Think of shooting an arrow into a bullseye.  Most of us can see the bullseye; that's not the problem.  The problem is getting the arrow to hit the right spot.  Anxiety makes this problem worse.  Most of us are not paralyzed by an overwhelming shame and self-consciousness if our arrow misses the target.  But singers do feel ashamed when they have trouble hitting the right note.  Sometimes this shame is so potent it makes it almost impossible for the singer to concentrate on what they are hearing.  Furthermore, when each arrow gets closer to the center of the target, we can clearly see that we are doing better.  But when we are singing the wrong note and moving closer to the right one, the sound will actually seem like it is getting worse until the moment when we actually find the right note.   This has to do with the way the different frequencies interact.  What is more frustrating for the would be singer than recognizing that they note they are singing seems lower than the right note, but as they move higher, it seems to be getting worse instead of better?  This is how people get stuck.  

My method for teaching people to match pitch involves separating "listening to sound" from "creating sound."  It is also designed to free you from the paralysis created by singing the wrong note.  First you will listen to me play a note on the piano and then sing a note.  I will ask you if they are the same note or two different notes.  Some people are confused by the many ways in which sounds differ besides pitch (loudness, color, etc.)  Then we will isolate which note is higher and which is lower.  You will hear what happens as I move closer to the target note.  I will have you identify the moment when the two sounds become the same.  Then we will practice having you produce sounds with your voice and identifying whether they are the same, higher, or lower.  We will practice sliding the voice up and down and learn how not to let the clashing sounds stop us from moving our voices.  We will identify the moment when the two sounds become the same and will practice getting to that moment faster.  Once you can match pitch reliably, we will learn how the different notes in a scale relate to each other and you will be on your way!

This method works best when it is approached like a game.  It can be tedious and requires patience on the part of the student as well as the teacher!  It works the same way for children as for adults.  Sometimes children learn faster because they don't have to contend with a lifetime of shaming messages that they can't sing.  Adults who are willing to sing things wrong without shame can eventually learn to sing things right.  Take basketball this time.  If the first time you missed a basket, you experienced as something "ugly", would you have tried again?  How much more willing would you be to practice and learn if missing the basket was just part of the process, and not something to be ashamed of?  Hitting the wrong note is just like missing the basket, except most people are so ashamed at missing that they give up.  But if you don't give up and keep trying, your aim will improve.