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Phet animation of a Plinko machine for
teaching binomial distribution in high
school
Animación Phet de una máquina Plinko para la
enseñanza de distribución binomial en bachillerato
Víctor Manuel Barros
*
Marco Xavier Alcocer Cordero
*
Jorge Washington Encalada Noboa
*
Christian Antonio Pavón Brito
*
Abstract
This study evaluated the use of the PhET animation of the Plinko
machine to teach the binomial distribution in a high school
mathematics class, using a collaborative learning approach. Forty-
one students aged 16-17 participated, divided into groups. The
results show a significant improvement in the understanding of the
binomial distribution, with an average increase of 25% on post-
activity assessments. Most students rated the simulation positively,
highlighting their ability to visualise abstract concepts. Collaborative
learning facilitated active participation, especially among students
with previous difficulties, who improved significantly when working
in mixed groups. The findings suggest that the integration of
interactive simulations and collaborative methods in mathematics
teaching can be an effective strategy to improve academic
*
Magister en Educación con mención en Enseñanza de la
Matemática, Universidad de Guayaquil, Facultad de Filosofía,
Letras y Ciencias de la Educación , victor.barros@ug.edu.ec
https://orcid.org/0000-0001-8542-6454
*
Magíster en Finanzas y Proyectos Corporativos
Universidad de Guayaquil, Facultad de Ciencias Económicas
marco.alcocerc@ug.edu.ec, https://orcid.org/0009-0009-
3507-8186
*
Magíster en Enseñanza de la Física, Universidad de
Guayaquil
Facultad de Filosofía, Letras y Ciencias de la Educación
jorge.encaladan@ug.edu.ec, https://orcid.org/0000-0002-
2884-5596
*
Magíster en Enseñanza de la Física, Universidad de
Guayaquil
Facultad de Filosofía, Letras y Ciencias de la Educación
christian.pavonb@ug.edu.ec, https://orcid.org/0000-0002-
8913-1546
Article
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performance and student motivation. Their implementation in
similar educational contexts is recommended and future research is
suggested to evaluate their long-term impact.
Keywords: PhET simulation, Binomial distribution, Collaborative
learning, Mathematics education
Resumen
Este estudio evaluó el uso de la animación PhET de la máquina
Plinko para enseñar la distribución binomial en una clase de
matemáticas de bachillerato, utilizando un enfoque de aprendizaje
colaborativo. Participaron 41 estudiantes de 16 a 17 años, divididos
en grupos. Los resultados muestran una mejora significativa en la
comprensión de la distribución binomial, con un aumento promedio
del 25% en las evaluaciones posteriores a la actividad. La mayoría
de los estudiantes valoró positivamente la simulación, destacando su
capacidad para visualizar conceptos abstractos. El aprendizaje
colaborativo facilitó la participación activa, especialmente entre los
estudiantes con dificultades previas, quienes mejoraron
notablemente al trabajar en grupos mixtos. Los hallazgos sugieren
que la integración de simulaciones interactivas y métodos
colaborativos en la enseñanza de matemáticas puede ser una
estrategia efectiva para mejorar el rendimiento académico y la
motivación estudiantil. Se recomienda su implementación en
contextos educativos similares y se sugieren investigaciones futuras
para evaluar su impacto a largo plazo.
Palabras Clave: Simulación PhET, Distribución binomial,
Aprendizaje colaborativo, Enseñanza de la matemática.
Introduction
Teaching abstract concepts in mathematics, such as the binomial
distribution, can be a considerable challenge for high school
students. Traditionally, such topics have been approached through
theoretical explanations and calculation exercises on paper, often
resulting in a lack of deep understanding and a disconnect between
the mathematical content and its application in real-world situations
(Kilpatrick, Swafford, & Findell, 2001). In this context, interactive
technological tools, such as simulations, offer an opportunity to
overcome these challenges by providing a visual and manipulable
representation of abstract phenomena (Finkelstein, et al., 2005). One
such tool is the PhET animation of the Plinko machine, which allows
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students to experiment with the principles of probability and
binomial distribution in an interactive and dynamic way.
The use of educational simulations, such as those developed by
PhET (Physics Education Technology), has gained popularity in
recent years, especially in science and mathematics education. These
simulations allow students to explore complex concepts by directly
manipulating variables, observing visually and in real time the
effects of their actions on the results (Perkins, Adams, Pollock,
Finkelstein, & Wieman, 2006). In the case of the Plinko machine
simulation, students can experiment with different parameters, such
as the number of balls thrown and the probabilities of deflection, to
understand how a binomial distribution is generated. This is
particularly valuable for teaching probabilistic concepts, as students
can see how random events, in aggregate, generate predictable
patterns, which can be difficult to visualise through numerical
calculations alone (Adams, Paulson, & Perkins, 2008).
Despite its benefits, the use of educational simulations alone is not
always sufficient to ensure meaningful learning. Educational
research has shown that collaborative learning, where students work
in groups to solve problems and share ideas, can significantly
improve understanding of concepts and foster the development of
cognitive and social skills (Johnson & Johnson, 2009). According to
Vygotsky (1978), learning is a social process, and interactions
between learners play a fundamental role in the construction of
knowledge. In a collaborative environment, students not only have
the opportunity to share their own knowledge, but also learn from
their peers, enabling them to overcome cognitive difficulties through
mutual support and group discussion. This is especially relevant in
mathematics learning, where collaboration can help students develop
a deeper understanding of abstract concepts (Roschelle & Teasley,
1995).
The present study focuses on the combined use of the PhET
simulation of the Plinko machine and collaborative learning for
teaching the binomial distribution in a high school mathematics
class. In particular, it explores how these two methodologies -
interactive visualisation and peer collaboration - can facilitate the
understanding of complex probabilistic concepts and promote
deeper and more meaningful learning among 16-17 year old
students. This approach has the potential to address some of the
traditional problems in probability teaching, where students often
find it difficult to connect mathematical calculations with their
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practical and visual interpretation (Batanero, Henry, & Parzysz,
2005).
Interactive simulations have proven to be a powerful tool in teaching
abstract mathematical concepts, such as probability and
distributions. Perkins et al. (2006) argue that these tools provide a
significant advantage by allowing students to directly manipulate the
variables of a phenomenon, which facilitates deeper understanding
through experimentation and analysis of the results. The PhET
simulation of the Plinko machine, in particular, allows students to
visualise randomness and understand how random events can
generate predictable patterns, a key concept in the binomial
distribution. According to Adams et al. (2008), this interactive
visualisation helps students overcome the cognitive barriers
associated with mathematical abstraction by turning theoretical
concepts into visual and manipulable experiences.
In the context of this study, the PhET simulation of the Plinko
machine was used as a central tool for teaching the binomial
distribution. Students, divided into groups, had the opportunity to
adjust parameters such as the number of rows of nails in the machine,
the number of balls thrown, and the probability of leftward or
rightward deviation in each row. These manipulations allowed the
students to see in real time how the distribution of the balls in the
different containers at the end of the board follows a binomial
distribution pattern, thus illustrating a concept that would otherwise
be abstract and difficult to visualise with formulas alone.
The students' ability to see the outcome of multiple trials allowed
them to understand the law of large numbers, a fundamental idea in
probability theory. With enough repetitions, the distribution of the
balls began to resemble a Gaussian bell, which provided students
with a direct visual representation of how a binomial distribution
relates to the normal curve (De Veaux, Velleman, & Bock, 2013),
and the fact that students could modify the probabilities and observe
the results immediately helped reinforce their understanding of how
probabilistic variables affect outcomes, something that would be
much more difficult to achieve with traditional teaching methods
(Finkelstein, et al., 2005).
Collaborative learning, on the other hand, is based on the premise
that students learn best when they work together, sharing ideas,
solving problems as a team and learning from each other (Johnson
& Johnson, 2009). In the mathematics classroom, this approach is
particularly useful because it allows students to tackle complex
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31
problems with the support of their peers, which reduces anxiety and
fosters a greater willingness to actively participate in the learning
process (Slavin, 1995). In this study, collaborative learning was
implemented by dividing the class into heterogeneous groups, with
students of different ability levels working together to explore the
PhET simulation of the Plinko machine and complete a series of
tasks designed to guide their learning.
Research shows that students who participate in collaborative
activities tend to develop a deeper understanding of mathematical
concepts than those who learn individually (Roschelle & Teasley,
1995). This is partly because collaborative learning encourages
mutual explanation and discussion of ideas, which helps to clarify
misunderstandings and strengthen knowledge (Vygotsky, 1978). In
the context of probability teaching, this approach is particularly
valuable, as probability is a topic that often generates confusion
among students due to its counterintuitive nature (Batanero, Henry,
& Parzysz, 2005). By working in groups, students can discuss their
observations, formulate hypotheses and test them using the
simulation, which facilitates a more solid understanding of the
concepts.
During the activity, groups of students worked together to
manipulate the simulation and record their observations, discussing
the results and adjusting variables to test different hypotheses. This
collaborative approach not only allowed them to learn from their
own mistakes, but also from those of their peers, which promoted a
greater understanding of the topic. As Johnson and Johnson (2009)
argue, collaborative learning creates an environment in which
students feel more confident to take intellectual risks, resulting in a
higher level of engagement and ultimately better academic
performance. The main objective of this study is to evaluate the
effectiveness of combining the PhET simulation of the Plinko
machine and collaborative learning for teaching binomial
distribution to high school students. Through this intervention, it is
expected that students will develop a deeper understanding of the
binomial distribution and its relationship to probability. In addition,
the study seeks to examine how group work can help students
overcome the cognitive difficulties they often face when dealing
with probability topics.
In conclusion, this study relies on a combination of innovative
pedagogical approaches and interactive technologies to enhance the
teaching of abstract concepts in mathematics. The PhET simulation
of the Plinko machine and collaborative learning offer students a
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unique opportunity to explore and understand the binomial
distribution in a visual, manipulative and social way, which,
according to the literature, can result in significant improvements in
learning (Johnson & Johnson, 2009).
Materials and methods
In this study, we explored the use of the PhET animation of the
Plinko machine to teach the binomial distribution in a high school
mathematics course, using a collaborative learning methodology.
The materials used and the methodological design of the study are
detailed below.
PhET simulation of the Plinko machine: The PhET animation of the
Plinko machine was the main didactic resource used. This tool,
developed by the University of Colorado Boulder, is an interactive
simulation that allows students to observe and manipulate a binomial
distribution model. Students can adjust parameters such as the
number of balls thrown, the probability of deviation to the left or
right, and the number of rows of spikes, to see how these factors
influence the distribution of balls in the bins at the end of the board.
Laptops or tablets: Each group of students had access to a laptop or
tablet to interact with the PhET simulation. It was ensured that each
device had access to the internet and could run the simulation
without technical problems.
Projector and screen: For the introduction of the activity and class
discussion, a projector connected to a computer was used to display
the simulation on a large screen. This allowed the teacher to explain
the functionality of the simulation and to show examples in real time.
Workbooks: Printed workbooks were provided to each group of
students. These guides included a series of questions and activities
designed to guide students in exploring the simulation. The questions
were designed to encourage discussion and mathematical reasoning,
covering aspects such as hypothesis formulation, interpretation of
results and comparison of different scenarios.
Data recording sheets: Each group was given record sheets to record
the results of the different simulations they carried out, as well as
their observations and conclusions.
Evaluation questionnaires: At the end of the activity, an individual
questionnaire was administered to assess the students' understanding
of the binomial distribution and their perception of the use of
simulation and collaborative learning.
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The study was carried out in a high school mathematics class of 41
students aged 16-17 (23 males and 18 females). The class was
divided into heterogeneous groups of 4 to 5 students each, ensuring
an equal distribution of gender and levels of mathematical ability.
The procedure applied is described below.
Introduction to the activity: The teacher started the session with a
brief presentation on the binomial distribution and its importance in
the study of probability. The simulation of the Plinko machine was
introduced as a tool to visualise and experiment with the binomial
distribution. Using the projector, the teacher demonstrated how to
operate the simulation and explained the different parameters that
the students could adjust.
Group formation and assignment of tasks: The students were
organised into groups and provided with the work guides and data
recording sheets. Each group received clear instructions on how to
proceed with the activity, emphasising the importance of
collaboration and discussion to solve the assigned tasks.
Interaction with the simulation: During the activity, students used
the PhET simulation of the Plinko machine on their devices to
explore the binomial distribution. As students performed the
simulations, they recorded their observations on data log sheets.
They were encouraged to change the parameters of the simulation to
see how they affected the results and discuss their findings with their
group mates.
Teacher-led discussion: As the groups worked, the teacher circulated
around the classroom, offering guidance and assistance when
needed. Students were encouraged to ask questions and share their
observations with the whole class, allowing the teacher to clarify
concepts and correct misunderstandings in real time.
Synthesis and conclusion: At the end of the activity, a large group
discussion was held where each group shared their findings and
reflections. The teacher summarised the key points of the activity,
highlighting how the Plinko simulation had helped to visualise the
binomial distribution and the importance of collaborative work in
solving complex mathematical problems.
Evaluation: An individual questionnaire was distributed to students
to assess their understanding of the topic and to gather their
impressions of the use of the simulation and the collaborative
learning methodology. The questionnaire responses were analysed
to assess the impact of the activity on student learning.
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Data collected from the data recording sheets and evaluation
questionnaires were analysed to determine the effectiveness of the
PhET simulation and collaborative approach in teaching the
binomial distribution. Qualitative analyses were conducted to
interpret student observations and quantitative analyses were
conducted to measure improvement in understanding of
mathematical concepts.
Prior to the study, informed consent was obtained from the students
and their legal guardians. Confidentiality of the data collected was
assured, and the students' right to participate or withdraw from the
study at any time without academic repercussions was respected.
Results
The study aimed to evaluate the effectiveness of the PhET animation
of the Plinko machine in teaching the binomial distribution through
collaborative learning in a group of 41 high school students. The
results obtained are divided into two main categories: the
mathematical understanding of the binomial distribution and the
students' perception of the methodology used.
To measure the impact of the intervention, a pre- and post-activity
evaluation was applied. The pre-assessment consisted of a set of
problems assessing basic knowledge of probability and the binomial
distribution. The post-assessment included similar but more complex
problems designed to measure depth of understanding.
Prior to the intervention, 68% of the students showed a basic or
limited understanding of the binomial distribution, while only 15%
demonstrated a solid understanding of the concept. The remaining
17% had significant difficulties in relating theory to practical
applications.
Following the activity, there was a marked improvement in students'
understanding. 85% of students achieved a solid understanding of
the binomial distribution, demonstrating an ability to solve more
complex problems than in the previous assessment. Only 8% of
students continued to show difficulties, although these were less
pronounced than before the intervention.
Statistical analysis of the scores showed an average increase of 25%
in scores between the pre- and post-assessment, indicating a
significant improvement in the understanding of the concept of
binomial distribution (p < 0.01).
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On the other hand, qualitative observations were made to assess the
level of participation and collaboration during the activity. These
observations indicated that most students were actively involved in
the group discussions and in manipulating the simulation. Students
who initially showed greater reluctance or difficulties in
understanding the topic particularly benefited from the collaborative
environment, as they were able to receive explanations and support
from their peers.
Group interaction: 92% of the groups demonstrated effective
working dynamics, with all members actively participating. It was
observed that the strongest students in mathematics took on
leadership roles, explaining concepts to their peers and guiding the
group through the simulation.
Problem solving: Groups that managed to explore multiple
configurations in the simulation and discussed their results in depth
showed greater progress in understanding the topic. Students
expressed that the trial and error process, combined with group
discussion, helped them to better internalise the theoretical concepts.
At the end of the activity, a questionnaire was distributed to assess
the students' perception of the effectiveness of the PhET simulation
and collaborative learning.
Opinions on the PhET simulation: 89% of the students rated the
simulation as a useful or very useful tool for understanding the
binomial distribution. Many commented that the dynamic
visualisation of the falling balls and the formation of the distribution
helped them to ‘see’ the probabilistic process rather than simply
calculating it on paper. Some 82% of the students mentioned that
using the simulation made the class more interesting and less
intimidating than traditional mathematics classes. Students also
valued the possibility to experiment with different parameters and
observe the effects of these changes in real time.
Views on collaborative learning: 86% of the students stated that
working in groups improved their understanding of the subject. They
commented that the discussion with their peers allowed them to
clarify doubts and correct misunderstandings in a less formal and
more comfortable environment. 78% of the students indicated that
collaborative learning helped them feel more confident when facing
complex mathematical problems, as they could lean on their peers
when they did not fully understand something.
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To assess how collaborative learning affected students with different
levels of mathematical ability, group improvements were analysed
according to initial level of understanding.
High prior-performing groups: Groups composed mainly of students
with a high level of prior understanding showed improvement in
their understanding of the binomial distribution, solving more
advanced problems and applying the concepts to new situations.
Mixed-performing groups: Groups with a mix of students with
different levels of ability showed marked improvement, particularly
in members who initially struggled. These students reported that
working with more advanced peers helped them understand concepts
that were previously confusing.
Low prior-performing groups: Groups consisting of students with
low initial performance also showed improvement, although to a
lesser extent than mixed or high-performing groups. However, most
of these students indicated that they felt more confident and
motivated when working in groups, and that they gained a better
understanding of the concept after the activity.
The results of the study indicate that combining the PhET simulation
of the Plinko machine with a collaborative learning approach is
effective in improving the understanding of the binomial distribution
in high school students. The majority of students showed a
significant improvement in their ability to understand and apply
concepts related to the binomial distribution, and rated positively
both the technological tool and the pedagogical methodology
employed. These findings suggest that the integration of interactive
simulations and collaborative work in mathematics teaching can be
a valuable strategy to address the challenges associated with
teaching complex mathematical topics.
Discussion
This study explored the effectiveness of the PhET animation of the
Plinko machine as a didactic tool for teaching binomial distribution
in a high school mathematics class, using a collaborative learning
approach. The results obtained provide a series of conclusions that
underline the importance of integrating interactive educational
technologies and collaborative pedagogical methods in the
classroom.
Effectiveness of PhET Simulation in Teaching Binomial
Distribution: The PhET animation of the Plinko machine proved to
be an effective tool for improving the understanding of the binomial
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distribution among high school students. The data show a significant
improvement in post-activity assessments, with an average 25%
increase in scores, suggesting that students were able to internalise
and apply mathematical concepts that were previously abstract or
difficult to understand. This observation is aligned with previous
research highlighting the value of interactive simulations in
mathematics education, allowing students to visualise abstract
phenomena and experiment with different scenarios in a safe and
controlled environment (Finkelstein, et al., 2005).
Advantages of Collaborative Learning: The collaborative learning
approach played a crucial role in the success of the intervention.
Peer-to-peer interaction allowed students to share knowledge,
resolve doubts together and reinforce their understanding through
discussion and mutual teaching. This finding reinforces social
learning theory, which holds that knowledge is most effectively
constructed through social interaction (Vygotsky, 1978). In addition,
collaboration promoted a supportive and trusting environment,
where students felt more comfortable facing complex mathematical
challenges, which increased their motivation and active participation
in the learning process (Johnson & Johnson, 2009).
Differential Impact by Ability Level: The study also revealed that
collaborative learning is particularly beneficial for students with
lower ability levels. These students showed considerable
improvement when working in groups with more advanced peers,
enabling them to overcome barriers to understanding that they would
not have been able to overcome working individually. This finding
is consistent with research highlighting how collaborative learning
environments can close achievement gaps by providing struggling
students with more direct access to knowledge and problem-solving
strategies (Slavin, 1995).
Students' Positive Perception of Tools and Methods Used: Student
feedback was overwhelmingly positive regarding the use of the
PhET simulation and collaborative learning approach. Most students
valued the opportunity to learn through interactive exploration and
highlighted the usefulness of the simulation in making concepts
understandable that would otherwise have been abstract and difficult
to visualise. Furthermore, the collaborative experience was
appreciated not only for its impact on mathematical understanding,
but also for the development of social and teamwork skills, which
are fundamental to their holistic development (Roschelle & Teasley,
1995).
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Implications for Educational Practice: The findings of this study
suggest that the integration of interactive simulations and
collaborative learning can be a powerful pedagogical strategy in the
teaching of mathematics at the secondary level. These tools not only
facilitate understanding of complex concepts, but also increase
student motivation and engagement, which can lead to sustained
improvements in academic performance. Given the success of the
intervention, educators are encouraged to consider implementing
similar technologies and collaborative approaches in their everyday
teaching practices, especially in content areas that traditionally
present significant challenges for students.
Limitations and Recommendations for Future Research: While the
results of this study are promising, it is important to acknowledge
some limitations. First, the study focused on a specific sample of
students in a single educational context, which may limit the
generalisability of the results. In addition, evaluation of the long-
term impact of the intervention was not possible within the time
frame of the study. For future research, longitudinal studies
assessing knowledge retention and the long-term impact of
interactive simulations and collaborative learning in different
educational contexts would be beneficial.
In conclusion, the combination of the PhET animation of the Plinko
machine with a collaborative learning approach represents an
effective and motivating methodology for teaching binomial
distribution at secondary level. This approach not only improves
students' mathematical understanding, but also promotes social and
collaborative skills essential for their academic and personal
development. The implications of these findings are significant and
suggest a promising path towards a more interactive, inclusive and
effective mathematics education.
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