I would start with blocks, maybe. Then pennies and dimes (reflecting our decimal system). I wrote a flash card like program for a young neighbor that seemed to work but it really needed verbal along with the visual. That's what you should be looking for.
How old is your kid? Start with teaching your kid to count. 1, 2, 3, 4, ...: Note that 1+1=2, 2+1=3, 3+1=4. The number that follows some other number in the sequence 1,2,3, ... is the first number plus one. For your benefit, this is the successor operator (the successor of one is two, etc.) Don't confuse your child with this. Just teach your child that counting and adding by one are very closely related. What if you aren't adding by one? As you know, multiplication in the integers is just repeated addition, (e.g, 2*5 = 2+2+2+2+2). Addition in the integers is just repeated adding by one. For example, to compute 5+4 you find the add by one (find the successor) four times: 5+1=6, 6+1=7, 7+1=8, 8+1=9. Similarly, 4+5 is adding by one five times, starting with four. You might want to work with tally marks here. To compute 5+4, make five tally marks, then make four more. Count the result: Nine. Using tally marks will get around the problem of carries. To compute 6+5 (or 5+6; it is a good idea to show that addition is commutative at some point) put down six tally marks, then five more, then count. At some point you want to move beyond tally marks and counting. You will need to introduce the concept of the addition table. This is rote memorization, and doing that rote memorization is absolutely essential. (The same applies to the multiplication tables). Some math educators have tried to do away with that rote memorization. The result: Suppose you buy something for that costs $13.31. Not wanting a pocketful of change, but not having exact change, you take a penny from the ashtray that holds extra pennies and give the kid cashier that penny plus a ten, a five, and two quarters. Nowadays you have to tell the kid to enter $15.51 into the cash register. Kids don't know how to make change because math educators think rote memorization is evil.
I don't think you need a computer for this. The way I introduced this to my kid was originally by counting. Limit yourself to addition first, I'd say, and make a difference between operational techniques on one hand, and a conceptual understanding. First he/she needs to grasp conceptually what "addition" means. However, before introducing addition, your kid must be able to count fluently. He/she must be able to count 1,2,3,4,.... up to 20 at least, without any help. Your kid must also be able to "count objects". Count blocks, count apples, count pencils. You put 3 pencils on the table, and you ask your kid to count them. You put 4 apples on the table and you ask your kid to count them. You put 7 blocks on the table and you ask your kid to count them. It is important that you can do this with different kinds of objects (blocks, apples...) and that this works fluently without a problem. Once your kid can count objects fluently, you can make him/her count 3 blocks, and 4 blocks. Then you move the 3 blocks and the 4 blocks together, and you have your kid count the whole (the 7 blocks). You can do the same with 3 apples and 4 apples, and then let him/her count the 7 apples. Repeat this a lot. Order the 7 apples or blocks or whatever in different ways. Let him/her discover that each time you had "3" and you had "4", you get "7", whether it are apples or blocks, and no matter HOW you put them together. Call this "addition". Let your kid make his own addition table for all additions of one-digit numbers. First you help him/her a lot, then you let him/her do it more and more by itself, with the help of blocks or ... (this is why your kid must be able to count without problems up to 20). Let your kid make an "addition table" several times (it is pretty long of course). In the end, it shouldn't need any "objects" any more.
This is much the way it is done at school. Counting starts with numbers of objects. The it is a matter of grouping objects into smaller numbers. http://en.wikipedia.org/wiki/Cuisenaire_rods http://educationalsolutions.com/visible-a-tangible-math/mathematical-situations?menuId=79&ms=2 http://www.amazon.com/Cuisenaire-Rods-Intro-Set-Wood/dp/B000FFWCOW/ http://www.teachingsupplystore.com/math_cuisenairereg-c-1011135_1032816.html
Remember using those in first grade ! However, I think they come AFTER the conceptual understanding of what addition is: namely "how many" we have when we "put the heaps together and count again". Those rods are great tools because the bars already represent "heaps". They are even better tools to introduce the subtraction "what rod is missing ?" In fact, there are several conceptual ways to look upon addition and although for us they are of course "equivalent", for youngsters, they represent different conceptual ideas. The concept that D H introduces (I guess coming from the logical definition of the natural numbers :-) ) is a different concept for a kid than what I mention. It should also be introduced (but I'm not sure it should be introduced *first* - we could discuss about that). D H talks about addition as repeated "is the successor of". I talk about "counting the two heaps, and then counting them as one heap". The rods could be seen as pre-defined heaps, OR as abstract numbers and a puzzle which defines an abstract addition independent of "counting" (a bit like "if you mix yellow paint and blue paint, you get green paint" - if you put together the violet rod (4) and the green rod (3) you get the "same as" the black rod (7). violet + green = black. 4 + 3 = 7
I don't think the Cuisenaire rods were around when I was in 1st grade back in the early 50s, but when I first heard of them, they seemed like great devices to teach all four arithmetic operations, making somewhat abstract operations much more concrete. Another possibility is an egg carton with pennies or marbles in the holes. You can demonstrate visually what 2 + 2 + 2 or 8 - 3 represent, or what 12/4 and 1/3 * 6 are.