The article describes the features of the evolution of the cloud-based system 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 (formerly called SageMathCloud) are considered in concern to the selection criteria. The main pe-culiarities of the CoCalc comparing to SageMathCloud are considered. The re-sults of the experimental testing of CoCalc use in the process of pre-service 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 pro-spects for further research are revealed.