I would say that if you asked my students that it means to do mathematics they would say things like find examples, look for patterns and solve problems. I am not so sure they would say the ones I have highlighted - "take chances, make mistakes", "discover more elegant solutions" and "explain, validate, convince". Why would they not say these? Here are my thoughts.
Take Chances, Make Mistakes - Although I try to have a culture where mistakes are seen as part of the learning process, students often don't want to make mistakes for fear of "losing face" in front of their peers. However, taking chances and making mistakes are a huge part of learning. Desmos has helped my students to take chances, try things out, make mistakes, revise their answers. Desmos gives feedback in a non-judgemental way. It says your wrong, but without using the word "wrong". I don't use the word "wrong", but what do I say specifically? I try to say things like "Why do you think that?" and "Can you explain your thinking?" Do I really value mistakes? I'll need to be more cognizant of this next year.
Discover More Elegant Solutions - Rarely do we take the time to have students discover more solutions, let alone more elegant solutions. This is especially true in the public school system. At times this has also been true in my classroom. To have more elegant solutions, you need to have problems that are interesting and can be solved by multiple perspectives. As I look to next year, can I make this part of my classroom culture? As I plan for my classes for the fall, my goal will be to find one or two problems per unit that lend themselves to discovering more elegant solutions.
Explain, Validate, Convince - This is something that should be happening in every classroom (not just math classrooms) daily. "Convince us." and "How can you tell if your answer is right/reasonable?" needs to also happen more frequently in my classroom. In the name of "covering" content, we often rush past this part of a lesson to do more examples. The answer is "right" and for many teachers that are pressured by state testing, a "right" answer is all that matters. However, I believe that explain, validate and convince will have a stronger and more lasting impact on understanding than drill and kill in the name of "right" answers.