I have no opinion on the pedagogical method. I just enjoy how nicely the game lets you play your way through some of symbolic algebra. Similar to how Nico Disseldorp's game Ancient Greek Geometry (http://sciencevsmagic.net/geo/) lets you play your way through its sacred namesake. I wonder if anyone has collected games that let you play your way through other parts of math. Personally, I'd love to play a game of group theory.

Just goes to illustrate that relatively advanced concepts can actually just be used as another "game mechanics"

I believe that getting a "feel" for abstract ideas in an almost physical way is how I experience maths and logic myself. Formulae or equations or expressions aren't abstract, they have their own laws of nature, just like picking up a screwdriver and easing in a screw to just the right torque, or getting solder flowing cleanly. There's a mental tension when things aren't done, relieved by a simplifying or cleaning up process. Maybe I'm not expressing that well: anyone else know what I'm getting at? 😊

I suppose my main frustration with DragonBox Algebra is that it's teaching algebra in much the same way I learned it: as a more or less arbitrary system of rules. Fun as far as it goes, but there are better puzzle game mechanics to be had. (Don't want to get into what makes for a good game just now. Though I was guest lecturing about it last week. I mean it's a topic I have more than a bit to say about. Allotted two words, I'll go with "The Witness" and leave it at that.)

The quote https://en.wikiquote.org/wiki/Algebra is apparently:

Algebra is the offer made by the devil to the mathematician. The devil says: `I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.' Michael Atiyah (2004). Collected works. Vol. 6. Oxford Science Publications. The Clarendon Press Oxford University Press. ISBN 978-0-19-853099-2.

Should we be selling our children's souls so early? In my education at least, Algebra's "arbitrary" manipulations are sort of a hoop jumping meta-game that culminates in Freshman Calculus (or AP Calc BC I think it was).

I suppose the most charitable read is that it's all a bunch of My. Miyagi chores in preparation for Physics.

For me, personally, I had a long journey feeling early on that the grand book of philosophy stands open written in the language of mathematics. Took a degree in mathematics (insufficient) with a followup in philosophy (better) to get oriented (Cartesian Closed Categories helped, see https://www.amazon.com/Conceptual-Mathematics-First-Introduction-Categories/dp/052171916X).

Going back to Algebra, I would like to see games connect notions a bit better. Montessori materials have an excellent design in this respect. (A Montessori classroom is an excellent example of game design by the way.) Most every activity has a mathematical element more or less developed. To explore group theory @Duncan Cragg, turtle graphics do have some nice affordances. For something more game-like, Engare http://www.engare.design/ is a good choice.

I suspect one key difference between all these examples is the way in which they guide you through the play space. (As you mentioned above — degrees of freedom.) DragonBox and Engare are very much “on rails”. 4D Toys is a total sandbox (and you have to tie one arm behind your back). I think there might be a happy medium.

For instance, the tutorial area, without talking at you, shows you all there is to the primary puzzle system (start at a circle, end at rounded nub, a puzzle can have multiple entrances, multiple exits, they light up wires to follow around the world). You do this all to open a gate. Later you can climb back around above the gate, to hear quote from Arthur Eddington. It serves as a sort of epigraph explaining in words what you have experienced more directly:

"I am standing on the threshold about to enter a room. It is a complicated business. In the first place, I must shove against an atmosphere pressing with a force of fourteen pounds on every square inch of my body. I must make sure of landing on a plank traveling at twenty miles a second round the sun — a fraction of a second too early or too late, the plank would be miles away. I must do this whilst hanging from a round planet head outward into space, and with a wind of aether blowing at no one knows how many miles a second through every interstice of my body. The plank has no solidity of substance. To step on it is like stepping on a swarm of flies. Shall I not slip through? No, if I make the venture one of the flies hits me and gives a boost up again; I fall again and am knocked upwards by another fly; and so on. I may hope that the net result will be that I remain about steady, but if, unfortunately, I should slip through the floor or be boosted too violently up to the ceiling, the occurrence would be, not a violation of the laws of Nature, but a rare coincidence. These are some of the minor difficulties. I ought really to look at the problem four-dimensionally as concerning the intersection of my world-line with that of the plank. Then again, it is necessary to determine in which direction the entropy of the world is increasing in order to make sure that my passage over the threshold is an entrance, not an exit. Verily, it is easier for a camel to pass through the eye of a needle than for a scientific man to pass through a door. And whether the door be barn door or church door it might be wiser that he should consent to be an ordinary man and walk in rather than wait till all the difficulties involved in a really scientific ingress are resolved."

Overall, there are some forty quotes, maybe 46, that help put you in the headspace of what the experience is driving at.

DragonBox uses game mechanics to do some algebra. The Witness uses game mechanics to do some philosophy.

I think that the general shape of “find one solution that produces this outcome” is a decent way to put learning into a game. Zachtronics games are another example of this – in those games, there are leaderboards for optimizing your solution, which I think is a neat addition.

I’m finding it difficult to think of a learning system which isn’t based around arbitrary rules. Learning how to navigate social situations seems like an extremely wholesome and valuable set of rules in that they enable you to connect with other people, but doesn’t seem intrinsically more or less valuable than another set of rules in terms of learning. (Is internalizing how Mario moves, jumps, slides more valuable than internalizing how to solve a puzzle from the Witness? After all, you could apply Mario’s physics to the physical world.)

His Boss keys series about Zelda is excellent too https://www.youtube.com/playlist?list=PLc38fcMFcV_ul4D6OChdWhsNsYY3NA5B2.

The Witness itself draws attention to how playing the line puzzles is a bit silly. In this this beautiful world, why are you walking around looking at screens? (And why have screens been placed here so haphazardly to begin with?)

When mentioning platformers above, you described them as having a "groupish" dynamic. I didn't follow that point — would you mind expanding on it?

(I'm very fond of platformers, metroidvanias, and the like. Celeste and Hollow Knight were two of my favourites this year. So whatever sense you were getting at, I've probably felt it, but might not know how to capture it in language. Curious to hear what you meant.)

@Ivan Reese about groups. You often have a few platforms moving at different rates where you have to wait until they line up. The individual platforms are like cyclic groups. The combined system is a direct product. It's a great way to get an intuitive feel for the least common multiple.

Have I 100%ed the Witness? Oh, dear friends, several times. The Witness is my favorite game by far. Watched a few others play through too, have a YouTube backlog. These days, I'm working on 200%ing so to speak. The Witness consists of a series of levels of abstraction, spheres if will – spheres with spheres, bubbles in a stream. There are six worlds https://en.wikipedia.org/wiki/Mysterium_Cosmographicum. The first two give you explicit rewards, that makes 100%. After that, none: this isn't Fez. The first is a tutorial for the second. The second is a tutorial for the third. Jonathan Blow has talked about the three levels and alluded to others. They're on prominently display on the walls of the Monastery. The third level is made from perspective Easter eggs. In the fourth, the previous three levels combine into vignettes. The fifth is a grand epic. It includes the history of the world. And the sixth is unattainable.

For an example vignette, visit the Chapel in the Quarry area. You can find in installation of The March of Progress extended to a Gnostic conclusion under the watchful gaze of TJ Eckleburg. I like to call the Witness the Ulysses of video games because they are similarly silly and obtuse and a literary gift that keeps on giving.

They also took roughly the same amount of time to complete, which is surely a coincidence as so many things are.

A challenge with making geometric construction (or anything with its own rules) into a game is that you're constrained by those rules, which may not lend themselves to puzzling things out. The trick is in teaching through practice, not just having the practice.