Here is a synopsis of about 30 minutes of a lesson on some math symbols. We had also done this several times before. This student is Autistic which doesn't mean can't learn, it means takes more time and impressions to get info into the part of the brain where it can be recalled. This one is often missed by students who have made the rule hard and fast in their heads two negatives equal a positive: -|-3| because they don't understand the concept of absolute value and fail to read the math symbols correctly.
When doing absolute value I often just talk about the block itself doesn't matter what side it's on a four block is "four" long. In her case though this explanation didn't sit as well because she wanted to be sure the block was on the positive side since all absolute values are positive. Other students seem to get the idea of "it's four from here to here" no matter which way you go. This also explains why the difference of two negative numbers is still positive.
Bright young students might inquire why it is that -22 - -28 = 6 where all the numbers are negative and we're even subtracting! The answer somehow ends up positive.
The difference can be explained in terms of direction on a number line, but as far as I have seen number lines and direction help confuse things, not clarify them. It's not about direction, it's about distance. The amount of "space" between the numbers, delineated by numbers themselves. Most kids don't ask. Might be wise not to bring it up especially with students who have poor self confidence in math. For the inquisitive ones and older students it can quickly devolve into a discussion on philosophy. How do you know the distance from here to here is four...? Keep it simple instead.
Explain what the math symbols mean and use blocks to bring the points home. Integers are no problem if presented properly. Look for more lessons on integers here on this blog and at the house of math where Sarah has her own page. The way I see it if you can explain it to Autistic students or little kids in a way they can understand then adults and older students as well as "gifted" students should be duck soup. Most of the time this is true. Often you have to get through math anxiety and other problems first with the older students.
Bottom line keep it simple.
“If you can't explain it simply, you don't understand it well enough”. ~Albert Einstein
I have made 100's of dollars re-explaining integers and reading symbols to all manner of students and have almost universally heard that using blocks makes it easier. Of late I have run into a lot of students who just want something to memorize rather than wanting to understand...understanding concepts is better.