Early childhood math: first problems with numbers
The first number problems that children can understand and figure out are ones they can act out and use counting skills to solve. We teach what addition and subtraction mean by showing how to act out and count to solve addition and subtraction problems. We invite childrens prior knowledge and intuition to help them when we ask contextualized problems (word problems) that are easy to understand and act out. Word problems are important for children to internalize and build understanding of number operations (addition, subtraction, etc.). We introduce number operations and problems through word problems that are appropriate for the childrens understanding--beginning with problems that are easy to solve by acting out and counting. This process of acting out a problem to understand it (usually with manipulatives of some sort) is called direct modeling.
What problems are easiest to direct model?
The easiest problems to direct model are ones where there is an amount of something, and some other amount is added to it or deducted from it through a process that happens in the problem. Problems where you are adding an amount to a set are called Join problems.
Sample Join problem (easier):
Mary had 8 erasers. Her best friend gave her 2 more erasers. How many erasers does she have now?
Two things about a Join problem make it easy to model and understand. One is that there is only a single set involved in the problem (in this example, the set is Mary's erasers). The second is there is an obvious action to show. She was given more erasers. When you get more erasers, that makes your eraser set larger--put more counters in the pile that represents Mary's erasers.
Compare this to a 2-set problem:
Sample Part-Part-Whole problem (harder):
Mary has 8 pink erasers and 2 yellow erasers. How many erasers does Mary have?
The distinction between the Join problem and this Part-Part-Whole problem probably isn't important for anyone teaching 2nd grade and up, and possibly not even for first grade, but it's vitally important for kindergartern teachers, because part of the learning and progressing that happens at about that age is the mental and verbal sophistication that lets children understand the second problem easily. A child who understands the part-part-whole problem will figure out that Mary has 10 erasers, a child who does not understand the part-part-whole problem will answer that she has 8 pink and 2 yellow erasers. The difficulty isn't a problem with the numbers, it's a problem with understanding what to do with the problem. For many children at the kindergarten level, the part-part-whole problem is fundamentally more difficult than the join problem, and the difficulty is in understanding the problem. The part-part-whole problem is more difficult because there are two sets in the problem: the set of pink erasers, and the set of yellow erasers, and the children don't have a mental map for combining two sets. Some children will even have this difficulty without the problem framing: if you show them 4 blue counters and 3 red counters and ask them how many counters there are, they will tell you that there are 4 blue counters and 3 red counters. Small changes in wording such as: "How many erasers does Mary have all together?" or "How many erasers does Mary have total?" aren't helpful for children at this level because those words are verbal cues that we use to remind ourselves and each other to put together the sets, but kindergarten children don't have "put the sets together" as a mental construct. To help children build a mental construct for putting sets together, we can add an action to the problem:
Modified Part-Part-Whole problems (with action):
Mary has 8 pink erasers, and 2 yellow erasers. If she puts all of the erasers in a long line and counts them all, how many will she have?
Mary had 8 pink erasers and 2 yellow erasers. She put them in a bag and mixed all her erasers together. How many erasers does she have?
These additions of actions are rather awkward--they don't fit well with the problem, but they do give children some mental images they can use to imagine what combining the sets would look like.
Take-away problems are straightforward to solve by direct modeling. These problems are called Separate problems:
Sample Separate problem:
Mary had 8 erasers. She gave 2 of her erasers to her best friend. How many erasers does she have now?
This problem is simple for the same reasons the Join problem is: first, there's only 1 set to worry about. That set changes, but you don't have to work with two sets at any time. Second, there's a clear action: giving something away is an action that children are familiar with, and so they can readily transfer that knowledge into knowing how to act out the problem. From a third grade perspective, subtraction is much harder than addition, but from a direct modeling perspective it isn't. Problems that may be harder later on because they are subtraction rather than addition or multiplication are hard because those operations (subtracting, adding, multiplying) are things that children are doing by figuring out or counting without acting out the problem with counters. For children who are direct modeling: acting out the problem with counters, and then counting to find the answer, the Join (getting more) and the separate (giving away) problems are approximately equal in how easy they are to solve.
When creating word problems with the goal of being easily understood and helpful to children at a kindergarten level, some things that make problems easier (more approachable and understandable) are: