Much of what students do in mathematics across the years involves four basic operations: addition, subtraction, multiplication, and division. Within these four operations we really can consider two pairs of operations since
- addition and subtraction are inverse operations (each “undoing” the other)
- multiplication and division are inverse operations.
As students’ mathematics skills increase, so does what we ask them to do in the operations. (NOTE: As I write, I shall concentrate on the process of addition, but please keep in mind that what I am saying applies equally to subtraction since it is the inverse operation of addition.)
Young children begin by adding whole numbers such as 2 + 5. Put into a context, it might be a question such as this: There were two birds in a tree. Five more birds came and perched in the same tree. Then how many birds were in the tree?
Later children begin adding multi-digit whole numbers. They might be given a problem such as this: 257 + 48. If children write this problem in the typical “stacked” format, we teach them to line up the numbers, beginning on the right.
Decimal addition follows soon after. This time the problem may look like this: 2.57 + 4.8. Again, such a problem is typically “stacked”, and we teach students a rule that has them lining up the decimal points.
Adding fractions brings about a new situation, and typically a new rule. If students are given a question such as 2/5 + 1/2, we teach them about finding a common denominator, changing one or both fractions (depending on the particular question) into equivalent fractions with the same denominators, and then proceeding to add the numerators. In our example we find the common denominator of 10, change the problem to be 4/10 + 5/10, then add to get 9/10. The common denominator rule must be followed if students are to arrive at a correct solution.
Students soon encounter algebra and its accompanying variables. Now they see problems such as 3x + 4y + 2x and we teach another rule: you always add “like terms”. Thus the example can be simplified to 5x + 4y, but these two terms cannot be combined because they are not alike.
I propose that the idea of adding “like terms” or “like things” is not new to algebra. In fact, it is exactly the same rule we have been using in every situation. We just have not been calling it that, but maybe we should!! How different would students’ understanding be of all of the addition and subtraction situations if we continually pointed out that we always, always, always add and subtract like things.
For young children, we must add things like apples to apples, birds to birds, pencils to pencils, etc.. I once worked with a young girl and asked her to make up a word problem for 2 + 5. She looked out the window from where we were sitting and she saw some birds in an apple tree. She began her story, “There were 2 birds in a tree and 5 apples…”.Then she paused, looked at me, and stated: “That doesn’t make sense!” I asked her what would make sense, and she responded, ” I think I should have apples and apples.” She was right. It only makes sense to add things that are the same. Even when we add 12 boys in a class to 14 girls, we end up saying there are 26 children or students. We actually had to find a “same name” or a “like term” for them if we were going to add them.
When it comes to adding multi-digit whole numbers, why is it important to “line up the right side”? Because in writing the problem in that manner we are setting it up so that we can add ones to ones, tens to tens, hundreds to hundreds, etc. This principle is the same for why we “line up the decimal points” when we add decimal numbers. We are ensuring we add ones to ones, tenths to tenths, hundredths to hundredths, etc. In our place-value-based number system, we must add like values, or like terms.
Students are often confused when doing an operation on fractions. They have trouble remembering when they have to change the denominators and when they can go “straight across”. I feel there would be less confusion regarding fractions if they knew the big idea about addition and subtraction: namely, that you always add and subtract like things.
By the time students begin doing algebra, it should be easy understanding about adding “like terms” since that is what they have been doing the whole time in addition, if only we would help them see that!
Whatever the grade you teach, whatever the level of mathematics, I encourage you to help your students understand this big idea about addition and subtraction: We ALWAYS add or subtract LIKE THINGS!