual inspection of the data (see Gast
& Spriggs, 2010) clearly showed that Belma’s performance continually improved over the course of
the treatment.
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8s
9s
Fig.1: Number of correctly answered multiplication problems for the 8s and the 9s.
The Percentage of Data Exceeding the Median (PEM) was used to represent the effect size of the
outcomes. It identifies the percentage of measurement points exceeding the median of the baseline
phase and varies between 0 and 1 (Ma, 2006). For both fact sets, the PEM was 1, reflecting a highly
effective intervention (Alresheed, Hott, & Bano, 2013). To determine the statistical significance of the
differences between baseline and intervention phase measures, a randomization test was applied
(Dugard, 2014). Because the two treatment points were selected randomly within a certain range, this
method could be used to assess whether the improvements were likely due to chance. With the help of
a specific Microsoft® Excel macro for AB multiple-baseline designs (developed by Dugard, File, &
Todman, 2012, and downloadable at https://www.routledge.com/products/9780415886932), it was
determined that the differences between the phases were statistically significant within a standard 95%
confidence interval.
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Discussion
In this study, skip counting turned out to be a highly effective intervention by all measures commonly
used in single-case research (visual inspection, effect size, and inferential statistics). In fact, the results
could hardly have been more positive. The participant improved her multiplication fluency skills in the
two target fact sets to the maximum level in the course of 10-15 sessions. Thus, the present case report
confirms findings from previous research, thus suggesting that skip counting is another potent means
to help break the negative spiral of insufficient multiplication fluency skills, difficulties with
apprehending higher-level mathematical concepts, and eventually problems with independent living
(Patton, Cronin, Bassett, & Koppel, 1997).
Nevertheless, the findings are subject to certain limitations. First, as with all single-subject designs, the
generalizability of the results is relatively problematic. Claims about whole populations are always
based on limited samples. Thus, even the findings from large-group experiments cannot be viewed as
universally applicable. However, generalizability is a bigger issue if a study involves only one
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participant (Kächele, Schachter, & Thomä, 2009). Second, no follow-up data were collected. Thus, it
is not possible to make claims about the long-term sustainability of the treatment effects. The
shortness of the data collection period was due to the beginning of school vacation. Even though it
seems unlikely that the participant would quickly forget what she had learned during the intervention,
it would have been desirable to have collected real follow-up data. The number of correctly solved
multiplication problems on worksheets can be counted very objectively and reliably. Thus, the way the
dependent variable was determined does not constitute a major threat to the explanatory power of the
study.
In summary, the present experiment confirms the usefulness of skip counting as a means to foster
multiplication fluency facts in children. Future research should focus on confirming these results.
According to Chambless and Ollendick (2001), four single-case studies with positive outcomes are not
enough for a certain treatment approach to qualilfy as being evidence-based. At least nine are needed.
The Council for Exceptional Children (CEC, 2014) proposes in its standards for evidence-based
practices in special education the need for 3 methodologically sound single-subject studies with
positive effects (meaningful change in the dependent variable for at least 75% of the cases) and a
minimum of 10 total participants. In addition, 2 methodologically sound group comparison studies
with positive effects and at least 60 participants are mandatory. Hence, group studies are needed to
verify the findings of the present and the three previous experiments in terms of the benefits of skip
counting. In addition, it would be helpful to investigate whether this technique is appropriate for peer
tutoring. Thus, being able to offer invididualized support to struggling learners by experienced
students under the supervision of a teacher would help to make assistance more readily available for
all children in need of support.
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