## Tuesday, July 17, 2007

### MKN2(a) & M1N3(d) - Meaning of Addition & Subtraction

MKN2. Students will use representations to model addition and subtraction.
a. Use counting strategies to find out how many items are in two sets when they are combined, separated, or compared.
M1N3. Students will add and subtract numbers less than 100 as well as understand and use the inverse relationship between addition and subtraction.
d. Understand a variety of situations to which subtraction may apply: taking away from a set, comparing two sets, and determining how many more or how many less.

In the previous two posts, we discussed the meanings of multiplication. In this post, I want to go back even further to discuss the meanings of addition and subtraction. Although the GPS does not explicitly state what the meanings of addition and subtraction are, MKN2(a) suggests the meanings that the GPS writers had in mind. According to MKN2(a), and also K1N3(d), addition and subtraction are operations that describe situations in which items in two sets are “combined, separated, or compared.” It is absolutely critical that we keep in mind that human beings give meaning to a new idea in context. Therefore, it is very important that the initial instruction on these operations utilize word problems in which two sets are “combined, separated, or compared.”

Addition is an arithmetic operation used to determine the total amount when two sets are combined. There are two slightly different situations when two sets are combined. In one case, the two sets are actually combined to make one set – that is, the number of items in a set increases. In the other case, two sets are “combined” in our mind, but not necessarily physically. For example, when we ask: Tom has 4 marbles and Carey has 3 marbles. How many marbles do they have in all? we are simply changing our perspective and looking at those marbles irregardless of who they belong to. However, neither Tom nor Carey is losing or gaining any marble. The Cognitively Guided Instruction (CGI) calls the first case as “Join” (or “Combine”) while the second as “Part-Part-Whole.”

There are two major situations for subtraction. The first is when you are trying to find out how many items will be left when a set is removed from another set – this is the take-away situation. Many people equate subtraction as “take away,” but this is not the only case where subtraction is used. Another important meaning of subtraction is to find out the difference between two sets. In fact, the formal term for the answer to subtraction is “difference,” suggesting that this “comparison” meaning of subtraction is perhaps more significant mathematically.

Although we use addition and subtraction when two sets are “combined, separated, or compared,” when we use which operation in each situation depends on what quantity is unknown. That is, even though we have a combining situation, if we don’t know the starting amount or the added amount, we must use subtraction to find the answer. Some people may call this as the missing-addend meaning of subtraction. In the same manner, if we don’t know the starting amount in a take-away situation, we use addition. The important idea we want children to understand, therefore, is how addition and subtraction are related to each other, and this is suggested by M1N3 statement – “the inverse relationship between addition and subtraction.”

So, here is the summary of the important ideas we want Kindergarteners and 1st graders to understand:

- join/combine
- part-part-whole
* meaning of subtraction
- take-away
- compare (to find the difference)
* the inverse relationship between addition and subtraction

In Kindergarten, our emphasis should be more on the meanings of these operations (using numbers to 10). As 1st graders continue their study of addition and subtraction, with increasingly larger numbers, they will also have to understand the relationship between addition and subtraction. However, I believe it is important for us to focus on one idea at a time. That is, if you want students to understand addition with sums greater than 10, let’s focus on that aspect, using only the familiar situations (combining two sets). However, if you want students to understand the relationship between two operations, let’s use numbers with which children are comfortable. A focused lesson should not expect students to learn two new ideas simultaneously.