The article describes the features of the evolution of the cloud-based systems in the context of mathematics and science teachers training. The notion of the systems of computer mathematics (SCM) is considered as the special kind of learning software. The lists of the special mathematical operations, functions and methods used within the structure of modern SCM are examined. Web-SCM properties are considered as general and specific. The cloud-based SCM are taken as the special stage of Web-SCM development. The criteria for an appropriate cloud service selection to support the process of mathematics disciplines learning are revealed. The key features of the CoCalc cloud service use are considered in concern to the proposed selection criteria: the available computing capacities, the diversity of tools for training and its control, the possibility of increasing the computing capability, the openness of the program code, the availability of tools for joint editing. The results of the experimental testing of CoCalc use in the process of mathematics teachers training are described. Besides, the discussion covering the issues of integration and compatibility with the other tools and services used for mathematics learning (on the example of Octave) is presented. The main trends of the CoCalc cloud service introduction into the training process and the prospects for further research are revealed.
The study aims to outline the state of the art and the current progress in the sphere of the cloud-based systems of computer mathematics development, to reveal the key features of the structure and functions of the cloud service CoCalc, relevant for its application as a tool for learning mathematics disciplines and to evaluate the promising trends of its application in the process of mathematics and science teachers training. 1.1
E. C. Granado, E. D. Garcia [
The components of the author's educational mobile environment elaborated by
M. A. Kislova are the following:
─ the tools for organizing educational processes and mathematical calculations,
─ the mobile tools to support the communication of a group of listeners,
─ the tools to support the process of teaching higher mathematics and personal
interaction between students and teachers [
Except just mobile learning the researchers have been paying much attention to the use
of cloud services for learning. For example, M. A. Kislova, K. I. Slovak [
Substantial achievements in terms of research opportunities of the cloud technologies use in education were made by the following scientists: G. A. Aleksanyan, V. Yu. Bykov, M. Yu. Kademiya, V. M Kobysya, O. G. Kuzminskaya, V. M. Kukharenko, S. G. Lytvynova, N. V. Morse, V. S. Mkrtchyan, A. S. Sviridenko, Z. S. Seydametova, S. O. Semerikov, O. M. Spirin, L. V. Rozhdestvenskaya, Y. V. Trius, M. P. Shyshkina, B. B. Yarmakhov, M. I. Zhaldak, and others. These issues were also investigated by M. Armbrust, R. Griffith, Y. Khmelevsky, M. Miller, K. Subramanian, N. Sultan, W. Chang, P. Thomas, A. Fox, et al.
The use of the cloud services has a number of advantages (according to
N. V. Rashevskaya) [
There is a separate group is research studies devoted to the problems of cloud technologies use for teachers training. Among them, there are the works of T. L. Arkhipova, N. V. Bakhmat, T. V. Zaitseva, Yu. G. Lotyuk, N. V. Soroko, M. A. Shynenko et al.
The use of the Sage computer algebra system for learning mathematics was studied
by P. Zimmermann, A. Casamayou, N. Cohen, G. Connan, T. Dumont, L. Fousse,
F. Maltey, M. Meulien, M. Mezzarobba, C. Pernet, N. M Thiery, E. Bray, J. Cremona,
M. Forets, A. Ghitza, H. Thomas [
In the study of A. B. Liecharlie [
M. D. Ruiz and F. Torralbo [
As there are new facilities and modes of using the cloud services in education due to the latest achievements and developments in this area, in particular, the current progress of CoCalc applications, the issues of innovative use of this kind of services need further attention. Among them, there are issues of the search for the new forms, methods and possible models of CoCalc use in the process of training mathematics teachers to support the teaching of mathematical disciplines. 2 2.1
The modern SCM can be of different types, application areas and architecture; still,
they appear to have similar key features of their structure and composition [
In the process of SCM evolution the special kind of them such as Web-SCM emerged along with their gradual transformation into cloud-based systems. In recent years, the cloud-based versions of mathematical packages from major providers, such as Maple Net, MATLAB web-server, WebMathematica and others have been supplied. A variation of this type of systems is CoCalc, the cloud-based version of Web-SCM Sage.
SCM has undergone some changes over time. During the 60's and 90's of the XX century SCM were mostly local, ie required installation on a local computer.
In the 90s of the XX century and until the first decade of the XXI century, WebSCM appeared, which allows you to work with SCM directly via the Internet and configure a web browser. It was the second stage of SCM evolution. The main features of such SCM are: ─ the computing core does not require pre-installation on the user's device; ─ all calculations take place on the web server (user standby capacities may be low); ─ all calculations can be organized using a web browser. ─ Web-SCM also has specific functions: ─ does not depend on the technical characteristics of the user's personal device; ─ you can use any browser; ─ intuitive ease of introduction; ─ constant access to educational materials anywhere and anytime, etc. The most commonly used are Web-SCMs such as MapleNet, MathCAD Application Server (MAS), webMathematica, Matlab Web Server (MWS), Sage and wxMaxima. Web-SCM has an intuitive interface, powerful graphics packages for building and visualizing mathematical objects, functions and methods.
Specific characteristics of modern SCM are: ─ the use of the elements of mathematical functions and programming languages; ─ integration with other software; ─ the capability of editing and printing of mathematical texts.
The use of Web-SCM had certain advantages: the availability of powerful tools for building and researching mathematical models for various mathematical disciplines.
The first version of SAGE emerged in 2006. The first cloud systems were developed in 2009, it caused the third stage in the development of SCM. CoCalc (new name) is a cloud service that combines all the features and tools of SAGE and Web-SCM SAGE. 2.3
The following selection criteria may be used for selecting a cloud service, appropriate for the certain educational needs: ─ computing power provided for the work of an individual user, as well as how much data may be processed simultaneously; ─ whether there are tools for organizing and monitoring the learning process (for certain disciplines, setting different levels of learning tasks, reviewing the process of implementation and evaluation of learning results, both for groups of users and individual students); ─ availability of different tariff plans to expand the computing capabilities of the cloud service (availability of additional functions); ─ open source code and free access to the installation or integration of additional modules and applications (including author's tools) along with the basic set of tools being protected; ─ availability of tools for creating, storing, joint editing of resources or files in various formats (the most common and compatible with modern devices). 2.4
Sage is a software that implements mathematical algorithms in different contexts. To
begin with, it can be used as a scientific pocket calculator and can manipulate all sorts
of numbers, from integer and rational numbers to numerical approximations of real and
complex numbers with arbitrary precision. However, mathematical calculations go far
beyond numbers: Sage is a system of computer algebra; for example, it can be used to
support solving linear equations or search for solutions, simplify expressions. In the
process of analysis, Sage can manipulate by the expressions relating to square roots,
exponents, logarithms, or trigonometric functions: integration, calculating limits,
simplifying sums, series, solving certain differential equations, and more. In linear algebra
operations with vectors, matrices, and subspaces can be performed. It can also help
illustrate individual tasks in probability theory, statistics, and combinatory [
Sage service will provide the user with consistent access to functions across a wide area of mathematics - from group theory to numerical analysis, and further to two- and three-dimensional visualization, animation, networking, databases. Using all the necessary functions within the unified cloud service is convenient for a user who needn’t transfer data between multiple tools and learn the syntax of several programming languages.
The specifics of using CoCalc is primarily the capability of projects creation (both group and individual) and uploading training materials to them. After that users will be able to share them. There is a chronology of user actions with individual training material and with the project as a whole. These features are useful in the process of evaluating each user individually.
CoCalc is the online computing environment [
The freeware software that is installed is much diversified, covers many
programming languages, and can be easily expanded by installing other programs on Linux in
user space. CoCalc user personal space is organized into different projects, each with
its files that can be generated on the platform or downloaded from a user's computer.
On the other hand, it also has tutorials on different aspects of both platform use and
programming in different languages. Regarding the course management, the teacher
can add students by their email address, assign tasks, automatically form a specific
folder for all course participants, keep track of the edits to these assignments, and return
a corrected copy to all students with the one click of a button. Another useful feature of
CoCalc as for a teacher point of view is that it allows simultaneous editing of Jupyter
Notebook documents in the Google Documents style, thanks to the proprietary version
of the Jupyter Notebook web interface. This add-on, not included in the default Jupyter
notepad, greatly facilitates group work on the task and is complemented by integrated
chat on the platform itself. All of these options make CoCalc a complete and
easy-touse solution for integrating Jupyter files into an academic course. Also, the openness of
the code led to the emergence of a local version of the platform, which could be installed
on any computer. This local version can be downloaded from the GitHub development
page and, if the local computer is powerful enough, used for teaching a Jupyter
Notebook course [
CoCalc (formerly called SageMathCloud) is a cloud computing and math
calculations management platform. CoCalc directly supports the Sage worksheet, which
interactively evaluates the Sage code. SageMath (formerly Sage or SAGE, an algebra and
geometry system for numerical operations and research) is a computer algebra system
that has various functions that cover many aspects of mathematics, including algebra,
combinatorics, graph theory, numerical analysis, calculus, and statistics. The new name
CoCalc, which stands for Collaborative Calculations, reflects the evolution of the
conventional cloud computing and calculation platform. Spreadsheets in CoCalc support
Markdown and HTML as design and R, Octave, Cython, Julia and others for
programming other than Sage [
CoCalc is an open-source software provided by SageMath Inc. The creator and principal developer of CoCalc is William Stein, a professor of mathematics at Washington University who also created the Sage software system. The initial development of CoCalc was funded by Washington University and grants from the National Science Foundation and Google. At the moment, CoCalc is largely funded by paid users.
To test the effectiveness of using CoCalc as a tool for teaching mathematical disciplines for pre-service teachers of mathematics the pedagogical experiment was conducted in 2017, at the Pedagogical Institute of the Kryvyi Rig National University (Kryvyi Rig).
The experimental group and the control group were formed according to the
following principle: in the experimental group there were students who studied the author's
method of using CoCalc (EG), in the control group there were students who studied the
traditional method (CG). The obtained results proved that the level of teachers’
professional competencies formation in the experimental and control groups coincided with
the level of significance α = 0.05 before the formation stage [
Comparing the levels of professional competences formation in control and experimental groups it was possible to observe the increase of a share of students with average and high level of professional competence after the formation stage of the experiment.
Comparison of the distribution of the experimental and the control groups of students after the formation stage of the experiment (scale levels): high – from 16% to 20%, sufficient – from 26% to 36%, average – from 27% to 28%, low – from 31% to 16%. During the study only some components of subject, professional and practical and technological competencies have been compared at four levels (high, sufficient, medium and low), among them: subject-pedagogical, mathematical and informationtechnological competencies. According to Fisher's test, the data at the beginning of the experiment and at the end were compared.
The analysis of the results of the forming stage of the pedagogical experiment
showed that the distribution of the levels of the formation of professional competencies
in the experimental and control groups of mathematics trainee teachers has statistically
significant differences due to the implementation of the developed method of using the
cloud service CoCalc, which confirms the hypothesis of the study [
During the various teaching experiences, several shortcomings of the cloud service
use were identified. The first disadvantage is the inability to prepare self-assessment
questionnaires (simulators, tests). This tool, available on many training platforms such
as Moodle, is not currently available on CoCalc. This tool would allow a teacher to
learn the level of theoretical content knowledge of a student on a subject before the
practical tasks. However, the CoCalc extension is planed [
The third limitation that was found concerns to the assessment of group work, since there is no option to create or evaluate activities for a group of users in a whole, which also allows you to evaluate the participation of each small group within it. 4
There is a way to examine the difference between working with software in local-based and cloud-based versions comparing the use of different programming tools of CoCalc cloud service (by the example of Octave).
Octave is available in CoCalc along with other kinds of languages such as Python, Java, C, and so on. Octave is an interpreted language designed for mathematical computing, it is largely compatible with MathLab. This is the reason that this tool is becoming more and more popular and useful in the process of math teachers training. In particular, in the Taras Shevchenko National University of Kyiv (Ukraine) there is a separate laboratory course devoted to it.
The use of Octave based on CoCalc enables the user to work with Octave in a cloud and also to use it in collaboration. Sagews may provide the mathematical functions (for working with matrices, vectors, graphs, mathematical constants, equations), but the syntax may be different and not compatible with MathLab. The source code for Octave is freely distributable under the terms of the GNU General Public License (GPL) it makes possible to use free all of its functions in the learning process.
In the cloud version, there are the same commands as in the local one (Fig. 1), and also there are some additional possibilities. There are tools for easy collaboration such as video chat, chat commands, code history of each user, the ability to save files (Fig. 2).
The advantage of the cloud version is that it can be accessed from different computers, even from a smartphone.
The cloud-based interface of Octave used through CoCalc may slightly differ from its local version, though not very significant, as its capabilities remain. When you create a file with the extension .m, you open an editor where you can edit the text, you can cut, copy, paste, there is a search in the text. The '> _Shell' key lets you run the code. The 'file' key enables you to work with the file (Fig.3). There are different ways to work with Octave in CoCalc. You can simply create a file with the extension .m, then you can open the terminal and run the file in it.
The .term file is used to use Linux terminal commands. In the terminal, you can run Octave or create a file with extention .m of .x11.
In the file with extention .x11 you can run different applications. There is possible to open Libre OfficeWriter, the environment for Python and so on and also use Octave as in the local version.
Now the Octave language is penetrating the math teachers’ education process. This language is a useful tool for teachers training as it enables learners to perform different mathematical calculations and operate with functions. Comparison of using the local version of Octave and the cloud one (based on CoCalc) is a prospective trend for further research. The emergence and development of new forms of learning, focused on the collaboration of groups of students and joint activities in the online mode was caused by the availability of cloud services (including specialized). It is justified that the cloud services may be purposely used in the process of science and mathematics teachers training as tools for: ─ group communication of listeners (synchronous and asynchronous); ─ collaboration in real time; ─ processing, systematization and storage of educational data.
Among the areas of CoCalc use in the science and math teachers training process are the next:
The described teaching experience with the use of CoCalc demonstrates its potential to become a ground for a variety of learning methods.
The teacher, through the use of the control and monitoring tools offered by the system, can supply and control students performance that is an added benefit of using this platform. Finally, this service is also suitable for use as the research collaborative platform.