My teaching approach is rather abstract, but I try to get to the nitty-gritty of Go and let the learner apply this knowledge on his/her own.
When I teach the game to a new learner, I explain the 3 rules - placement, capture, and ko - after which I proceed to illustrate the rule of liberties and capture by creating different stone group shapes, from 1 to 4 stones, and place them in different positions in the center, edge, and corner. This way I show that the more liberties a group has, the more turns needed to capture it. Relative positioning changes the number of liberties for a given shape of group. Also, I subsequently show that it is easier to capture a group with few liberties and no potential for growth than to capture one that, regardless of the number of actual liberties, has the potential to grow in size and therefore get more liberties. I know that some will say that ko should not be mentioned till the new learner has discovered it during play, but in my opinion it is better to mention and illustrate ko, then leave it alone till it arises in a game.
Once the rule of liberties and capture has been understood some, then I introduce life & death. Life & death is simply assessing whether a group can be legally captured or not and, if this status is undecided, how to proceed toward a desired result. If a group can be legally captured and therefore is subject to capture at any time, it is considered dead. Otherwise, if it is immune to capture, it is considered alive. A group with two or more complete (solid) eyes is considered alive as it cannot be legally captured; the only way to capture such a group is to play two or more stones on the same turn and that is not allowed. On the other hand, a group consisting of only a complete eye and an incomplete (false) eye can be made subject to capture; the stones making up the incomplete eye can be captured, thus reducing the entire group - including the part with the complete eye - to one liberty, which puts it into atari or at least subjects it to capture at any time. If a group is big enough that it can make two solid eyes at any time and becomes too large to capture, it is also considered alive. Keep in mind that a group must be completely surrounded and isolated from other groups in order for it to be killed (made subject to capture at any time). If the surrounding chain can be broken and the group under attack be connected to safety, then it becomes alive.
Everything in a game of Go is ultimately based on the rule of liberties and capture. In the opening stage of the game, one seeks to lay claim to the largest areas for developing the potential to produce the groups with the largest amount of liberties for a given number of stones. This need for efficiency is why the traditional order of play is corners, sides, and center; it is easiest to establish a living group in the corner with the fewest moves. In the middle game, there are more stones on the board, but the number of points on the board has not increased (i.e., still 361 intersections minus all currently occupied intersections), so of course there is fighting, namely to increase the potential for growth of one's own groups and reduce the same for that of the opponent's groups, through which available liberties are increased and more liberties means greater chances of staying on the board. Sometimes this means killing groups outright or reducing their potential maximum size. The killed groups (groups that had been unconditionally subject to capture for much of the game) are removed from the board at the end of the game and are either not counted (in area scoring) or counted as captures plus lost empty points (in territory scoring), whereas living groups are counted as points for the side holding them. In the endgame, liberties for groups are increased or decreased in any part of the board where 2 enemy groups are in contact and on a smaller scale than in the middle game. However, games are often won or lost through an accumulation of said small-scale gains or losses. When both sides conclude that it is no longer possible to increase one's gains (i.e., safely occupy more intersections), then play stops and both players pass.
This is like when two groups of people fight over a piece of land. As both sides grow in populations, there is less and less for everyone, so sooner or later fighting breaks out and large numbers of people die, but it is always possible to survive by uniting with groups that have the greatest potential to survive till the end of the conflict.
I often prefer to start new learners off on capture Go if they have not had any previous experience with the game. The idea of territory as occurs in the regular game is often hard to understand and apply. Go is of course a game of territory and there are two ways to make it. Either one surrounds by demarcation with stones or one kills or captures stones outright, so that the empty points on which there were once the opponent's stones now belong to the side that just captured them. And it is not possible to make any territory if one's stones keep getting captured. Thus, I tend to introduce capture Go as a way to allow new learners to gain experience in capturing and protecting stones, a very basic skill in the game. Another thing I teach is to count liberties. By counting liberties for each of the groups on the board at a given point in the game, the player can determine which group to try capturing first. It is easier to capture a small group that is almost entirely surrounded and has no access to space for further growth and thus getting more liberties, than to try capturing a small group that has ready access to space in which to gain more liberties. At the same time, though, one must prevent one's groups from being surrounded and captured; otherwise it's just making the game easier for the opponent. Why make things easy for the opponent?
Some do say that capture Go is not advisable, but I think that if attention is drawn to protecting one's own groups in addition to capturing, instead of merely trying to capture, then instead of being merely obsessed with capturing, both players get to learn how capturing works and thus conclude that there will be times when capture just isn't possible. Another problem is when new learners try to capture the biggest group possible without first paying attention to the integrity of their own groups. The chain of stones doing the capturing has to be intact or it will be impossible to capture anything, like when a fisherman uses a net to capture fish, but the net breaks and the fish escape. In capture Go, it's easier to make a few captures at a time than to try capturing a lot at once.
When capture Go becomes boring because it is no longer possible to capture X number of stones due to both players being proficient in protecting their own groups, or if a player does not wish to play the capture game, then I move over to the regular game. One problem with new learners when they play the regular game is that they never know when to pass. So what I do is that I play some games of regular Go on 9x9 board, but instead of passing we play on inside enclosed territory till one side runs out of liberties. This does two things. One, the importance of having more territory than the opponent is inculcated; if you have more territory, you have more liberties so that while your opponent has run out of liberties, you still have some. Two, when it is not possible to increase the size of one's groups by even 1 liberty and all that can be done is to play inside one's own territory, it's time to stop playing and call pass.
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