Three things I would like to see happen with the practice of mathematics in the next ten years are:
Computer proof assistants…
…that are easy to use for the typical mathematician. Why? Mathematics is expanding tremendously and some areas are getting quite abstract. Some areas are so complex that I even wonder if anyone in some of these fields has time to verify important details. Proof assistants will probably become necessary, not just for the creation of new mathematics but to help us organize the amazing amount of existing mathematics.
Actually, there are several proof assistants that exist are and currently maintained. I've tried ones like Isabelle, Mizar, and Coq, but have not found them easy to use. Unfortunately making it possible to input typical mathematics into a computer so that it can be verified mechanically is a pretty difficult problem.
Blockchain Paper Publishing
This isn't just for mathematics, but I think if anyone could adopt blockchain technology for journals, it would be mathematicians. Actually compared to other fields I think math has done pretty well with creating independent, open-access journals. However, we could go further and adopt blockchain technology.
If mathematicians used a single blockchain system (or maybe a few), every peer-review process could be recorded on the blockchain ledger. Traditional journals could still operate on this system, and in this case the name of the journal could be recorded in the blockchain ledger. In fact, this would be a benefit to journals because the book-keeping records would be in the blockchain and external hosting would not be necessary.
But one wouldn't need a traditional 'journal unit'; someone who writes a paper could send it off to a willing independent editor and that editor could just send the paper to be reviewed as usual. The result of this independent transaction would be recorded on the blockchain.
The upshot is that all peer-reviewed transactions would be recorded on a distributed blockchain and all paper-reviewing activities would receive a quantitative measure of credit.
De-emphasis of proving new results in mathematics
This relates back to the idea that mathematics today is progressing so quickly that theorems are being proved more quickly than people can understand them. So today you have hundreds of fields, some of them with only a few people inside them knowing what's going on. That's kind of a shame because mathematics would be enriched if there were a little more cross-disciplinary understanding.
Thus it would be great if more emphasis and credit were placed upon simplifying existing research and making it more accessible. This would help the next generation of mathematicians enter the field. But it would also alleviate the pressure that exists for many existing research to output theorem after theorem, which is probably not all that useful for developing the understanding of mathematics as a whole anyway.
I still think we should prove theorems, and there are many cool things that haven't been done yet. But with a more varied approach to math, we could concentrate on fewer, more quality theorems rather than pure quantity.