Improving at Chess Tactics - train the visual and the logical
There is both a visual and a logical piece to chess, and both need to be trained.
Beginners need to know basic tactical motifs -- both what they look like (visual), and how they work (logical). For the visual piece they need to know, for example, that a bunch of pieces in a line could be a tactic (pin, skewer, discovery, fork, etc), pieces on the same color square as the knight can be forked, or what loose pieces look like. Then they learn the logical piece that generate candidate ideas like "attack pinned pieces", and "loose pieces drop off".
Once a beginner has these basic concepts down, and has been taught to be aware of/look for (the visual piece) these things in their games, the next piece is the calculation step (or what some people call "looking ahead").
Say a beginner who has been taught to be aware of basic tactical patterns "sees" a bunch of pieces in a line, and thinks it might be a pin. First of all that beginner needs to stop right there and do a happy dance! Visually seeing patterns in games is a huge step for beginners. It is frustrating as a more experienced player to watch beginners play, and you "see" lots of potential (even though you might not have calculated any actual moves), but the beginner is completely clueless as to what opportunities are right in front of him. As you improve, you learn to see more and more, and hopefully miss less and less. There is a great quote about Kasaprov's ability to see things:
"I thought I was playing the World Champion - not some 27 - eyed monster who sees everything." - (on losing May 1986 match 5-1/2 - 1/2) - Tony Miles
Beginners have problems "seeing" things on the board in front of them. More advanced players have problems "seeing" things several ply into the future -- for example, a series of captures could lead to a mate pattern that would be completely obvious if on the board in front of you, but is much harder to "see" deeper into al ine. Exact same problem though.
Ok, so back to the example...our beginner finished his happy dance, and goes back to looking at the pieces in a line. What now? This is now the "logical" piece, and where training often fails for beginners. But this process is simply calculation and every beginner can learn some basic stuff. The process is to generate a list of candidate moves, calculate each one to the end, evaluate the final position and pick the move that leads to the best evaluation. It's not actually that hard to do, and there is a very specific logic used to calculate that anyone can learn.
Continuing with the same example, our beginner needs to compile a list of candidate moves. In order to do that, he needs candidate "ideas". Luckily he remembers something he heard once: "attack pinned pieces". So he starts to search for moves that attack the pinned piece. Most beginners can get this far -- and it is at this point they begin to fail. They find one move that attacks the pinned piece, do another happy dance, and make the move...wondering how in the world his opponent is going to reply to this vicious tactical blow.
And then he gets mated.
The beginner needs to learn to do a few things. First, identify as many candidate moves as possible based off of the candidate "ideas" you found when looking at the position. Look wide first, not deep. You should always try to force yourself to find more than one candidate move.
Second, calculate each candidate move to the end. Look for your opponent's refutations first. I have a blog on how to do line calculation (I taught my young daughter how to do this -- I call it the "logic of forced variations").
Third, evaluate the final position. Beginners should just stick with a basic material-only evaluation, and as you get more advanced you can add "activity" to your list. But keeping track of what happens to your pieces and pawns during an exchange is really hard for beginners to do accurately. Evaluation is a skill that will come with practice, and evaluating future positions can be pretty daunting.
Finally, having evaluated the final positions you either scrap both candidate moves because they suck, and you have to start all over in looking for candidate ideas, or find you get something good in one (or more) of the candidate moves and make that move.
The visual and logical are indeed very different skills, and beginners should take the time to learn how to calculate variations as part of learning tactics.