Two ways of looking at addition

Consider the following problems:

Mike had 5 marbles. His friend gave him 3 more marbles. How many marbles does he have now?

Sam has 5 red marbles and 3 blue marbles. How many marbles does he have in all?

You're all over age 8, and so these problems look to you as if they are of the exact same difficulty: the numbers are the same, they're both addition, and the situations are easy to imagine and visualize. But, if you were 5, there's a good chance (maybe about 20%) that you'd get one problem right and the other one wrong. So, what's going on here? If you slow way down to imagine both situations, you'll see that in one case there's an action--a process--that's explicitly described, and in the other case there isn't an action at all--or at least the action is hidden in the code words "in all".

Look at the first problem: Mike has 5 marbles (ooooo). Then he gets 3 more marbles. Where do those 3 marbles go? Why into Mike's bag/pocket/toy box of course: (oooooooo). Now the set we started out with (Mike's marbles) is bigger. If I have something to act out/keep track of the marbles for me, I should be able to count them up and tell you how many he has now.

OK, on to the second problem: Sam has 5 red marbles and 3 blue marbles (ooooo)(ooo). How many marbles does he have in all? Well, you just told me--he has 5 red ones and 3 blue ones. (How many is that all together?) Well, it's 5 red and 3 blue. (Don't you know what all together means?) ... All together and in all are code phrases, they mean: think of those two little sets as being one big set. For some students that's an easy concept, and for others it's not--they need a context in which the two smaller sets would be a big set. The context could be: if Sam mixes all of his marbles up together, how many will he have in all? A completely different situation (5 girls and 3 boys in the classroom, how many children?) where the relationship of the smaller sets to the larger set was more familiar might help. As a potential teacher of kindergarten children, it's important for you to be aware that notion of reordering sets to think of them as a big set is an obstacle for some children to understand how to solve the problem.

One way to think about what makes it easy or difficult is the process/object understanding dichotomy. Understanding addition as a process (adding something into an existing set) is an inherently easier way of thinking of addition than as considering a set as being made of two smaller subsets, which is a static (no process) relationship. In the first example there's an obvious process to solve it, and in the second example you have to work back from an object presentation of addition (the set of all the marbles) to create a process.

Word problems where there is an action of some new things joining an already existing set are called Join problems. In particular, the problem about Mike's marbles is a Join, Result Unknown problem, because in the Joining story, the question asks about the amount at the end of the story: what happens after the new marbles join Mike's set of marbles.

Word problems where there's a bigger set that is made out of smaller sets is called a Part-Part-Whole problem. In a Part-Part-Whole problem, the whole that is made out of the two parts is a set that is different from either of the part sets: there's the set that's red marbles, and the set that's blue marbles, and the set that's all of the marbles. There isn't any action in a problem of this type unless you add in something extra to help students create an action that goes with the situation.

It's possible to write a problem that's like a hybrid of the two problem types. For example:

Alice has 5 red marbles in a bag. She puts 3 blue marbles in the bag. How many marbles are in the bag now?

In this case it's not clear whether a student would find the action or the color separation more compelling. I'd predict that in this case the action would usually be more compelling because the sets aren't "red marbles" or "blue marbles" and "all marbles", but instead there's just the one set: "marbles in the bag"--at least that's the way the sentence is structured. But I can't be sure that it would strike a child that way necessarily. Without any words in the problem at all, if you present some children with a set of 5 red counting cubes, and 3 blue counting cubes, and ask them how many counting cubes they have in all, some children will count up and tell you that they have 8, and other children will tell you that they have 5 red ones and 3 blue ones, and will persist in seeing only that answer unless they are given an action to perform, such as mixing up the cubes.