# 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: You have 3 toy cars and I have 2 toy cars. How many toy cars do we have altogether? Hard Addition: You have 3 toy cars. I have 2 more toy cars than you have. How many toy cars do I have?

## 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 short-cutting 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.

 Counting on is usually the first shortcut that children learn. Here Rose solves an Easy Addition problem by counting on - and then I suggest another method. In this example, Eva solves a Hard Addition word problem using two different shortcuts. In this example Eva solves an Easy Addition word problem involving 2-digit numbers. Along the way she uses two different methods for computing 9 + 7. 