make it equal to the row of 7 and write their addition 5 + 2. This 2 is the difference between 5 and 7, it is the amount extra 7 has, so such exercises help children begin to see this third quantity in the comparison situation.

Writing Equations

There is not sufficient evidence to indicate the best time for teachers to start writing addition and subtraction problems in equations or for students to do so. The equation form can be confusing to some students even in Grade 1, and students may confuse the symbols + = and −. This confusion and limited meanings for the = sign often continue for many years and are of concern for the later learning of algebra. Because the fundamental aspect of an equation is that the sides are equal to each other, it is important for children to learn to conceptually chunk each side. Thus, some children may need extensive experience just with expressions, such as 3 + 2 or 7 − 5, before these are used in equations. These forms might be introduced before the full equation is introduced, perhaps even with 4-year-olds. It may also help for the teacher to circle or underline these expressions to indicate that this group of symbols is a chunk that represents a single number. Future research directed at such issues of when and how to write such pre-equation forms would be helpful.

The other issue with equations is the form of the equation to write. As mentioned earlier, it is important for later algebraic understanding of acceptable forms of equations for children to see equations with only one number on the left, such as 6 = 4 + 2 to show that 6 breaks apart to make 4 and 2. This equation form can be written for take apart situations in which the total is being separated into two parts, for example, Grammy has 6 flowers. She put four flowers in one vase and two flowers in the other vase. Children can show this situation with objects or fingers (Count out 6 objects and then separate them into 4 and 2) or make a math drawing of it while the teacher records the situation in an equation. This form can also be used in practice activities with objects in which children find all of the partners (addends) of a given number. For example, children can make 5 using two different colors of objects, and each color can show the partners. The teacher can record all of the partners that children find: 5 = 1 + 4, 5 = 2 + 3, 5 = 3 + 2, 5 = 4 + 1. This can be in a situation (Let’s find all of the ways that Grammy can put her 5 flowers in her 2 vases) or just an activity with numbers (Let’s find all of the partners of 5).

Change plus and change minus situations can be recorded by equations with only one number on the right because that is the action in these situations (see Box 2-4), for example, 3 + 1 − 4 or 5 − 2 = 3. In these equations the = sign is really more like an arrow, meaning gives or results in. As dis-

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