I know a fair few teachers of various different subjects, and as such have followed the ongoing discussion concerning (post-)modernising education with the massive digital toolset available for use nowadays. Games seem to come up every now and then (granted, they've been on the table for a while, but have not been discussed this actively ever before), which obviously serves to increase my otherwise mild interest tenfold. I also have some personal insight into the topic, since I spent last winter running game design workshops for 7-14 year olds in local schools and merit games with having taught me all sorts of things as a child.
First of all, I think the whole concept of ”educational games” approaches the beast from the wrong direction: downwind. If you start making a game with education in the forefront of your approach, especially for children (who can smell teachers a mile away), you're bound to get it wrong. Indeed, most educational games take a topic from a curriculum, and attempt to camouflage it into a fun game. This doesn't work. However, games can and do educate, they just do it in a different way.
When running the workshops, I often asked the kids whether they had played any educational games, such as the ones found in various study books. Most classes had every hand in the air at this point. Then I inquired who thought these kind of games were good, and the sound of tiny hands slamming into desks was almost deafening. Children the world over know that a spoonful of sugar does not make the medicine taste any better, but rather is a consolation prize for suffering something unpleasant. Children also know that a good game is fun and entertaining, something which educational games tend to universally fail at.
The next question that I posed my captive (very literally, it wasn't recess yet...) audience was whether they felt they'd learned anything from playing games. The answers were at least as varied as the different classrooms I visited, but there was a general pattern. English language was one of the major trends that came up every time. Communication and co-operation in a group were not far behind, surprisingly. There were all sorts of other answers, like patience, manual dexterity and resource management, but the main thing is that there were lots of different things that the children were themselves aware of having learned through games they played for fun.
This clearly shows that games and learning are not anathema to each other, but rather exist in a pseudo-symbiosis of sorts. All games require the player to acquire a certain set of skills in order to do well. Some of these skills translate poorly to the real world, such as learning enemy behaviour or memorising level maps. But a fair number of skills needed in games do translate into real world abilities, for example learning the English language is a necessity for various types of games (at least locally, nobody translates games into Finnish).
The relevant thing to notice here is that kids learn things in order to play their games better, but they don't play games in order to learn. If we return to my silly hunting analogy this means that if we design with education in mind we must approach the game from upwind (see what I did there?). The thing you're teaching cannot be the goal of the game, but rather part of the skill set needed to advance to the goal. The difference is subtle but relevant, and some topics accommodate this change in modality better than others.
As for personal experience, I've mentioned before that a large portion of my own English skills have been acquired through gaming. When I was a kid Indiana Jones and the Fate of Atlantis had me glued to the screen with a dictionary in one hand and the mouse in the other. Learning English was necessary to advance, so I did my best with what resources I had to muster. Had algebra or calculus been similarly necessary to access a riveting play experience I might be better with numbers now. Incidentally, I'm rather bad at mathematics apart from probabilities, which I had to learn in order to handle tabletop-RPG rules and understand dice (d100 all the way!). Thus my personal observations entirely support the idea of what we might call incidental learning in gaming.
In short: kids learn for games, but not so much from games. Also, I think most adult gamers and designers would be surprised with the level of understanding and introspection going on in young minds. Children nowadays analyse games at a very advanced level very early on, and some of the game concepts they can come up with could very easily compete with actual developed and published games, had they the chance to do so. We shouldn't underestimate children, they're just small versions of us.