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Astrid Beckmann
University of Education (PH) Schwäbisch Gmünd (GERMANY)
Hybrid education is a concept which overcomes traditional borders and dissolves dichotomies of
typical learning concepts while connecting physical and digital learning rooms, analog and digital
learning materials, online and offline work, individual and collective experiences, as well as learning
and teaching activities of the students. The MathEdu Digital-teaching concept, developed at the
University of Education Schwäbisch Gmünd/Germany, can be easily used to implement hybrid
education. The fruitfulness of this implementation is shown on the example „concept of function in
maths teacher education . As a result, hybrid education offers new and advanced opportunities, is
flexible and opens doors for deeper and low-threshold participation and free discussions, allows to
include physical activities into digital learning environments and therefore to implement more relevant
and important contents. It can create interdependencies and, as an outcome, can lead to more
students´ activities and to more comprehensive experiences, and it allows almost unlimited
attendances and the students to choose their own way of learning.
Keywords: hybrid education, digitized learning environments, blended-learning, maths teacher
education, concept of function.
Hybrid education is a concept which overcomes traditional borders and dissolves dichotomies of
typical learning concepts. It connects physical and digital learning rooms, analog and digital learning
materials, online and offline work, individual and collective experiences, as well as learning and
teaching activities of the students. Some authors emphasize the combination of synchronous and
asynchronous learning phases or expand the definition by including the mergence of certainty and
uncertainty and of endedness and open-endedness etc. (cf. [1], [2], [3], [4], [5], [6]). However, hybrid
education is open for own learning and teaching arrangements with diverse hybridities.
We explain hybrid education on three examples:
First example: The traditional blended-learning concept differentiates strictly between (1) the
digital learning phase, mostly fulfilled self-dependent at home, and (2) the physical learning
phase, mostly implemented as a face-to-face session at the university. Hybrid education
combines both (1) and (2) in exactly the same learning phase; e.g.: students are able to follow
the activities at the university, digital as well as physical.
Second example: The physical learning phase differentiates strictly between (1) individual
experiences, e.g. working on mathematics tasks for your own, and (2) collective experiences,
e.g. working and discussing in groups. Hybrid education combines both (1) and (2); e.g. a group
of students participate in the experiences of individuals (see example below: concept of
Third example: The digital learning phase is (normally) strictly digital. Hybrid education
overcomes the border between digital learning and analog or physical learning; e.g.: the
students use the material of a digital library including LOs (learning objects) like videos, tasks
for physical activities, literature in PDF format, apps for digital activities etc. provided in a LMS
(learning management system like Moodle etc).
Background idea of hybrid education is that overcoming the borders and dissolving dichotomies
combine the best of all learning environments. This should lead to more student-oriented teaching,
more learner-centred experiences, more social-emotional learning, and more personalized and
relevant learning contents.
Proceedings of INTED2021 Conference
8th-9th March 2021
ISBN: 978-84-09-27666-0
The MathEdu Digital-teaching concept was developed at the University of Education (PH) Schwäbisch
Gmünd/Germany as part of the MathEdu Digital-project, concerning digitization in maths teacher
education ([7]). It has been regularly taught on Master´s-level Mathematics teacher training courses
for secondary education at the Universities in Schwäbisch Gmünd and in Ulm since 2018. A total of
more than 100 students have participated in the 7 courses held to date. This teaching concept
consists of a range of teaching elements including digital tools and physical formats, like video demos,
apps, online forums, webinars respectively online-live-sessions, LMS providing e-books, tasks and
literature, and face-to-face sessions. The course structure is presented to all students in a clear and
comprehensible manner from the start using the LMS (in this case, moodle). Its format largely
corresponds to the blended-learning concept ([8], [9]), with students independently working their way
into the respective topic in learning phase 1 and students engaging in an exchange, interacting and
communicating with other students and the lecturer in learning phase 2. Each topic (about 9 to 11
topics) has a learning phase 1 and a learning phase 2 (for more detail, see [10]).
In contrast to traditional blended-learning formats, the MathEdu Digital-teaching concept includes
digital and physical learning formats in both of the learning phases. That means: There are inherent
hybrid elements in the concept, i.e.:
Hybrid elements in learning phase1: digital library, provided in a LMS: including LOs like video
demos (audio slide show presentations), tasks for physical activities or students learning groups
(face-to-face), Emails, Forum exchange etc.
Hybrid elements in learning phase 2 (mandatory): formats are not only face-to-face sessions,
but also webinars (virtual meeting; location can be freely selected), forum discussion, e.g. after
uploading solutions to student folder in LMS (detailed feedback from the lecturer), and app-
based surveys spontaneously integrated into the face-to-face sessions etc.
But in contrast to hybrid education the digital and physical formats of learning phase 2 were mostly
divided. Implementing hybrid education means therefore the continuous combination of the digitized
and physical formats in learning phase 2. Using the MathEdu Digital-teaching concept, this can be
realized easily.
The fruitfulness of implementing hybrid education will now be shown on the example „concept of
function in maths teacher education“.
3.1 The concept of function
The concept of function is one of the most important and most complex mathematical concepts.
Functions form the basis of many mathematical fields, they are the key means of mathematization and
modelling, they help us establish, understand, and describe relationships, making them important
parts of everyday life. Understanding the concept of function is a process that extends throughout the
entire duration of school education. It can succeed only if we first develop adequate fundamental
approaches and an active grasp of the aspects of the concept as represented in various forms (cf.
[11], [12], [13], [14], also [15]).
The three core aspects of the concept of function are:
Aspect of correspondence (action): Every x of a set X corresponds precisely to one element y of
a set Y. Simple correspondence considers only one x and its corresponding y, whereas
continuous correspondence involves (continuously) considering many/all x of X in succession.
Example Car travels: Each distance s corresponds precisely to a time t.
Aspect of covariation (process): As x changes, so the corresponding y changes accordingly. In
the context of discrete covariation, the process of changing x is clearly separated from the
subsequent x (i.e. the change is discrete). Continuous covariation involves allowing each x to
run through set X and observing the continuous changes.
Example Car travels: As certain time t passes, the distance travelled by the car increases
Aspect of object: Understanding a function as an object means comprehending the function as
a whole. This requires us to be familiar with aspects such as simple and continuous
correspondence, discrete and continuous covariation in all forms of representation, potential
shifts between these, and types of shifts.
The forms of representation/depiction are image or verbal description, spreadsheet, graph, and
algebraic expression or term/formula.
3.2 The concept of function in teacher education
Against the background of the importance and complexity, the concept of function is one of the most
fundamental and significant ones not only in mathematics, but also in mathematics education ([11],
[16]). For many decades the concept understanding has been a huge object of research ([12], [17],
[18], [19], [20], [21], [22], etc.). Several studies identified pupils´ often limited formal understanding and
shed light on the difficulties encountered when shifting between different forms of representation.
Various solutions were developed in response to the unsatisfying findings, with the move away from a
purely mathematical treatment of functions and toward an application-based approach proving to be a
decisive step. The importance of the real world with respect to understanding the concept of function
has been mentioned in several papers (e.g. [23], [24]). Also, it became clear that understanding the
concept of function requires a systematic structure. Mathematics teaching should therefore (first)
concentrate on mastering the aspect of correspondence (action), then subsequently on mastering the
aspect of covariation (process) in all its forms of representation and for all the shifts between these
forms ([25]).
Ganter ([26]) undertook a more large-scale study demonstrating that real-world experiments have a
significant beneficial impact on understanding of the aspects of correspondence and covariation.
These findings correspond with those by the European ScienceMath-Project ([27]) based on trials
conducted in a teaching environment, which indicated that experiments succeed in engaging even
lower-achiever learners in prolonged mathematical discussions. Real-world respectively natural
science experiments are particularly suitable for promoting understanding the concept of function,
because ([25]):
the steps in an experiment correspond to the aspects of the concept of function,
the particular attributes of the experimental activities help generate authentic experiences,
experiments simultaneously address multiple principles involved in mathematics and support a
wide range of references to the real world.
In consequence, real-world experiments for supporting the learning of the concept of function has
come to be an approved part in maths teacher education. On the other side, the digital age brought
with it a series of digital tools that can have a beneficial impact on mathematics teaching. As shown in
many scientific papers, learning the concept of function could be supported by: dynamic geometry
systems, computer algebra systems, spreadsheets, simulations, video of real-life experiments,
selected apps (e.g. like padlet etc.), (possibly smart speakers (like Hey-Google, Alexa etc.),
augmented reality etc. ([28], [29], [20], [30], [31], [32], [33], see summarizing this scientific discussion
of advantages and disadvantages in [25]). Teacher education must acquaint the future teachers with
all the different and promising methods for learning the concept of function. And it should submit
proposals for the most beneficial way.
There have already been several studies aiming to compare the merits of digitized and non-digitized
teaching. In order to be adequately prepared for the digital world to come, it is essential for pupils to
amass considerable amounts of digital experience. These pupils should be just as familiar with the
opportunities of digitization as they are with its limits and hazards. In particular, they should receive a
digital education to ready them for the digital future. And while this contribution has addressed the fact
that there are numerous proposed methods for understanding the concept of function with the support
of digital resources, it has also been emphasized the strength of real-world experiments. Research
results show that experiments are particularly suitable for beginners in a unique way: they permit wide-
ranging analysis of the real-world phenomenon in question, enable specific and explicit links to the
aspect of correspondence, and ultimately make it possible to experience the concept of function in all
its facts. Based on that the variety of digital tools and virtual environments are suitable for
consolidating the knowledge that is acquired, with new innovations appearing on a daily basis. In
conclusion, real-world experiments should be furthermore an important learning subject in maths
teacher education, also in our digital era and in our digitized lectures at the universities.
3.3 Implementing hybrid education in maths teacher education, concept of
The issue of digitization in university teaching has acquired real significance, not least because
universities are central to shaping our future in a digitized world. This is especially true for the field of
teacher training education, as teachers adopt a crucial multiplier role with regard to digital education in
our society. Digitization is relevant not only as a topic to be taught, but also as a means of tapping into
new and innovative opportunities for teaching itself. However, digitization has become an important
aspect of university teaching and took over a special role, also beyond the background of equal
opportunities in education, e.g. for handicapped students or students with duties for children or high-
maintenance persons, and especially in a pandemic situation.
On the other hand, learning mathematics strictly needs face-to-face activities time and again. As
described above, beginners need to act with real-world experiments for understanding the concept of
function; and therefore teacher students should gather this real-world experience, too. This could be
provided in face-to-face sessions, or where not possible in a combination of real-world and
digitized formats. Here, the answer is hybrid education!
Implementing hybrid education in maths teacher education can be realized easily using the MathEdu
Digital-teaching concept: In learning phase 2 we offer simultaneously meetings at home and at the
university. The real-world experiments are performed during a physical-digital meeting at the
university, including a permanent digital exchange and common activities between the students in the
digital and physical rooms. The teaching concept is characterized by interaction as kind of overcoming
borders between digital and physical rooms, and (positive) interdependence between these rooms,
which leads to more students´ activities ([34]). The format includes group activities, common
exchange, meaningful division of work, free discussions, low-threshold participation and student-
orientation. Table 1 gives an insight into the digital and physical aspects and the connections using
this hybrid teaching concept.
Table 1. Hybrid teaching concept in learning phase 2 (MathEdu Digital).
Digital meeting (at home)
Connection/ common activities
Physical-digital meeting at the
Students experience the
experiments live and are able to
interact and discuss simultaneously
discussing the phenomenon,
recherche in the internet,
developing experimental situations,
doing measurements (live recorded),
evaluation of data,
documentation of measured data,
discussing results,
presentation of results
Students are doing the real-world
experiments by themselves (live
video recorded)
Students are able to change
between individual and group
Students are in an exchange with
all students in the physical and
digital rooms
Students support the experimental
process using digital tools etc.
Students use digital tools and can
change from learning to teaching
The format of hybrid education offers the opportunity for live experiences and live interaction between
physical and digital rooms. In the case of the concept of function the aspect of correspondence is
experienced directly through live experimentation. This experience is based on physical activities of an
individuum or a group and on digital interaction during experimentation and evaluation of the data etc.
Organizing (positive) interdependence between the rooms, e.g. students at home evaluate the real-
world experiments using digital tools, can lead to more students´ activities.
In conclusion, hybrid education is seen as a specific chance for integrating important (physical
founded) contents in digitized lecturers. The experiences concerning the topic “concept of function in
maths teacher education” lead to the following (first) observations:
- Hybrid education offers new and advanced opportunities.
- Hybrid education is flexible and opens doors for deeper and low-threshold participation and free
- Hybrid education allows to include physical activities into digital learning environments and
therefore to implement more relevant and important contents.
- Hybrid education can create interdependence and therefore can lead to more students´ activities.
- Hybrid education can lead to more comprehensive experiences.
- Hybrid education allows students to choose their own way of learning.
- Hybrid education allows almost unlimited attendances.
Hybrid education can be easily implemented using the MathEdu Digital-teaching concept; learning the
concept of function in teacher education is a fruitful example.
It will be our future task to develop and test more teaching tools for different mathematical topics for a
fruitful hybrid learning environment using the MathEdu Digital-teaching concept.
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