Roy writes, "How do you approach the movement of math ed participants across different degrees of freedom?.."
First of all, I have smaller units of analysis: individual activities, which may take literally a few mintues, not whole courses. Or maybe individual game mechanics and mini-games: one boss encounter rather than the whole World of Warcraft.
Each person needs a balance between activities that are at different points on each spectrum: give-observe-take, freedom-structure-restraint, exercises-problems-inquiry... Different people have different personal preferences for each of these mixtures, and these preferences change from day to day, or from minute to minute. For example, take problem-solving: a student may need some chaos (no problems! hakuna matata!), some inquiry (open problem-posing), and some grindy meditative exercises. A very anxious person who's been repeatedly burned may prefer a lot of easy exercises and will shy from inquiry, especially if observed or tested. Toddlers often, but not always, need a lot of inquiry and chaos. It's not like people strive, fight, and work toward more and more freedom, in all circumstances.
The main problem I meet with math is that so many people have PTSD, more or less. It's almost a tautalogy that self-regulation requires freedom. For example, it's very difficult to me to explain to parents why kids need the freedom to shift their attention - "to become distracted" - during math activities. How do you know if the activity is right for you if someone directly manages your attention?
I try to address most of these balancing issues with experience design. Maker tasks are especially good for helping people to self-balance, because people have healthier metaphors about making things: following other people's recipes, or remixing them, or experimenting more openly.
"How do you (and the participants) work with the homologies between maths and gaming?" - a game is a type of experience, so game design is a type of experience design. It can be just a tool for time and task management (a gamification of some other activity), or it can be an intrinsic way to engage content. I am very interested in intrinsically mathematical game mechanics - in games like Set or Nim. It's very hard to design a game intrinsic to a given math concept.
About the "Synaesthesia" piece: I do think the whole idea of "abstraction" needs a good shake-up. In particular, I work with very young kids on relatively advanced math ideas, such as algebra with three year olds. This Fall I started a series of math circles called Inspired by Calculus for kids ages four to ten. Conventional thinking tells us numbers are concrete, algebra is an abstraction of numbers, and calculus is an abstraction of algebra. But it isn's so if you take the embodied, grounded approach. Kids who can't reliably count to twenty already love infinity, and function machines, and fractals, and covariation grids, and zooming through powers, and...