=Paper=
{{Paper
|id=Vol-3781/paper02
|storemode=property
|title=Using STACK to support adaptive mathematics learning in LMS Moodle
|pdfUrl=https://ceur-ws.org/Vol-3781/paper02.pdf
|volume=Vol-3781
|authors=Mariia M. Astafieva,Oksana M. Hlushak,Oksana S. Lytvyn
|dblpUrl=https://dblp.org/rec/conf/3lperson/AstafievaHL24
}}
==Using STACK to support adaptive mathematics learning in LMS Moodle==
https://ceur-ws.org/Vol-3781/paper02.pdf
Mariia M. Astafieva et al. CEUR Workshop Proceedings 30–41
Using STACK to support adaptive mathematics learning in
LMS Moodle
Mariia M. Astafieva, Oksana M. Hlushak and Oksana S. Lytvyn
Borys Grinchenko Kyiv Metropolitan University, 18/2 Bulvarno-Kudriavska Str., Kyiv, 04053, Ukraine
Abstract
The article describes some practices of using Systems for Teaching and Assessment using Computer Algebra
Kernel (STACK) in LMS Moodle to organize and support adaptive mathematics learning at the university. It
highlights the potential opportunities of this technology in the educational process to ensure effective teaching
and learning in mixed or distance learning environments. Shifting the educational focus from providing static
content, the same for all students, to a didactically motivated adaptive design of content and activities is an urgent
task and goal of modern education. STACK is one of the leading technologies that allows the creation of dynamic,
variable tests with automatic evaluation of student answers and individual feedback. However, there is no
widespread practice of its use in Ukraine. Therefore, the author’s examples focus on the advantages of interactive
math tests using the potential response tree, particularly during formative assessment, and step-by-step tasks
with separate prompts for self-study. Examples of STACK integration with LMS Moodle functions, such as Quiz
and Grouping users, demonstrate the technology’s capa-bilities to create personalized learning trajectories during
the study of mathe-matical disciplines. Below are some feedback examples from students about the positive
aspects of using STACK.
Keywords
adaptive mathematics learning, STACK, LMS Moodle
1. Introduction
Based on [1, 2, 3, 4, 5], adaptive learning means educational technology that ensures the personalization
of the educational process, enables the construction of a flexible individual learning trajectory for each
student that meets his/her needs, abilities, and pace of learning the material. Adaptive learning tools
react in real time to the actions (answers) of the student, providing him/her with individual support.
Adaptive learning is based on the following key principles:
• personalization, which means providing each student with pedagogical support on the way to
knowledge, taking into account his/her educational needs and individual characteristics (level of
previous educational achievements, cognitive characteristics, temperament, etc.);
• real-time feedback and dynamic correction, which allow the student to quickly receive information
about his/her achievements and mistakes, on the basis of which the educational trajectory is
corrected;
• analysis of data on student progress, which allows the teacher and the administration of the
educational institution to identify weak points, draw conclusions about the effectiveness of the
used learning strategy, make corrections in the educational process, predict future results.
It is obvious that the implementation of these principles in the educational process of any educational
institution without modern digital technologies, “by hand”, so to speak, is an unattainable task. That
3L-Person 2024: IX International Workshop on Professional Retraining and Life-Long Learning using ICT: Person-oriented Approach,
co-located with the 19th International Conference on ICT in Education, Research, and Industrial Applications (ICTERI 2024)
September 23, 2024, Lviv, Ukraine
" m.astafieva@kubg.edu.ua (M. M. Astafieva); o.hlushak@kubg.edu.ua (O. M. Hlushak); o.lytvyn@kubg.edu.ua
(O. S. Lytvyn)
~ https://fitm.kubg.edu.ua/struktura1/kafedry/kafedra-matematyky-i-fizyky/sklad/832-1.html (M. M. Astafieva);
https://fitm.kubg.edu.ua/struktura1/kafedry/kafedra-kompiuternykh-nauk/sklad/820-1.html (O. M. Hlushak);
https://fitm.kubg.edu.ua/pro-fakultet/komanda/812-1.html (O. S. Lytvyn)
0000-0002-2198-4614 (M. M. Astafieva); 0000-0001-9849-1140 (O. M. Hlushak); 0000-0002-5118-1003 (O. S. Lytvyn)
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
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Mariia M. Astafieva et al. CEUR Workshop Proceedings 30–41
is why adaptive learning is digital and is most effective in blended or remote learning. According to
G. Borich, it is the use of digital tools that allows, “apply different instructional strategies to different
groups of learners so the natural diversity prevailing in the classroom does not prevent any learner
from achieving success” [1, p. 37].
In recent years, new and new adaptive educational web systems have been created, research into
the possibilities and problems of their integration into the educational process has been intensified
[6, 7, 8, 9, 10, 11, 12, 13]. One of the most common practices is the implementation of adaptive technology
in LMS Moodle [14, 15, 16], in particular, in the teaching of mathematics [17, 18]. Among the main LMS
Moodle tools used to support adaptive learning is the Quiz module. It provides a palette of flexible test
settings, including an adaptive mode, when the student can see the correct answers and comments before
completing the attempt. This mode makes it possible not so much to control the level of knowledge
as to help in the assimilation of knowledge or the development of skills. However, the preparation of
many options for test tasks, all comments, on incorrect answers, falls on the teacher, which, in the case
of mathematical disciplines, is too time-consuming a task. This problem is well solved by interactive
tests based on STACK. STACK can generate feedback automatically and adaptively, based on so-called
“potential response tree”. In addition, it completely solves the problem of the number of test items of
the same type (which is important for teaching mathematics), since the questions can be formulated
with randomized variable parameters.
But, unfortunately, the use of STACK has not yet become part of the widespread practice of Ukrainian
mathematics teachers. In March 2024, we interviewed 27 teachers of mathematics disciplines from 7
universities in different regions of Ukraine. To the question “Do you have experience using the STACK
system for educational purposes?” 23 of respondents answered that they do not know about such a
system, the rest ( 13 ) – have heard about it, but do not use it.
The purpose of our article is to highlight some of the possibilities and advantages of the STACK
system for supporting adaptive learning of mathematics in LMS Moodle, which, in particular, can
increase the interest of university lecturers in its use.
2. Methodology
This study employed a mixed-methods approach to investigate the effectiveness of using STACK (System
for Teaching and Assessment using Computer Algebra Kernel) in supporting adaptive mathematics
learning within the LMS Moodle environment. The research was conducted at Borys Grinchenko Kyiv
Metropolitan University as part of the scientific topic “Mathematical methods and digital technologies
in education, science, technology” (state registration number: 0121U111924).
2.1. Research design
The study utilized a combination of qualitative and quantitative methods to provide a comprehensive
understanding of the implementation and impact of STACK in adaptive mathematics learning.
1. An extensive review of existing literature on adaptive learning, mathematics education, and the
use of digital tools in education was conducted to establish the theoretical framework for the
study.
2. In March 2024, we conducted a survey of 27 mathematics teachers from 7 universities across
different regions of Ukraine to assess their familiarity and experience with the STACK system.
3. The researchers developed a series of adaptive mathematics tasks using STACK, integrated within
the LMS Moodle environment. These tasks covered various topics in mathematics and were
designed to support different levels of learning.
4. Students majoring in mathematics at the bachelor’s level (1st and 2nd year) were engaged in
completing the STACK-based tasks as part of their e-learning courses. Their interactions with
the system were observed, and feedback was collected through open-ended questionnaires.
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Mariia M. Astafieva et al. CEUR Workshop Proceedings 30–41
2.2. Data collection
Data was collected through multiple sources:
1. Quantitative data from the survey of mathematics teachers regarding their knowledge and use of
STACK.
2. Data on student interactions with the STACK-based tasks, including attempts, scores, and time
spent on tasks.
3. Qualitative data from student responses to open-ended questions about their experiences with
the STACK-based tasks.
4. Qualitative data from teachers’ observations of student engagement and performance with the
STACK-based tasks.
2.3. Data analysis
The collected data was analyzed using both quantitative and qualitative methods:
1. Descriptive statistics was used to analyze the survey responses from mathematics teachers and
quantitative data from STACK system logs.
2. Thematic analysis has been applied to the qualitative data from student feedback and teacher
observations to identify key themes and patterns in the experiences and perceptions of using
STACK for adaptive mathematics learning.
3. Detailed examination of specific examples of STACK implementation to provide in-depth insights
into the adaptive learning process.
3. Research results
3.1. Features of adaptive mathematics learning and STACK
The specificity of mathematics as a science and as an educational discipline (hierarchical structure of
knowledge, close relationship and interdependence of different sections and even different mathemat-
ical disciplines, complex logical structure of many mathematical concepts, high level of abstraction,
interdisciplinary connections, universality of methods) not only dictates the need for adaptive learning
of mathematics, but also causes certain features and requirements for such learning. This in particular:
• the need to identify gaps in knowledge and understanding and take timely measures to fill them
in order to avoid obstacles to further learning, which should be focused on understanding;
• the need to use visualizations, interactive simulations and other multimedia resources not only
to explain complex abstract concepts, but also to organize students’ independent research and
experimentation, which lead the student to his/her own discoveries of new knowledge for him/her;
• the need for constant feedback, which allows the student to quickly understand his/her mis-
takes and correct them; providing, if necessary, step-by-step instructions and explanations for
completing the task;
• the exceptional importance of formative assessment, i.e. constant monitoring and assessment in
the learning process and for learning;
• the need to solve many practical tasks with a gradual increase in the level of their complexity to
develop stable procedural skills.
To meet the specified didactic requirements, we use Systems for Teaching and Assessment using
Computer Algebra Kernel (STACK) in the process of teaching and learning mathematics [19]. STACK
is an open-source computer-based automatic assessment system, inter alia compatible with the LMS
Moodle. Mostly used in the study of mathematics with an emphasis on formative assessment. The
system can automatically classify correct, half-correct and incorrect responses using a potential response
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Mariia M. Astafieva et al. CEUR Workshop Proceedings 30–41
tree (PRT) and provide appropriate feedback for each response. PRT is an acyclic directed graph, formed
based on a series of true/false tests of student answers.
It should be noted that creating high-quality, useful interactive questions that would perform an
adaptive function, help students manage their learning and develop metacognitive skills is not an easy
task for a teacher. Creating questions requires skills in various areas, in particular:
• mathematics to design tasks, appropriately randomize and differentiate them;
• teaching mathematics to predict common errors and formulate adequate comments;
• programming using the Maxima computer algebra system to create problems, program a tree of
potential answers, and etc.;
• LATEX for creating mathematical texts;
• LMS Moodle for storing questions, creating a test and organizing the learning process in general.
3.2. Examples of using STACK to implement an adaptive approach to teaching
mathematics in LMS Moodle
Here are some examples of the use of interactive test tasks in STACK to support adaptive mathematics
learning.
Example 1. The complex task on the topic “Derivative” (figure 1) requires the student to understand
the concept of derivative and its geometric meaning, to have certain procedural skills.
One of the advantages of STACK is the ability to automatically create a huge number of tasks of
the same type. In our example, these are tasks related to the derivative of the quadratic function
𝑓 (𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 with different randomly selected numerical values of parameters 𝑎, 𝑏, 𝑐. In the
settings, the teacher sets Question variables:
a:rand_with_prohib(-3,3,[0]);
b:-3+rand(6);
c:-3+rand(6);
exp:a*x^2+b*x+c;
pt:rand(5);
ta1:diff(exp,x);
ta2:subst(x=pt,ta1);
ta3:remainder(exp,(x-pt)^2);
Separately, the teacher writes the text of the question in the Question text field and embeds the
Geogebra Applet in the test task using the appropriate code (item 4 of the code):
1. Differenctiate {@exp@} with respect to \(x\).[[input:ans1]] [[validation:
ans1]] [[feedback:prt1]]
2. Evaluate your derivative at \(x={@pt@}\).[[input:ans2]] [[validation:ans2]]
[[feedback:prt2]]
3. Hence, find the equation of the tangent line. \(y=\) [[input:ans3]]
[[validation:ans3]] [[feedback:prt3]]
4. Check your answer in point 3 using geogebra [[geogebra set="a,b,c"
watch="ans1"]] params["material_id"]="wzmqgshr"; var params = {"appName":
"graphing", "width": 800,"height": 600, "showToolBar": true, "showAlgebraInput":
true, "showMenuBar": true };var applet = new GGBApplet(params, true);
window.addEventListener("load", function() { applet.inject(’ggb-element’);});
[[/geogebra]]
Another useful function of the system is automated checking of the student’s answer and error
diagnosis using PRT, pre-configured by the teacher (figure 2). In this way, the teacher does not need to
“guess” after each answer of the student why he/she gave such an answer. STACK does it for him/her.
For example, if a student incorrectly found the form of the derivative at a variable point x but based on
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Mariia M. Astafieva et al. CEUR Workshop Proceedings 30–41
Figure 1: Screenshot of the task with the student’s answers.
“its” form of the derivative correctly calculated its value at a specific point, then STACK automatically
recognizes this error and generates a corresponding comment (see comment to question 2 in figure 1).
The GeoGebra window prompts the student to check whether he/she has correctly written the tangent
equation (item 3 figure 1) and draw a conclusion (item 4 figure 1).
Example 2. The task on the topic “Irrational equations” is an example of a task differentiated by levels
of complexity: I – reproductive level (the ability to apply the theory in standard situations to solve typical
problems); II – the level of establishing connections (the ability to perform additional transformations,
solve problems that require knowledge of various sections of mathematics and problems with variable
conditions); III – search and research level (ability to conduct the necessary research, reveal a creative
approach).
Using the Grouping users in Moodle function, based on previous knowledge, the teacher divides
students into groups by level. Different groups are offered tasks with a corresponding level of cognitive
requirements. The groups are dynamic and the student, according to his/her progress (or vice versa),
migrates from one group to another, trying to achieve the best possible results.
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Mariia M. Astafieva et al. CEUR Workshop Proceedings 30–41
Figure 2: PRT setup page.
√
In our example, the I level group is asked to solve an equation of the type: 𝑎𝑥 + 𝑏 = 𝑐𝑥 + 𝑑 (by
randomizing the values of parameters 𝑎, 𝑏, 𝑐 and 𝑑, √ STACK generates
√ a set of specific equations); for
the II level group – to solve the type equation 𝑚 𝑎𝑥 + 𝑏 + 𝑛 𝑐𝑥 + 𝑑 = 𝑝; for group III level – an
equation with a parameter.
Figure 3 presents a screenshot of the student’s answer to the II level task with a comment on his/her
incorrect answer. Please note that the comment does not directly indicate what the error is (of course,
the author of the comment understands the reason for the error, he/she programmed this error), but
encourages the student to figure it out him/herself. That is, we see the so-called delayed (timed) feedback,
which is very useful in teaching mathematics.
Figure 4 shows a screenshot of the answer to the III level task. Here the student is asked to complete
two tasks. The second of them involves an own experimental study, which the student performed with
the help of GeoGebra. The student writes down his/her thoughts in the window provided for this, and
the teacher checks them, which once again emphasizes digital technologies are only a good assistant
for the teacher, but they cannot (and should not!) completely replace him/her.
Example 3. The task of finding the convergence set of a series (figure 5) is intended for self-
learning, as it offers a step-by-step implementation of it: find the radius of convergence, the interval of
convergence, check the convergence of the series at the ends of the interval and, finally, write down the
convergence set of the series.
The task page also contains help in the form of questions, the answers to which you need to know to
solve the task, and a hyperlink to a separate Help page, which contains the answers to these questions
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Mariia M. Astafieva et al. CEUR Workshop Proceedings 30–41
Figure 3: A page with the student’s answer to the II level task.
in the form of hidden text (figure 6). The student can, as needed, use this help, not use it at all, or check
him/herself after answering the questions him/herself.
Since the proposed task is step-by-step, the student’s answer is checked at each step and the student
receives a corresponding comment for each wrong answer. Because the task is intended for self-study,
comments on incorrect answers encourage the student to find the correct answer on his/her own
(figure 7).
3.3. Student feedback
We asked students majoring in mathematics, who practice performing test tasks in the STACK system
in an e-learning course on the Moodle platform, to share their impressions of testing. Here are some
fragments of these reviews.
Student 1 (bachelor’s level, 1st year of study): “Due to certain circumstances, I missed many classes
on mathematical analysis, in fact the entire topic “Limits”. But with the help of tests in the STACK
system, I learned to calculate the limits of functions. Automatic prompts and instant feedback after
each answer were especially valuable. It helped me quickly correct mistakes and understand where I
made a wrong step. And the good thing is that I could train as much as needed to achieve the desired
result, because STACK generated new and new exercises for each type of limits.”
Student 2 (bachelor’s level, 1st year of study): “I enrolled in the first year of the mathematics major
after almost a year of studying at another university, majoring in physics, which I left because I realized
that this major was “not for me”. That is why I am studying the course of mathematical analysis for the
second time. I used to perform test tasks in Moodle, but here I met STACK-based tests for the first time.
And I want to demonstrate their advantages on the example of test tasks on finding indefinite integrals
(antiderivative). I will point out only two of them. Since the primitive integral is not a number, but a
set of functions, most often in the test tasks that I performed before, it was suggested to choose the
correct answer from several given ones or to establish correspondences between integrals and their
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Mariia M. Astafieva et al. CEUR Workshop Proceedings 30–41
Figure 4: The page with the student’s answer to the III level task.
corresponding functions. Such a task did not put me in front of the need to find the integral, because
you can simply check by differentiation whether the indicated function is primitive. A similar test task
in STACK is open-ended, it requires writing down the answer, and not choosing (guessing) it. That
is, it is necessary to carry out the indefinite integration process, which is already much more difficult.
In addition, as is known, the form of an antiderivative is not unambiguous, it all depends on which
method of integration is used. We have repeatedly encountered the fact that the integration result we
received differs from the printed (paper) collection of problems specified in the answer. STACK, on the
other hand, “accepts” the answer in any form and checks whether it is correct or not.”
Student 3 (bachelor’s level, 2nd year of study): “What I liked about the math tests on the STACK
platform was the variety of tasks. Here you can find both simple tasks to consolidate basic skills, and
more complex ones that really make you think. The presence of hints, as well as the ability to familiarize
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Mariia M. Astafieva et al. CEUR Workshop Proceedings 30–41
Figure 5: Tasks and questions that should help in solving. The Help hyperlink is a link to a separate page
(figure 6) with answers to these questions.
Figure 6: Hints page with hidden text created in Moodle in HTML code editing mode.
yourself with detailed solutions, are especially useful, because they allow you to better understand the
material and learn from your own mistakes.”
Student 4 (bachelor’s level, 2nd year of study): “Working with mathematical tests on the STACK
platform was a real discovery for me. Tasks with different levels of difficulty helped me gradually
improve my skills. Explanations of errors, their causes, and advice on what I should do to eliminate
gaps in knowledge were very helpful.”
These reviews confirm the effectiveness of tests in STACK for improving the results of learning
mathematics, developing a dynamic educational environment, supporting students’ efforts on the way
to knowledge.
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Mariia M. Astafieva et al. CEUR Workshop Proceedings 30–41
Figure 7: A fragment of the student’s answer, which contains a comment on the wrong answer.
4. Conclusion
1. Based on the understanding of the specificity of mathematics as a science and educational
discipline, special requirements for adaptive teaching of mathematics and adaptive tools are
formulated to support self-regulated acquisition of knowledge by students and their achievement
of optimal educational results.
2. Specific examples illustrate the expediency of using STACK in adaptive teaching of mathematics
for creating and conducting interactive tests in the LMS Moodle environment. Using computer
algebra, STACK automatically checks student answers, providing detailed feedback. This allows
you to create interactive tests that not only check the correctness of the answer, but also consider
different ways of solving the problem, which is important for an adequate assessment of knowledge
in mathematics.
3. Student reviews show that the use of tests in STACK improves understanding of the material,
helps to gradually increase the level of knowledge and the formation of skills, allows you to learn
at your own pace. They also note the convenience and usefulness of instant and meaningful
feedback.
4. As part of this study, a full-fledged experiment is planned, which will include statistical or other
mathematical processing of the results to evaluate the effectiveness of using STACK tests to
support adaptive mathematics learning.
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