This lesson is a discussion of the differences between Join, Result Unknown problems and Part Part Whole, Whole Unknown problems. In the Common Core Math Standards, Join problems are called Add To problems, so a Join, Result Unknown problem can also be called an Add To, Result Unknown problem. Part Part Whole problems are called Put Together/Take Apart problems in the Common Core. Part Part Whole, Whole Unknown problems correspond to the Put Together/Take Apart, Total Unknown problem type.
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 (my readers) are 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. In particular the problem about Sam's marbles is a part part whole, whole unknown problem because you are told how many are in the two parts (the red marbles and the blue marbles), and the question asks how many are in the whole (all of the marbles).
Summary:
A Join problem has an action--a change over time. A join problem has a set of some objects, and other objects join the set, and at the end the set is larger. A join result unknown problem (JRU) has a set of some objects, other objects join the set, and the problem asks how many are in the set at the end?
Examples:
Mike had 5 marbles. His friend gave him 3 more marbles. How many marbles does he have now?
Janice had 4 paper butterflies. After lunch she made 5 more paper butterflies. How many butterflies does Janice have now?
A Part-Part-Whole problem explains a relationship of smaller sets that make up a larger set. A part part whole problem has a small set and another small set and togther they make a big set. A part part whole, whole unknown (PPW-WU) problem tells how many are in the smaller sets, and asks how many are in the larger set.
Examples:
Sam has 5 red marbles and 3 blue marbles. How many marbles does he have in all?
There are 11 boys and 4 girls in Mrs. Triangle's class. How many children are in the class?
Join problems are easier to understand and solve than part part whole problems for many children. You are most likely to see this difficulty when working with kindergarten students. The implication of this is that join problems are more appropriate for early work with addition.
The difficulty in part part whole problems is the lack of an action to guide the problem solving process, and a lack of experience with the idea of combining two sets into a larger set. The implications of this is that children, especially in kindergarten and first grade, may need support and practice with solving part part whole problems.
Appropriate ways to provide support and practice include:
When discussing what the phrases "in all" and "all together", it is best to explain this with a description of the two sets getting put together and mixed up (possibly with manipulatives to show the putting together idea). It is unwise to tell children that the words "in all" and "all together" mean to add, and I will be very disappointed in you if that is your strategy. It is a) much too easy to write a subtraction problem that uses the words "in all", and b) it deprives children of an understanding of what "in all" means and replaces it by a process of how to solve the problem.