Addition
Your child is
likely to see two types of addition problems in school. I call one of
those types Easy Addition and the other Hard Addition.
The two types are distinguished from one another by the way in which
each is modeled. The two types are discussed below; but first, here is
an example of each. 
Easy Addition
1. What Is The Model For Easy Addition?
Here is the simplest way to model the Easy Addition problem above:
 Count out 3 objects to represent the cars that "you" have,
 then count out 2 objects to represent the cars that "I" have,
 then join together the two sets of objects that represent the toy cars,
 and finally count those two collections altogether.
2. How Do You Teach Easy Addition?
You can help children with Easy Addition by posing simple problems for them to solve and then helping them make models and count.
 Here, at 2 years and 8 months, is an example of me helping Trixie to understand what she is supposed to do to solve an Easy Addition problem.
 By the time that she is 3 years and 4 months Trixie knows how to model Easy Addition word problems  she has stopped making the kind of mistakes illustrated above. Click here to see an example.
Hard Addition
1. What Is The Model For Hard Addition?
Here is an example of a Hard Addition problem:
Suppose that you have 3 toy cars and that I have 2 more toy cars than you have. How many toy cars do I have?
The simplest way to model this problem is this:
 Count out 3 objects to represent the toy cars that you have,
 count out another 3 objects to represent the toy cars that I would have if I had the same number of toy cars that you have,
 count out an additional 2 toy cars for me,
 and finally count how many toy cars I have altogether.
2. How Do You Teach Hard Addition?
Here is an example of me helping Trixie learn to model a Hard Addition problem. Although at 3 years and 4 months she had a pretty good understanding of Easy Addition (see the example above), she had little or no understanding of Hard Addition. 
Summary and Additional Comments
Easy Addition and Hard Addition are not the same
thing. Easy Addition
always involves two sets (for example, a set of 3 toy cars that you
have and a set of 2 toy cars that I have) and the problem involves
joining those two sets and finding the number of members that the join
contains (in this case, the total number of toy cars).
Hard Addition always involves one set whose size is known (for example,
3 toy cars that you have) and a second set (the toy cars that I have)
that is described only in terms of its relationship to the first set
(for example, I have 2 more toy cars than you have). The problem is to
find the number of members in that second set (the number of toy cars
that I have).
Click here
to download a printable PDF summary, in a new window, of the models for
addition word problems.
Click here for a
further discussion of the classification of word problems.
Shortcuts
Once your children have learned to model Easy
Addition and Hard
Addition they will start, on their own, to use shortcuts. For example,
they may realize that they can find 8 + 2 by “counting on” (8, (pause)
, 9, 10). Or they may reason “5 plus 4 must equal 9 because I remember
that 4 plus 4 equals 8.” When children do this kind of work they are
doing what mathematicians do – they are solving problems by reasoning
logically.
Mathematicians are not people who are good at memorizing a
whole lot of facts and formulas and procedures. What they are good
at is solving problems by reasoning. When children are taught
arithmetic by having them remember facts and formulas and procedures
they are not learning mathematics, they are not learning to think. They
will be much better off if they are encouraged to use their own
shortcutting procedures and helped to do so. My experience has been
that children like to reason and that they are surprisingly good at it.
Once children know how to model word problems they can (with help) use
their own ability to reason to find better ways to solve these
problems.
The examples below illustrate the approach.
