What would you consider "understanding" addition and subtraction?

[nature girl]

My DD does simple addition and subtraction up to 20 using manipulatives, and counting on her fingers. We've fiddled around with Miquon, and she can also add with the rods. So she understands the basic idea, but doesn't have anything memorized, and can't do any of the sums without counting (other than the very basic sums up to 4, and adding 1 to any number up to 10.)  We're working on counting on, and she's getting reasonably good at it, so once she has that mastered what would your next step in teaching be? Is there any real benefit in memorizing at a young age, or is it just as pointless as counting 1-20 by rote without really understanding what they mean?

[Korrale]

Play lots of games using addition. Facts will come in time. I wouldn't drill her for them yet. Maybe if she doesn't have them down by the time she is 1st or 2nd grade.

 

Work on place value. But even before place value is learnt she can still do multi digit addition by using expanded form or columns as long as there is not regrouping. But I like to teach place value first just because it makes it go more smoothly.

[fdrinca]

I'd play a lot of dice games at this point. There are lots of variations (the website Lets Play Math comes to mind). We began playing a lot of the card game 21 at this point, too. 

[Coco_Clark]

I would consider that understanding the concept of addition and subtraction but not actually understanding addition and subtraction.  It IS understanding counting, which is in itself an achievement :)

She should at LEAST memorize the numbers that add to 10 (1 and 9, 2 and 8, ect).  There are schools of thought that say she should memorize all the addition and subtraction facts within 10.  I think that depends on the child.  Some children are just terrible at memorizing or just terrible at math and that would be barking up the wrong tree (IMO).  It is endlessly useful for children capable of such.

My next step with a child that is counting on toes and fingers would be to understand number families.  IE, 2+3=5, 3+2=5, 5-2=3, 5-3=2.

I would also be solving for x.  IE, 2+x=5.  Some people are very uncomfortable with x in the elementary years and insist on writing such problems as 2+___=5.  I personally think that's silliness but whichever way you do it, it is an important that she be able to fill in that blank.  It will prove she understands number families ;)

I would finally be working on "regrouping".  IE, 8+5 = 10+3.  If she is using rods I'm assuming she understands the concept of "making a 10", but she should eventually be moving on to doing it without the rods.  This is where memorizing the numbers that add to 10 come in handy.     

 

After that I would take a break with addition and subtraction and work on things like shapes, measuring (length and weight), telling time, and money.