  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.
