Analyzing Interactions in Automatic Formative Assessment Activities
for Mathematics in Digital Learning Environments
Alice Barana
and Marina Marchisio
Department of Molecular Biotechnology and Health Sciences, University of Turin, Via Nizza 52, 10126 Turin, Italy
Keywords: Automatic Formative Assessment, Digital Learning Environment, Interactions, Mathematics Education,
Online Learning.
Abstract: This paper discusses the theme of the analysis of the interactions in a Digital Learning Environment (DLE)
to study formative assessment processes. We propose a definition for a Digital Learning Environment based
on the concept of a learning ecosystem, and we provide a model to analyze the interactions occurring among
the components of a DLE during automatic formative assessment activities for Mathematics. Using the model,
we qualitatively analyze two different activities of symbolic computations, carried out by 396 students of
grade 8 in different contexts, to identify the interactions through which formative assessment strategies are
developed. In the conclusions, we suggest ways to adopt this model for learning analytics, to analyze the
interactions in large online courses.
According to Wilson (1995), a learning environment
is a place where learning is fostered and supported”.
It includes at least two elements: the learner, and a
setting or space wherein the learner acts, using tools
and devices, collecting and interpreting information,
interacting perhaps with others, etc.” (Wilson, 1995).
The traditional learning environment that everyone
knows is the classroom, where the teacher teaches,
students learn, individually or with their peers, using
tools such as paper, pen, and a blackboard. The
diffusion of technology transformed this traditional
learning environment by adding digital tools, as
tablets or computers, and the IWB (Interactive White
Board). Besides equipping physical places with
technologies, the technological revolution brought to
the creation of a new learning environment, situated
in a non-physical dimension: that of the Internet,
accessible from everywhere via computers, tablets, or
even smartphones. This is the essence of the “Digital
Learning Environment” (DLE); besides the learner
and a setting, which can be virtual, a device is needed
to access the activities.
Today, due to the COVID-19 pandemic, online
platforms have known increasing popularity,
supporting smart-schooling, and class attendance
from home (Giovannella et al., 2020). They have been
invaluable to permit students from all social and
cultural backgrounds to carry on their education. The
interest in DLEs in the research has increased
accordingly, making different theories and models
come to life (Fissore et al., 2020).
This paper intends to contribute to the discussion
about the essence of DLEs providing a definition and
a model for analyzing learning interactions in a DLE.
The theoretical framework includes a review of
various studies on DLEs and a proposal of definition.
Particular characteristics of DLEs for Mathematics
are considered, based on theories on formative
assessment and Automatic Formative Assessment
(AFA). Then, a model for the interactions among the
members of a DLE is proposed, to highlight the
interactions during AFA activities. In the following
sections, an AFA activity for grade 8 Mathematics in
a classroom context is presented. Some episodes
involving students working on this activity are
analyzed using our models, to show what kinds of
interactions can support formative assessment
strategies. The conclusions suggest how these
findings could be used in learning analytics research.
Barana, A. and Marchisio, M.
Analyzing Interactions in Automatic Formative Assessment Activities for Mathematics in Digital Learning Environments.
DOI: 10.5220/0010474004970504
In Proceedings of the 13th International Conference on Computer Supported Education (CSEDU 2021) - Volume 1, pages 497-504
ISBN: 978-989-758-502-9
2021 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
2.1 Definition of Digital Learning
The concept of “Digital Learning Environment” has
a long history, and it has known several developments
and many different names over the years: Virtual
Learning Environments (Wilson, 1996), Online
Learning Environments (Khan, 1997), Computerized
Learning Environments (Abdelraheem, 2003), and
Digital Learning Environments (Suhonen, 2005). The
common factor among all these definitions is the use
of the Internet and its tools to provide an environment
where learning is supported, generally represented by
a Learning Management System (LMS). An LMS,
according to Watson and Watson (2007), is the
infrastructure that delivers and manages
instructional content, identifies and assesses
individual and organizational learning or training
goals, tracks the progress towards meeting those
goals, and collects and presents data for supervising
the learning process of an organization as a whole.”
While similar environments are mainly used to
support online educational processes, we are
convinced and have proof of the fact that web-based
platforms can also be successfully adopted in
classroom-based settings: in our conception, DLEs
should not only be confined to distance education
(Barana, Marchisio, & Miori, 2019; Barana &
Marchisio, 2020; Borba et al., 2018).
More recently, many authors have developed an
interest in conceptualizing digital learning
environments as ecosystems, borrowing the term
from ecology (García-Holgado & García-Peñalvo,
2018; Giovannella et al., 2020; Guetl & Chang, 2008;
Uden et al., 2007; Väljataga et al., 2020). According
to Encyclopaedia Britannica (,
an ecosystem is “a complex of living organisms, their
physical environment, and all their interrelationships
in a particular unit of space.” The natural ecosystem,
constituted by a biological community in a physical
environment, is the fundamental example; however,
this definition can be applied to any domain, even
artificial environments, by specifying the living
community, environment, and space unit.
There are several models of learning or e-learning
ecosystems in the literature, which vary for the
components included based on the theoretical
assumptions considered. In general, they contemplate
individuals, computer-based agents, communities,
and organizations in a network of relations and
exchanges of data that supports the co-evolutions and
adaptations of the components themselves (Guetl &
Chang, 2008). Following this trend, in this thesis, we
chose to use the term “Digital Learning Environment”
to indicate a learning ecosystem in which teaching,
learning, and the development of competence are
fostered in classroom-based, online or blended
settings. It is composed of a human component, a
technological component, and the interrelations
between the two. The human component consists of
one or more learning communities whose members
can be: teachers or tutors, students or learners, and
their peers, the administrators of the online
environment. The technological component includes:
A Learning Management System, together
with software, other tools, and integrations
which accomplish specific purposes of
learning (such as web-conference tools,
assessment tools, sector-specific software, and
many others);
Activities and resources, static or interactive,
which can be used in synchronous or
asynchronous modality;
Technological devices through which the
learning community has access to the online
environment (such as smartphones,
computers, tablets, IWB);
Systems and tools for collecting and recording
data and tracking the community's activities
related to learning (such as sensors, eye-
trackers, video cameras.
The interrelations between the two components
can include the interactions and learning processes
activated within the community and through the use
of the technologies as well as pedagogies and
methodologies through which the learning
environment is designed.
Independently of the fact that the DLSs are based
on a web-based platform, teaching and learning can
occur in one of the following modalities:
Face to face, in the classroom or a computer
lab, with students working autonomously or in
groups through digital devices, or solving
tasks displayed on the Interactive White Board
with paper and pen or other tools;
Entirely online, using the DLE as the only
learning environment in online courses or
In a blended approach, using online activities
to integrate classroom work, such as asking
students to complete them as homework.
These three modalities can be adapted to different
situations, grades, aims, and needs. For example, the
face-to-face modality can be suitable with students of
the lowest grades and in scholastic situations where
CSEDU 2021 - 13th International Conference on Computer Supported Education
the classroom work is predominant. The blended
approach can offer useful support to the face-to-face
lessons at secondary school or university (Marchisio
et al., 2020). Online courses are generally used for
training and professional courses, university courses,
or learning in sparse communities, where face-to-face
meetings are difficult to organize (Abdelraheem,
2003; Marchisio et al., 2020).
In this conceptualization, the DLE is not limited
to technological artifacts, even if they play a crucial
role. The learning community takes a prominent
place: it can include, according to the kind of DLE,
students and peers, teachers and tutors (who are
facilitators of learning activities), designers of
educational materials, and administrators of the
digital environment. The use of these technologies,
such as web-based platforms, assessment tools, and
other systems such as sensors or eye-trackers, allows
for collecting, recording, and using learning data.
These data can be elaborated within the DLE to
provide information useful to make decisions and
take action. In the following paragraphs, we will
explain how these data can be used to improve
learning, teaching, and the development of
2.2 Formative Assessment
Formative assessment is one of the key principles
which, according to the majority of scholars, should
be included in the design of a learning environment,
being it physical or virtual (Barana & Marchisio,
2016, Barana, Fissore, & Marchisio, 2020; Gagatsis
et al., 2019). In this study, we refer to Black and
Wiliam’s definition and framework of formative
assessment (Black & Wiliam, 2009). According to
them, a practice in a classroom is formative to the
extent that evidence about student achievement is
elicited, interpreted, and used by teachers, learners,
or their peers, to make decisions about the next steps
in instruction that are likely to be better, or better
founded, than the decisions they would have taken in
the absence of the evidence that was elicited”. They
identified three agents that are principally activated
during formative practices: the teacher, the student,
and peers. Moreover, they theorized five key
strategies enacted by the three agents during the three
different processes of instruction:
KS1: clarifying and sharing learning
intentions and criteria for success;
KS2: engineering effective classroom
discussions and other learning tasks that elicit
evidence of student understanding;
KS3: providing feedback that moves learners
KS4: activating students as instructional
resources; and
KS5: activating students as the owners of their
own learning.
2.3 DLEs for Mathematics
In this paper, we consider particular DLEs for
working with Mathematics through suitable
technologies and methodologies. The LMS that we
use is based on a Moodle platform and it is integrated
with an Advanced Computing Environment (ACE),
which is a system for doing Mathematics through
symbolic computations, geometric visualization, and
embedding of interactive components (Barana,
Brancaccio, Conte, et al., 2019), and with an
Automatic Assessment System based on the ACE
engine. In particular, we chose Maple ACE and
Moebius AAS. Through this system, we create
interactive activities for Mathematics based on
problem solving and Automatic Formative
Assessment (AFA), which are the main
methodologies used in the DLE, and that we have
better defined and characterized in previous works
(Barana, Conte, et al., 2018; Fissore et al., 2020). In
detail, the characteristics of the Mathematics
activities that we propose are the following:
Availability of the activities for a self-paced
use, allowing multiple attempts;
Algorithm-based questions and answers, so
that at each attempt different numbers,
formulas, graphs, and texts are displayed,
computed on the base of random parameters;
Open mathematical answers, accepted for its
Mathematical equivalence to the correct one;
Immediate feedback, returned when the
student is still focused on the task;
Interactive feedback, which provides a sample
of a correct solving process for the task, which
students can follow step-by-step;
Contextualization in real-life or other relevant
2.4 Modelling Interactions in a Digital
Learning Environment
The technological apparatus of a DLE, particularly
when the LMS is integrated with tools for automatic
assessment, has a mediating role in the learning
processes. We can identify the following functions
through which it can support the learning activities
(Barana, Conte, Fissore, et al., 2019):
Analyzing Interactions in Automatic Formative Assessment Activities for Mathematics in Digital Learning Environments
Creating and Managing: supporting the
design, creation, editing, and managing of
resources (e.g., interactive files, theoretical
lessons, glossaries, videos), activities (e.g.,
tests, chats for synchronous discussions,
forums for asynchronous discussions,
questionnaires, submission of tasks) and more
generally of the learning environment by
teachers, but also by students or peers;
Delivering and Displaying: making the
materials and activities available to the users;
Collecting: collecting all the quantitative and
qualitative data concerning the actions of the
students (such as movements and dialogues),
the use of the materials (for example, if a
resource has been viewed or not, how many
times and how long), and the participation in
the activities (such as given answers, forum
interventions, number of tasks delivered,
number of times a test has been performed,
evaluations achieved);
Analyzing and Elaborating: analyzing and
elaborating all the data collected through the
technologies related to teaching, learning, and
the development of competences;
Providing Feedback: giving the students
feedback on the activity carried out and
providing teachers, as well as students, with
the elaboration of learning data.
To schematize these functions, we propose the
diagram shown in Figure 1. The external cycle
represents the five functions; the black dashed arrows
represent how data are exchanged within the DLE
through automatic processes. The technologies of a
DLE, to accomplish one function, uses the data or the
outputs resulting from the previous one: the learning
materials, created through the LMS or other sector-
specific software through the “creating and managing
function”, are displayed via devices through the
“delivering and displaying function”. Information
about the students’ activities is collected by the LMS,
other software, or tools through the “collecting
function” and it is analyzed by these systems, which
may use mathematical engines, learning analytics
techniques, algorithms of machine learning, or
artificial intelligence, through the “analyzing and
elaborating” function. The results of the analysis are
feedback in the sense of Hattie’s definition (i.e.,
information provided by an agent regarding aspects
of one’s performance or understanding) (Hattie &
Timperley, 2007). They can be returned to students
and teachers through the “providing feedback”
function, and they can be used to create new activities
or edit the existing ones. This circle represents a
perfect adaptive system from the technological
perspective (Di Caro et al., 2018).
Figure 1: Diagram of the interactions among the
components of a DLE through the functions of the
In a human-centered approach, at the center of the
DLE, there is the learning community, composed of
students, teachers, and peers (who are the agents in
the Black and Wiliam’s theory of formative
assessment): they can interact with the DLE through
its functions receiving and sending information. The
blue dotted arrows represent the interactions between
the community and the digital systems that occur
through human actions, such as reading, receiving,
inserting, providing, digiting. For example, the
teacher, or designer, or tutor can create the digital
activities through the “creating and managing”
functions of the DLE; tasks are displayed (“delivering
and displaying” function) and received, seen, or read
by the students through some device. The students,
individually or with their peers, can insert their
answers or work. The technology collects them
through the “collecting” function. The system
analyzes the students’ answers and provides feedback
(“providing feedback” function) returned to the
student. Simultaneously, the information about the
students’ activity is returned to the teacher through
the “providing feedback” function; the teacher can
use it to edit the existing task or create new ones. The
continuous double-ended orange arrows represent the
interactions among students, teachers, and peers,
which in classroom-based settings can be verbal
while in online settings can be mediated by the
technology. This model allows us to identify some
outcomes that the adoption of a similar DLE with
AFA, through the functions previously shown, makes
it possible to achieve:
CSEDU 2021 - 13th International Conference on Computer Supported Education
To Create an Interactive Learning
Environment: all the materials for learning
and assessment can be collected in a single
environment and be accessible at any time.
They can activate the students who can be
engaged in the navigation of the learning path,
solve the tasks and receive feedback;
To Support Collaborative Learning,
through specific activities, delivered to groups
of students, which enhance the
communication and sharing of materials,
ideas, understanding;
To Promote Formative Assessment, by
offering immediate feedback to students about
their results, their knowledge and skills
acquired, and their learning level. Feedback
can also be returned to the teachers on the
students’ results and their activities,
supporting decision-making.
The identification and classification of a DLE's
functions can allow us to identify the interactions in a
DLE, to analyze their nature and the contribution of
technology that mediates them. In this sense, the
diagram in Figure 1 is a proposal of schematization of
the interactions among the components of a DLE. It
helps us understand how data are shared among the
components of a DLE, elaborated, and used. The
information gained is useful from a learning analytics
perspective since it allows us to identify the role of
data during the learning processes. Moreover, this
model helps us identify and separate the functions and
outcomes of technology in learning processes, which
is necessary to have a clear frame and find causal
connections, especially when analyzing large data
In this study, we aim at showing how the diagram of
the interactions among the components of a DLE can
be used to model learning processes, and in particular
to understand how formative assessment can be
enacted in a DLE for Mathematics. To this purpose,
we analyzed an AFA activity concerning symbolic
computations for students of grade 8, experimented in
a classroom-based context. The task (Figure 2) asks
students to formulate, represent, and compare
different formulas derived from a geometrical shape.
The shape is non-standard and students are asked to
find as many formulas as they can to express its area.
Figure 2: Part of the activity on symbolic computations.
In the first section, they have 3 attempts to write a
formula for the area. In the second one, they are asked
to fill other 4 response areas with different formulas
expressing the same area. The intent is to make them
explore the symbolic manipulation of an algebraic
formulas through the geometric context, to confer a
more concrete meaning to the technical operations. In
the last part, students have to substitute the variable
with a given value and compute the area. This activity
was tested in a classroom-based setting in an
experiment involving 97 students of 4 different
classes of grade 8. In the classrooms there were the
teacher and 2 researchers of our research group; the
students worked in pairs using a computer or a tablet.
The work and discussions of some pairs of students
were recorded through a video camera. Data from the
platform were analyzed through the diagram of the
interactions in a DLE presented in the previous
section; the video recordings were analyzed as well.
We analyzed the videos realized during the activity in
the classrooms, to understand how the interactions
among the components of the DLE changed and how
the formative assessment strategies took place during
Analyzing Interactions in Automatic Formative Assessment Activities for Mathematics in Digital Learning Environments
a group activity. We choose some episodes which we
considered most significant. Here, the learning
community includes a class of students and a teacher;
the digital activities are created in a LMS integrated
with an AAS, and the devices used to access them are
an IWB and computers.
The first episode involves the teacher who
illustrates the task to the class. The teacher was at the
IWB and was pointing at the figure shown.
TEACHER: Look at this figure. Write the formula
which expresses how the area of this figure varies when a
varies. That is, [pointing at one of the sizes of the yellow
triangles] how long is this side?
TEACHER: Well, you have to calculate the area of this
figure using a. Those sides measure a. What does it mean?
What is a?
STUDENTS: A variable.
In this excerpt, the teacher introduced the activity
and explained to the students what their task was. The
explanation took the form of a dialogue, as he
engaged the students with questions to make sure that
they were following the discourse. The teacher
exploited the “delivering and displaying” function of
the technology to display the task and, in particular,
the figure; then, she interacted with the students. If we
consider the diagram, we are in the right part; the
parts of the model involved in this excerpt are shown
in yellow in Figure 3. While explaining the tasks, she
developed the KS1 “clarifying and sharing learning
intentions and criteria for success”. The KS2
“engineering effective classroom discussions and
other learning tasks that elicit evidence of student
understanding” was accomplished during the phase of
the creation of this activity by the researchers (that we
can include in the “Teacher” subject of our analysis)
through the “creating and managing” function of the
technologies; it is also activated when the teacher
asks questions to the class aimed at making students
reason in the correct direction.
The second episode involves Marco (M) and
Giulia (G), two students of medium level who were
trying to solve the first part of the activity, working
together. In the beginning, they observed the figure
displayed on the screen of their computer and tried to
understand the task.
M: We have to compute the area, but we don’t have any
G: But we have a.
M: But a is not a number!
G: Ok, but we can compute the area using a.
M: Teacher, how can we compute the area without
numbers? Can we use a?
T: Yes, it is like a generic number.
G: We have to write a formula using a, isn’t it?
T: That’s right.
Figure 3: Diagram of the formative assessment strategies
enacted in the first episode of activity 2 through the
interactions in the DLE.
The two students started reasoning together on the
figure trying a way to compute the area. After about
15 minutes, they came up with a quite complex
formula, built subtracting the area of the inner white
square to that of the external square. They used the
Pythagorean theorem to compute the length of the
white square’s side. They inserted the formula in the
response area and the system returned a green tick
with positive feedback. They passed to the following
part, which asked them to find other 4 formulas for
the same area. For the first two formulas, they
reasoned algebraically, manipulating the original
formula. For the other two, they reasoned
geometrically, developing new ways to compute the
area. The peer discussion allowed them to correct
mistakes before entering the formulas in the response
areas, so their answers were marked as correct at their
first attempt.
In this episode, the students look at the task
displayed on the screen through the “delivering and
displaying” function, then interact among them
discussing the task. They also interact with the teacher
asking questions about their doubts. Then they insert
their answers in the system, which collects them
through the “collecting” function, analyzes them, and
returns feedback. They repeat the same cycle several
times. The students activate KS4 “activating students
as instructional resources” when discussing in pair.
KS5 “activating students as the owners of their own
learning” is enacted when they insert their answers in
the AAS, and KS3 is developed when they receive
feedback from the AAS, but also by the teacher. The
yellow parts in Figure 11 schematize the interactions
CSEDU 2021 - 13th International Conference on Computer Supported Education
that occurred in this episode and the formative
assessment strategies developed.
Figure 4: Diagram of the formative assessment strategies
enacted in the second episode of activity 2 through the
interactions in the DLE.
The episode presented in the previous section helps
clarify how the interactions among the members of a
DLE occur during AFA Mathematics activities in a
classroom-based setting. The main feedback is
provided by the social interactions within the learning
community and especially among peers; in fact,
Marco and Giulia reasoned more time on the tasks
and they tended to answer correctly at the first
attempt. In other cases, we could observe that the
computerized interactive feedback has a key role in
providing a feedback and in engaging the students.
The design of the activity enables KS3 and KS5,
which keep students engaged with the task until its
full comprehension, demonstrated by the repeated
attempts. Similar activities can lead to a deep
understanding of fundamental Mathematics concepts;
the technologies and methodologies used in
particular, an AAS based on a mathematical engine
and AFA – supported the design and implementation
of interesting activities for the development of
mathematical competences.
The diagram used for the analyses helped clarify
what functions of the technologies and through which
kinds of interactions the formative assessment
strategies are elicited in different situations. In
particular, we can see that all the Black and Wiliam’s
strategies of formative assessment can be enacted
through AFA activities, and all of them are identified
and located along the arrows of our diagram, that is
during the interactions among the human components
of the DLE or between human and technological
components. Thus, we can include a fourth agent in
Black and Wiliam’s framework: in the DLEs that we
consider, the technology is also an agent of the
formative assessment strategies, especially for
providing feedback and engaging students (KS3 and
Through the analysis of the interactions among
the members of these DLEs, we can also point out that
the three outcomes mentioned in our framework are
achieved. In particular: the analyzed learning
environments are interactive, since students are
actively engaged in the activities, they are stimulated
to reflect and have the opportunity to achieve
important understanding; the formative assessment is
promoted by the activities, as all the 5 key strategies
are enacted; collaboration among students is
supported, especially in the classroom-based settings
where students are asked to work together.
The diagram used in the analyses allows us to
conceptualize the DLE as an ecosystem: we can see
that the human and technological components are
strictly related, and the interrelations among them
cause the development of the learning community, in
terms of learning processes, knowledge, and
competences gained; but also an improvement of the
learning activities on the base of the results obtained.
The analyses conducted in this study have a
qualitative nature: they are aimed at showing how the
schema of the interactions among the components of
a DLE can be used to model formative assessment
practices, especially when the AFA is adopted.
However, they can be a starting point for the research
about learning analytics for formative assessment.
This model can be used to create a taxonomy of the
interactions occurring in a DLE, identifying which
support formative assessment or other learning
processes. Since interactions in a DLE can be
described using log data, this model can also be used
with extensive learning data to identify the formative
assessment strategies or other learning processes
occurring in large online courses. This would allow
us to identify the learning activities which are better
related to the development of knowledge, abilities,
and competences or the elicitation of interactions and
engagement. The results of similar analyses could
help adjust and improve the digital materials in online
courses. Using other technologies and different
learning methodologies to build suitable activities,
this model of analysis could also be adapted to other
Analyzing Interactions in Automatic Formative Assessment Activities for Mathematics in Digital Learning Environments
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