Bachelor’s Thesis to earn the
BEd of Education
Beneficiary effects of skip-counting on
attitude towards multiplication in
international primary school students
Name: Vlad Coraci
Word Count: 8316
Supervising lecturer: Cody Buye
Student number:1522481
Research question: Does skip-counting change the attitude towards learning multiplication in
students of a lower primary international school?
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Table of Contents
Abstract ................................................................................................................................ 3
Introduction .......................................................................................................................... 4
Context and Objectives................................................................................................................. 4
Motivation.................................................................................................................................... 4
Significance................................................................................................................................... 5
Research Question........................................................................................................................ 5
Literature Review ......................................................................................................................... 6
Research question and sub-questions .......................................................................................... 8
Methodology ........................................................................................................................ 8
Research positionality .................................................................................................................. 8
Research design ............................................................................................................................ 9
Participants ................................................................................................................................ 10
Methods of data collection ......................................................................................................... 10
Methods of data analysis............................................................................................................ 11
Quality ........................................................................................................................................ 12
Ethics .......................................................................................................................................... 13
Results ................................................................................................................................ 14
RQ1: What determines a student’s attitude towards the subject of mathematics? ................... 14
RQ2: How can skip-counting be implemented into a teacher’s daily routine?............................ 17
RQ3: Does skip-counting change students’ attitudes towards mathematics? ............................ 18
Conclusion and Discussion ......................................................... Error! Bookmark not defined.
Limitations .......................................................................................................................... 26
References .......................................................................................................................... 28
Appendices ......................................................................................................................... 32
Appendix A: Survey for students ................................................................................................ 32
Appendix B: Teacher Interviews ................................................................................................. 34
Appendix C: Classroom Observations ......................................................................................... 40
Appendix D: Focus Group Discussions ........................................................................................ 44
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Abstract
Skip-counting is an instructional strategy to enable young students to grasp the concept of
multiplication. Several studies have evaluated the benefit of skip-counting on mathematical
proficiency, however, there is a lack of research on whether skip-counting can influence a student’s
attitude towards the subject of mathematics and multiplication specifically. The attitude a student has
towards a given subject has been shown to be a strong predictor for later academic success and as
such it is necessary to explore this gap in knowledge in order to improve current teaching strategies.
The present study performed a two-week skip-counting intervention in which 24 primary school
students were exposed to skip-counting lessons in the form of mental strategies, interactive games,
songs and chants, and artistic projects. Quantitative data was gathered before and after the
intervention in the form of a Likert-scale questionnaire to gauge changes in students’ attitudes
towards the subject of multiplication. Additionally, qualitative data was gathered in the form of
classroom observations, teacher interviews, and focus group discussions. The questionnaire results
suggest that the skip-counting intervention improved the student’s attitude towards the subject,
assessed by observing globally improved attitudes in enjoyment and confidence of students to solve
multiplication tasks, perceived usefulness of the subject, and enjoyment of working on mathematical
operations with their peers. These results were further confirmed by qualitative means, with students
stating feeling less overwhelmed with multiplication tasks, experiencing more enjoyment when
learning multiplication, and developing a deeper comprehension of mathematical concepts, as
described in focus group discussions and interviews. Together these findings suggest that skipcounting can be effectively integrated into daily instruction and may improve students’ motivation
and engagement in learning multiplication to foster a positive learning environment.
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Introduction
Context and Objectives
The purpose of this study is to examine how skipping numbers affects students' attitudes
toward multiplication in a primary international school. As primary school students develop their basic
math skills, it is important to identify effective instructional strategies related to skip-counting that
can foster a positive attitude towards mathematics to enhance learning outcomes. Overall, this study
aims to contribute to the field of mathematics education by closely looking at how skip-counting as a
daily routine and teaching strategy can impact and affect students’ attitudes towards mathematics,
particularly learning multiplication.
Mathematics plays a crucial role in the early development of problem-solving skills and visual
thinking in young students. However, many students struggle with mathematics, which has been
correlated to harboring negative attitudes towards the subject. Consequently, there is a need to
identify effective strategies that can enhance students' attitudes towards mathematics, more
specifically towards learning multiplication. Skip-counting has been identified as a strategy that can
improve students' understanding of mathematical concepts, however, it is not well understood if skipcounting may also improve students’ attitude towards the subject. This research aims to explore the
impact of skip-counting on students' attitudes toward mathematics in a primary international school
setting. The International School of Bucharest (ISB) in Bucharest, Romania, is a private international
school running the Cambridge curriculum and leading in education with over 700 international
students aged 2-18. The present study will be conducted on Year 4 students aged 8-10 years old.
Motivation
As a primary school teacher, my personal motivation for conducting this study is to improve
my students' attitudes toward mathematics and enhance their learning outcomes in multiplication.
Multiplication is a foundational math skill that is essential for future math learning and everyday life.
However, many students may struggle with multiplication and develop negative attitudes toward
math, which can hinder their learning progress (Zhang et al, 2014).
Through this study, I aim to identify effective instructional skip-counting strategies that can
help my students enhance their attitude towards mathematics, more specifically towards learning
multiplication. I believe that skip-counting has the potential to make multiplication more engaging
and accessible for my students, and I am interested in exploring its impact on students' attitudes and
learning outcomes.
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Overall, my personal motivation for conducting this study is to improve my instructional
practice when it comes to skip-counting and help my students develop the math skills and positive
attitudes they need to succeed in their future academic and personal pursuits.
Significance
The present study focused on the impact of a skip-counting method on lower primary school
students' attitudes toward learning multiplication, in order to provide evidence for the effectiveness
of skip-counting as a pedagogical strategy in teaching multiplication (Franke & Kazemi, 2010). This
research encourages teachers and educators to consider incorporating skip-counting activities into
their daily routines, fostering a positive learning environment.
Understanding the impact of skip-counting on students' attitudes contributes to a deeper
understanding of how students perceive and approach mathematical concepts. By positively
influencing students' attitudes, skip-counting may lead to improved learning outcomes and
mathematical proficiency. This research emphasizes the importance of incorporating varied
instructional strategies to enhance students' conceptual understanding and mathematical fluency.
The findings also have implications for curriculum design and development. By incorporating
skip-counting activities, curricula can be designed to promote a more holistic and student-centered
approach to mathematics education. This can cater to diverse learning needs and foster a positive
attitude towards mathematical concepts.
Additionally, this research opens avenues for further investigations and exploration in the field
of mathematics education. Future studies can delve deeper into the specific aspects of skip-counting
that contribute to attitude change and explore its long-term impact on students' mathematical
abilities (Douglas & Sarana, 2020). Research can also examine the transferability of skip-counting skills
to other mathematical domains and investigate the potential benefits of integrating skip-counting in
different educational settings and grade levels.
In summary, the significance of this research lies in its potential to inform educational practice,
enhance student learning outcomes, shape curriculum development, and inspire further research in
the field of mathematics education.
Research Question
The research question of the present study was designed to be specific, measurable,
achievable, relevant, and time-bound (SMART). It is specific by focusing on the impact of students'
attitudes towards multiplication. It is measurable with both students' and teachers’ attitudes being
quantified using surveys, interviews, and group discussions. It is achievable because the study can be
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conducted within a reasonable time frame and with available and limited resources. It is relevant
because it seeks to address a problem that is relevant to primary international school students and
currently poorly understood in literature. Lastly, it is time-bound having a clear deadline for
completion.
The research question is significant because it addresses the need to identify effective
strategies that can improve students' attitudes toward mathematics. By exploring the impact of skipcounting on students' attitudes, this research can contribute to the knowledge base on effective
teaching strategies and promote positive attitudes and effective learning towards mathematics
among primary school students.
Literature Review
It is well established that early mathematical competency is a strong predictor for not only
later mathematical but overall academic success, even surpassing reading, and attention skills
(Duncan et al., 2007; Claessens et al., 2009; Claessens & Engel, 2013; Romano et al., 2010), promoting
the idea of early mathematical education. Multiplication is a fundamental mathematical concept
taught within the primary school curriculum that is essential not only by itself but also in the
acquisition of more advanced mathematical skills. However, arithmetic approaches to teaching
multiplication show poor performance and prove as a memorization challenge to many children
(Dotan & Zivran-Ginat, 2022). These findings showcase the need for the development of new effective
teaching strategies for young students.
Skip-counting is a mathematical technique taught most notably in newer reform mathematic
textbooks such as the TERC in the United States. These textbooks have been created based on
developmental psychology research as to how children best learn mathematics. Skip-counting is
defined as adding the same number each time to the previous number in the counting sequence. For
example, when forward skip-counting by 3, starting from 0, the sequence will be 0+3=3, 3+3=6, 6+3=9,
and so on. Similarly, backward skip-counting can be used by subtracting instead of adding the same
number in the counting sequence. In this way, young students can make use of already acquired
mathematical skills, namely addition and subtraction, and learn to see patterns in numbers through
repeated addition, which lays the foundation for learning multiplication and division early in education
(Carpenter et. al., 2015).
Research on the effectiveness of skip-counting has been predominantly explored in children
with learning disabilities. Several studies have found skip-counting to be an effective strategy in
teaching multiplication and division, enhancing the students understanding and mastery of
mathematical concepts. In a recent case-control study by Grünke (2016) a 7-year-old girl with
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difficulties learning the multiplication table, especially for counting sequences of 8 and 9, was able to
reach proficiency of these sequences following 15 interventions using skip-counting. This study
concluded that skip-counting was an effective tool in multiplication fluency skills and apprehending
higher-level mathematical concepts. In a similar case-control study by DuVall, McLaughlin, and
Sederstrom (2003), the effects of skip-counting on accuracy and fluency of see-to-write math facts in
three middle school students with learning disabilities were assessed. After five sessions of practicing
skip-counting before completing a 2-minute math probe sheet, two of the three students were able
to significantly improve their accuracy and fluency through skip-counting, with the third showing little
improvement. However, despite the positive results in these studies, it is important to note the
limitations of a low sample size and lack of follow-up data. In one of the most recent and
comprehensive case-control studies on skip-counting, Gibbs, Hinton, and Flores (2018) investigated
the effects of a skip-counting learning strategy in 15 third- and fourth-grade students with learning
disabilities struggling with learning multiplication. Following 11 lessons involving solving multiplication
problems using skip-counting, students demonstrated a higher mathematical proficiency, and the
authors describe students developing an improved number sense. However, as no follow-up data was
collected and the effectiveness of other teaching strategies was not compared, the validity of these
findings needs to be investigated further.
It is well established that a positive relationship exists between a student’s attitude towards
school and their academic success (Sünbul et al., 2013). More specifically, the attitude a student
showcases toward a subject such as mathematics can have a significant impact on their performance
and engagement, with studies having described students with positive attitudes toward mathematics
as being more likely to perform well and enjoy the subject (Sen, 2022). Conversely, students having
negative attitudes toward mathematics have been linked to lower academic performance in the
subject (Schunk & Usher, 2019). Although attitude is highly subjective, it is relatively easy to
quantitively and qualitatively measure, for instance through the use of surveys and interviews,
respectively (ElÇi, 2017; León-Mantero et al., 2020). Multiplication is an essential mathematical
concept for acquiring further mathematical skills as well as problem-solving and visual thinking skills,
however, many young students struggling with multiplication may develop negative attitudes towards
it, which can hinder their learning and further education (Fyfe & Rittle-Johnson, 2017). Based on this,
employing teaching strategies to improve not only subject proficiency but additionally attitude
towards the subject may have potent and long-lasting learning effects in young students. However,
whereas studies have predominantly displayed the effectiveness of skip-counting on mathematical
proficiency, there is lacking evidence on the impact of this teaching strategy on the students' attitudes
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towards mathematics. To fill this gap in research, the present study aims to answer whether skipcounting influences student’s attitudes towards mathematics.
Research question and sub-questions
Based on the gap in literature described previously, the present study investigated the primary
research question of “How does skip-counting impact students’ attitude towards learning
multiplication in a lower primary international school?”. To investigate this primary research question,
the research is further divided into three sub-questions. The first sub-question, “What factors impact
a student’s attitude towards learning?”, aims to investigate underlying mechanisms through which a
student’s attitude towards mathematics can be altered and provides a starting point for this study.
The second sub-question, “How can skip-counting be integrated into a teacher’s daily routine?”,
investigates the degree to which skip-counting can be integrated into the daily teaching routine. Lastly,
the third subquestion, “Does skip-counting change students’ attitudes towards mathematics”,
determines whether a skip-counting intervention can yield potential learning benefits by improving
the perceived attitude towards mathematics.
Methodology
Research positionality
This study aims to objectively investigate the attitudes of students toward mathematics
following a skip-counting intervention. Quantitative data allows us to investigate trends in a
population by compiling unique experiences and converting them to measurable numbers. As such, it
is suitable to investigate the overall attitude toward mathematics in a student population to
determine the overall effects of skip-counting. Nonetheless, participants of this study have diverse
backgrounds, experiences, and identities that shape their attitudes towards mathematics and learning.
Qualitative data does not represent data as numbers but rather shifts the focus to unique experiences
and reports, making it less statistically significant and reliable, however providing valuable insights
into subjective experiences and context that go missing when grouping quantitatively (Rahman, 2016;
Cleland, 2017). Furthermore, attitude is a highly subjective experience, why it would be ill-advised to
solely focus on quantitative data and qualitative data may still be used to approximate effects and
results. Due to the aforementioned reasoning, this study will apply a mixed-methods design, collecting
both quantitative and qualitative data. Quantitative data is gathered to produce reliable results in
changes of attitude towards mathematics in young students, whereas qualitative data is used to
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highlight unique experiences of skip-counting in both students and teachers as well as perceived ease
of implementation of skip-counting into the teachers’ daily routine.
Regarding the positioning of the researcher to the data, an interpretivist research positionality
was adopted for this study, which, unlike positivist research paradigms, prioritizes the subjective
nature of social phenomena and the importance of considering context (Kivunja & Kuyini, 2017). This
paradigm is suited to the context of the present study due to a higher focus on qualitative data
collection and the focus of the study investigating attitude and as such human behavior and emotion
as a highly subjective experience. However, as such it is acknowledged that the researcher's own
experiences and perspectives as a teacher may influence the research process and outcomes. In this
study, an open and reflective attitude towards assumptions and biases will be maintained to be
receptive to the diverse perspectives and experiences of the students.
Research design
The quantitative aspect of the study will involve administering pre-and post-tests to measure
students' attitudes towards mathematics before and after the use of skip-counting. The pre-test will
establish a baseline measurement of the students’ attitudes toward mathematics, and the post-test
will evaluate the effectiveness of the instructional strategies in enhancing the students' attitudes.
The qualitative aspect of the study will involve classroom observation, two teacher interviews,
and focus group discussions with the students to gain an in-depth understanding of their experiences
and perspectives regarding the use of skip-counting (Wellington, 2015). The focus group discussions
will allow for the exploration of the students' perceptions of the instructional strategies and their
impact on their attitudes toward mathematics.
To ensure the validity and reliability of the research findings, triangulation will be used,
combining the results from both the quantitative and qualitative data. The mixed-methods design will
provide a comprehensive understanding of the impact of skip-counting on primary school students'
attitudes toward mathematics.
Ethical considerations will be addressed by obtaining informed consent from the parents or
guardians of the students, ensuring confidentiality, and obtaining approval from the relevant
institutional ethics committee.
Overall, the mixed-methods design will provide a rich and comprehensive understanding of
the research question and objectives, enabling the development of effective instructional strategies
for enhancing primary school students' attitudes toward mathematics.
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Participants
At the time this study was conducted, participants were students attending year 4 (grade 3)
of the international school of Bucharest following the Cambridge curriculum. Participants consisted of
23 students total of which 18 were girls aged 8 (n=10) or 9 (n=8), and 6 were boys, aged 9 (n=6). No
students had a history of learning disabilities, and the participants were appropriate in the context of
the present study as the students had been first introduced to multiplication shortly previously to the
intervention.
Methods of data collection
For this research study, a combination of quantitative and qualitative data collection methods
was utilized to provide a comprehensive understanding of the impact of skip-counting on primary
school students' attitudes toward mathematics. Prior to the intervention, students had been taught
basic multiplication, including counting by twos and threes partially using skip-counting strategies,
however, they had not been explained the concept of skip-counting previously and had not received
instruction using only skip-counting strategies, as with this intervention. For the intervention, the
researcher attended 10 mathematics lessons of the students for 2 weeks. Over the course of the
intervention, the classroom teacher used various skip-counting exercises with the students, including
digital interactive games, group activities, skip-counting songs, and puzzles.
For quantitative data, a 20-question questionnaire was created by the author to evaluate the
attitude of the students towards mathematics, with items of the questionnaire including whether
students enjoy working on multiplication problems, how easy they experience these tasks, and
whether they enjoy working on mathematical tasks with other students (see Appendix A). Students
evaluated the questions on a 5-level Likert scale ranging from strongly agree, agree, neutral, disagree
and strongly disagree and the nominal data was collected and analyzed as previously described. The
students completed this questionnaire as a pre-test on the first day of the intervention before being
exposed to skip-counting strategies and once more as a post-test on the last day of the intervention.
To evaluate how skip-counting can be integrated into a teacher’s routine and which factors
impact a student’s attitude towards learning, qualitative data was gathered before, during, and after
the intervention. Classroom observations of the instruction strategies and student engagement were
documented by the author during each lesson in the intervention period and can be found in Appendix
C. Additionally, qualitative data was gathered in the form of two one-on-one interviews held after the
intervention between the author and the classroom teacher, which can be found in Appendix B. The
first interview served the purpose of providing a general picture of the classroom atmosphere and
attitude when it comes to mathematics and more specifically multiplication. The second interview
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focused on the teachers’ opinions on skip-counting and strategies that a teacher could use to integrate
skip-counting into their daily routine. Lastly, at the end of the intervention period, a focus group
discussion was held in which students and teachers could openly share their experiences and
perspectives regarding the skip-counting intervention. Students were split into three groups of eight
students each. Interactions of this discussion were documented by the author and can be found in
Appendix D.
Methods of data analysis
Quantitative data obtained from questionnaire answers was analyzed and plotted using Excel.
Qualitative data collected from classroom observations, focus group discussions, and teacher
interviews were analyzed using thematic analysis. Preference ranking is a method that can be used to
justify and determine the attitudes of students before and after using the skip-counting technique
(Carpenter et. al, 2015). This involves identifying patterns and themes within the data, coding the data
accordingly, and categorizing them into different themes. The aim is to identify common themes and
patterns that emerge from the data that can provide insights into the impact of skip-counting on
students' attitudes toward mathematics.
The combination of both quantitative and qualitative data analysis methods will enable a
more comprehensive analysis of the data collected, allowing for a more in-depth understanding of the
impact of skip-counting on students' attitudes toward mathematics. Triangulation of the data
collected from multiple sources will also allow for a more robust and valid interpretation of the
findings (Wellington, 2015).
Data analysis of the quantitative data consisted of documenting the numerical data of the
Likert-scale questions in the student questionnaire and plotting answer frequencies and trendlines to
identify changes in attitude. For qualitative data, detailed notes from classroom observations were
taken, focusing on capturing and interpreting students' behaviors, interactions, and engagement
during skip-counting activities. Observations were then reviewed to identify recurring patterns or
noteworthy observations that could provide insights into students' attitudes toward learning
multiplication. For teacher interviews, the interaction was transcribed, and responses were examined
and categorizing for common themes. Similarly, the focus group discussion data was transcribed and
common themes, opinions, and experiences were identified. By delving into the perspectives shared
by the participating students, the analysis aimed to uncover their perceptions of skip-counting and its
impact on their attitude towards multiplication. The goal was to identify any shared consensus,
divergent viewpoints, or noteworthy insights that emerged during the discussion. Triangulating the
findings from multiple perspectives allows for robust conclusions and evidence-based interpretations
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to be drawn regarding the impact of skip-counting on students' attitudes. The analysis conducted in
this study is justified by its alignment with existing literature on the topic of skip-counting and its
impact on students' attitudes toward learning multiplication. The inclusion of relevant literature
strengthens the credibility and validity of the analysis by providing a theoretical framework and
contextual background to support the findings and interpretations. The incorporation of existing
research also helps establish connections between the study's findings and the existing body of
knowledge, contributing to the advancement of the field and opening avenues for further research.
Quality
In this research, several measures were taken to promote credibility, transferability,
dependability, and confirmability, enhancing the overall quality and trustworthiness of the study.
Credibility refers to confidence in the truthfulness and authenticity of the research findings.
To promote credibility, various data collection methods were employed, including surveys, interviews,
observations, and focus group discussions. These multiple methods allowed for triangulation of data,
cross-validation of findings, and a more comprehensive understanding of the research topic. The use
of peer-reviewed literature throughout the research process also contributed to credibility by
grounding the study in established theories, concepts, and empirical evidence.
Transferability relates to the extent to which the findings can be applied or generalized to
other settings or contexts. To enhance transferability, efforts were made to ensure diversity and
representativeness in the sample. This approach aimed to capture a broad spectrum of experiences
and increase the likelihood of findings being applicable to similar educational settings.
Dependability refers to the consistency and reliability of the research process and findings. To
establish dependability, clear and detailed documentation of the research procedures, including the
development of questionnaires, interview protocols, and observation guidelines, was maintained. The
use of standardized data collection instruments and clear instructions in the responses ensured
consistency across the data. Additionally, the researcher's adherence to ethical considerations, such
as maintaining confidentiality and providing informed consent, further enhanced the dependability of
the study.
Confirmability refers to the objectivity and neutrality of the research, minimizing bias or
personal influence on the findings. In this research, efforts were made to maintain confirmability by
using a systematic and structured approach to data collection, analysis, and interpretation.
Transparency in reporting the research process and decisions made throughout the study also
contributed to confirmability.
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In conclusion, through the adoption of multiple research methods, adherence to ethical
considerations, use of several data sources, referencing existing literature, and maintaining
transparency in reporting, this research promotes credibility, transferability, dependability, and
confirmability. These measures enhance the trustworthiness and reliability of the study's findings,
allowing for meaningful interpretation and potential application in educational contexts.
Ethics
Ethical considerations according to British Education Research Association (BERA) guidelines
were carefully upheld throughout the entire research process to safeguard the participants' well-being,
privacy, and rights (BERA, 2018). Key measures were implemented to ensure ethical standards were
maintained.
Participants were provided with clear and comprehensive information about the study's
purpose, procedures, potential risks, and benefits. Informed consent was obtained from all
participants, allowing them to make voluntary decisions on participation and the freedom to withdraw
without consequences.
Strict confidentiality measures were employed to protect the participants' identities and
personal information. Any identifying details were removed or altered in the responses, and
participants' identities were kept confidential. Data were securely stored and accessible only to the
researcher.
Comprehensive data protection measures were implemented to prevent unauthorized access,
use, or disclosure of collected data. Digital data were stored in secure, password-protected systems,
while physical data were securely stored in locked locations. Data were used solely for research
purposes and were kept confidential.
Participants were assured of the voluntary nature of their involvement, with no coercion or
pressure to participate. They were informed of their right to withdraw from the study at any point
without negative consequences. Their decision to participate or withdraw did not impact their
academic or professional standing.
The researcher maintained ethical integrity throughout the study, conducting research with
honesty, transparency, and respect for the participants. Personal biases or conflicts of interest were
avoided to ensure objectivity. Data analysis and reporting were conducted objectively and accurately,
representing participants' perspectives faithfully.
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Results
RQ1: What determines a student’s attitude towards the subject of mathematics?
In order to answer the primary research question of the present study, “How does skipcounting impact students’ attitude towards learning multiplication in a lower primary international
school?”, it is first necessary to determine the factors that contribute to a student’s attitude towards
a subject. Concerning the development of a specific attitude of a student toward a subject, two main
theories may be of relevance, namely the Social Cognitive (Bandura, 1986) and the Expectancy Value
Theory (Eccles & Wigfield, 2000). The Social Cognitive Theory, developed by Albert Bandura (1986), is
a psychological theory that emphasizes the reciprocal interaction between individuals, their
environment, and their cognitive processes. It suggests that individuals learn and develop attitudes,
behaviors, and beliefs by observing others, modelling their behaviour, and evaluating the outcomes
of their actions. In the present research on students' attitudes towards multiplication, the Social
Cognitive Theory can provide insights into how students' attitudes are formed and influenced in the
context of three key concepts: Observational Learning, Self-Efficacy, and Outcome Expectations.
In Observational Learning, students can acquire attitudes towards multiplication by observing
others, such as teachers, peers, or role models. In the context of the experiment, students may have
observed their teachers or classmates demonstrating a positive attitude towards multiplication, which
can influence their own attitudes. If students see their peers enjoying multiplication or receiving praise
for their multiplication skills, they are more likely to develop a positive attitude towards it themselves.
Self-Efficacy refers to an individual's belief in their own ability to successfully perform a
specific task. In the context of multiplication, students who perceive themselves as capable of
mastering multiplication and solving multiplication problems are more likely to develop a positive
attitude toward it. This belief in their own abilities can be influenced by their past experiences with
multiplication, the support they receive from teachers or parents, and their perceived level of
competence in mathematical tasks. Students who have experienced success in multiplication or who
have received positive feedback are more likely to have higher self-efficacy beliefs and, consequently,
more positive attitudes toward multiplication.
Outcome expectations refer to an individual's beliefs about the potential outcomes or
consequences of their actions. In the context of the experiment, students' attitudes towards
multiplication may be influenced by their expectations of the outcomes associated with mastering
multiplication. For example, if students believe that mastering multiplication will lead to academic
success, improved problem-solving skills, or a sense of achievement, they are more likely to develop
a positive attitude toward it. These outcome expectations can be shaped by various factors, including
personal experiences, societal influences, and educational messages.
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The Expectancy-Value Theory, initially proposed by Eccles (1983), suggests that individuals'
attitudes and motivation toward a task are influenced by their expectations of success and the value
they place on the task. In the context of the experiment on students' attitudes toward multiplication,
the Expectancy-Value Theory can provide insights into how students' attitudes are shaped by their
expectations and valuations of multiplication.
According to the Expectancy-Value Theory, students' attitudes toward multiplication can be
influenced by their expectations of success in mastering multiplication skills. Students who believe
they have the ability to succeed in multiplication are more likely to have positive attitudes toward it.
These expectations can be influenced by past experiences, feedback from teachers, peers, and selfperception of their mathematical abilities. If students perceive themselves as capable of succeeding
in multiplication, they are more likely to approach the task with a positive attitude.
Task value refers to the importance or personal significance that students attach to
multiplication. It includes intrinsic value (enjoyment derived from the task itself), utility value
(perceived usefulness of multiplication in real-life contexts), attainment value (desire for academic
success), and cost value (perceived effort or sacrifices required to excel in multiplication). Students'
attitudes towards multiplication are influenced by their perceived task value. If students find
multiplication to be meaningful, useful, or enjoyable, they are more likely to have positive attitudes
toward it.
To further determine which factors may influence a student’s attitude towards mathematics,
a one-on-one interview with the classroom teacher of the participants was conducted following the
skip-counting intervention (see Appendix B1). The teacher states that the majority of students exhibit
positive attitudes towards multiplication, implicated by active participation, asking questions, and
engaging in discussions (see Appendix B1, 3.). This observation suggests an intrinsic curiosity and
enjoyment towards mathematics, in line with the task value described Eccles and Wigfield (2000) and
with research that suggests primary school students generally have a positive disposition towards
mathematics (Wilkie & Sullivan, 2017). This positive inclination can be further nurtured and sustained
through effective teaching practices and instructional strategies (Attard et. al, 2016). The teacher's
observations confirm the existing literature, indicating that students are generally curious and
enthusiastic about learning multiplication.
Furthermore, the teacher argues that attitude is impacted by several factors, most notably
through the instructional strategies used, including puzzles, digital and physical games, songs, handson activities, the use of real-life examples and art projects in order to teach students mathematics
(see Appendix B1, 4.). According to the teacher, students naturally develop more positive attitudes
towards the subject through these approaches and these strategies could also be observed in the skip-
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counting intervention as described in Appendix C. Similarly, studies have shown that using hands-on
activities, visual representations, and real-life examples enhances students' understanding and
positive attitudes towards multiplication (DuVall et. al., 2003). This supports the teacher's approach
in incorporating various instructional strategies to engage students and make multiplication
meaningful.
According to the teacher, ‘’the classroom environment and the teacher's attitude towards
multiplication greatly impact student attitudes’ (see Appendix B1, 4.)’. This observation directly
correlates to Observational Learning described by Bandura (1986) and is consistent with research that
highlights the importance of a positive classroom climate in fostering positive attitudes toward
mathematics. A classroom characterized by a sense of belonging, mutual respect, and opportunities
for active engagement promotes a positive mindset toward learning multiplication (Sen, 2022).
The identification of instructional strategies, classroom environment, and peer interactions as
influential factors aligns with research on promoting positive attitudes towards mathematics.
Research emphasizes the significance of a supportive classroom climate, including teacher attitudes
and praise for effort, in fostering positive attitudes toward mathematics (Fyfe & Rittle-Johnson, 2017).
The teacher's focus on creating a supportive and encouraging atmosphere demonstrates this idea and
speaks toward the quality of teaching. Peer interactions and collaborative learning have also been
found to contribute to positive attitudes and engagement in mathematics (Van de Walle, et. al 2013),
which aligns with the teacher's recognition of the benefits of peer collaboration in promoting positive
attitudes toward multiplication.
The recognition of skip-counting as an effective strategy aligns with research on the
importance of conceptual understanding in mathematics. Skip-counting allows students to make
connections between repeated addition and multiplication, promoting a deeper understanding of the
concept (Carpenter et al., 2015). Research has shown that when students have a strong conceptual
foundation, they are more likely to develop positive attitudes toward mathematics (Wen & Dube,
2022). The teacher's use of skip-counting as a strategy supports this research, as it helps students
establish the underlying patterns and relationships in multiplication.
Overall, the interview findings align with existing literature, highlighting the importance of
creating a positive classroom environment, utilizing effective instructional strategies, and fostering
conceptual understanding to promote positive attitudes toward multiplication. The integration of
these factors, as supported by research, can contribute to a more positive and engaging learning
experience for students. The teacher's approach demonstrates alignment with the research and
reflects an understanding of the factors that influence students' attitudes towards multiplication.
16
By considering the principles of the Social Cognitive and Expectancy-Value Theory in the
present study, as well as insights obtained from the teacher interview, we can summarize that
students’ attitudes towards multiplication are likely to be shaped by their intrinsic enjoyment of the
task, their expectancy to succeed, the perceived usefulness of multiplication and their interpersonal
environment. By examining and understanding these factors, we can determine the attitude of
students toward multiplication. Beyond the scope of this research, this understanding can also help
educators and researchers identify strategies to enhance students' expectations of success and
promote the value and relevance of multiplication, thereby fostering more positive attitudes and
motivation toward learning multiplication.
RQ2: How can skip-counting be implemented into a teacher’s daily routine?
To answer the second sub-question, ‘’How can skip-counting be implemented into a teacher’s daily
routine’’, qualitative data in the form of a second teacher interview focusing on this specific research
question and classroom observations was gathered and analyzed.
According to the teacher’s experience, skip-counting has been realized as an integral part of
their instruction and is deemed a powerful strategy in teaching students’ multiplication (see Appendix
B, 2.). More specifically, the teacher highlights the ability of skip-counting to grasp the repetitive
nature of multiplication counting, enhancing students’ efficiency and accuracy in calculations, and
adding more interactive elements to the classroom (see Appendix B2, 4.). These observations align
with scholars such as Jo Boaler (2016), who highlights the significance of providing students with rich,
engaging mathematical experiences that involve problem-solving, collaboration, and real-world
connections. Similarly, Fuson et. al. (1983) emphasizes the importance of developing a strong number
sense and recognizing patterns in mathematics learning and the acquisition of early number word
meanings as a crucial aspect of mathematical development.
Considering the integration of skip-counting into a teacher’s daily routine and lessons, several
strategies were explored over the course of the intervention and recorded (see Appendix C). Skipcounting was initially taught to the students in a group setting in which the teacher modeled the skipcounting process before students were asked to join in (see Appendix C, Observation 1). Through this
observational instruction strategy, recognition of the number patterns was quickly learned by the
students, and the activity elicited a positive response. This confirms concepts proposed by the social
cognitive theory (Schunk & Usher, 2016), emphasizing the importance of observational learning and
the influence of peers and teachers. For this reason, the act of engaging in skip-counting activities
through group activities is also consistent in the majority of strategies. Other group activities made
use of digital or physical interactive games and puzzles in which students were required to fill out the
17
missing numbers in a skip-counting number sequence (see Appendix C, Observation 2, 5, 9, 10). This
approach was found to engage discussion in students and positively influenced teamwork, social
interaction, and active participation and not only improved the students’ mathematical capabilities
but also trained motor and problem-solving skills and their technological understanding. Furthermore,
many of these games implemented physical objects into the skip-counting exercises as mathematical
applications in real life, such as the number of pencils in a box or counting cubes (Appendix C,
Observation 3 & 6). By presenting skip-counting as a practical and relevant skill, the teacher in the
interview taps into students' expectancy and value, enhancing their motivation to participate actively
in skip-counting activities. Lastly, skip-counting could also be implemented using a memorable song
(Appendix C, Observation 4), which yielded positive reactions from the students. A frequent challenge
for students is the memorization of multiplication factors, however, a teacher may integrate skipcounting using ‘’mnemonic devices, interactive games, and frequent practice opportunities’’ (see
Appendix C, 6.), which can ease memorization and improve student engagement.
In conclusion, the concept behind skip-counting was easily taught to the students and further
adapted into the student’s curriculum and instruction. The emphasis on aligning skip-counting
activities with the curriculum and learning objectives aligns with research on effective curriculum
design and instruction. The teacher's approach to reviewing curriculum guidelines and standards
demonstrates a commitment to ensuring that skip-counting activities serve as a purposeful
component of the broader mathematics curriculum. This approach is in line with the principles of
coherence and progression outlined in resources like Principles to Actions (NCTM, 2014), which stress
the importance of a well-aligned curriculum that builds on students' prior knowledge and provides a
clear pathway for learning. Furthermore, skip-counting can be disguised in games or manipulatives
that relate to a real-life context, making it more attractive for students.
RQ3: Does skip-counting change students’ attitudes towards mathematics?
To answer the final research question, of whether skip-counting changes the attitude of
students towards multiplication, quantitative data in the form of a 20-question questionnaire was
gathered before and after the intervention (see Appendix A). The pre-test before the intervention
serves as a baseline to determine the general attitude toward multiplication. Although answers to all
questions were recorded, answers to the following questions were chosen to be represented in this
research based on their relevance to core principles of the previously mentioned Social Cognitive and
the Expectancy Value Theory; ‘’I enjoy learning and practicing multiplication’’ and ‘’multiplication is
an interesting mathematical operation’’ were used to represent the intrinsic enjoyment a student
feels towards multiplication, ‘’I find multiplication to be useful in everyday life’’ gauges the perceived
18
usefulness of multiplication, ‘’I feel confident when solving mathematical operation‘’ determines the
student’s expectancy to succeed, and lastly ‘’I enjoy working on multiplication activities with my
classmates’’ considers the interpersonal environment of the student.
A
B
C
D
E
Figure 1: Pre-test results showcasing number of respondents for each of the displayed questionnaire questions.
As described in literature, the Likert scale questionnaire quantified the qualitative data of the
students’ subjective experiences and revealed trends in attitude (ElÇi, 2017; León-Mantero et al.,
2020). Results of the pre-test, administered prior to the skip-counting intervention, reveal a neutralto slightly negative attitude of students towards their enjoyment of multiplication and their interest
19
in multiplication as a mathematical operation (Fig. 1A, 1B)). The largest shift towards a negative
attitude could be observed for the perceived usefulness of multiplication (Fig. 1C), with the large
majority of students stating that they strongly disagree with the statement. Similarly, the student’s
expectancy to succeed and confidence in solving multiplication equations (Fig. 1D) and their
enjoyment in solving multiplication equations with their classmates (Fig. 1E) revealed overall negative
attitudes of the students, with the majority of students disagreeing with these statements. Although
the academic performance of the students was not directly assessed in the present study, the overall
negative attitudes towards mathematics reported by the students may suggest a lower academic
performance (Schunk & Usher, 2019).
A
B
C
D
E
20
Figure 2: Post-test results showcasing number of respondents for each of the displayed questionnaire questions.
Students were presented with the same questionnaire as a post-test, following the two-week
skip-counting intervention (Fig. 2). Responses to the questions indicate that the majority of students
had a neutral-to-positive attitude towards multiplication, as evidenced by their agreement that they
enjoy learning multiplication and find it an interesting mathematical operation (Fig. 2A, 2B).
Furthermore, the majority of students agreed that multiplication is valuable in everyday life and
expressed increased confidence in their multiplication abilities (Fig. 2C, 2D). Students also reported
strong enjoyment in working on multiplication activities with their classmates (Fig. 2E). Comparing
post-test with pre-test questionnaire results, we can observe globally increased enjoyment of
multiplication tasks, perceived usefulness of multiplication tasks, expectancy to succeed, and
enjoyment in solving mathematical problems in groups. The largest shift was observed in the
perceived usefulness of multiplication, with the pre-test having the strongest negative relationship
with number of responses to usefulness and following the skip-counting intervention a positive
relationship, with the majority of students agreeing with the statement (Fig. 2C). These results are
indicative of an improved attitude towards multiplication, in line with concepts of the social cognitive
and expectancy value theory (Bandura, 1986; Eccles & Wigfield, 2000) and may suggest improved
academic success (Sen, 2022).
Furthermore, qualitative data regarding the student’s attitudes towards mathematics and
multiplication specifically obtained from the focus group discussions (see Appendix D) was gathered
and analyzed. Students reported a notable shift in their attitudes towards multiplication after being
exposed to skip-counting strategies. Students expressed feelings of being less overwhelmed and more
motivated to engage with multiplication tasks. This finding suggests that skip-counting has the
potential to positively influence students' perceptions of multiplication, transforming it from a
challenging and daunting task to a more manageable and enjoyable one.
Additionally, the participants highlighted the increased interest and enjoyment they
experienced while learning multiplication through skip-counting. The variety of skip-counting activities,
such as games, real-life applications, and interactive exercises, contributed to a more engaging and
interactive learning experience. This finding aligns with the literature that emphasizes the importance
of incorporating hands-on and meaningful activities to enhance student motivation and engagement
in mathematics (Lambdin et al., 2018).
Skip-counting appeared to enhance students' understanding and confidence in multiplication,
with participants reporting that by visualizing skip-counting patterns and making connections
between numbers, they developed a deeper comprehension of multiplication concepts. This finding
21
is consistent with research that suggests that visual representations and pattern recognition support
students' conceptual understanding of multiplication (Fuson & Briars, 1990).
Moreover, the practical relevance of skip-counting was evident in students' responses. By
applying skip-counting strategies to real-life scenarios, such as calculating quantities or prices,
participants recognized the practical applicability of multiplication. This finding aligns with previous
research that highlights the importance of connecting mathematical concepts to real-world contexts
to enhance students' understanding and motivation (Hiebert & Lefevre, 1986) and is consistent with
questionnaire results of the observed shift in students’ attitudes toward the usefulness of
multiplication following the skip-counting intervention.
Lastly, the participants emphasized the development of problem-solving skills through skipcounting. By engaging in mental math exercises and solving skip-counting-related problems, students
reported an improvement in their problem-solving abilities and overall mathematical confidence. This
finding aligns with the cognitive theory of problem-solving, which suggests that practicing skipcounting fosters flexible thinking and mental computation skills (Mayer, 1992).
In conclusion, the analysis suggests that integrating skip-counting strategies in teaching
multiplication has a positive impact on students' attitudes toward learning. Skip-counting enhances
student engagement, understanding, confidence, and problem-solving skills. These findings are
consistent with existing literature that supports the use of concrete and visual representations, handson activities, and real-world applications to promote positive attitudes and conceptual understanding
in mathematics (Deci & Ryan, 1985). By leveraging skip-counting techniques and incorporating them
into daily instruction, teachers can create a learning environment that fosters students' motivation,
engagement, and conceptual understanding of multiplication. However, it is essential to consider
individual differences in learning preferences and provide ongoing support and scaffolding to ensure
the optimal integration of skip-counting strategies in the classroom.
Discussion and Conclusion
Skip-counting has been shown to be an effective strategy for developing multiplication skills
in primary school students (Hurst & Hurell, 2016). However, it is currently poorly understood whether
skip-counting can improve multiplication skills and student engagement by improving their attitude
toward the subject. To fill this gap in research, the present study explored the impact of skip-counting
on students' attitudes toward learning multiplication in a lower primary international school. Through
a comprehensive investigation using multiple data collection methods, including a questionnaire,
interviews, classroom observations, and focus group discussions, insights were gained regarding the
relationship between skip-counting and students' attitudes towards multiplication.
22
In order to explore the effects of incorporating skip counting into daily teaching, this research
was divided into three research questions. The first research question explored factors which could
determine a student’s attitude toward mathematics, providing a theoretical background for assessing
subjective attitude changes and identifying the research-relevant data. The findings of this research
question included that student’s attitudes concerning a step counting intervention may be
represented by considering core concepts of the Social Cognitive (Bandura, 1986) and the Expectancy
Value Theory (Eccles & Wigfield, 2000). These theories state that student’s attitudes towards a subject
are influenced by the student’s intrinsic enjoyment of the subject, their confidence in the subject and
their expectancy to succeed when confronted with (multiplication) tasks, the perceived usefulness of
the subject, and the interpersonal environment in which they are confronted with tasks, including
peer and teacher feedback. Many studies evaluating attitude in academic settings have been
published, in which synonymous concepts to those mentioned in the Social Cognitive and Expectancy
Value Theory can be identified. A large-scale study of 200 Romanian secondary students reported that
student’s attitudes towards learning are impacted by involvement, purpose, and achievement
(Anghelache, 2013), consistent with ideas of intrinsic enjoyment, perceived usefulness, and
expectancy to succeed proposed by the aforementioned theories. In a similar study investigating
student’s attitudes towards mathematics in over 200 English secondary school children aged 11-15
(Syyeda, 2016), it was found that the intrinsic enjoyment and the perceived usefulness of mathematics
can influence behavioural and cognitive aspects of attitude and motivate students to study the subject.
Furthermore, it has been reported that the emotional support a teacher provides proved is a
significant determinant for the student’s attitudes towards the subject, consistent with ideas of the
interpersonal environment affecting student motivation. These studies showcase the common and
overlapping behavioural aspects that are observed in studies focusing on attitude, and in the context
of the present study and first research question, support the use of the previously mentioned core
concepts of the social cognitive and expectancy value theory for determining attitude.
The second research question aimed to investigate how skip-counting could be implemented
into a teacher’s daily routine. Previous studies have already showcased strategies in which skipcounting could be integrated into daily teaching activities. Grünke (2016) performed a skip-counting
intervention in which a researcher repeatedly read index cards showing the skip counting sequence
for 8s and 9s to a student before encouraging the student to join the counting sequence and
facilitating memorization of the sequence in the form of a rap song. In the present study, students
were initially introduced to the skip counting method through a simple verbal repeating exercise.
However, once students understood the concept and rules behind skip counting, the teacher
showcased that skip counting could be incorporated beyond memorization sheets in creative and
23
interactive ways, most often in group settings. In some lessons, skip-counting was incorporated as a
hands-on activity using physical objects by instructing students to determine the number of pencils in
multiple boxes or using counting cubes to build towers by skip-counting. Other lessons facilitated
sequence memorization through songs or digital and interactive games by filling out missing numbers
in a skip-counting sequence. These instruction strategies revealed increased enthusiasm, active
involvement, and a sense of enjoyment among students when exposed to skip-counting. Students
exhibited enjoyment, collaboration, and problem-solving skills, indicating a positive impact on their
attitude toward learning multiplication. Furthermore, the teacher interview emphasized the
importance of integrating skip-counting into the daily routine and the teacher recognized skipcounting as a valuable tool for promoting conceptual understanding, building fluency, and fostering a
positive learning environment in the classroom (see Appendix B2, 4.). The teacher highlighted the
benefits of incorporating skip-counting through various instructional strategies, such as visual aids,
hands-on activities, and real-life applications. This practical teaching strategy aligns with effective
teaching practices outlined in resources like the National Council of Teachers of Mathematics (NCTM,
2014). The NCTM emphasizes the use of manipulatives, games, and real-life contexts to make
mathematics engaging and meaningful, showcasing a need to understand and apply mathematics
beyond the classroom. The teacher's integration of skip-counting exercises during warm-ups,
collaborative group work, and connecting skip-counting to real-life contexts resonates with these
recommendations. The implementation strategies of skip-counting in the present study are supported
by literature, with the use of songs or chants also aligning with research on the use of mnemonics and
multisensory techniques to enhance learning (Van de Walle et al., 2013). Additionally, Tomlinson
(2017) advocates for instructional strategies that accommodate diverse learning styles and
preferences, as it leads to increased student engagement and motivation. By offering hands-on
activities, mental math exercises, technology-based games, and creative art projects, the teacher
caters to different learning modalities and supports students in developing a positive attitude towards
skip-counting and mathematics. Another study by Kaufmann (2019) explored the use of games as a
way to engage students and promote their learning of multiplication. The study found that games
were an effective way to help students develop a conceptual understanding of multiplication, which
in turn positively impacted their attitudes toward learning multiplication. In the context of the second
research question, the skip-counting method could be implemented into the teacher’s instruction in
interactive and diverse ways, including making use of interactive and digital games, as an artistic
activity, implemented into songs or chants, and making use of real-life objects (see Appendix C). This
implementation resonates both with official guidelines and research on effective teaching strategies.
Nonetheless, we suggest future research focus on examining pedagogical approaches and
24
instructional strategies employed by teachers to integrate skip-counting into their daily routines.
Investigating the specific techniques, materials, and scaffolding methods used by teachers could
provide valuable evidence-based insights into practices to implement skip-counting most effectively
into the curriculum.
Lastly, this research aimed to identify overall changes in attitude before and after the skipcounting intervention. Skip counting is regarded as an effective strategy for teaching multiplication in
literature as it allows young students to identify patterns in numbers through repeated addition
(Carpenter et. al., 2015). However, it is currently poorly understood whether skip-counting may
impact a student’s attitude towards the subject of mathematics, despite a positive relationship being
described between students attitudes and academic performance (Sen, 2022). In order to determine
attitude towards mathematics, qualitative data was collected on the core concepts of the social
cognitive and expectancy value theory described previously, in the form of a Likert scale questionnaire
ranging from strongly disagree to strongly agree with questions addressing each of the concepts. This
form of data collection is easy to use, allows for obtaining quantitative values that can be analyzed
from qualitative data, and has also been used for similar studies assessing student’s attitude towards
a subject (Syyeda, 2016; Blazar & Kraft, 2017). Five questions were selected to represent each of the
five central concepts, namely the student’s intrinsic enjoyment, subject confidence, expectancy to
succeed, perceived usefulness and interpersonal environment. Students answered the questionnaire
once prior to the skip-counting intervention as a pre-test, and after the intervention as a post-test.
Comparing the test results revealed that after the intervention, students reported an improved
attitude in all five of the assessed concepts. The largest shift in attitude following the intervention was
observed in the perceived usefulness of multiplication. This observation may be explained through the
several practical and real-life contexts through which the skip-counting method was integrated, such
as interactive games or with the use of objects, showcasing the use of in everyday life. Furthermore,
the skip-counting intervention enhanced their overall enjoyment of learning multiplication, increased
their motivation and confidence in tackling multiplication problems, and improved their enjoyment of
solving multiplication problems with their peers. Furthermore, focus group discussions with the
students indicated that the students directly viewed the skip-counting method as positive and
perceived it as an engaging and effective strategy for learning multiplication. Students expressed a
shift in their perception of multiplication, considering it more approachable and enjoyable due to the
exposure to skip-counting strategies. They shared experiences of increased confidence, improved
mathematical skills, and a greater sense of achievement as a result of incorporating skip-counting into
their learning. Together these results indicate an improved attitude of students toward multiplication
following the skip-counting intervention. Considering the positive relationship that exists in literature
25
between student’s attitudes and academic performance in the subject (Sen, 2022), the present
findings provide evidence that skip-counting may ultimately improve students’ academic performance
by improving attitude towards mathematics and supports the integration of skip-counting into daily
instructional practices. By engaging students in meaningful and interactive ways, skip-counting fosters
a positive learning environment, enhances conceptual understanding, and may promote motivation
and enjoyment in the learning process. It is advisable for educators and curriculum developers to
recognize this value for future teaching practices.
Limitations
While the research conducted was designed to provide insights into the impact of skipcounting on students' attitudes towards learning multiplication, it is important to acknowledge
limitations that may affect the generalizability and interpretation of the findings.
The sample size of 24 students may not fully represent the diversity and characteristics of the
broader student population. The findings have been interpreted with caution, and further studies with
larger and more diverse samples would be beneficial. Furthermore, the research was conducted in the
context of a lower primary international school, and the findings may be influenced by specific factors
such as the school's curriculum, teaching methods, and student demographics. Generalizing the
findings to different educational settings or grade levels is only correct in certain circumstances and if
these are not followed, results may lose their validity.
The skip-counting intervention took place over the course of two weeks, with daily exposure
of the students to skip-counting in the classroom. The length of time and frequency of exposure to
skip-counting may impact the degree of attitude change among students. Future research could
explore the optimal duration and dosage of skip-counting interventions to maximize the impact on
students' attitudes. Similarly, repeating the present study with an appropriate control group is
necessary to determine the degree to which skip-counting improves can improve a students’ attitude,
compared to alternative strategies.
The assessment of students' attitudes towards learning multiplication relied on self-report
measures, such as surveys and focus group discussions. These measures are subject to potential biases,
such as social desirability bias or limited self-awareness. Additionally, the research did not account for
external factors that may influence students' attitudes toward learning, including home environment,
parental support, or other educational interventions. Considering and controlling for these external
factors in future studies would provide a more comprehensive understanding of the impact of skipcounting on students' attitudes.
26
Furthermore, it is important to acknowledge the variations in the obtained data and areas of
potential concern. Neutral-to-negative responses following the skip-counting intervention hint
towards some students finding multiplication challenging or confusing and retaining a negative
attitude towards the subject. This indicates a need for additional support or alternative instructional
strategies to address these learning difficulties.
27
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Appendices
Appendix A: Survey for students
This Appendix contains the questions that students were asked to answer using a Likert scale both
before and after being exposed to the two-week period of skip-counting methods in various classes.
Multiplication is an interesting mathematical operation.
Strongly Agree | Agree | Neutral | Disagree | Strongly Disagree
I feel confident when solving multiplication problems.
Strongly Agree | Agree | Neutral | Disagree | Strongly Disagree
I enjoy learning and practicing multiplication.
Strongly Agree | Agree | Neutral | Disagree | Strongly Disagree
I find multiplication to be useful in everyday life.
Strongly Agree | Agree | Neutral | Disagree | Strongly Disagree
I believe that mastering multiplication will help me in other areas of math.
Strongly Agree | Agree | Neutral | Disagree | Strongly Disagree
I feel motivated to improve my multiplication skills.
Strongly Agree | Agree | Neutral | Disagree | Strongly Disagree
I find multiplication challenging.
Strongly Agree | Agree | Neutral | Disagree | Strongly Disagree
I enjoy using different strategies to solve multiplication problems (e.g., skip-counting, arrays,
number lines).
Strongly Agree | Agree | Neutral | Disagree | Strongly Disagree
Multiplication is boring.
Strongly Agree | Agree | Neutral | Disagree | Strongly Disagree
I feel proud when I solve multiplication problems correctly.
32
Strongly Agree | Agree | Neutral | Disagree | Strongly Disagree
I believe that practicing multiplication regularly is important for improvement.
Strongly Agree | Agree | Neutral | Disagree | Strongly Disagree
I find multiplication to be confusing.
Strongly Agree | Agree | Neutral | Disagree | Strongly Disagree
I think multiplication is a valuable skill to have.
Strongly Agree | Agree | Neutral | Disagree | Strongly Disagree
I enjoy working on multiplication activities with my classmates.
Strongly Agree | Agree | Neutral | Disagree | Strongly Disagree
I feel bored or disinterested when learning multiplication.
Strongly Agree | Agree | Neutral | Disagree | Strongly Disagree
I feel anxious or stressed when confronted with multiplication problems.
Strongly Agree | Agree | Neutral | Disagree | Strongly Disagree
I believe that multiplication is important for future academic and career success.
Strongly Agree | Agree | Neutral | Disagree | Strongly Disagree
I feel that my multiplication skills are improving over time.
Strongly Agree | Agree | Neutral | Disagree | Strongly Disagree
I enjoy exploring patterns and relationships in multiplication.
Strongly Agree | Agree | Neutral | Disagree | Strongly Disagree
I feel motivated to learn new multiplication strategies and techniques.
Strongly Agree | Agree | Neutral | Disagree | Strongly Disagree
33
Appendix B: Teacher Interviews
Appendix B1: Teacher Interview 1
1. Introduction
Interviewer: Hello, thank you for participating in this interview. The purpose of this interview is to
gather insights into students' attitudes towards multiplication. Your participation is highly appreciated,
and all information shared will remain confidential. Let's begin!
2. Background and Classroom Context
Interviewer: Could you please provide a brief overview of your classroom dynamics and the grade
level you teach?
Teacher: I teach a Grade 2 class in an international school with a diverse group of students from
various cultural backgrounds. In our classroom, we have a warm and inclusive learning environment
where students feel safe to take risks and express their ideas. We have a balanced mix of individual,
small-group, and whole-class activities to cater to different learning styles and preferences. The
classroom is equipped with resources such as manipulatives, visual aids, and technology to support
learning.
Interviewer: How do you approach the teaching of multiplication at this grade level?
Teacher: At the Grade 2 level, we introduce multiplication as repeated addition and emphasize the
concept of equal groups. I integrate hands-on activities, visual representations, and real-life examples
to make multiplication meaningful and relatable for students. We also explore skip-counting as a
strategy to help students understand the patterns and relationships between numbers. By
incorporating a variety of instructional approaches, we aim to engage students and foster a positive
attitude towards multiplication.
3. Perceived Student Attitudes
Interviewer: Based on your observations, what are the general attitudes of your students towards
multiplication?
34
Teacher: Overall, I find that most of my students exhibit positive attitudes towards multiplication. They
are curious and enthusiastic about learning new concepts. During multiplication lessons, they actively
participate, ask questions, and engage in discussions. I often see smiles on their faces and hear
expressions of excitement when they discover patterns or successfully solve multiplication problems.
However, there might be a few students who initially find multiplication challenging and exhibit some
hesitation or uncertainty. It is important for me to provide the necessary support and encouragement
to help them develop a positive attitude over time.
4. Factors Influencing Attitudes
Interviewer: From your experience, what factors do you think influence students' attitudes towards
multiplication?
Teacher: Several factors influence students' attitudes towards multiplication. Firstly, the instructional
strategies and approaches used in the classroom play a significant role. When I incorporate hands-on
activities, visual aids, and real-life examples, students tend to develop more positive attitudes. These
approaches make multiplication more accessible and meaningful to students, helping them see its
relevance in their lives.
Secondly, the classroom environment and the teacher's attitude towards multiplication greatly impact
student attitudes. Creating a supportive and encouraging atmosphere where students feel safe to take
risks and make mistakes is crucial. Celebrating students' efforts and highlighting their progress can
boost their confidence and motivation. Lastly, peer interactions and group work can also influence
attitudes towards multiplication. When students collaborate, share strategies, and learn from each
other, it fosters a sense of community and creates a positive learning environment. Students are more
likely to develop positive attitudes when they feel supported by their peers and experience the
benefits of working together.
5. Impact of Skip-Counting
Interviewer: How do you perceive the impact of skip-counting as an instructional strategy for
multiplication on students' attitudes?
Teacher: Skip-counting has proven to be an effective strategy in introducing multiplication concepts
to students. By skip-counting, students recognize the patterns and relationships between numbers,
which lays a strong foundation for multiplication. I have observed that when students grasp skipcounting, their confidence in approaching multiplication problems increases, which positively
35
influences their attitudes towards the subject. They feel more equipped to tackle multiplication
challenges and see it as an approachable and logical process. Skip-counting provides a sense of
structure and familiarity, making multiplication less intimidating and more engaging for students.
The use of skip-counting as a steppingstone towards multiplication allows students to develop a
conceptual understanding of multiplication as they see the connection between repeated addition
and skip-counting sequences. It also enhances their number sense and mental math skills, which
further contributes to positive attitudes towards multiplication. By addressing these questions in more
detail, the interview provides a comprehensive understanding of the background and classroom
context, students' perceived attitudes, factors influencing attitudes, the impact of skip-counting, and
how instructional strategies can shape students' attitudes towards multiplication.
6. Challenges and Support
Interviewer: Can you share any specific challenges you have encountered in promoting positive
attitudes towards multiplication?
Teacher: One challenge I have encountered is that some students may struggle with memorizing
multiplication facts. This can lead to frustration and affect their attitudes towards the subject. To
address this, I implement various strategies such as mnemonic devices, interactive games, and
frequent practice opportunities to make memorization more engaging and enjoyable for students. I
also emphasize the importance of understanding the concepts behind multiplication rather than solely
relying on memorization.
Another challenge is addressing the diverse learning needs within the classroom. Students come from
different backgrounds and have varying levels of prior knowledge or mathematical abilities. To ensure
all students develop positive attitudes, I provide differentiated instruction and personalized support.
This includes small-group instruction, one-on-one interventions, or peer tutoring, depending on
individual students' needs. Creating a supportive and inclusive learning environment where students
feel comfortable asking questions and seeking help is crucial in overcoming these challenges.
7. Assessment and Feedback
Interviewer: How do you assess students' attitudes towards multiplication? Do you provide any
feedback to students regarding their attitudes?
Teacher: Assessing attitudes can be challenging as it involves understanding students' emotions and
perceptions. To assess students' attitudes towards multiplication, I observe their behaviours,
36
engagement levels, and their verbal and nonverbal expressions during multiplication activities. For
example, I pay attention to their level of participation, their enthusiasm, and their willingness to tackle
challenging problems.
In addition to observational assessments, I use formative assessments to gain deeper insights into
students' attitudes. This can include open-ended questions or reflection activities where students can
express their thoughts and feelings about multiplication. Through these assessments, I can identify
students who may have more negative attitudes and provide additional support or intervention if
needed. When providing feedback to students, I acknowledge their effort, progress, and positive
attitudes towards multiplication. I emphasize the importance of perseverance, problem-solving skills,
and a growth mindset. Positive reinforcement and specific feedback regarding their attitudes and
approach to multiplication can further motivate students and reinforce positive attitudes.
8. Long-term Impact
Interviewer: In your experience, how do positive attitudes towards multiplication impact students'
overall mathematical abilities and learning outcomes in the long run?
Teacher: Positive attitudes towards multiplication have a significant impact on students' overall
mathematical abilities and learning outcomes in the long run. When students have a positive attitude
towards multiplication, they are more motivated to engage in mathematical activities and explore
other mathematical concepts. They develop problem-solving skills, critical thinking abilities, and a
sense of mathematical confidence.
Students with positive attitudes towards multiplication are more likely to persist through challenging
problems, seek help when needed, and apply their understanding to real-life situations. This positive
mindset extends beyond multiplication and influences their overall mathematical skills.
37
Appendix B2: Teacher Interview 2
1. Introduction
Thank you for having me. I'm thrilled to have the opportunity to discuss how skip-counting can be
integrated into my daily teaching routine to enhance students' understanding and attitudes towards
multiplication. It's an important topic as skip-counting serves as a foundational skill that supports
students' development of multiplication concepts. By exploring effective strategies and practical
approaches, we can create a rich learning environment that promotes mathematical fluency and
engagement.
2. Background and Experience
I have been teaching multiplication to lower primary students for the past five years. Throughout my
teaching experience, I have witnessed the power of skip-counting in helping students build
connections between addition and multiplication. By utilizing skip-counting as a regular part of my
instruction, I have observed improvements in students' ability to grasp multiplication concepts and
apply them confidently. Through ongoing professional development and collaboration with colleagues,
I continue to deepen my understanding of skip-counting and explore innovative ways to integrate it
into my teaching practice.
3. Understanding of Skip-Counting
Skip-counting involves counting by a specific number, such as counting by twos, fives, or tens. It serves
as a fundamental strategy for developing students' understanding of multiplication by enabling them
to identify patterns and relationships between numbers. By skip-counting, students can observe the
repetitive nature of counting, which lays the foundation for recognizing the concept of equal groups
in multiplication. As students engage in skip-counting activities, they develop a sense of numerical
sequencing and gain a visual and conceptual understanding of multiplication as repeated addition.
4. Benefits of Skip-Counting
Integrating skip-counting into my daily routine offers several benefits to students. Firstly, it helps
students develop a strong conceptual understanding of multiplication. By engaging in skip-counting
activities, students are able to observe and internalize the patterns that emerge. This understanding
allows them to make connections between repeated addition and multiplication, which are key
concepts in developing fluency and problem-solving skills. Students learn to identify the relationships
between numbers and recognize the systematic nature of multiplication. This deep conceptual
38
understanding becomes the foundation for more complex mathematical concepts as students
progress in their mathematical journey.
Secondly, skip-counting enhances students' ability to mentally calculate multiplication facts. By
practicing skip-counting, students develop strategies for quickly determining the products of numbers,
leading to increased efficiency and accuracy in calculations. Skip-counting helps students internalize
multiplication facts, making them readily available for retrieval during problem-solving situations. This
mental math skill not only saves time but also builds students' confidence and mathematical fluency.
Lastly, skip-counting makes the learning process more engaging and enjoyable for students. It adds a
dynamic and interactive element to the classroom, fostering curiosity and excitement for exploring
the numerical patterns inherent in multiplication. Students actively participate in skip-counting
activities, experiencing the joy of discovery as they uncover patterns, predict the next numbers, and
explore various skip-counting sequences. This engagement enhances their motivation to learn and
develop a positive attitude towards multiplication.
5. Integrating Skip-Counting
I incorporate skip-counting into various activities throughout the day to ensure its integration into my
daily teaching routine. During morning routines, we engage in skip-counting exercises as part of our
mental math warm-up. Students count aloud or in their heads by different increments, such as twos,
fives, or tens. This activity not only reinforces skip-counting skills but also helps students develop a
sense of number patterns and sequencing. It sets a positive tone for the day, activating students'
mathematical thinking and preparing them for more complex multiplication tasks.
In addition to morning routines, I integrate skip-counting into group work and collaborative learning
experiences. For example, I design partner or small-group exercises.
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Appendix C: Classroom Observations
Observation 1
Date: May 3, 2023
Time: 9:00 AM - 9:30 AM
Description: During the math lesson, the teacher introduces skip-counting to the students. The
students gather around a large number line displayed at the front of the classroom. The teacher starts
by skip-counting by twos, modelling the process and encouraging students to join in. The students
eagerly participate, taking turns to call out the next number in the sequence. As the skip-counting
continues, students start recognizing the pattern and anticipate the next number. Their faces light up
with excitement as they successfully complete the skip-counting sequence.
Observation 2
Date: May 4, 2023
Time: 10:15 AM - 10:45 AM
Description: In today's math lesson, the teacher incorporates skip-counting into a group activity. The
students are divided into small groups of four, and each group is given a set of manipulatives. The
teacher instructs the students to use the manipulatives to skip-count by fives. The students
collaborate and engage in lively discussions as they arrange the manipulatives in groups of five,
counting and skip-counting together. They apply their skip-counting skills to solve a series of skipcounting puzzles provided by the teacher. The classroom atmosphere is buzzing with active
participation and a sense of accomplishment as each group successfully completes the skip-counting
puzzles.
Observation 3
Date: May 5, 2023
Time: 1:30 PM - 2:00 PM
Description: In today's math class, the teacher integrates skip-counting into a real-life context. The
students are given a set of word problems that require them to use skip-counting to find the solutions.
For example, the teacher presents a scenario where students need to determine the number of pencils
in multiple boxes, with each box containing a specific quantity of pencils. The students enthusiastically
engage in skip-counting to solve the problem, utilizing their skip-counting skills to quickly determine
the total number of pencils. As they share their solutions with the class, they realize the efficiency and
accuracy of skip-counting in solving such problems.
40
Observation 4
Date: May 6, 2023
Time: 11:45 AM - 12:15 PM
Description: During today's warm-up activity, the teacher incorporates a skip-counting song. The
students gather in a circle, and the teacher leads them in singing a catchy skip-counting song for the
threes multiplication table. The students energetically sing along, moving their bodies to the rhythm
of the song. As the song progresses, the students internalize the skip-counting pattern and confidently
join in, singing and skip-counting in unison. The classroom atmosphere is filled with enthusiasm, and
the students display a sense of pride in their ability to recall the skip-counting sequence.
Observation 5
Date: May 7, 2023
Time: 2:00 PM - 2:30 PM
Description: In today's math lesson, the teacher introduces skip-counting through a digital interactive
game. The students gather around the interactive whiteboard, where the teacher displays a skipcounting game. The game prompts the students to fill in missing numbers in a skip-counting sequence.
The students take turns coming up to the board to solve the challenges. They eagerly work together,
discussing the patterns and supporting each other in identifying the missing numbers. The interactive
game enhances their concentration and problem-solving skills, as they actively engage with skipcounting in a dynamic and technology-driven environment.
Observation 6
Date: May 10, 2023
Time: 9:45 AM - 10:15 AM
Description: During today's math class, the teacher incorporates skip-counting into a hands-on activity
using physical objects. The students are given sets of counting cubes and are instructed to build towers
by skip-counting. For instance, they are asked to create a tower by skip-counting by fours. The
students eagerly arrange the cubes in groups of four, counting aloud and skip-counting to determine
the number of cubes needed. They compare their towers and discuss the patterns they observe. The
tactile nature of the activity enhances their understanding of skip-counting and helps them visualize
the multiplication process.
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Observation 7
Date: May 11, 2023
Time: 11:00 AM - 11:30 AM
Description: In today's math lesson, the teacher introduces skip-counting as a mental math strategy.
The students practice skip-counting by tens, starting from different numbers. The teacher presents a
series of mental math problems, such as "What is 30 more than 50?" or "What is 20 less than 80?" The
students quickly engage in skip-counting to find the answers mentally. They demonstrate improved
mental calculation skills and an ability to apply skip-counting to solve real-life math problems
efficiently. The activity boosts their confidence in mental math and strengthens their overall number
sense.
Observation 8
Date: May 12, 2023
Time: 1:15 PM - 1:45 PM
Description: In today's math class, the teacher integrates skip-counting into a creative art project. The
students are provided with various art materials and asked to create skip-counting number lines. They
illustrate skip-counting patterns by creating colourful number line displays that incorporate visuals,
patterns, and skip-counting sequences. The students take pride in their artistic creations and actively
engage in discussions, explaining their skip-counting patterns to their peers. This hands-on and visual
approach to skip-counting enhances their understanding of the concept and allows for artistic
expression and creativity.
Observation 9
Date: May 13, 2023
Time: 9:30 AM - 10:00 AM
Description: During a collaborative group activity, students are given a set of skip-counting puzzles to
solve together. The puzzles consist of missing numbers in skip-counting sequences that the students
need to fill in. As the students work collaboratively, they engage in lively discussions, sharing strategies,
and supporting each other in finding the correct missing numbers. The atmosphere is filled with
enthusiasm, as the students demonstrate teamwork, critical thinking, and problem-solving skills. This
collaborative approach to skip-counting not only enhances their multiplication understanding but also
promotes positive social interactions and a sense of achievement.
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Observation 10
Date: May 14, 2023
Time: 2:00 PM - 2:30 PM
Description: In today's math class, the teacher incorporates technology to reinforce skip-counting
skills. The students are provided with tablets or laptops loaded with an interactive skip-counting game.
The game presents various skip-counting sequences, and the students are tasked with filling in the
missing numbers by selecting the correct options. As the students engage with the game, their level
of focus and concentration is evident. They eagerly interact with the digital interface, demonstrating
fluency in skip-counting and a competitive spirit. The game-based approach not only reinforces skipcounting skills but also appeals to the students' technological interests, making the learning
experience more engaging and enjoyable.
43
Appendix D: Focus Group Discussions
Appendix D1: Focus Group Discussion 1
Teacher : Thank you all for sharing your thoughts on the skip-counting methods. Now, let's shift our
focus towards evaluating any changes in your attitude towards multiplication as a result of using skipcounting strategies. Do you feel that skip-counting has influenced your attitude towards learning
multiplication? Please share your thoughts and experiences.
Participant 1: I definitely feel a positive change in my attitude towards multiplication since we started
using skip-counting. Before, multiplication seemed complicated and overwhelming, but skip-counting
has helped me understand the patterns and relationships between numbers. It has made
multiplication more accessible and less intimidating.
Participant 2: I agree. Skip-counting has made multiplication feel more approachable. By seeing the
patterns and sequences through skip-counting, it has become easier to grasp multiplication concepts.
It has boosted my confidence and made me more interested in exploring multiplication further.
Participant 3: I used to find multiplication boring and repetitive, but skip-counting has added a fun and
engaging element to it. The different skip-counting activities have made me see multiplication as a
puzzle to solve rather than just memorizing tables. It has made learning multiplication more enjoyable
and interesting.
Participant 4: Skip-counting has helped me see the practical side of multiplication. By relating skipcounting to real-life scenarios, I understand how multiplication is used in everyday situations. It has
made me appreciate the relevance of multiplication and its applications, which has positively
impacted my attitude towards learning it.
Participant 5: For me, skip-counting has changed my mindset about multiplication. It has shifted my
focus from rote memorization to understanding the underlying concepts. Skip-counting has allowed
me to see the connections between numbers and how they interact in multiplication. It has made me
more curious and motivated to explore different multiplication strategies.
Moderator: Thank you all for sharing your experiences. It's great to hear that skip-counting has
influenced your attitudes towards learning multiplication. Can you provide any specific examples of
44
how skip-counting has made multiplication more engaging or helped you better understand
multiplication concepts?
Participant 6: One specific example is when we used skip-counting to solve real-life problems, like
calculating the number of objects in a set or determining the total cost of multiple items. Skip-counting
helped me break down the problem into manageable parts, and it made the multiplication process
more meaningful and relatable.
Participant 7: I found skip-counting particularly helpful when learning multiplication with larger
numbers. By skip-counting in larger intervals, it became easier to visualize the multiplication process
and comprehend the magnitude of the product. It made the whole multiplication process less
daunting.
Teacher: Thank you for sharing those specific examples. It's evident that skip-counting has had a
positive impact on your understanding and engagement with multiplication. These changes in attitude
can contribute to a deeper and more meaningful learning experience. Your feedback will be valuable
in refining our approach to integrating skip-counting strategies further.
45
Appendix D2: Focus Group Discussion 2
Teacher : Thank you all for sharing your thoughts on the skip-counting methods. Now, let's shift our
focus towards evaluating any changes in your attitude towards multiplication as a result of using skipcounting strategies. Do you feel that skip-counting has influenced your attitude towards learning
multiplication? Please share your thoughts and experiences.
Participant 1: Skip-counting has made a huge difference in how I perceive multiplication. It used to be
a daunting task for me, but skip-counting has helped me see patterns and relationships between
numbers. Now, I find multiplication more manageable and less overwhelming.
Participant 2: I completely agree. Skip-counting has made multiplication more interesting and
engaging. It has given me a new perspective on how numbers interact and how multiplication works.
I feel more motivated to learn and explore multiplication because of skip-counting.
Participant 3: Before skip-counting, multiplication felt like a series of random calculations. But now,
skip-counting has shown me the structure and logic behind multiplication. It has made me appreciate
the beauty and patterns in mathematics. I genuinely enjoy learning multiplication through skipcounting.
Participant 4: Skip-counting has helped me build a solid foundation in multiplication. By understanding
skip-counting sequences, I can quickly recall multiplication facts and solve problems more efficiently.
It has boosted my confidence in math overall.
Participant 5: I used to find multiplication boring and repetitive, but skip-counting has added
excitement to the process. The various skip-counting activities, such as games and interactive
exercises, have made multiplication more enjoyable. I look forward to skip-counting as a way to learn
and practice multiplication.
Participant 6: Skip-counting has made me realize the real-life applications of multiplication. By using
skip-counting in scenarios like calculating quantities or finding patterns in everyday life, I see the
practicality of multiplication. It has made me more interested in learning multiplication and its
relevance.
46
Participant 7: I feel that skip-counting has given me a deeper understanding of multiplication concepts.
By seeing the relationships between numbers through skip-counting, I can make connections and
grasp multiplication better. It has made me more confident in tackling multiplication problems.
Moderator: Thank you all for sharing your experiences and insights. It's fantastic to hear that skipcounting has positively impacted your attitudes towards learning multiplication. Can you provide
specific examples of how skip-counting has made multiplication more engaging or helped you in
solving multiplication problems?
Participant 1: Sure! When we played skip-counting games, it was challenging yet fun. It made me think
creatively and apply skip-counting to solve problems quickly. It improved my mental calculation skills
and made multiplication more exciting.
Participant 2: During skip-counting activities, we explored skip-counting patterns and sequences. It
helped me understand how numbers relate to each other and how they can be used in multiplication.
It made multiplication more visual and easier to comprehend.
Participant 3: I found skip-counting particularly helpful when multiplying larger numbers. By skipcounting in larger intervals, I could break down the multiplication process into smaller steps, making
it less overwhelming. It also helped me with estimating and approximating the product.
Participant 4: Skip-counting has made multiplication more interactive. Through group activities, we
collaborated and shared strategies, which made learning multiplication a social experience. It created
a supportive environment where we could learn from one another.
Participant 5: I found skip-counting helpful in understanding multiplication as repeated addition. By
skip-counting, I could see how adding groups of numbers together resulted in the multiplication
product. It helped me bridge the connection between addition and multiplication.
Participant 6: Skip-counting activities involving real-life scenarios were eye-opening. We used skipcounting to calculate quantities and prices, which made multiplication more relatable. It showed me
how multiplication is used in practical situations and enhanced my understanding of its applications.
47