If you child isn't understanding the standard definition of subtraction (going backwards or taking away), don't fret. There is nothing wrong with you as teacher, and there is nothing wrong with your child as a learner. Most likely, they just think about numbers differently. This is a good thing. All great mathematicians have thought about numbers differently - that is why they were great.
How your child thinks is sometimes very apparent. In other kids it takes some trial and error. Don't give up and don't try to force something that isn't working. It is just as important to look at what works as what doesn't. Does your child like sports, wiggling constantly, dancing, or movement; explain with manipulatives you have to move around or have them step backward and forward as if they are personally moving in space. Is your child good at English, words, definitions, vocabulary, stories; explain it by relating it to words and the nuance of meaning. Is your child good with pictures, drawing, artistic expression, and the relationship of concepts through art; explain it visually with manipulatives which show the concept. Is your child a big picture thinker, needs to know how it all works, likes to tinker, wants to see how thing relate to one another on a larger scale; explain it by relating the numbers to one another in the space on a number line.
Explaining things for my son's different learning styles was definitely a growth space for me as a parent. It has done wonders for my understanding of the subject and also really helped me develop many areas of my brain which I never used.
Explanations of Subtraction
Standard definition: Subtraction is counting down or going backward on the numberline. You subtract three, you go backwards 3, you count down 3. This is the one most people always think of about subtraction; it is when you lose something or give it away. 8-3 =5 looks like this:
English definition: Subtraction is like saying "or." You are separating a group into its pieces. Instead of wanting this and that (addition), you want this or that. You are separating the group up and isolating part of it. You start with an amount, then separate part away, keeping one part for yourself. You want either one part or the other. In a story problem you should use subtraction if they only want to know about part of the group. 8-3 = 5
Movement definition: This is done the same as with the English, just have them physically separate three items from the group. Some kids need to actually move the three manipulatives. 8-3 = 5
Big Picture Thinkers definition: Subtraction is the difference in between two numbers. It is describing how many things are in between. Eight minus three is the same thing as saying how many numbers are there if you start at eight and go till three. In a story problem, you use subtraction when people want to know how much more or less of something - when they want to know the space in between two items. 8-3 = 5
Visual definition: Subtraction is a way of doing reciprocal algebra (doesn't that sound fancy! Most every adult I know does this regularly; it isn't as fancy as it sounds). 8 minus 3 is the same as saying "what do you have to add to three to get eight?" Later on, everyone does this in algebra with 3+x=8
If you think about subtraction in terms of adding you are basically saying: You had two groups of numbers and added them together (this would be the top number in a subtraction problem), now you are taking one of them away (the bottom number), how many are left (the answer)? In example, eight minus three is like saying, "You added three and five to make eight. Now you are taking the three away, how many are left?"
Since subtraction is splitting a group up into parts, another way to say it is: if eight is the answer and one of the pieces is three, what is the other piece? If a child has a strong foundation of addition, this causes a very logical jump to how subtraction is the opposite (reciprocal) of addition. It also helps later when working with algebra because the concept is basically 3+x=8.
Manipulatives really helped when trying to explain to the Barracuda what I was talking about when we used the algebra method. I took two manipulatives, declared one 3 and one 5. I pointed to one, he said,"Three." I pointed to the other one, he said, "Five." We did this like four times. By putting the manipulatives together, he could see they made 8. We did this like four times. He would say, "eight." He thought I was crazy, we had covered this already and it had nothing to do with subtraction as far as he was concerned.
Next, we started with both manipulatives together. How many? "8," came from him. I pointed to one and asked him how many. "3," he would say. I pointed to the next one, "5," he would declare. I then took the 3 manipulative away. The answer was plainly 5. We had to do this demonstration a good six or so times with different numbers on different days. The concept seemed so simple to me, but requires quite a bit of knowledge chaining. It is a jump in concepts and you have to link them together, but it helps quite a bit later on.
I told him each time we did the demonstration that it was okay if he didn't understand. For the first three times it was obvious he had no clue what I was trying to say. By fourth and fifth time, the glimmer was there, he knew the answer but how he got it. He couldn't apply it at all. The sixth time, you could tell it all made sense in one big "A ha!" moment. He was obnoxiously proud of himself! Now he talks himself through subtraction problems by saying, "How many do I have to add to 3 to get to 8?"
Hopefully this is helpful to all those frustrated parents out there. Far too many times, kids learn to bluff their way through in school because they don't want to say they don't understand or they become frustrated with themselves. Unfortunately, by the time I would get most of these children in high school they had already made up their minds that they "just weren't good at math." I have yet to meet even one with which that was the case.